ASPECTS OF PART VS WHOLE RELATIONSHIPS IN QUANTUM INFORMATION PROCESSING A Thesis Submitted to the Faculty in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics and Astronomy by Peter Douglas Johnson DARTMOUTH COLLEGE Hanover, New Hampshire October 20, 2016 Examining Committee:
Lorenza Viola, Chair
Chandrasekhar Ramanathan
Miles Blencowe
F. Jon Kull, Ph.D. Fernando G.S.L. Brandão Dean of Graduate and Advanced Studies
Abstract
Since its inception in quantum theory, the phenomenon of quantum entanglement has evolved from Einstein’s enigmatic “spooky action at a distance” to a crucial resource for quantum information processing. Recent technological advances geared towards controlling quantum systems and harnessing quantum entanglement have borne new perspectives and challenges. One major challenge is the development of a complete theory of multi-partite entanglement. Quantum theory places highly non-trivial con- straints on how entanglement may be distributed among the parts of a whole com- posite quantum system. In the simplest example, the more entangled system A is with system B, the less entangled system B can be with system C. This principle, known as the “monogamy of entanglement”, is a uniquely quantum feature enabling, in particular, secure quantum key distribution protocols and having ramifications for control of many-body quantum systems. In the first half of this thesis, I describe our contributions toward understanding the principles governing the distribution of multipartite entanglement. In particular, we elucidate surprising connections between the underlying kinematic constraints and the dynamical constraints stemming from the “no-cloning” principle and the uncer- tainty principle for incompatible quantum observables. In the second half of the thesis, I describe our contributions towards developing methods to create and control multipartite entanglement under realistic resource con- straints. Thanks to a number of recent experimental realizations, dissipative control of quantum systems is garnering increasing attention, alongside traditional unitary approaches. We investigate the use of dissipative control for driving a quantum sys- tem towards a target entangled state independently of initialization, a task known as “stabilization” – subject to the constraint that control resources be “quasi-local”. In particular, we develop mathematical tools for discovering hidden structures among the parts of a multi-partite entangled state which enable their stabilization.
ii
Preface
I cannot imagine a more ideal setting for balancing work and play than Dartmouth College in Hanover, New Hampshire. I am grateful to have spent such formative years in an environment that combines intense intellectual stimulation with striking natural beauty. I will truly miss the Norwich hills cycling, Connecticut river swimming, and granite peaks hiking. Certainly, this sense of connection to place is rooted in connections to friends. I am sincerely thankful to all of you who have helped me write such a vibrant chapter of my life. First and foremost, I thank my PhD adviser and mentor Lorenza Viola. You have shaped my development as a scientist in ways that I have yet to fully appreciate and have taught me lessons that extend far beyond physics. For the remainder of my career I will be drawing inspiration from your tireless demand of quality. I thank my local thesis committee Chandrasekhar Ramanathan and Miles Blencowe for guidance and encouragement throughout my PhD. I also thank my outside ex- aminer Fernando Brandão for participating in my defense and encouraging me as a scientist. I owe thanks to two other members of my Italian academic family. Thanks to my academic older brother, Francesco Ticozzi, for friendship, for expanding my mathe- matical tool set, and for supporting me in many ways. Thanks to my academic uncle, Roberto Onofrio, for thoughtful, timely guidance over the past six years. My working days were made brighter by the companionship of two wonderful friends Abhijeet Alase and Salini Karuvade. I look forward to the evolution of our friendship and collaboration for years to come. A number of physicists have been responsible for steering my course at various points over the past six years. Thanks to: Sandu Popescu for a conversation over billiards that blossomed into an undying pursuit; Stephon Alexander for helping me to see some some beautiful connections between my passions of physics and music; Carlton Caves for some memorable anecdotes, your encouragement, and for being a role model. Finally, I want to give special thanks to Ben Schumacher for setting my course and to Bill Wootters for sustaining it. Outside of academia, I thank my Hanover friends Dan Reeves, Ian Adelstein, Billy Braasch, and Mana Francisquez for pulling me out of the office for adventures and
iv then still talking shop along the way. Thanks to Sam, Russ, Andrew, and Ben for making deep thinking a fond passtime. Lastly, I want to especially thank my parents, brothers, and Ariana for your love and for putting up with (and even encouraging!) my pursuit of physics.
v Contents
Abstract ...... ii Preface ...... iv
1 Introduction 1
2 Quantum marginals: sharability and joinability 9 2.1 Introduction ...... 10 2.2 Joining and sharing classical vs. quantum states ...... 12 2.2.1 Joinability ...... 13 2.2.2 Sharability ...... 17 2.3 Joining and sharing Werner and isotropic states ...... 20 2.3.1 Werner and isotropic qudit states, and their classical analogues 20 2.3.2 Classical joinability limitations ...... 23 2.3.3 Joinability of Werner and isotropic qudit states ...... 24 2.3.4 Isotropic joinability results from quantum cloning ...... 30 2.3.5 Sharability of Werner and isotropic qudit states ...... 31 2.4 Further remarks ...... 33 2.4.1 Joinability beyond the three-party scenario ...... 33 2.4.2 Sharability of general bipartite qubit states ...... 35
3 Joinability of causal and acausal relationships 37 3.1 Introduction ...... 38 3.2 General quantum joinability framework ...... 40 3.2.1 Homocorrelation map and positive cones ...... 41 3.2.2 Generalization of joinability ...... 45 3.3 Three-party joinability settings with collective invariance ...... 48 3.3.1 Joinability limitations from state-positivity and channel-positivity 48 3.3.2 Joinability limitations from local-positivity ...... 53 3.4 Agreement bounds for quantum states and channels ...... 57
vi 4 Towards an alternative approach to joinability: enforcing positivity through purification 63 4.1 Introduction ...... 64 4.2 Joinability limitations from hard constraints ...... 66 4.3 Joinability limitations from soft constraints ...... 70
5 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics 77 5.1 Introduction ...... 78 5.2 Preliminaries ...... 79 5.2.1 Notation and background ...... 79 5.2.2 Fixed points of quantum dynamical semigroups ...... 82 5.2.3 Quasi-local semigroup dynamics ...... 86 5.2.4 Quasi-local stabilizability: prior pure-state results and frustration- free semigroup dynamics ...... 88 5.3 Frustration-Free Stabilizable States: Necessary Conditions ...... 90 5.3.1 Linear-algebraic tools ...... 90 5.3.2 Invariance conditions for quasi-local generators ...... 92 5.3.3 From invariance to necessary conditions for stabilizability . . . 94 5.4 Frustration-Free Stabilizable States: Sufficient Conditions ...... 96 5.4.1 A key result on frustration-free Markovian evolutions . . . . . 96 5.4.2 Sufficient conditions for full-rank target states ...... 98 5.4.3 Sufficient conditions for general target states ...... 100 5.5 Illustrative Applications ...... 102 5.5.1 Some notable failures of quasi-local stabilizability ...... 102 5.5.2 Quasi-local stabilization of graph product states ...... 103 5.5.3 Quasi-local stabilization of commuting Gibbs states ...... 107 5.5.4 Quasi-local stabilization beyond commuting Hamiltonians . . 111 5.5.5 Approximate FFQLS ...... 120
6 Finite-time stabilization of quantum states with discrete-time quasi- local dynamics 125 6.1 Introduction ...... 126 6.2 Preliminaries ...... 127 6.2.1 Quasi-local discrete-time dynamical semigroups ...... 127 6.2.2 Convergence notions ...... 129 6.2.3 Quasi-local parent Hamiltonians ...... 130 6.3 Finite-time stabilization ...... 132 6.3.1 Conditions for finite-time stabilization ...... 132 6.3.2 Efficiency of finite-time stabilization ...... 137 vii 6.4 Robust finite-time stabilization: necessary conditions ...... 140 6.5 Robust finite-time stabilization: sufficient conditions ...... 144 6.5.1 Non-operational sufficiency criteria ...... 144 6.5.2 Operational sufficiency criteria ...... 152 6.6 Efficiency of robust finite-time stabilization ...... 158 6.6.1 Circuit complexity ...... 158 6.6.2 Connection to rapid mixing ...... 161 6.7 Extension of results to mixed target states ...... 163 6.8 Proofs ...... 165
7 Towards finite-time dissipative quasi-local quantum encoders 187 7.1 Preliminaries ...... 188 7.2 Repetition code ...... 191 7.3 Toric code ...... 195 7.4 General stabilizer codes ...... 200 7.5 Further questions ...... 203
8 Summary and outlook 205
A List of abbreviations 213
References 215
viii
List of Tables
2.1 n-m sharability of Werner states ...... 34
x
List of Figures
1.1 Overlapping neighborhoods ...... 4
2.1 Werner and isotropic state joinability ...... 25 2.2 Werner and isotropic state projected joinability region ...... 29 2.3 Qubit Werner and isotropic state 1-n sharability ...... 33
3.1 State-channel duality commutativity diagram ...... 42 3.2 State-positive and channel-positive cone example ...... 44 3.3 Joinability problem homomorphism ...... 48 3.4 Brauer operator positivity ...... 51 3.5 Comparison of Werner state and depolarizing channel joinability . . . 53 3.6 Local-positive joining of Werner operators ...... 54 3.7 Comparison of separable joining and state-channel intersection joining 57
6.1 1-D nearest-neighbor neighborhood structure ...... 128 6.2 Cubic graph AKLT state: not finite-time stabilizable ...... 133 6.3 Finite-time stabilization scheme for AKLT state ...... 138 6.4 Non-factorized robust finite-time stabilizable state ...... 149 6.5 Generalized Bravyi-Vyalyi state ...... 153 6.6 Robust finite-time stabilizable state with non-commuting canonical frustration-free parent Hamiltonian ...... 157 6.7 Low-depth dissipative circuit for robust finite-time stabilization on lat- tice system ...... 158 6.8 CCZ-state on Kagome lattice ...... 160
7.1 Stabilizers and labeling scheme for 2-D toric code ...... 196 7.2 Logical operators for toric code ...... 197 7.3 Sequence of correction maps for finite-time encoding ...... 198
xii
Chapter 1
Introduction
1 Introduction
Quantum information processing provides a radically different means of comput- ing that shows promise for solving previously intractable problems. From a physical standpoint, one of the most compelling applications is to the simulation of quan- tum systems [1, 2]. A very practical and promising use for quantum simulation is, for instance, the calculation of molecular energies [3]. With a classical computer, such calculations are feasible for small systems, but they soon become too resource- intensive as larger molecules are considered. Quantum algorithms have been devel- oped which are believed to solve this problem by requiring exponentially fewer steps than the best known classical algorithms. A few months before the time of writing, Google Inc. and collaborators published work that demonstrated the first example of a scalable quantum simulation calculating molecular energies [4]. Besides its technological applications, quantum information science also offers a diverse set of mathematical tools and concepts. Many of these are actively being developed or, even, created. These tools and concepts are proving useful in areas of physics as diverse as condensed matter physics [5, 6], quantum field theory [7, 8], and quantum gravity [9, 10, 11]. Even with its technological potential and powerful mathematical toolbox, quan- tum information science has captivated many researchers, including the author, for a different reason. Some of the seminal contributions to the field were initiated by physicists asking foundational questions about quantum mechanics [12]. Quantum information science has given us deep insights into the workings of Nature. Certainly, more insights lie just beyond the horizon. In the most common setting, quantum information is processed with quantum bits, or qubits as opposed to bits. In principle, any two-level quantum system may serve as a qubit. Examples include the spin of an electron, the polarization of a photon, or the two lowest energy levels in a superconducting Josephson junction [13]. No single system has yet to triumph as the universally used qubit. In practice, certain systems (e.g. trapped ions or superconducting qubits) are more accurately controlled than others, and therefore, show more promise for becoming the standard qubit. Quantum bits obey more subtle rules than classical bits do. In particular, multiple quantum systems can exhibit a uniquely quantum type of correlation known as quan- tum entanglement. Although we do not yet fully understand its ramifications, it is thought that quantum entanglement plays a crucial role in providing the advantages of quantum computing over classical computing. Also, not all classical information processes translate to the quantum case. For instance, there is no quantum process which can perfectly copy, or clone, an unknown quantum bit [14]. This mentioning of quantum entanglement and the no-cloning principle anticipate some of the nuances of quantum information processing that we address throughout the thesis. The field of quantum information science, which gained traction in the 1990s, is now a firmly established research field [15]. Many of the most pressing challenges in
2 Introduction
quantum information science demand improvements in hardware engineering. Much of the theoretical groundwork is, by now, relatively well-established. The important concepts from classical information theory have been appropriately repurposed for use in quantum information theory [16]. There are a number of well-established models of quantum computing such as the circuit model [17], measurement-based quantum computing [18], and, towards solving certain optimization problems, quantum an- nealing [19]. The theory of quantum error correction, necessary for building scalable, fault-tolerant quantum computers, is relatively well-understood [20]. Despite these advances, many theoretical challenges remain and are being actively pursued. We still lack a complete understanding of the principles ensuring “quantum supremacy” over classical information processing. And, furthermore, we cannot antic- ipate the breakthroughs in understanding and unforeseen applications that are likely to result from a deeper inquiry into the quantum. The theory of bipartite entanglement, quantum correlations between two systems, is well-understood, including mixed-state entanglement. However, even in the sim- plest setting where subsystems are distinguishable, multipartite entanglement de- mands a much richer tool-kit of mathematical concepts, many of which are currently being developed. Some of these tools are already finding diverse applications, such as towards exploring space-time as an emergent phenomenon based on quantum entan- glement [11]. Nevertheless, questions remain as for how best to describe multipartite entanglement, to characterize its features, and to use it as a resource. In the circuit model approach to quantum information processing, one applies a sequence of unitary quantum gates to some fiducial input state, seeking to maintain sufficient coherent control of the relevant parts of the system. However, even within the quantum circuit model, access to suitable non-unitary control is crucial for proper initialization or “entropy removal” in fault-tolerant architectures, as well as for uni- versal “digital” simulation of open-system dynamics [21]. Control theoretic advances in open quantum systems have further revealed new avenues for utilizing incoherent control by means of engineered dissipation. Such non-unitary control resources have been shown to be beneficial for the tasks of robust preparation of resource states [22], rapid quantum state purification [23], and engineered dissipative quantum memories [24]. The last few years, in particular, have seen a surge of interest in proposals for dissipatively preparing strongly-correlated and topological phases of matter. Recent experimental advances have used engineered dissipation to autonomously drive a two- qubit system towards an entangled resource state. This feat has been achieved with superconducting qubits [25] as well as with ion traps [26]. With these advances comes a demand for developing mathematical tools which can fully describe the capabilities and limitations of controlled dissipation. This thesis contributes to the problems identified above. The unifying theme tying this work together is the quantum part vs whole relationship. Although the content
3 Introduction
Figure 1.1: Three quantum systems ABC with delineated subsystems AB and BC.
is varied in motivation and mathematical techniques, nearly all of the problems we consider admit a simple example on three subsystems as pictured in Fig. 1.1. Many questions we consider involve confronting the interaction, or “overlap”, of subsystems (e.g. subsystem AB overlaps with subsystem BC). Like discussing the weather, the ubiquitous difficulty of “the interacting case” gives even physicists with differing backgrounds a topic with which to mutually relate. It may be useful for the reader to keep this picture in mind throughout. The thesis naturally divides into two parts. The first half explores a concept that we developed known as quantum joinability. The notion of joinability addresses the question of whether or not there exists a description of the whole (e.g. ABC) which is consistent with constraints on the parts (e.g AB and BC); equivalently, we ask if the constrained parts can be “joined” into some physically allowed whole. Due to the phenomenon of entanglement, quantum theory places non-trivial constraints on the ways that the parts of a quantum system may be correlated among one another. As an example, the more entangled qubit A is with B, the less entangled qubit B can be with C. This phenomenon, known as the monogamy of entanglement, is uniquely quantum in that classical correlations are not limited by such a constraint. As the number of systems increases, the complexity of such constraints grows drastically. The study of such constraints is often referred to as the quantum marginal problem [27]. This problem originated from the field of quantum chemistry [28], motivated by the attempt to simplify calculations of atomic and molecular ground state energies. More recently, this problem has gained attention from the quantum information com- munity since it addresses the nature of entanglement distribution in a multipartite quantum system. Researchers have drawn on and developed diverse mathematical tools towards tackling this very difficult problem. We have contributed to this effort by analyzing, in detail, a number of tractable cases which bear insight on the more general problem. Furthermore, we have established rigorous connections among the concepts of monogamy of entanglement, the no-cloning principle, and incompatible measurements. These concepts are ubiquitous in quantum information and are find- ing application in other areas of physics such as black-hole thermodynamics [29]. We develop the framework of quantum joinability in order to put these notions on equal footing and elucidate their common origin.
4 Introduction
• Chapter 2 defines the problems of quantum sharability and quantum joinability for bipartite quantum states. We investigate some basic scenarios in which a given set of bipartite quantum states may consistently arise as the set of re- duced states of a global N-partite quantum state. We restrict the discussion to bipartite reduced states that belong to the paradigmatic classes of Werner and isotropic states in d dimensions, and focus on two specific versions of the quantum marginal problem which we find to be tractable. The first is Alice- Bob, Alice-Charlie joining, with both pairs being in a Werner or isotropic state. The second is m-n sharability of a Werner state across N subsystems, which may be seen as a variant of the N-representability problem to the case where subsystems are partitioned into two groupings of m and n parties, respectively. By exploiting the symmetry properties that each class of states enjoys, we de- termine necessary and sufficient conditions for three-party joinability and 1-n sharability for arbitrary d. Our results explicitly show that although entangle- ment is required for sharing limitations to emerge, correlations beyond entan- glement generally suffice to restrict joinability, and not all unentangled states necessarily obey the same limitations. The relationship between joinability and quantum cloning as well as implications for the joinability of arbitrary bipartite states are discussed. In particular, the observations regarding quantum cloning lead naturally into the investigation of the subsequent chapter. • Chapter 3 develops the framework of quantum joinability which unifies seem- ingly different joinability problems for bipartite quantum states and channels. This includes well known problems such as optimal quantum cloning and quan- tum marginal problems as special instances. Central to our generalization is a variant of the Choi-Jamiolkowski isomorphism between bipartite states and dynamical maps which we term the “homocorrelation map”: while the former emphasizes the preservation of the positivity constraint, the latter is designed to preserve statistical correlations, allowing more direct contact with entan- glement. In particular, we define and analyze state-joining, channel-joining, and local-positive joining problems in three-party settings exhibiting collective U ⊗ U ⊗ U symmetry, obtaining exact analytical characterizations in low di- mension. Suggestively, we find that bipartite quantum states are limited in the degree to which their measurement outcomes may agree, while quantum channels are limited in the degree to which their measurement outcomes may disagree. Loosely speaking, quantum mechanics enforces an upper bound on the extent of positive correlation across two subsystems at a single time, as well as on the extent of negative correlation between the state of a single system across two instants of time. We argue that these general statistical bounds inform the quantum joinability limitations, and show that they are in fact sufficient for the three-party U ⊗ U ⊗ U-invariant setting.
5 Introduction
• Chapter 4 delves deeper into underpinnings of quantum joinability. We adopt a different approach to the joinability problem, ensuring positive-semidefiniteness or complete positivity from the outset. With this, we formalize the “composition law” of correlations from the previous two chapters. We incorporate the notion of incompatibility of observables as another example of quantum joinability. Furthermore, we draw parallels between the quantum and classical problems, finding a intuitive common cause for their respective joinability constraints. The mathematical techniques developed here are intended to elucidate some quantum peculiarities such as the distinction between causal and acausal quan- tum relationships and the origin of monogamy of entanglement, no-cloning, and measurement incompatibility. The second half of this thesis explores engineering dissipation to stabilize quantum states both asymptotically and in finite time. Towards preparing the quantum re- source of many-body entanglement in a realistic setting, one must address the issue of constrained control capabilities. This becomes increasingly relevant as the number of parts in the systems is scaled up, as is needed for quantum information processing to deliver its full potential. In particular, a possible realistic constraint on one’s control of the system is that only parts of the system can be addressed at a time. Much like a logical circuit, we may assume that manipulations of the whole are achieved by a series of manipulations of the parts. Such an implementation is said to be quasi-local. Thus, in quasi-local stabilization, the part-whole relationship features, in that we seek a preparation of the whole (e.g. ABC) by addressing only its parts (e.g. AB and BC). The task of engineering quasi-local dissipation to drive a quantum system, all-to- one, into a target quantum state has been initiated and explored in [30, 31, 32, 33, 34]. These works consider a continuous-time dynamics generated by an engineered quasi-local Markovian master equation, giving examples and exploring conditions for preparing, or “stabilizing”, a target pure state. In practice, pure quantum states are never available and natural dissipative dynamics exhibit mixed steady states. Accordingly, we contribute to this line of research by exploring quasi-local stabilization in the case of mixed target states. Some implementations of engineered dissipation are best modeled by discrete-time dynamics [35]. Constraining the dynamical maps to act quasi-locally, we can view the sequence of maps as a dissipative quantum circuit. An advantage in this case, compared to that of continuous-time, is that a target state may be exactly stabilized in a finite time. We contribute to the field of dissipative quantum control by determining conditions for finite-time stabilization of a target state. We describe how well-known resource states, such as graph states for measurement based quantum computing, may be stabilized with these schemes. To conclude, we present some preliminary work geared towards achieving quantum encoding using quasi-local resources.
6 Introduction
• Chapter 5 builds off of previous work by L. Viola and F. Ticozzi to investi- gate the engineering of dissipative continuous-time dynamics to render a target mixed quantum state as the unique global attractor of the dynamics. In par- ticular, we determine necessary and sufficient conditions for whether or not a given target state can be the unique steady state of frustration-free quasi-local continuous-time Markovian dynamics. We investigate under which conditions a mixed state on a finite-dimensional multipartite quantum system may be the unique, globally stable fixed point of frustration-free semigroup dynamics subject to specified quasi-locality constraints. Our central result is a linear- algebraic necessary and sufficient condition for a generic (full-rank) target state to be frustration-free quasi-locally stabilizable, along with an explicit procedure for constructing Markovian dynamics that achieve stabilization. If the target state is not full-rank, we establish sufficiency under an additional condition, which is naturally motivated by consistency with pure-state stabilization re- sults yet provably not necessary in general. Several applications are discussed, of relevance to both dissipative quantum engineering and information process- ing, and non-equilibrium quantum statistical mechanics. In particular, we show that a large class of graph product states (including arbitrary thermal graph states) as well as Gibbs states of commuting Hamiltonians are frustration-free stabilizable relative to natural quasi-locality constraints. Likewise, we provide explicit examples of non-commuting Gibbs states and non-trivially entangled mixed states that are stabilizable despite the lack of an underlying commuting structure, albeit scalability to arbitrary system size remains in this case an open question.
• Chapter 6 complements the work of Chapter 5 by investigating the discrete- time analog of quantum state stabilization with quasi-local dynamics. While continuous-time Markovian dynamics cannot exactly stabilize a target state in finite time, discrete-time dynamics can, in principle. We develop necessary and sufficient conditions for establishing if a given target state can be stabilized by a finite sequence of quasi-local dynamical maps. Then we investigate the efficient scheme of robust finite-time stabilization, whereby the target state is stabilized regardless of the implementation order of the dynamical maps. A main theme in this chapter is the role that certain “commuting structures” play in facilitating robust stabilization.
• Chapter 7 turns to the task of preparing quantum information in a quantum error correcting code. As in the previous chapter, we explore several examples whereby this task can be achieved exactly, in principle, by a finite sequence of quasi-local dynamical maps. We develop a number of principles which aid the construction of such finite-time dissipative encoders utilizing the quantum
7 Introduction
stabilizer formalism.
8 Chapter 2
Quantum marginals: sharability and joinability
9 Quantum marginals: sharability and joinability
This chapter presents material that appeared in Physical Review A, 88:032323 (2013), in an article titled “Compatible quantum correlations: Extension problems for Werner and isotropic states”, which is joint work with Lorenza Viola.
2.1 Introduction
Understanding the nature of quantum correlations in multiparty systems and the distinguishing features they exhibit relative to classical correlations is a central goal across quantum information processing (QIP) science [17], with implications rang- ing from condensed-matter and statistical physics to quantum chemistry, and the quantum-to-classical transition. From a foundational perspective, exploring what different kinds of correlations are, in principle, allowed by probabilistic theories more general than quantum mechanics further helps to identify under which set of physical constraints the standard quantum framework may be uniquely recovered [36, 37]. In this context, entanglement provides a distinctively quantum type of correla- tion, that has no analogue in classical statistical mechanics. A striking feature of entanglement is that it cannot be freely distributed among different parties: if a bi- partite system, say, A(lice) and B(ob), is in a maximally entangled pure state, then no other system, C(harlie), may be correlated with it. In other words, the entan- glement between A and B is monogamous and cannot be shared [38, 39, 40, 41, 42]. This simple tripartite setting motivates two simple questions about bipartite quan- tum states: given a bipartite state, we ask whether it can arise as the reduced state of A-B and of A-C simultaneously; or, more generally, given two bipartite states, we ask if one can arise as the reduced state of A-B while the other arises as the reduced state of A-C. It should be emphasized that both of these are questions about the existence of tripartite states with given reduction properties. While formal (and more general) definitions will be provided later in the chapter, these examples serve to in- troduce the notions of sharing (1-2 sharing) and joining (1-2 joining), respectively. In its most general formulation, the joinability problem is also known as the quantum marginal problem (or local consistency problem), which has been heavily investigated both from a mathematical-physics [27, 43, 44] and a quantum-chemistry perspective [45, 46] and is known to be QMA-hard [47]. Our choice of terminology, however, facilitates a uniform language for describing the joinability/sharability scenarios. For instance, we say that the joinable correlations of A-B and A-C are joined by a joining state on A-B-C. The limited sharability/joinability of entanglement was first quantified in the sem- inal work by Coffman, Kundu, and Wootters, in terms of an exact (CKW) inequality obeyed by the entanglement across the A-B, A-C and A-(BC) bipartitions, as mea- sured by concurrence [38]. In a similar venue, several subsequent investigations at- tempted to determine how different entanglement measures can be used to diagnose
10 2.1 Introduction
failures of joinability, see e.g. [48, 49, 41]. More recently, significant progress has been made in characterizing quantum correlations more general than entanglement [50, 51], in particular as captured by quantum discord [52]. While it is now established that quantum discord does not obey a monogamy inequality [53], different kinds of limitations exist on the extent to which it can be freely shared and/or communicated [54, 55]. Despite these important advances, a complete picture is far from being reached. What kind of limitations do strictly mark the quantum-classical correlation boundary? What different quantum features are responsible for enforcing different aspects of such limitations, and how does this relate to the degree of resourcefulness that these correlations can have for QIP? While the above are some of the broad questions motivating this work, our spe- cific focus here is to make progress on joinability and sharability properties in low- dimensional multipartite settings. In this context, reference [56] has obtained a nec- essary condition for three-party joining in finite dimension in terms of the subsystem entropies, and additionally established a sufficient condition in terms of the trace- norm distances between the states in question and known joinable states. For the specific case of qubit Werner states [57], Werner himself established necessary and sufficient conditions for the 1-2 joining scenario [58]. With regards to sharability, necessary and sufficient conditions have been found for 1-2 sharing of generic bipar- tite qubit states [59], as well as for specific classes of qudit states [60]. To the best of our knowledge, no conditions that are both necessary and sufficient for the joinability of generic states are available as yet. In this chapter, we obtain necessary and suffi- cient conditions for both the three-party joinability and the 1-n sharability problems, in the case that the reduced bipartite states are either Werner or isotropic states on d-dimensional subsystems (qudits). Though our results are restricted in scope of applicability, they provide key in- sights as to the sources of joinability limitations. Most importantly, we find that standard measures of quantum correlations, such as concurrence and quantum dis- cord, do not suffice to determine the limitations in joining quantum correlations. Specifically, we find that the joined states need not be entangled or even discordant in order not to be joinable. Further to that, although separable states may have join- ability limitations, they are, nonetheless, freely (arbitrarily) sharable. By introducing a one-parameter class of probability distributions, we provide a natural classical ana- logue to qudit Werner and isotropic quantum states. This allows us to illustrate how classical joinability restrictions carry over to the quantum case and, more inter- estingly, to demonstrate that the quantum case demands limitations which are not present classically. Ultimately, this feature may be traced back to complementarity of observables, which clearly plays no role in the classical case. It is suggestive to note that the uncertainty principle was also shown to be instrumental in constraining the sharability of quantum discord [54]. It is our hope that further pursuits of more
11 Quantum marginals: sharability and joinability
general necessary and sufficient conditions may be aided by the methods and findings herein. The content is organized as follows. In Sec. 2.2 we present the relevant mathemat- ical framework for defining the joinability and sharability notions and the extension problems of interest, along with some preliminary results contrasting the classical and quantum cases. Sec. 2.3 contains the core results of our analysis. In particular, after reviewing the defining properties of Werner and isotropic states on qudits, in Sec. 2.3.1 we motivate the appropriate choice of probability distributions that serve as a classical analogue, and determine the resulting classical joinability limitations in Sec. 2.3.2. Necessary and sufficient conditions for three-party joinability of quantum Werner and isotropic states are established in Sec. 2.3.3, and contrasted to the clas- sical scenario. Sec. 2.3.4 shows how the results on isotropic state joinability are in fact related to known results on quantum cloning, whereas in Sec. 2.3.5 we establish simple analytic expressions for the 1-n sharability of both Werner and isotropic states, along with discussing constructive procedures to determine m-n sharability properties for m > 1. In Sec. 2.4, we present additional remarks on joinability and sharability scenarios beyond those of Sec. 2.3. In particular, we outline generalizations of our analysis to N-party joinability, and show how bounds on the sharability of arbitrary bipartite states follow from the Werner and isotropic results. For ease and clarity of presentation, we have omitted the technical proofs of the results in Sec. 2.3 from this thesis. These proofs can be found in the appendix of [61].
2.2 Joining and sharing classical vs. quantum states
Although our main focus will be to quantitatively characterize simple low-dimensional settings, we introduce the relevant concepts with a higher degree of generality, in order to better highlight the underlying mathematical structure and to ease connections with existing related notions in the literature. We are interested in the correlations among the subsystems of a N-partite composite system S. In the quantum case, we thus require a Hilbert space with a tensor product structure:
N (N) O (1) (1) H ' Hi , dim(Hi ) ≡ di, i=1
(1) where Hi represents the individual “single-particle” state spaces and, for our pur- poses, each di is finite. In the classical scenario, to each subsystem we associate a sample space Ωi consisting of di possible outcomes, with the joint sample space being
12 2.2 Joining and sharing classical vs. quantum states
given by the Cartesian product:
(N) Ω ' Ω1 × ... × ΩN .
Probability distributions on Ω(N) are the classical counterpart of quantum density operators on H(N).
2.2.1 Joinability The input to a joinability problem is a set of subsystem states which, in full generality, may be specified relative to a “neighborhood structure” on H(N) (or Ω(N)) [31, 34]. That is, let neighborhoods {Nj} be given as subsets of the set of indexes labeling individual subsystems, Nk ( ZN . We can then give the following:
Definition 2.2.1. [Quantum Joinability] Given a neighborhood structure {N1, N2, (N) ..., N`} on H , a list of density operators
(ρ1, . . . , ρ`) ∈ (D(HN1 ),..., D(HN` ))
is joinable if there exists an N-partite density operator w ∈ D(H(N)), called a joining state, that reduces according to the neighborhood structure, that is,
Tr ˆ (w) = ρ , ∀k = 1, . . . , `, (2.1) Nk k ˆ where Nk ≡ ZN \Nk is the tensor complement of Nk.
The analagous definition for classical joinability is obtained by substituting corre- ˆ sponding terms, in particular, by replacing the partial trace over Nk with the corre- sponding marginal probability distribution. As remarked, the question of joinability has been extensively investigated in the context of the classical [62] and quantum [27, 63, 64, 56] marginal problem. A joining state is equivalenty referred to as an extension or an element of the pre-image of the list under the reduction map, while the members of a list of joinable states are also said to be compatible or consistent. Clearly, a necessary condition for a list of states to be joinable is that they “agree” on any overlapping reduced states. That is, given any two states from the list whose neighborhoods are intersecting, the reduced states of the subsystems in the intersec- tion must coincide. From this point of view, any failure of joinability due to a dis- agreement of overlapping reduced states is a trivial case of non-compatible N-party correlations. We are interested in cases where joinability fails despite the agreement on overlapping marginals. This consistency requirement will be satisfied by construc- tion for the Werner and isotropic quantum states we shall consider in Sec. 2.3.
13 Quantum marginals: sharability and joinability
One important feature of joinability, which has recently been investigated in [65], is the convex structure that both joinable states lists and joining states enjoy. The set of lists of density operators satisfying a given joinability scenario is convex under component-wise combination; this is because the same convex combination of their joining states is a valid joining state for the convex combined list of states. Similarly, the set of joining states for a given list of joined states is convex by the linearity of the partial trace. As mentioned, one of our goals is to shed light on limitations of quantum vs. classical joinability and the extent to which entanglement may play a role in that respect. That quantum states are subject to stricter joinability limitations than classical probability distributions are, can be immediately appreciated by considering two density operators ρAB = |ΨBihΨB| = ρAC , where |ΨBi is any maximally entangled Bell pair on two qubits: no three-qubit joining state wABC exists, despite the reduced state on A being manifestly consistent. In contrast, as shown in [56, 62], as long as two classical distributions have equal marginal distributions over A, p(A, B) and p(A, C) can always be joined. This is evidenced by the construction of the joining state: w(A, B, C) = p(A, B) p(A, C)/p(A). As pointed out in [56], although the above choice is not unique, it is the joining state with maximal entropy and represents an even mixture of all valid joining distributions. Although any two consistently-overlapped classical probability distributions may be joined, limitations on joining classical probability distributions do typically arise in more general joining scenarios. This follows from the fact that any classical probability assignment must be consistent with some convex combination of pure states. Consider, for example, a pairwise neighboorhood structure, with an associated list of states p(A, B), p(B,C), and p(A, C), which have consistent single-subsystem marginals. Clearly, if each subsystem corresponds to a bit, no convex combination of pure states gives rise to a probability distribution w(A, B, C) in which each pair is completely anticorrelated; in other words, “bits of three can’t all disagree”. In Sec. 2.3.3, we explicitly compare this particular classical joining scenario to analogous quantum scenarios. While all the classical joining limitations may be expressed by linear inequalities, the quantum joining limitations are significantly more complicated. The limitations arise from demanding that the joining operator be a valid density operator, namely, trace-one and non-negative (which clearly implies Hermiticity). This fact is demon- strated by the following proposition, which may be readily generalized to any joining scenario:
Proposition 2.2.2. For any two trace-one Hermitian operators QAB and QAC which obey the consistency condition TrB (QAB) = TrC (QAC ), there exists a trace-one Her- mitian joining operator QABC . Proof. Consider an orthogonal Hermitian product basis which includes the identity
14 2.2 Joining and sharing classical vs. quantum states
for each subsystem, that is, {Ai ⊗ Bj ⊗ Ck}, where A0 = B0 = C0 = I. Then we can construct the space of all valid joining operators QABC as follows. Let dABC be the dimension of the composite system. The component along A0 ⊗ B0 ⊗ C0 is fixed as 1/dABC , satisfying the trace-one requirement. The components along the two- body operators of the form Ai ⊗ Bj ⊗ I are fixed by the required reduction to QAB, and similarly the components along the two-body operators of the form Ai ⊗ I ⊗ Ck are determined by QAC . The components along the one-body operators of the form Ai ⊗ I ⊗ I, I ⊗ Bi ⊗ I, and I ⊗ I ⊗ Ci are determined from the reductions of QAB and QAC . This leaves the coefficients of all remaining basis operators unconstrained, since their corresponding basis operators are zero after a partial trace over systems B or C. Thus, requiring the joining operator to be Hermitian and normalized is not a limiting constraint with respect to joinability: any limitations are due to the non- negativity constraint. Understanding how non-negativity manifests itself is extremely difficult in general and far beyond our scope here. We can nevertheless give an ex- ample in which the role of non-negativity is clear. Part of the job of non-negativity is to enforce constraints that are also obeyed by classical probability distributions. For example, in the case of a two-qubit state ρ, if hX ⊗ Iiρ = 1 and hI ⊗ Xiρ = 1, then hX ⊗ Xiρ must equal 1. More generally, consider a set of mutually commuting ob- k servables {Mi}i=1 and any basis {|mi} in which all Mi are diagonal. Any valid state must lead to a list of expectation values (Tr (ρM1) ,..., Tr (ρMk)), whose values are element-wise convex combinations of the vertexes {(hm|M1|mi,..., hm|Mk|mi)|∀m}. The interpretation of this constraint is that since commuting observables have simul- taneously definable values, just as classical observables do, probability distributions on them must obey the rules of classical probability distributions. We call on this fact when we compare the quantum joining limitations to the classical analogue ones in Sec. 2.3.3. Non-negativity constraints that do not arise from classical limitations on compat- ible observables may be labeled as inherently quantum constraints, the most familiar being provided by uncertainty relations for conjugate observables [66, 67]. Although complementarity constraints are most evident for observables acting on the same system, complementarity can also give rise to a trade-off in the information about a subsystem observable vs. a joint observable. This fact is essentially what allows Bell’s inequality to be violated. For our purposes, the complementarity that comes ~ ~ into play is that between “overlapping” joint observables (e.g., between S1 · S2 and ~ ~ S1 · S3 for three qubits). We are thus generally interested in understanding the inter- play between purely classical and quantum joining limitations, and in the correlation trade-offs that may possibly emerge. Historically, as already mentioned, a pioneering exploration of the extent to which quantum correlations can be shared among three parties was carried out in [38],
15 Quantum marginals: sharability and joinability yielding a characterization of the monogamy of entanglement in terms of the well- known CKW inequality: 2 2 2 min CAB + CAC ≤ (C )A(BC), where C denotes the concurrence and the right hand-side is minimized over all pure- state decompositions. Thus, with the entanglement across the bipartition A and (BC) held fixed, an increase in the upper bound of the A-B entanglement can only come at the cost of a decrease in the upper bound of the A-C entanglement. One may wonder whether the CKW inequality may help in diagnosing joinability of reduced states. If a joining state wABC is not a priori determined (in fact, the existence of such a state is the entire question of joinability), the CKW inequality may be used to obtain a necessary condition for joinability, namely, if ρAB and ρAC are joinable, then
2 2 CAB + CAC ≤ 1. (2.2)
However, there exist pairs of bipartite states – both unentangled (as the following Proposition shows) and non-trivially entangled (as we shall determine in Sec. III.B, see in particular Fig. 2.2a) – that obey the “weak” CKW inequality in Eq. (2.2), yet are not joinable. The key point is that while the limitations that the CKW captures are to be ascribed to entanglement, entanglement is not required to prevent two states from being joinable. In fact, weaker forms of quantum correlations, as quantified by quantum discord [52], are likewise not required for joinability limitations. Consider, specifically, so-called “classical-quantum” bipartite states, of the form
X i X ρ = pi|iihi|A ⊗ σB, pi = 1, i i
i where {|iiA} is some local orthogonal basis on A and σB is, for each i, an arbitrary state on B. Such states are known to have zero discord [68]. Yet, the following holds:
Proposition 2.2.3. Classical-quantum correlated states need not be joinable.
Proof. Consider the two quantum states
ρAB = (|↑X ↑X ih↑X ↑X | + |↓X ↓X ih↓X ↓X |)/2,
ρAC = (|↑Z ↑Z ih↑Z ↑Z | + |↓Z ↓Z ih↓Z ↓Z |)/2, on the pairs A-B and A-C, respectively. Both have a completely mixed reduced state over A and thus it is meaningful to consider their joinability. Let wABC be a joining state. Then the outcome of Bob’s X measurement would correctly lead him to predict Alice to be in the state | ↑X i or | ↓X i, while at the same time the outcome of Charlie’s Z measurement would correctly lead him to predict Alice to be in the state | ↑Z i or
16 2.2 Joining and sharing classical vs. quantum states
| ↓Z i. Since this violates the uncertainty principle, wABC cannot be a valid joining state. The existence of separable but not joinable states has been independently reported in [56]. While formally our example is subsumed under the more general one presented in Thm. 4.2 therein (strictly satisfying the necessary condition for joinability given by their Eq. (2.2)), it has the advantage of offering both a transparent physical interpretation of the underlying correlation properties, and an intuitive proof of the joinability failure.
2.2.2 Sharability As mentioned, the second joinability structure we analyze is motivated by the concept of sharability. In our context, we can think of sharability as a restricted joining (2) (1) (1) scenario in which a bipartite state is joined with copies of itself. If H 'H1 ⊗H2 , (1) consider a N-partite space that consists of m “left” copies of H1 and n “right” (1) copies of H2 , with each neighborhood consisting of one right and one left subsystem, respectively (hence a total of mn neighborhoods). We then have the following:
Definition 2.2.4. [Quantum Sharability] A bipartite density operator ρ ∈ D(HL⊗ ⊗m ⊗n HR) is m-n sharable if there exists an N-partite density operator w ∈ D(HL ⊗HR ), called a sharing state, that reduces left-right-pairwise to ρ, that is,
Tr (w) = ρ, ∀i = 1, . . . , m, j = 1, . . . , n, (2.3) LˆiRˆj where the partial trace is taken over the tensor complement of neighborhood ij. Each m-n sharability scenario may be viewed as a specific joining structure with the additional constraint that each of the joining states be equal to one another, the list being (ρ, ρ, . . . , ρ). In what follows, we shall take arbitrarily sharable to mean ∞-∞ sharable, whereas finitely sharable means that ρ is not m-n sharable for some m, n. Also, each property “m-n sharable” (sometimes also referred to as a “m-n extendible”) is taken to define a sharability criterion, which a state may or may not satisfy. It is worth noting the relationship between sharability and N-representability. The N-representability problem asks if, for a given (symmetric) p-partite density operator (1) ⊗p ρ on (H1 ) , there exists an N-partite pre-image state for which ρ is the p-particle reduced state. N-representability has been extensively studied for indistinguishable bosonic and fermionic subsystems [69, 45, 46] and is a very important problem in quantum chemistry [70]. We can view N-representability as a variant on the shara- bility problem, whereby the distinction between the left and right subsystems is lifted, and m + n = N. Given the p-partite state ρ as the shared state, we ask if there exists a sharing N-partite state which shares ρ among all possible p-partite subsystems. In
17 Quantum marginals: sharability and joinability
the setting of indistinguishable particles, the associated symmetry further constrains the space of the valid N-partite sharing states. Just as with 1-2 joinability, any classical probability distribution is arbitrarily sharable [37]. Likewise, similar to the joinability case, convexity properties play (1) an important role towards characterizing sharability. If dim(H1 ) = d1 ≡ dL and (1) dim(H2 ) = d2 ≡ dR, then it follows from the convexity of the set of joinable states lists that m-n sharable states form a convex set, for fixed subsystem dimensions dL and dR. This implies that if ρ satisfies a particular sharability criterion, then any mixture of ρ with the completely mixed state also satisfies that criterion, since the completely mixed state is arbitrarily (∞-∞) sharable. Besides mixing with the identity, the degree of sharability may be unchanged under more general transformations on the input state. Consider, specifically, completely- positive trace-preserving bipartite maps M(ρ) that can be written as a mixture of local unitary operations, that is,
X i i i† i† X M(ρ) = λiU1 ⊗ V2 ρU1 ⊗ V2 , λi = 1, (2.4) i i
i i where U1 and V2 are arbitrary unitary transformations on HL and HR, respectively. These (unital) maps form a proper subset of general Local Operations and Classical Communication (LOCC) [17]. We establish the following:
Theorem 2.2.5. If ρ is m-n sharable, then M(ρ) is m-n sharable for any map M that is a convex mixture of unitaries.
Proof. Let M(ρ) be expressed as in Eq. (2.4). By virtue of the convexity of the set of m-n sharable states (for fixed subsystem dimensions), it suffices to show that each term, UV ρU †V †, in M(ρ) is m-n sharable. Let w be a sharing state for ρ, and define
0 † † † † w = U1 ...UmVm+1 ...Vm+n w U1 ...UmVm+1 ...Vm+n .
Then, for any left-right pair of subsystems i and j, it follows that
0 † † † † Tri,j (w ) = UiVjTri,j (w) Ui Vj = U ⊗ V ρU ⊗ V = ρUV .
0 Hence, w is an m-n-sharing state for ρUV , as desired.
This result suggests a connection between the degree of sharability and the en- tanglement of a given state. In both cases, there exist classes of states for which these properties cannot be “further degraded” by locally acting maps (or any map for that matter). Obviously, LOCC cannot decrease the entanglement of states with no entanglement, and convex unitary mixtures as above cannot increase the sharability
18 2.2 Joining and sharing classical vs. quantum states
of states with ∞-∞ sharability (because they are already as sharable as possible). These two classes of states can in fact be shown to coincide as a consequence of the fact that arbitrary sharability is equivalent to (bipartite) separability. This result has been appreciated in the literature [40, 36, 71, 37] and is credited to both [72] and [73]. We reproduce it here in view of its relevance to our work:
Theorem 2.2.6. A bipartite quantum state ρ on HL ⊗ HR is unentangled (or sepa- rable) if and only if it is arbitrarily sharable. L R Proof. (⇐) Let ρ be separable. Then for some set of density operators {ρi , ρi }, it P L R P can be written as ρ = i λiρi ⊗ ρi , with i λi = 1. Let n and m be arbitrary, and let the N-partite state w, be defined as follows:
X L ⊗m R ⊗n w = λi(ρi ) ⊗ (ρi ) , i with N = m + n. By construction, the state of each L-R pair is ρ, since it follows straighforwardly that Eq. (2.3) is obeyed for each i, j. Thus, w is a valid sharing state. (⇒) Since ρ is arbitrarily sharable, there exists a sharing state w for arbitrary values of m, n. In particular, we need only make use of a sharing state w for m = 1 and arbitrarily large n, whence we let n → ∞. Given w, let us construct another sharing state w˜, which is invariant under permutations of the right subsystems, that is, let 1 X † w˜ = Vπ wVπ, |Sn| π∈Sn ⊗n where Sn ≡ {π} is the permutation group of n objects, acting on H via the natural Q R n-fold representation, Vπ( i |ψii) = ⊗i|ψπ(i)i, i = 1, . . . , n. It then follows that w˜ shares ρ:
1 X † TrL,ˆ Rˆ (w ˜) = TrL,ˆ Rˆ Vπ wVπ |Sn| π∈Sn 1 X 1 X = Tr (w) = ρ = ρ. L,πˆ (Rˆi) |Sn| |Sn| π∈Sn π∈Sn
⊗∞ Having established the existence of a symmetric sharing state w˜ ∈ D(HL ⊗ HR ), Fannes’ Theorem (see section 2 of [73]) implies the existence of a unique representation P i i i of w˜ as a sum of product states, w˜ = i λiρL ⊗ ρR ⊗ ρR ⊗ .... Reducing w˜ to any L-R pair leaves a separable state. Thus, if ρ is 1-n sharable it must be separable. As we alluded to before, a Corollary of this result is that in fact 1-∞ sharability implies ∞-∞ sharability. In closing this section, we also briefly mention the concept
19 Quantum marginals: sharability and joinability
(1) ⊗p of exchangeability [74, 75]. A density operator ρ on (H1 ) is said to be exchangeable if it is symmetric under permutation of its p subsystems and if there exists a symmetric (1) ⊗(p+q) state w on (H1 ) such that the reduced states of any subset of p subsystems is ρ for all q ∈ N. Similar to sharability, exchangeability implies separability. However, the converse only holds in general for sharability: clearly, there exist states which are separable but not exchangeable, because of the extra symmetry requirement. Thus, the notion of sharability is more directly related to entanglement than exchangeability is.
2.3 Joining and sharing Werner and isotropic states
Even for the simplest case of two bipartite states with an overlapping marginal, a general characterization of joinability is extremely non-trivial. As remarked, no con- ditions yet exist which are both necessary and sufficient for two arbitrary density op- erators to be joinable; although, conditions that are separately necessary or sufficient have been recently derived [56]. In this Section, we present a complete characteriza- tion of the three-party joining scenario and the 1-n sharability problem for Werner and isotropic states on arbitrary subsystem dimension d. We begin by introducing the relevant families of quantum and classical states to be considered.
2.3.1 Werner and isotropic qudit states, and their classical analogues
The usefulness of bipartite Werner and isotropic states is derived from their simple analytic properties and range of mixed state entanglement. For a given subsystem dimension d, Werner states are defined as the one-parameter family that is invariant under collective unitary transformations [57] (see also [75]), that is, transformations of the form U ⊗ U, for arbitrary U ∈ U(d). The parameterization which we employ is given by d I 1V ρ(Ψ−) = (d − Ψ−) + Ψ− − , d2 − 1 d2 d d where V is the swap operator, defined by its action on any product ket, V |ψφi ≡ |φψi. This parameterization is chosen because Ψ− is a Werner state’s expectation value with respect to V , Ψ− = Tr[V ρ(Ψ−)]. Non-negativity is ensured by −1 ≤ Ψ− ≤ 1, and the completely mixed state corresponds to Ψ− = 1/d. Furthermore, the concurrence of Werner states is simply given by [76]
C(ρ(Ψ−)) = −Tr V ρ(Ψ−) = −Ψ−, Ψ− ≤ 0. (2.5)
20 2.3 Joining and sharing Werner and isotropic states
For Ψ− > 0, the concurrence is defined to be zero, indicating separability. Werner states have been experimentally characterized for photonic qubits, see e.g. [77]. In- terestingly, they can be dissipatively prepared as the steady state of coherently driven atoms subject to collective spontaneous decay [78]. Isotropic states are defined, similarly, as the one-parameter family that is invariant under transformations of the form U ∗ ⊗ U [79]. We parameterize these states as
d I 1 ρ(Φ+) = (d − Φ+) + Φ+ − |Φ+ihΦ+| , d2 − 1 d2 d where |Φ+i = p1/d P |iii. The value of the parameter is given by the expectation i h i + TA + value with respect to the partially transposed swap operator, Φ = Tr V(AB)ρ(Φ ) , and is related to the so-called “singlet fraction” [80] by Φ+ = dF . Non-negativity is now ensured by 0 ≤ Φ+ ≤ d, whereas the concurrence is given by [81], s 2 C(ρ(Φ+)) = (Φ+ − 1), Φ+ ≥ 1, (2.6) d(d − 1)
and is defined to be zero for Φ+ ≤ 1. Before introducing probability distributions that will serve as the analogue classi- cal states, we present an alternative way to think of Werner states, which will prove useful later. First, the highest purity, attained for the Ψ− = −1 state, is 2/[d(d − 1)], with the absolute maximum of 1 corresponding to the pure singlet state for d = 2. Second, collective projective measurements on a most-entangled Werner state return only disagreeing outcomes (e.g., corresponding to |1i ⊗ |3i, but not |1i ⊗ |1i). The following construction of bipartite Werner states demonstrates the origin of both of these essential features. For generic d, the analogue to the singlet state is the following d-partite fully anti-symmetric state:
− 1 X |ψ i = √ sign(π)Vπ|1i|2i ... |di, (2.7) d d! π∈Sd
where, as before, Sd ≡ {π} denotes the permutation group and {|`i} is an orthonor- mal basis on H(1) ' Cd. The above state has the property of being “completely disagreeing”, in the sense that a collective measurement returns outcomes that dif- fer on each qudit with certainty. The most-entangled bipartite qudit Werner state − is nothing but the two-party reduced state of |ψd i. Thus, we can think of general bipartite qudit Werner states as mixtures of the completely mixed state with the − two-party-reduction of |ψd i. The inverse of 2/[d(d − 1)] (the purity) is precisely the number of ways two “dits” can disagree. Understanding bipartite Werner states to
21 Quantum marginals: sharability and joinability
− arise from reduced states of |ψd i will inform our construction of the classical analogue states, and also help us understand some of the results of Sec. 2.3.3 and 2.3.5.
For Werner states, increased entanglement corresponds to increased “disagree- ment” for collective measurement outcomes. For isotropic states, increased entangle- ment corresponds to increased “agreement” of collective measurements, but only with respect to the computational basis {|ii} relative to which such states are defined. It is this expression of agreement vs. disagreement of outcomes which carries over to the classical analogue states, which we are now ready to introduce. The relevant probabil- ity distributions are defined on the outcome space Ωd × Ωd = {1, . . . , d} × {1, . . . , d}. To resemble Werner and isotropic quantum states, these probability distributions should have completely mixed marginal distributions and range from maximal dis- agreement to maximal agreement. This is achieved by an interpolation between an even mixture of “agreeing pure states”, namely, (1, 1), (2, 2),..., (d, d), and an even mixture of all possible “disagreeing pure states”, namely, (1, 2),..., (1, d), (2, 1),..., (d, d − 1). That is: α 1 − α p(A = i, B = j) = δ + (1 − δ ), (2.8) α d i,j d(d − 1) i,j
where α is the probability that the two outcomes agree.
To make the analogy complete, it is desirable to relate α to both Ψ− and Φ+. We define α in the quantum cases to be the probability of obtaining |ki on system A, conditional to outcome |ki on system B for the projective measurement {|ijihij|}. For Werner states, this probability is related to Ψ− by
Ψ− + 1 p(|ki | |ki ) = ≡ α , (2.9) A B W d + 1 W and, similarly for isotropic states, we have
Φ+ + 1 p(|ki | |ki ) = ≡ α . (2.10) A B I d + 1 I We may thus re-parameterize both the Werner and isotropic states in terms of their respective above-defined “probabilities of agreement”, namely:
d I 1V ρ(α ) = (1 − α ) + α − , (2.11) W d − 1 W d2 W d d d I 1 ρ(α ) = (1 − α ) + α − |Φ+ihΦ+| , (2.12) I d − 1 I d2 I d
22 2.3 Joining and sharing Werner and isotropic states
subject to the conditions 2 1 0 ≤ α ≤ , ≤ α ≤ 1. W d + 1 d + 1 I
For Werner states, αW can rightly be considered a probability of agreement be- cause it is independent of the choice of local basis vectors in the projective measure- † † ment {U ⊗ U|ijihij|U ⊗ U }. For isotropic states, αI does not have as direct an interpretation. We may nevertheless interpret α as a probability of basis-independent agreement if we pair local basis vectors on A with their complex conjugates on B. In other words, αI can be thought of as the probability of agreement for local projective measurements of the form {U ∗ ⊗ U|ijihij|U ∗† ⊗ U †} 1.
2.3.2 Classical joinability limitations In order to determine the joinability limitations in the classical case, we begin by not- ing that any (finite-dimensional) classical probability distribution is a unique convex combination of the pure states of the system. In our case, there are five extremal three-party states, for which the two-party marginals are classical analogue states, as defined in Eq. (2.8). These are
1 X p(A, B, C agree) = (i, i, i), d i 1 X p(A, B agree) = (i, i, j), d(d − 1) i6=j 1 X p(A, C agree) = (i, j, i), d(d − 1) i6=j 1 X p(B,C agree) = (j, i, i), d(d − 1) i6=j 1 X p(all disagree) = (i, j, k), d(d − 1)(d − 2) i6=j6=k
1In principle, one might question the validity of our analogy on account of the quantum states being more pure than the mixed classical states in Eq. (2.8), e.g., the classical analogue with α = 1 may seem closer to the separable quantum state (|11ih11|+...+|ddihdd|)/d. We note, however, that any POVM gives a mapping from a quantum state to a probability distribution; we have chosen here two families of quantum states for which the resulting probability distribution is minimally dependent on the choice of measurement, as argued. Furthermore, we aim to understand which features of joinability limitations are of a quantum origin and which are simply due to classical P 2 limitations. In contrast, the family of “classical-like” quantum states p i |iiihii|/d + (1 − p)I/d would be un-interesting, as manifestly only constrained by classical joining limitations.
23 Quantum marginals: sharability and joinability
where (i, j, k) stands for the pure probability distribution p(A, B, C) = δA,iδB,jδC,k. The first four of these states are valid for all d ≥ 2 and each corresponds to a vertex of a tetrahedron, as depicted in Fig. 2.1(left). The fifth state is only valid for d ≥ 3 and corresponds to the point (αAB, αAC , αBC ) = (0, 0, 0) in Fig. 2.1(right). Any valid three-party state for which the two-party marginals are classical analogue states must be a convex combination of the above states. Therefore, the joinable-unjoinable boundary is delimited by the boundary of their convex hull. For the d = 2 case, the inequalities describing these boundaries are explicitly given by the following:
p(A, B, C agree) ≥ 0 ⇒ αAB + αAC + αBC ≥ 1,
p(C disagrees) ≥ 0 ⇒ −αAB + αAC + αBC ≤ 1,
p(B disagrees) ≥ 0 ⇒ αAB − αAC + αBC ≤ 1,
p(A disagrees) ≥ 0 ⇒ αAB + αAC − αBC ≤ 1,
where each inequality arises from requiring that the corresponding extremal state has a non-negative likelihood. In the d ≥ 3 case, the inequality p(A, B, C agree) ≥ 0 is replaced by αAB, αAC , αBC ≥ 0.
2.3.3 Joinability of Werner and isotropic qudit states We now present our results on the three-party joinability of Werner and isotropic states and then compare them to the classical limitations just found in the previous section. While, as mentioned, the proofs have been relegated to the published version of the paper [61], the basic idea is to exploit the high degree of symmetry that these classes of states enjoy. Consider Werner states first. Our starting point is to observe that if a tripartite state wABC joins two reduced Werner states ρAB and ρAC , then the “twirled state” w˜ABC , given by Z † w˜ABC = (U ⊗ U ⊗ U) wABC (U ⊗ U ⊗ U) dµ(U), (2.13)
is also a valid joining state. In Eq. (2.13), µ denotes the invariant Haar measure on U(d), and the twirling super-operator effects a projection into the subspace of operators with collective unitary invariance [82]. By invoking the Schur-Weyl duality [83], the guaranteed existence of joining states with these symmetries allows one to narrow the search for valid joining states to the Hermitian subspace spanned by representations of subsystem permutations, that is, density operators of the form X w = µπVπ, w ∈ WW , (2.14)
π∈S3
24 2.3 Joining and sharing Werner and isotropic states
(a) Werner joinability limitations (d=2) (b) Werner joinability limitations (d=5)
Figure 2.1: Three-party quantum and classical joinability limitations for Werner and isotropic states, and their classical analogue, as parameterized by Eqs. (2.11), (2.12), (2.8), respectively. a) Qubit case, d = 2. The Werner state boundary is the surface of the darker cone with its vertex at (2/3, 2/3, 2/3), whereas the isotropic state boundary is the surface of the lighter cone with its vertex at (1/3, 1/3, 2/3). The classical boundary is the surface of the tetrahedron. b) Higher-dimensional case, d = 5. The Werner state boundary is the surface of the bi-cone with vertices at (0, 0, 0) and (1/3, 1/3, 1/3), whereas the isotropic state boundary is the flattened cone with its vertex at (1/6, 1/6, 1/3). The classical boundary is the surface of the two joined tetrahedra. In both panels the grey line resting on top of the cones indicates the colinearity of the cone surfaces along this line segment.
25 Quantum marginals: sharability and joinability
∗ where Hermiticity demands that µπ = µπ−1 . Given wABC which joins Werner states, each subsystem pair is characterized by the expectation value with the respective − swap operator, Ψij = Tr[wABC (Vij ⊗ Iij)], where i, j ∈ {A, B, C} with i 6= j. Hence, − − − the task is to determine for which (ΨAB, ΨBC , ΨAC ) there exists a density operator wABC consistent with the above expectations. Our main result is the following:
− − − Theorem 2.3.1. Three Werner qudit states with parameters ΨAB, ΨBC , ΨAC are join- − − − able if and only if (ΨAB, ΨBC , ΨAC ) lies within the bi-cone described by
2 − − 2 − 1 ± Ψ− ≥ Ψ + ωΨ + ω Ψ , (2.15) 3 BC AC AB for d ≥ 3, or within the cone described by
2 − − 2 − 1 − Ψ− ≥ Ψ + ωΨ + ω Ψ , Ψ− ≥ 0, (2.16) 3 BC AC AB for d = 2, where 1 − − − i 2π Ψ− = (Ψ + Ψ + Ψ ), ω = e 3 . (2.17) 3 AB BC AC
Similarly, if a tripartite state wABC joins isotropic states ρAB and ρAC , then the “isotropic-twirled state” w˜ABC , given by Z ∗ ∗ † w˜ABC = (U ⊗ U ⊗ U) wABC (U ⊗ U ⊗ U) dµ(U), (2.18)
is also a valid joining state. A clarification is, however, in order at this point: although we have been referring to the isotropic joinability scenario of interest as three-party isotropic state joining, this is somewhat of a misnomer because we effectively consider the pair B-C to be in a Werner state, as evident from Eq. (2.18). Compared to Eq. (2.14), the relevant search space is now partially transposed relative to subsystem A, that is, consisting of density operators of the form
X TA w = µπVπ , w ∈ Wiso. (2.19) π∈S3 Our main result for three-party joinability of isotropic states is then contained in the following:
Theorem 2.3.2. Two isotropic qudit states ρAB and ρAC and qudit Werner state + + − + + − ρBC with parameters ΦAB, ΦAC , ΨBC are joinable if and only if (ΦAB, ΦAC , ΨBC ) lies within the cone described by
+ + − ΦAB + ΦAC − ΨBC ≤ d , (2.20)
26 2.3 Joining and sharing Werner and isotropic states
+ + − 1 + ΦAB + ΦAC − ΨBC ≥ (2.21) r − 2d iθ + −iθ + d(Ψ − 1) + (e Φ + e Φ ) , BC d − 1 AB AC e±iθ = ±ip(d + 1)/(2d) + p(d − 1)/(2d), or, for d ≥ 3, within the convex hull of the above cone and the point (0, 0, −1). The results of Theorems 2.3.1 and 2.3.2 as well as of Sec. 2.3.2 are pictorially sum- marized in Fig. 2.1. We now compare these quantum joinability limitations to the joinability limita- tions in place for classical analogue states. As described in Sec. 2.3.2, the non- negativity of p(A, B, C agree) and p(A disagrees) is enforced by the two inequalities αAB + αAC + αBC ≥ 1 and −αAB − αAC + αBC ≥ 1, respectively. We expect the same requirement to be enforced by the analogue quantum-measurement statistics. For d = 2, the bases of the Werner and isotropic joinability-limitation cones are − − − + + − determined by ΨAB + ΨAC + ΨBC ≥ 0 and ΦAB + ΦAC − ΨBC ≤ 2, respectively. Writing down each of these parameters in terms of the appropriate probability of agreement α, as defined in Eqs. (2.9) and (2.10), we obtain αAB + αAC + αBC ≥ 1 and −αAB − αAC + αBC ≥ 1. Hence, for qubits, part of the quantum joining limita- tions are indeed derived from the classical joining limitations. This is also illustrated in Fig. 2.1(left). Of course, one would not expect the quantum scenario to exhibit violations of the classical joinability restrictions; still, it is interesting that states which exhibit manifestly non-classical correlations may nonetheless saturate bounds obtained from purely classical joining limitations. For d ≥ 3, the only classical boundary which plays a role is the one which bounds + + − the base of the isotropic joinability-limitation cone: ΦAB +ΦAC −ΨBC ≤ d. Again, in terms of the agreement parameters, this is (just as for qubits) −αAB −αAC +αBC ≥ 1. In the Werner case, the quantum joinability boundary is not clearly delineated by the classical joining limitations. We can nevertheless make the following observation. By the non-negativity of Werner states, the three-party joinability region in Fig. 2.1(right) is required to lie within a cube of side-length 2/(d + 1) with one corner at (0, 0, 0). Consider the set of cubes obtained by rotating from this initial cube about an axis through (0, 0, 0) and (2/(d + 1), 2/(d + 1), 2/(d + 1). It is a curious fact that the exact quantum Werner joinability region (the bi-cone) is precisely the intersection of all such cubes. Another interesting feature is that there exist trios of unentangled Werner states − − − which are not joinable. For example, the point (ΨAB, ΨAC , ΨBC ) = (1, 1, 0) cor- responds to three separable Werner states that are not joinable. This point is of particular interest because its classical analogue is joinable. Translating (1, 1, 0) into the agreement-probability coordinates, (αAB, αAC , αBC ) = (2/3, 2/3, 1/3), we see that
27 Quantum marginals: sharability and joinability this point is actually on the classical joining limitation border. Thus, these three sepa- rable, correlated states are not joinable for purely quantum mechanical reasons. Note that the point (αAB, αAC , αBC ) = (2/3, 2/3, 1/3) does correspond to a joinable trio of pairs in the isotropic three-party joining scenario: this point lies at the center of the face of the isotropic joinability cone, as seen in Fig. 2.1(left). The same fact holds for (2/3, 1/3, 2/3) or (1/3, 2/3, 2/3) when the Werner state pair in the isotropic joining scenario describes A-C or A-B, respectively; in both cases, we would have obtained yet another cone in Fig. 2.1 that sits on a face of the classical tetrahedron boundary. Having determined the joinable trios of both Werner and isotropic states, we are now in a position to also answer the question of what pairs A-B and A-C of states are joinable with one another. In the Werner state case, this is obtained by projecting − − the Werner joinability bicone down to the ΨAB-ΨAC plane, resulting in the following:
− − Corollary 2.3.3. Two pairs of qudit Werner states with parameters ΨAB and ΨAC − − 1 are joinable if and only if ΨAB, ΨAC ≥ − 2 , or if the parameters satisfy 1 (Ψ− + Ψ− )2 + (Ψ− − Ψ− )2 ≤ 1, (2.22) AB AC 3 AB AC
− − 1 or additionally, in the case d ≥ 3, if ΨAB, ΨAC ≤ 2 . For isotropic states, we may similarly project the cone of Eq. (2.21) onto the + + ΦAB-ΦAC plane to obtain the 1-2 joining boundary. This yields the following:
+ + Corollary 2.3.4. Two pairs of qudit isotropic states with parameters ΦAB and ΦAC are joinable if and only if they lie within the convex hull of the ellipse
(Φ+ /d + Φ+ /d − 1)2 (Φ+ /d − Φ+ /d)2 AB AC + AB AC = 1, (2.23) (1/d2) (d2 − 1)/d2
+ + and the point (ΦAB, ΦAC ) = (0, 0). Lastly, by a similar projection of the isotropic cone given by Eqs. (2.20)-(2.21), we may explicitly characterize the Werner-isotropic hybrid 1-2 joining boundary:
+ Corollary 2.3.5. An isotropic state with parameter ΦAB and a Werner state with − parameter ΨBC are joinable if and only if they lie within the convex hull of the ellipse (Φ+ /d + Ψ− /d − 1)2 (Φ+ /d − Ψ+ /d)2 AB BC + AB BC = 1, (2.24) (1/d2) (d2 − 1)/d2
+ − and the point (ΦAB, ΨBC ) = (0, 1), and, for d ≥ 3, within the additional convex hull + − introduced by the point (ΦAB, ΨBC ) = (0, 1).
28 2.3 Joining and sharing Werner and isotropic states
(a) Werner state joinability (b) Isotropic state joinability
Figure 2.2: Two-party joinability limitations for Werner and isotropic qudit states. (a) Werner states. The shaded region corresponds to joinable Werner pairs, with the lighter region being valid only for d ≥ 3. The rounded boundary is the ellipse determined by Eq. (2.22). This explicitly shows the existence of pairs of entangled Werner states that are within the circular boundary determined by the weak CKW inequality, Eq. (2.2), yet are not joinable. (b) Isotropic states. The three regions correspond to the joinable pairs of isotropic states for d = 2, d = 3 and d = 1000. This shows how, in the limit of large d, the trade-off in isotropic state parameters becomes linear, consistent with known results on d-dimensional quantum cloning [84].
29 Quantum marginals: sharability and joinability
The above results give the exact quantum-mechanical rules for the two-pair join- ability of Werner and isotropic states, as pictorially summarized in see Figs. 2.2a and 2.2b. A number of interesting features are worth noticing. First, by restricting to the − − line where ΨAB = ΨAC , we can conclude that qubit Werner states are 1-2 sharable if and only if Ψ− ≥ −1/2, whereas for d ≥ 3, all qudit Werner states are 1-2 sharable. As we shall see, this agrees with the more general analysis of Sec. 2.3.5. Second, some insight into the role of entanglement in limiting joinability may be gained. In the first quadrant of Fig. 2.2a, where neither pair is entangled, it is no surprise that no joinability restrictions apply. Likewise, it is not surprising to see that, in the third quadrant where both pairs are entangled, there is a trade-off between the amount of entanglement allowed between one pair and that of the other. But, in the second and fourth quadrants we observe a more interesting behavior. Namely, these quadrants show that there is also a trade-off between the amount of classical correlation in one pair and the amount of entanglement in the other pair. In fact, the smoothness of the boundary curve as it crosses from one of the pairs being entangled to unentangled suggests that, at least in this case, entanglement is not the correct figure of merit in diagnosing joinability limitations.
2.3.4 Isotropic joinability results from quantum cloning Interestingly, the above results for 1-2 joinability of isotropic states can also be ob- tained by drawing upon existing results for asymmetric quantum cloning, see e.g. [84, 85] for 1-2 and 1-3 asymmetric cloning and [86, 87, 88] for 1-n asymmetric cloning. One approach to obtaining the optimal asymmetric cloning machine is to exploit the Choi isomorphism [89] to translate the construction of the optimal cloning map to the construction of an optimal operator (or a “telemapping state”). This connection is made fairly clear in [86, 90]; in particular, “singlet monogamy” refers to the trade-off in fidelities of the optimal 1-n asymmetric cloning machine or, equivalently, to the trade-off in singlet fractions for a (1 + n) qudit state. We describe how the approach to solving the optimal 1-n asymmetric cloning problem may be rephrased to solve the 1-n joinability problem for isotropic states. The state |Ψi described in Eq. (4) of [86] is a 1-n joining state for n isotropic states characterized by singlet fractions F0,j (related to the isotropic state parameter + by F0,j = Φ0,j/d, as noted). The bounds on the singlet fractions are determined by the normalization condition of |Ψi, together with the requirement that |Ψi be an eigenstate of a certain operator R defined in Eq. (3) of [86]. That |Ψi is an isotropic joining state is readily seen from its construction, and that it may optimize the singlet fractions (hence delineate the boundary in the {F0,j} space) is proven in [90]. Our contribution here is the observation that this result provides the solution to the 1-n joinability of isotropic states. The equivalence is established by the fact
30 2.3 Joining and sharing Werner and isotropic states that optimality is preserved in either direction by the Choi isomorphism. Quantitatively, the boundary for 1-n optimal asymmetric cloning, is given by Eq. (6) in [86] in terms of singlet fractions. Specializing to the 1-2 joining case and rewriting in terms of Φ+, we have
1 q q 2 Φ+ + Φ+ ≤ (d − 1) + Φ+ + Φ+ . AB AC n + d − 1 AB AC As one may verify, this is equivalent to the result of Corollary 2.3.4. In light of this connection, the fact that, as d increases, the isotropic-joinability cone of Fig. 2.1(right) becomes flattened down to the αAB-αAC plane is directly related to the linear trade-off in the isotropic state parameters for the semi-classical limit d → ∞, as discussed in [84]. Within our three-party joining picture, we can give a partial explanation of this fact: namely, it is a consequence of the classical joining boundary in tandem with the upper limit on the agreement parameter αBC for the Werner state on B-C: αBC ≤ 2/(d + 1). In the limit of d → ∞, these two boundaries conspire to limit the (A-B)-(A-C) isotropic state joining boundary to a triangle, as explicitly seen in Fig. 2.2b(right). For the general 1-n isotropic joining scenario, the quantum-cloning results addi- tionally imply the following: + + Theorem 2.3.6. A list of n isotropic states characterized by parameters Φ0,1,..., Φ0,j is 1-n joinable if and only if the (positive-valued) parameters satisfy
n n 2 X 1 X q Φ+ ≤ (d − 1) + Φ+ . (2.25) 0,j n + d − 1 0,j j=1 j=1
Interestingly, similar to our discussion surrounding Eq. (2.2), the authors of [86] argue how the “singlet monogamy” bound can lead to stricter predictions (e.g., on ground-state energies in many-body spin systems) than the standard monogamy of entanglement bounds based on CKW inequalities [38, 41].
2.3.5 Sharability of Werner and isotropic qudit states We next turn to sharability of Werner and isotropic states in d dimension, beginning from the important case of 1-n sharing. For Werner states, a proof based on a representation-theoretic approach is given in [61]. Although we expect a similar proof to exist for isotropic states, we obtain the desired 1-n sharability result by building on the relationship with quantum cloning problems highlighted above. We then outline a constructive procedure for determining the more general m-n sharability of Werner states.
31 Quantum marginals: sharability and joinability
Our main results are contained in the following: Theorem 2.3.7. A qudit Werner state with parameter Ψ− is 1-n sharable if and only if d − 1 Ψ− ≥ − . (2.26) n Theorem 2.3.8. A qudit isotropic state with parameter Φ+ is 1-n sharable if and only if d − 1 Φ+ ≤ 1 + . (2.27) n Proof. Specializing Eq. (2.25) to the case of equal parameters for all n isotropic states, the above result immediately follows. As stated in [86], this is consistent with the well known result for optimal 1-n symmetric cloning. We depict the qubit case (d = 2) of the above result in Fig. 2.3. In the case of Werner state sharing, Eq. (2.26) implies that a finite parameter range exists where the corresponding Werner states are not sharable. In contrast, for d ≥ 3, every − Werner state is at least 1-2 sharable. This simply reflects the fact that |ψd i (recall Eq. (2.7)) provides a 1-(d − 1) sharing state for a most-entangled qudit Werner state. With isotropic state sharing, for all d there is, again, a finite range of isotropic states which are not sharable. The simplicity of the results in Eqs. (2.26)-(2.27) is intriguing and begs for intu- itive interpretations. Consider a central qudit surrounded by n outer qudits. If the central qudit is in the same Werner or isotropic state with each outer qudit, then Theorems 2.3.7 and 2.3.8 can be reinterpreted as providing a bound on the sums of concurrences. For Werner states, we have that the sum of all the central-to-outer concurrences cannot exceed the number of modes by which the systems may disagree (i.e., d−1). In the isotropic state case, the sum of the n pairwise concurrences cannot p exceed the maximal concurrence value given by Cmax,d = 2(d − 1)/d. These rules do not hold in more general joining scenarios, as we already know from Sec. 2.3.3. There, we found that the trade-off between A-B concurrence and A-C concurrence is not a linear one, as such a simple “sum rule” would predict; instead, it traces out an ellipse (recall Fig. 2.2a). Starting from the proof of Thm. 2.3.7 found in [61] in conjunction with similar representation-theoretic tools, it is possible to devise a constructive algorithm for determining the m-n sharability of Werner states. The basic observation is to re- alize that the most-entangled m-n sharable Werner state corresponds to the largest eigenvalue of a certain Hamiltonian operator Hm,n, which is in turn expressible in terms of Casimir operators. Calculation of these eigenvalues may be obtained using Young diagrams. Although we lack a general closed-form expression for max(Hm,n), the required calculation can nevertheless be performed numerically. Representative results for n-m sharability of low-dimensional Werner states are shown in Table 2.1.
32 2.4 Further remarks
Figure 2.3: Pictorial summary of sharability properties of qubit Werner and isotropic states, according to Eqs. (2.26) and (2.27). The arrow-headed lines depict the pa- rameter range for which states satisfy each of the sharability properties displayed to the right and left, respectively. The vertical ticks between end points of these ranges indicate the points at which subsequent 1-n sharability properties begin to be satisfied.
2.4 Further remarks
2.4.1 Joinability beyond the three-party scenario
In Sec. 2.3, we focused on considering joinability of three bipartite (Werner or isotropic) states in a “triangular fashion”, namely, relatively to the simplest over- (3) lapping neighborhood structure N1 = {A, B}, N2 = {A, C} on H . In a more general N-partite scenario, other neighborhood structures and associated joinability problems may naturally emerge. For instance, we may want to answer the follow- ing question: Which sets of N(N − 1)/2 Werner-state (or isotropic-state) pairs are joinable? The approach to solving this more general problem parallels the specific three-party case we discussed. If each pair is in a Werner state, then if a joining state exists, there must exist a joining state with collective invariant symmetry (that is, invariant under arbitrary collective unitaries U ⊗N ). Thus, we need only look in the set of states respecting this symmetry. Any such operator may be decomposed into a sum of operators, which each have support on just a single irreducible subspace. This is useful because posi-
33 Quantum marginals: sharability and joinability
Table 2.1: Exact results for n-m sharability of Werner states for different subsystem dimension, with m and n increasing from left to right and from top to bottom in each table, respectively. For each sharability setting, the value −Φ is given. Asterisks correponds to entries whose values have not been explicitly computed.
n, m 1 2 3 4 5 1 1 1/2 1/3 1/4 1/5 2 1/2 1/2 1/3 1/4 1/5 d = 2 3 1/3 1/3 1/3 1/4 1/5 4 1/4 1/4 1/4 1/4 1/5 5 1/5 1/5 1/5 1/5 1/5
n, m 1 2 3 4 5 1 1 1 2/3 1/2 2/5 2 1 1/2 1/2 1/2 2/5 d = 3 3 2/3 1/2 1/3 1/3 1/3 4 1/2 1/2 1/3 1/4 1/4 5 2/5 1/3 1/3 1/4 ∗
n, m 1 2 3 4 5 1 1 1 1 3/4 3/5 2 1 1 2/3 1/2 1/2 d = 4 3 1 2/3 5/9 1/2 ∗ 4 3/4 1/2 1/2 ∗ ∗ 5 3/5 1/2 ∗ ∗ ∗ tivity of the joining operator when restricted to each irreducible subspace is sufficient for positivity of the overall operator. The joining operators may then be decomposed into the projectors on each irreducible subspace and corresponding bases of traceless operators on the projectors. The basis elements will be combinations of permuta- tion operators and the dimension of each such operator subspaces is given by the square of the hook length of the corresponding Young diagram [91]. The remaining task is to obtain a characterization of the positivity of the operators on each irre- ducible subspace. In [92], for example, a method for characterizing the positivity of low-dimensional operator spaces is presented. As long as the number of subsystems remains small, this approach grants us a computationally friendly characterization of positivity of the joining states. The bounds on the joinable Werner pairs may then be obtained by projecting the positivity characterization boundary onto the space of Werner pairs, analogous to the space of Fig. 2.2a. While a complete analysis is beyond our scope, a similar method may in principle
34 2.4 Further remarks
be followed to determine more general joinability bounds for isotropic states. How- ever, a twirling operation that preserves the joining property only exists for certain isotropic joining scenarios. For instance, we took this issue into consideration when we required the B-C system to be in a Werner state while A-B and A-C were isotropic states; it would not have been possible to take the same approach if all three pairs were isotropic states.
2.4.2 Sharability of general bipartite qubit states For qubit Werner states, one can use the methods of the proof of Thm. III.6 to show that 1-n sharability does imply n-n sharability [cf. Table I.(a)]. This property neither holds for Werner qudit states nor bipartite qubit states in general. The simplest example of a Werner state which disobeys this property is the most-entangled qutrit − Werner state ρ(Ψ = −1)d=3. This state is 1-2 sharable, as evidenced by the point (−1, −1, −1) lying within the bi-cone described by Eq. (2.15). The corresponding sharing state is the totally antisymmetric state on three qutrits as given by Eq. (2.7). This is the unique sharing state because the collective disagreement between the subsystems of each joined bipartite Werner state forces collective disagreement among the subsystems of the tripartite joining state; the totally antisymmetric state is the only quantum state satisfying this property. Since the only 1-2 sharing state − for ρ(Ψ = −1)d=3 is pure and entangled, clearly there can exist no 2-2 sharing. Additionally, we present below a counter-example that involves qubit states off the Werner line:
Proposition 2.4.1. For a generic bipartite qubit state ρ, 1-n sharability does not imply n-n sharability.
Proof. We claim that the following bipartite state on two qubits, 1 i ρ = |00i + |11i h00| + h11| + |10ih10| ≡ ρ , 3 L1R1 is 1-2 sharable but not 2-2 sharable. To show that ρ is 1-2 sharable, direct calculation shows that the two relevant partial-trace constraints uniquely identify w3 ≡ |ψihψ| as the only valid sharing state, with 1 |ψi ≡ √ (|000i + |101i + |110i). 3 The above state may in turn be equivalently written as
1 r2 1 |ψi = √ |0i ⊗ |00i + |1i ⊗ √ (|01i + |10i) . 3 3 2
35 Quantum marginals: sharability and joinability
In order for ρ to be 2-2 sharable, a four-partite state w4 must exist, such that Tr (w ) = ρ, for i, j = 1, 2. Any state which 2-2 shares ρ must then 1-2 share LˆiLˆj 4 the pure entangled state w3. That is, in constructing the 2-2 sharing state for ρ, we bring in a fourth system L2 which must reduce (by tracing over L1 or L2) to w3. But, since w3 is a pure entangled state, it is not sharable. Thus, there cannot exist a 2-2 sharing state for ρ. We conclude by stressing that our Werner and isotropic state sharability results allow in fact to put bounds (though not necessarily tight ones) on the sharability of an arbitrary bipartite qudit state. It suffices to observe that any bipartite state can be transformed into a Werner or isotropic state by the action of the respective twirling map (either Eq. (2.13) or (2.18)). Theorem 2.2.5 proves that the sharability of a state cannot be decreased by a unitary mixture map, and hence twirling cannot decrease sharability. This thus establishes the following:
Corollary 2.4.2. A bipartite qudit state ρ is no more sharable than the Werner state Z † † ρ˜ ≡ U ⊗ UρV U ⊗ U dµ(U), and the isotropic state Z ∗ T † ρ¯ ≡ U ⊗ UρV U ⊗ U dµ(U),
† for any ρV = I ⊗ V ρ I ⊗ V , with V ∈ U(d). In the qubit case, for instance, any maximally entangled pure state can be trans- formed into |Ψ−i or |Φ+i by the action of some local unitary I ⊗ V . Thus, all maximally entangled pure qubit states and their “pseudo-pure” versions, obtained as mixtures with the fully mixed states, have the same sharability properties as the Werner/isotropic states.
36 Chapter 3
Joinability of causal and acausal relationships
37 Joinability of causal and acausal relationships
This chapter presents material that appeared in Journal of Physics A: Mathemat- ical and Theoretical, 48:035307 (2015), in an article titled “On state versus channel quantum extension problems: exact results for U ⊗ U ⊗ U symmetry”, which is joint work with Lorenza Viola.
3.1 Introduction
It has long been appreciated that many of the intuitive features of classical probability theory do not translate to quantum theory. For instance, every classical probability distribution has a unique decomposition into extremal distributions, whereas a gen- eral density operator does not admit a unique decomposition in terms of extremal operators (pure states). Entanglement is responsible for another distinctive trait of quantum theory: as vividly expressed by Schrödinger back in 1935 [93], “the best possible knowledge of a total system does not necessarily include total knowledge of all its parts,” in striking contrast to the classical case. Certain features of classi- cal probability theory do, nonetheless, carry over to the quantum domain. While it is natural to view these distinguishing features as a consequence of quantum theory being a non-commutative generalization of classical probability theory in an appropri- ate sense, thoroughly understanding how and the extent to which the purely quantum features of the theory arise from its mathematical structure remains a longstanding central question across quantum foundations, mathematical physics, and quantum information processing (QIP), see e.g. Refs. [94, 95, 96, 97]. In this chapter, we investigate a QIP-motivated setting which allows us to directly compare and contrast features of quantum theory with classical probability theory, namely, the relationship between the parts (subsystems) of a composite quantum sys- tem and the system as a whole. Specifically, building on our earlier work [98], we develop and investigate a general framework for what we refer to as quantum join- ability, which addresses the compatibility of different statistical correlations among quantum measurements on different systems. Arguably, the most familiar case of joinability is provided by the “quantum marginal” (aka “local consistency”) problem [27, 99]. In this case, we ask whether there exists a joint quantum state compatible with a given set of reduced states on (typically non-disjoint) groupings of subsys- tems. The quintessential example of a failure of joinability is the fact that two pairs of two-level systems (qubits), say, Alice-Bob (A-B) and Alice-Charlie (A-C), cannot simultaneously be described by the singlet state, |ψ−i = p1/2(|↑↓i − |↓↑i). A semi- nal exploration of this observation was carried out by Coffman, Kundu, and Wootters [38] and later dubbed the “monogamy of entanglement” [40]. In classical probability theory, a necessary and sufficient condition for marginal probability distributions on A-B and A-C to admit a joint probability distribution (or “extension") on A-B-C is that the marginals over A be equal [27, 62]. The analogous compatibility condition
38 3.1 Introduction remains necessary in quantum theory, but, as demonstrated by the above example, is clearly no longer sufficient. The identification of necessary and sufficient conditions in general settings with overlapping marginals remains an actively investigated open problem as yet [61, 56, 100]. Physically, standard state-joinability problems as formulated above for density op- erators, may be regarded as characterizing the compatibility of statistical correlations of two (or more) different subsystems at a given time. However, correlations between a single system before and after the action of a quantum channel – a completely posi- tive trace-preserving (CPTP) dynamical map – may also be considered, for example, in order to characterize the “location” of quantum information that one subsystem may carry about another [101] and/or the causal structure of the events on which probabilities are defined [96, 102]. The work in [102] thoroughly explores, in par- ticular, the idea of placing kinematic and dynamic correlations on equal footing, by introducing a formalism of “quantum conditional states” to represent the correlations of either bipartite quantum states or quantum channels as bipartite operators. With these ideas in mind, one may want to formulate a quantum marginal problem for quantum channels (see also Ref. [103]). For example, given two quantum channels MAB : B(HA) → B(HB) and MAC : B(HA) → B(HC ) (with B(H) denoting the space of bounded linear operators on H), one may ask whether there exists a quan- tum channel MABC : B(HA) → B(HB ⊗ HC ), whose reduced channels are MAB and MAC , respectively. A motivation for considering such channel-joinability problems is that questions regarding the optimality of paradigmatic QIP tasks such as quantum cloning [84, 85] or broadcasting [104] may be naturally recast as such. A fundamental tool here is the Choi-Jamiolkowski isomorphism [105, 89], which may been used to translate optimal cloning problems into quantum marginal problems [86, 106], and vice-versa [61]. Both monogamy of entanglement and the no-cloning theorem [14] have significant impli- cations for the behavior of quantum systems: the former effectively constrains the kinematics of a multipartite quantum system, while the latter constrains the dynam- ics of a quantum system (composite or not). As both of these fundamental concepts are closely related to respective quantum joinability problems, we are prompted to explore in more depth their similarities and differences. Identifying a general join- ability framework, able to encompass all such quantum marginal problems, is one of our main aims here. The content is organized as follows. In Section 3.2, we introduce and motivate the use of what we term the homocorrelation map as our main tool for representing quantum channels as bipartite operators. Despite the different motivation, this repre- sentation will share suggestive points of contact with the conditional-state formalism of [102]. Formally, we show how it enables a notion of quantum joinability that in- corporates all joinability problems of interest, and discuss ways in which different
39 Joinability of causal and acausal relationships
joinability problems may be (homomorphically) mapped into one another. In Section 3.3, we obtain a complete analytical characterization of some archetypal examples of low-dimensional quantum joinability problems. Namely, we address three-party join- ability of quantum states, quantum channels, and block-positive (or “local-positive”) operators, in the case that the relevant operators are invariant under the group of collective unitary transformations, that is, under the action of arbitrary transforma- tions of the form U ⊗ U ⊗ U. These examples allow us to distinguish the joinability limitations stemming from classical probability theory from those due to quantum theory and, furthermore, to contrast the joinability properties of quantum channels vs. states. In Section 3.4, we investigate a possible source for the stricter joinability bounds in quantum theory, as compared to classical probability theory. We introduce the notion of degree of agreement (disagreement), that is, the probability that a ran- dom local collective measurement yields same (different) outcomes, as given by an appropriate two-value POVM. We find that quantum theory places different bounds on the degree of agreement arising from quantum states than it does on that of quantum channels: while quantum states are limited in their degree of agreement, quantum channels are limited in their degree of disagreement. The differences in these bounds point to a crucial distinction between quantum channels and states. At least in the examples of Section 3.3 and a few others, these limitations suffice in fact to determine the bounds of joinability exactly. Possible implications of such bounds with regards to joinability properties of general quantum states and channels are also discussed.
3.2 General quantum joinability framework
We begin by reviewing the standard state-joinability (quantum marginal) problem, framing it in a language suitable for generalization. Given a composite Hilbert space (N) NN H = i=1 Hi, a joinability scenario is defined by a list of partial traces {Tr`k }, with each `k ⊆ [1,...,N], along with a set of allowed “joining operators,” W , which in this case is the set of positive trace-one operators acting on H(N); accordingly, we may
associate a joinability scenario with a 2-tuple (W, {Tr`k }). For a given joinability
scenario, the images of W under the Tr`k define a set of reduced states {Rk} = {Trk(W )}. For any list of states {ρk} ∈ {Rk}, the following definition then applies:
Definition 3.2.1. [State-Joinability] Given a joinability scenario described by the
pair (W ≡ {w| w ≥ 0}, {Tr`k }), the reduced states {ρk} ∈ {Rk} are joinable if there
exists a joining state w ∈ W such that Tr`k (w) = ρk for all k.
The first step toward achieving the intended generalization of the above definition to quantum channels is to represent the latter as bipartite operators. In the following subsection, we establish a tool to achieve this and highlight its broader utility.
40 3.2 General quantum joinability framework
3.2.1 Homocorrelation map and positive cones
One way to identify channels with bipartite operators is by use of the Choi-Jamiolkowski (CJ) isomorphism [89, 107]. This isomorphism, denoted J , identifies each map M ∈ L(HA, HB) with the state resulting from the map’s action on one member of a maximally entangled state:
1 X J (M) ≡ [I ⊗ M](|Φ+ihΦ+|) = |iihj| ⊗ M(|iihj|), (3.1) A d A ij √ + P where IA is the identity map on B(HA), |Φ i = i |iii/ dA and dA = dim(HA). + + TA We note that dA|Φ ihΦ | = V , where V is the swap operator on HA ⊗ HA and TA denotes partial transposition on subsystem A. The transformation is an isomorphism in that it preserves the positivity of the objects it maps to and from; namely, quantum channels (CPTP maps) are mapped to quantum states (positive trace-one operators). Consequently, J is a useful diagnostic tool for determining whether a map is CP 1. Here, we employ an alternative means of identifying quantum channels with bi- partite operators, building on an identification that was introduced for the special case of qubits in [108]. In this approach, basis-dependence is avoided by replacing the reference state with the normalized swap operator V/d. Since the latter is not a den- sity operator, this correspondence lacks an interpretation as a physical process. But, for our purposes, this comes at the greater benefit of yielding a resulting bipartite operator that bears the statistical properties of the corresponding channel. Formally, we define a homocorrelation map, H, which takes any map M ∈ L(HA, HB) (with L(HA, HB) being the set of linear maps, or “superoperators”, from B(HA) to B(HB)), to a “channel operator” MH ∈ B(HA ⊗ HB) according to
1 X H(M) ≡ [I ⊗ M](V/d ) = |iihj| ⊗ M(|jihi|), (3.2) A A d A ij
P 2 where V = i,j |ijihji| with respect to any orthonormal basis {|ii} . While the CJ isomorphism is a handy diagnostic tool, the homocorrelation map serves a different purpose. It does not take CP maps to positive operators. Instead, it takes each map to an operator which exhibits the same statistical correlations as that map:
1 + Note that J depends on a choice of local basis, needed to define |Φ i and TA. In order for the isomorphism to hold, the reference state (|Φ+ihΦ+| above) must be maximally entangled; again for d > 2, any such state reflects a choice of local bases. 2The homocorrelation map H is closely related to the causal conditional states defined in [102]. Namely, a channel operator resulting from the homocorrelation map is precisely the causal condi- tional state conditioned on ρA being the completely mixed state. Note that Eq. (3.2) above differs from Eq. (29) in [102] by a factor of dA.
41 Joinability of causal and acausal relationships
(a) (b)
Figure 3.1: (a) Commutativity diagram summarizing the relationship between the Choi-Jamiolkowski isomorphism and the homocorrelation map defined in Eqs. (3.2)- (3.1). In (b), the corresponding actions are given in terms of tensor network diagram notation [109]. Proposition 3.2.2 may be straightforwardly proved using this notation.
Proposition 3.2.2. A bipartite state ρ ∈ B(HA ⊗ HB) and a quantum channel M : B(HA) → B(HB) exhibit the same correlations, that is, 1 Tr[ρA ⊗ B] = Tr[M(A)B], ∀A ∈ B(HA),B ∈ B(HB). (3.3) dA if and only if the equality H(M) = ρ holds.
Proof. The two operators ρ and H(M) are equal if and only if their expectations Tr[ρA⊗B] = Tr[H(M)A⊗B] for all A, B. Thus, it suffices to show that Tr[H(M)A⊗ B] = 1 Tr[M(A)B] for all A, B. This equality may be established as follows: dA
Tr[H(M)A ⊗ B] = 1 P Tr[|iihj| ⊗ M(|jihi|)A ⊗ B] dA i,j = 1 P Tr[(|iihj|A) ⊗ (M(|jihi|)B)] dA i,j = 1 P hj|A|iiTr[M(|jihi|)B] = 1 Tr[M(A)B]. 2 dA i,j dA
Equation (3.3) may be taken as the defining property of the homocorrelation map. An example may explicitly demonstrate the utility of this representation. Consider the one-parameter family of qudit depolarizing channels [17], defined as
I D (ρ) = (1 − η)Tr(ρ) + ηρ. (3.4) η d
† The action of this channel commutes with all unitary channels in that Dη(UρU ) = † UDη(ρ)U . Under the homocorrelation map, the depolarizing channels are taken to
42 3.2 General quantum joinability framework operators with U ⊗ U symmetry, namely, I ⊗ I V H(D ) = (1 − η) + η , (3.5) η d2 d where V is, again, the swap operator. Trace-one, positive operators of this form are the well-known Werner states [57] (see also Sec. 3.3.1). Imagine that an observer does not know a priori whether her two measurements are made on distinct systems in a Werner state or if they are made on the same system before and after a depolarizing channel has been applied. If presented with a Werner state or depolarizing channel 1 1 having η = − d2−1 to d+1 , the observer will not be able to distinguish between the two cases. The homocorrelation map makes this operational identification explicit. To contrast, the CJ map takes the depolarizing channels to so-called isotropic states [80], I ⊗ I J (D ) = (1 − η) + η|Φ+ihΦ+|, (3.6) η d2 where as before |Φ+i is the maximally entangled state. The isotropic states are defined by their symmetry with respect to U ⊗ U T transformations. An observer in the scenario above would certainly be able to distinguish between the correlations of the depolarizing channel and the isotropic states, as long as η 6= 0. The distinction between the CJ isomorphism and the homocorrelation map can be further appreciated by contrasting the sets of operators they produce. The set of CP maps forms a cone in the set of superoperators L(HA, HB). Both the CJ isomorphism and the homocorrelation map are cone-preserving maps (by linearity) from L(HA, HB) to B(HA ⊗HB). While in the case of the CJ isomorphism, the resulting cone is exactly the cone of bipartite states, in the case of the homocorrelation map, the cone is distinct from the cone of states. One of the main findings of this chapter is that the correlations exhibited by bipartite states and the ones exhibited by quantum channels need not be equivalent. Furthermore, we find that this difference plays a role in their distinct joinability properties. The homocorrelation representation of channels provides us with a natural framework for exploring this difference: a channel and a state with differing correlations will be represented as distinct operators in the same operator space; as an example, the classes of bipartite Werner states and depolarizing channels are depicted in Fig. 3.2. These notions and their use in joinability are fleshed out in what follows. The cone of positive operators plays a central role in defining joinability of quan- tum states. Analogously, the cone of homocorrelation-mapped channels (or “channel- positive operators”) will play a central role in defining joinability of quantum channels.
Definition 3.2.3. [State-positivity] An operator M ∈ B(H) is state-positive if Tr(MP ) ≥ 0 for all Hermitian projectors P = P † = P 2 ∈ B(H). We notate this
43 Joinability of causal and acausal relationships
condition as M ≥st 0 and emphasize that the resulting set is a self-dual cone. Recall that a map M is a valid quantum channel if Tr[J (M)P ] ≥ 0 for all P = P 2 ∈ B(HA ⊗ HB) [89]. Using the relationships of Fig. 3.1, we translate this condition to one on the homocorrelation-mapped operator M = H(M). Specifically, we define:
Definition 3.2.4. [Channel-positivity] An operator M ∈ B(HA ⊗HB) is channel- positive with respect to the A-B bipartition if Tr(MP TA ) ≥ 0 for all Hermitian projec- † 2 tors P = P = P ∈ B(HA ⊗ HB). We notate this condition M ≥ch 0, and emphasize that the resulting set is, again, a self-dual cone.
V I
I – V/d
θ θ
0 T r M = 1 Figure 3.2: State- and channel-positive cones for two qudit Werner operators. The re- gion of the solid arc (blue) corresponds to state-positive operators, while the region of the dashed arc (pink) corresponds to channel-positive operators. The overlapping re- gion, seen as purple, corresponds to PPT operators; of these, the normalized operators are also unentangled state-positive operators. The self-dual nature of the state- and channel-positive cones is consistent with the right angles of each cone’s vertex. The 1 Young diagrams represent the corresponding projectors into the symmetric 2 (I + V ) 1 and antisymmetric 2 (I − V ) subspaces, respectively. For qudit dimension d, the angle θ is calculated to be cos θ = Tr[V (I + V )]/pTr[V 2]Tr[(I + V )2] = p(d + 1)/2d.
In the general case, we can give a characterization of the intersection of the two cones and their complements. This is aided by the fact that the CJ isomorphism and the homocorrelation map are related to one another by partial transpose. A commutivity diagram of these relationships is given in Fig. 3.1, where the tensor network diagram calculus [109] may be used to concisely demonstrate that, up to normalization, −1 −1 J ◦ H = H ◦ J = TA.
44 3.2 General quantum joinability framework
Proposition 3.2.5. A bipartite state ρ ∈ B(HA ⊗ HB) and a quantum channel M : B(HA) → B(HB) exhibit the same correlations if and only if the density operator (or equivalently, channel operator) has a positive partial transpose (PPT). Proof. By Prop. 3.2.2, if a bipartite state and a quantum channel exhibit the same correlations, then ρ = H(M). Since J is related to H by a partial trace, we also TA have H(M) = J (M)/dA. By the positivity preservation of J , M being CPTP TA implies that J (M) is a positive operator. Thus, we have that ρ = J (M)/dA is positive. This result may be used to directly connect quantum channels to entanglement:
Corollary 3.2.6. If the correlations of a bipartite state ρ ∈ B(HA ⊗ HB) cannot be exhibited by a quantum channel, then the state is entangled. Proof. Since the correlations cannot be exhibited by a quantum channel, the operator is not PPT, by Prop. 3.2.5. Then, by the Peres-Horodecki criterion [110], the state is necessarily entangled.
3.2.2 Generalization of joinability We are now poised to use the homocorrelation representation to define the joinability of channels. The channel-positive operators provide an alternative set with which to define the allowed joining operators W . As a warm-up, we rephrase the channel- joinability problem that was posed in the Introduction. Consider quantum channels from HA to HB ⊗HC . Under the homocorrelation map, these correspond to tripartite operators lying in the channel-positive cone, notated WA|BC . The partial traces TrC and TrB take channel-positive operators in WA|BC to channel-positive operators in WA|B and WA|C , respectively; that is, operators in WA|B and WA|C correspond to valid quantum channels via the homocorrelation map. The corresponding channel- joining scenario is then defined as (WA|BC , {TrC , TrB}). A channel-joinability problem presents two channel operators MAB ∈ WA|B and MAC ∈ WA|C and seeks to determine the existence of a channel operator MABC ∈ WA|BC which reduces to the two channel operators in question. In general, we thus have the following: Definition 3.2.7. [Channel-Joinability] Given a joinability scenario described by the pair (W ≥ch 0}, {Tr`k }), the reduced operators {Mk} ∈ {Rk} are joinable if there exists a joint operator M ∈ W such that Tr`k (M) = Mk for all k. We note that a channel joinability (or extension) problem can be stated using the CJ isomorphism instead of the homocorrelation map, as done in [103]. However, as we argued, the homocorrelation map provides a platform to directly compare the joinability of states and channels of equivalent correlations. For instance, it will allow
45 Joinability of causal and acausal relationships
us to simultaneously compare the joinability of local-unitary-invariant quantum states and channels, and consequently to compare these both to the joinability of analogous classical probability distributions (c.f. Fig. 3.5). Before proceeding to the general notion of joinability, we also remark that allowed joining operators in W have thus far been considered to be either state-positive or channel-positive. However, from a mathematical standpoint, a sensible joinability problem only needs W to be a convex cone. To investigate this generalization and (as motivated later) to meld state and channel joining, we consider a third type of positivity that we call local-positivity. This notion is equivalent to both block- positivity [111] and to map-positivity (not necessarily CP) [112, 113], in that by representing linear maps using the homocorrelation map, the cone of (transformed) positive maps is equal to the cone of bipartite block-positive operators. Formally:
Definition 3.2.8. [Local-positivity] An operator M ∈ B(HA⊗HB) is local-positive with respect to the A-B factorization if Tr(MPA ⊗ PB) ≥ 0 for all pure states PA = 2 2 PA ∈ B(HA) and PB = PB ∈ B(HB). We notate this condition M ≥loc 0. The set of channel-positive operators and state-positive operators are each sub-cones of the local-positive operators, as local-positivity clearly is a weaker condition. Local- positive operators are directly relevant to QIP, in particular because they may serve as an entanglement witnesses [114]. Moreover, in comparing quantum joinability limita- tions to analogous limitations stemming from classical probability theory, joinability scenarios defined with respect to W ≥loc 0 may allow the identification of quantum limitations in a “minimally constrained" setting, closer to the (less strict) classical boundaries. In Sec. 3.3.2, we find that local-positivity does nevertheless provide stricter-than-classical limitations on joinability. Another way of viewing the various definitions of positivity is to understand the subscript on the inequality to indicate the dual cone from which inner products with M must be positive. For M ≥st 0, M ≥ch 0, and M ≥loc 0, the respective dual cones are the positive span of rank-one projectors, the positive span of partially- transposed projectors, and the positive span of product projectors (from which the trace-one condition confines to the set of separable states). We note that the first two cones are self-dual (and, furthermore, symmetric [115]), while the local-positive cone is not. With several important examples of positivity established, each being a different convex set with which to define W , we are in a position to give the following: Definition 3.2.9. [General Quantum Joinability] Let W be a convex cone in (N) B(H ), and {Tr`k } be partial traces with `k ⊂ ZN . Given the joinability scenario
(W, {Tr`k }), the operators {Mk} ∈ {Rk} are joinable if there exists a joining operator
w ∈ W such that Tr`k (w) = Mk for all k. This general definition naturally encompasses the various joinability problems ref- erenced in the Introduction. Specifically, in the case where W is the set of quan-
46 3.2 General quantum joinability framework
tum states on a multipartite system, the joinability problem reduces to the quan- tum marginal problem, while if W consists of channel-positive operators describing quantum channels from one multipartite system to another, one recovers the channel- joining problem instead. Specific instances of this problem are the optimal asymmet- ric cloning problem [84, 85, 88], the symmetric cloning problem [116, 117], and the k-extendibility problem for quantum maps [118]. In addition to providing a unified perspective, our approach has the important advantage that different classes of join- ability problems may be mapped into one another, in such a way that a solution to one provides a solution to another. This is made formal in the following:
Proposition 3.2.10. Let W and W 0 be two positive cones of operators acting on (N) the space H , let {Tr`k } be a set of partial traces that apply to both cones, and let φ : W → W 0 be a positivity-preserving (homo)morphism, which permits reduced actions φk satisfying φk ◦ Tr`k = Tr`k ◦ φ. If {Mk} ∈ {Tr`k (W )} is joinable with 0 0 respect to W , then {φk(Mk)} ∈ {Tr`k (W )} is joinable with respect to W .
Proof. Assume that w is a valid joining operator for the operators {Mk} ∈ {Tr`k (W )}. 0 Then, the set of operators {φk(Mk)} ∈ {Tr`k (W )} is joined by the operator φ(w), 0 since Tr`k [φ(w)] = φk(Tr`k [w]) = φk(Mk) and φ(w) ∈ W .
This is shown in the commutative diagram of Fig. 3.3. We shall use a stronger corollary of this result in the remaining sections:
Corollary 3.2.11. Let φ be a one-to-one positivity-preserving map from W to W 0,
with invertible reduced actions φk satisfying φk ◦ Tr`k = Tr`k ◦ φ (and similarly for
their inverses). Then a set of operators {Mk} ∈ {Tr`k (W )} is joinable if and only if the set of operators φk(Mk) is joinable.
Proof. The forward implication follows from Proposition 3.2.10, while the backwards implication follows from the fact that φ and the φk are invertible, along with the contrapositive of Proposition 3.2.10.
The joinability-problem isomorphism we make use of is the partial transpose map. The latter permits a natural reduced action, namely, partial transpose on the remain- ing of the previously transposed subsystems. As explained, the partial transpose is a positivity-preserving bijection between states and channel operators (via H). Thus, if we determine the joinable-unjoinable demarcation for a class of states, we will determine the joinable-unjoinable demarcation for a corresponding class of channel- operators.
47 Joinability of causal and acausal relationships
Figure 3.3: Commutivity diagram showing a homomorphism of joinability problems.
3.3 Three-party joinability settings with collective invariance
In this Section, we obtain an exact analytical characterization of the state-joining, channel-joining, and local-positive joining problems in the three-party scenario, by taking advantage of collective unitary invariance. That is, we determine what trio of bipartite operators MAB, MAC , and MBC may be joined by a valid joining operator wABC , subject to the appropriate symmetry constraints. As noted, the most familiar case is state joinability, whereby the bipartite operators along with the joining tri- partite operator are state-positive. The next case considered is referred to as “1-2 channel joinability”: here, we specify a bipartition of the systems (say, A|BC) and consider the bipartite operators which cross the bipartition (MAB and MAC ), along with the joining operator, to be channel-positive with respect to the bipartition, while the remaining bipartite operator (MBC ) is state-positive. Since each of the three possible bipartition choices (A|BC, B|AC, and C|AB) constitutes a different channel joinability scenario, a total of four possibilities arise for three-party state/channel joinability. Lastly, motivated by the suggestive symmetry arising from these results and their relation to classical joining, we consider the weaker notion of local-positive joining, in which all operators involved are only required to be local-positive.
3.3.1 Joinability limitations from state-positivity and channel- positivity
We begin by describing the operators which are to be joined. The bipartite reduced operators inherit the collective unitary invariance from the tripartite operators from which they are obtained. Therefore, by a standard result of representation theory [119], the operators which are to be joined are of the following form:
I V ρ(η) = (1 − η) + η , (3.7) d2 d 48 3.3 Three-party joinability settings with collective invariance
where V is the swap operator defined earlier. The above operators are known to 1 1 be state-positive for the range − d−1 ≤ η ≤ d+1 , corresponding to the d-dimensional (qudit) Werner states we already mentioned. The parameterization is chosen so that η is a “correlation” measure: if d = 2, η = −1 corresponds to the singlet state, η = 0 to the maximally mixed state, while η = 1 is not a valid quantum state, but expresses perfect correlation for all possible collective measurements. Note that a value η = 1, for instance, does correspond to a valid quantum channel. Intuitively, channel-positive operators with U ⊗ U-invariance correspond to depolarizing channels. It is known that complete positivity (or channel-positivity) of the depolarizing channel is ensured 1 provided that − d2−1 ≤ η ≤ 1 [120]. However, we find it instructive to independently establish state- and channel- positivity bounds using the CJ isomorphism. To this end, we enlarge the above class of U ⊗ U-invariant operators to the class of operators with collective orthogonal invariance, namely, invariance under transfor- mations of the more general form O ⊗ O, belonging to the so-called Brauer algebra [121, 122]3. In addition to U ⊗ U-invariant operators, the Brauer algebra also con- tains U ∗ ⊗ U-invariant operators. The latter class of operators, which includes the well-known isotropic states, are spanned by the operators I and V TA . Thus, the set of O ⊗ O-invariant operators are of the form
I V V TA ρ(η, β) = (1 − η − β) + η + β . (3.8) d2 d d In particular, the operator ρ(0, 1) is a generic Bell state on two qudits, ρ(0, −1/(d−1)) is the maximally entangled Werner state (namely, the singlet state for d = 2), ρ(1, 0) is the identity channel, and ρ(0, 0) is the completely mixed state (or the completely depolarizing channel). We can then establish the following:
Proposition 3.3.1. A bipartite operator ρ(η) with collective unitary invariance is 1 channel-positive if and only if − d2−1 ≤ η ≤ 1. Proof. The Brauer algebra includes all the state-positive operators which are mapped, via the CJ isomorphism J , to the U ⊗ U-invariant channel-positive operators; under J , η and β in Eq. (3.8) are swapped with one another. Since J takes state-positive operators to channel-positive operators, we need only obtain the set of state-positive operators. State-positivity of these operators is enforced by the inner products with
3Operators in this algebra have been extensively analyzed in [123, 124], and recent work char- acterizing their irreducible representations may be found in [125, 126]. The Brauer algebra act- ing on N d-dimensional Hilbert spaces is spanned by representations of subsystem permutations {Vπ|π ∈ SN }, along with their partial transpositions with respect to groupings of the subsystems Tl {Vπ |π ∈ SN , l ⊆ {1,...,N}}. In terms of tensor network diagrams, each element of this basis is represented by a set of disjoint pairings of 2N vertices, with the vertices arranged in two rows, both containing N of them.
49 Joinability of causal and acausal relationships
respect to their (operator) eigenspaces being non-negative. Such eigenspaces are PA, P+, and PY , independent of η and β: the first is the anti-symmetric subspace, the second is the one-dimensional space spanned by |Φ+i, and the third is the space spanned by vectors |yi satisfying hy|(|yi)∗ = 04. The eigenvalues are as follows:
2 ρ(η, β)PA = [(1 − η − β)/d − η/d]PA, 2 ρ(η, β)P+ = [(1 − η − β)/d + η/d + β]P+, 2 ρ(η, β)PY = [(1 − η − β)/d + η/d]PY .
Hence, state-positivity of the bipartite Brauer operators is ensured by
1 ≥ (d + 1)η + β, 1 ≥ −(d − 1)η − (d2 − 1)β, 1 ≥ −(d − 1)η + β.
The inequalities bounding channel-positivity are obtained by swapping the ηs and βs. In particular, we recover that the state-positive range for U ⊗ U-invariant operators 1 1 1 is − d−1 ≤ η ≤ d+1 , whereas the channel-positive range is − d2−1 ≤ η ≤ 1. In a similar manner, we can also obtain the ranges of local-positivity: Proposition 3.3.2. A bipartite operator ρ(η) with collective unitary invariance is 1 local-positive if and only if − d−1 ≤ η ≤ 1. Proof. Local positivity is ensured by the non-negativity of expectation values with respect to the product vectors {|xxi, |xx¯i, |yyi, |yy¯i}, satisfying |xi∗ = |xi and |yi∗ = |y¯i, where the bar indicates a vector orthogonal to the original vector. In terms of η and β, these constraints read
2 0 ≤ hρ(η, β)ixx = (1 − η − β)/d + η/d + β/d, 2 0 ≤ hρ(η, β)ixx¯ = (1 − η − β)/d , 2 0 ≤ hρ(η, β)iyy = (1 − η − β)/d + η/d, 2 0 ≤ hρ(η, β)iyy¯ = (1 − η − β)/d + β/d.
More compactly, these boundaries are given by 1 − ≤ η + β ≤ 1, −(d − 1)η + β ≤ 1, η − (d − 1)β ≤ 1. d − 1 Thus, for bipartite Brauer operators, local-positive operators are equivalent to convex combinations of state- and channel-positive operators5. In particular, for the local-
4As noted, both the definition of |Φ+i and the use of complex conjugation are basis-dependent notions. It is understood that all usages of either refer to the same (arbitrary) choice of basis. 5This property is known as decomposability [112]. Interestingly, such an equivalence also holds for arbitrary bipartite qubit states [127].
50 3.3 Three-party joinability settings with collective invariance
(a) (b)
Figure 3.4: Positivity regions for bipartite Brauer operators: (a) d = 2. (b) d = 5. The solid triangle (blue) encloses the state-positive region, the dashed triangle (pink) encloses the the channel-positive region, and the outer boundary encloses the local- positive region.
1 positive range of U ⊗ U-invariant operators, it follows that − d−1 ≤ η ≤ 1, as stated.
A pictorial summary of the three positivity bounds is presented in Fig. 3.4. Having characterized all types of positivity for the (bipartite) operators to be joined, we now turn to characterize the positivity for the (tripartite) joining opera- tors. For each positive tripartite set (W ≥st 0, W ≥ch 0, and W ≥loc 0), we obtain the trios of joinable bipartite operators by simply applying the three partial traces (TrA, TrB, TrC ) to each positive operator. In more detail, our approach is to obtain an expression for the positivity boundary of the tripartite operators in terms of operator space coordinates, and then re-express this boundary in terms of reduced-state param- eters (the three Werner parameters in this case). For state- and channel-positivity, the desired characterization follows directly from the analysis reported in our previous work [61].
Corollary 3.3.3. With reference to the parameterization of Eq. (3.7), we have that: (i) Three Werner states with parameters (ηAB, ηAC , ηBC ) are joinable with respect to the (WABC ≥st 0, {TrA, TrB, TrC }) scenario if and only if
1 2 i2π/3 2 (1 − ηAB − ηAC − ηBC ) ≥ |ηAB + ωηAC + ω ηBC |, ω ≡ e , ηAB + ηAC + ηBC ≥ −1,
for d = 2, while for d ≥ 3 they need only satisfy
2 |ηAB + ωηAC + ω ηBC |.
51 Joinability of causal and acausal relationships
(ii) Three U ⊗ U-invariant operators with parameters (ηAB, ηAC , ηBC ) are channel- joinable with respect to the (WA|BC ≥ch 0, {TrA, TrB, TrC }) scenario if and only if r 1 2 2d iθ −iθ + ηAB + ηAC − ηBC ≥ + dηBC + (e ηAB + e ηAC ) , d − 1 d − 1 d − 1 eiθ ≡ p(d − 1)/2d ± ip(d + 1)/2d.
The channel-joinability limitations in the other two scenarios B|AC and C|AB may be obtained by permuting the ηs accordingly.
Proof. Result (i) corresponds to Theorem 3 in [61], re-expressed in terms of the parametrization of Eq. (3.7) (with reference to the notation of Eqs. (15)–(17) in [61], 2 − one has η` = (d/(d − 1))(Ψ` − 1/2), ` = AB, AC, BC). In order to establish (ii), note that J may be used to translate any U ∗⊗U-invariant state-positive joinability problem into a U ⊗ U-invariant channel-positive joinability problem, drawing on Corollary 3.2.11. Explicitly, under J (partial transpose in the ∗ case of operators), the U ⊗ U ⊗ U-invariant state-positive operators WA∗BC are in one-to-one correspondence with the U ⊗ U ⊗ U-invariant channel-positive operators WA|BC . Hence, by the joinability isomorphism induced by the partial transpose, the solution to a joinability problem of the scenario (WA∗BC , {TrA, TrB, TrC }) gives a solution to a corresponding joinability problem of (WA|BC , {TrA, TrB, TrC }). Thus, to obtain the depolarizing channel-joinability boundaries, we simply translate the isotropic state parameters of Eqs. (20)-(21) in [61] into η parameters.
The joinability limitations of all four scenarios are depicted in Fig. 3.5. As stressed in [61], the quantum joinability limitations must adhere to the analogous classical joinability limitations (seen as the tetrahedra in Fig. 3.5). In the qubit case, we find it intriguing that the inclusion of the quantum channel-joinability limitations allows us to regain the tetrahedral symmetry imposed by the classical limitations; whereas each scenario on its own expresses a continuous rotational symmetry that is not reflected classically. In other words, if we consider the joinability scenario defined by (span{WABC ,WA|BC ,WB|AC ,WC|AB}, {TrA, TrB, TrC }), the joinable bipartite operators respect the tetrahedral symmetry suggested by the classical joinability bounds. This amounts to asking the question: what trios of bipartite correlations – as derivable from either quantum states or channels, or from probabilistic combinations of the two – may be obtained from the measurements on three systems? Though the result expresses the tetrahedral symmetry of the classical joinability limitations, these classical joinability limitations do not suffice to enforce the stricter quantum joinability limitations, as manifest in the fact that the corners
52 3.3 Three-party joinability settings with collective invariance
(a) (b)
Figure 3.5: Joinability of operators on the β = 0 line for (a) d = 2 and (b) d = 5 (each from a different perspective). State-positivity, along with channel-positivity with respect to each of the three bipartitions, obtains the four cones depicted here. The joinability limitations for classical probability distributions are given by (a) the tetrahedron with black edges and (b) the union of the two tetrahedra with black edges. of the classical joinability tetrahedron are not reached by the quantum boundaries. We diagnose such limitations as strictly quantum features that do not have classical analogues – as we will discuss later in this work.
3.3.2 Joinability limitations from local-positivity
We now explore how local-positive joinability (a strictly weaker restriction, as noted) relates to the state/channel-joinability limitations above, as well as to the underlying classical limitations. As of yet, we only know that the local-positive limitations will lie between the classical and the quantum boundaries. Since obtaining a simple analytical characterization for arbitrary subsystem dimension d appears challenging in the local-positive setting, and useful insight may already be gained in the lowest- dimensional (qubit) setting, we focus on d = 2 in this section. Our main result is contained in the following:
Theorem 3.3.4. With reference to Eq. (3.7), three qubit Werner operators (con- strained by local-positivity) with parameters (ηAB, ηAC , ηBC ) are joinable by a local-
53 Joinability of causal and acausal relationships
Figure 3.6: Boundary of local-positive Werner operators which are joinable via local- positive operators, as described by the Roman surface, see Theorem 3.3.4.
positive tripartite Werner operator w if an only if the following conditions hold:
1 + ηAB + ηAC + ηBC ≥ 0, 1 + ηAB − ηAC − ηBC ≥ 0,
1 − ηAB + ηAC − ηBC ≥ 0, 1 − ηAB − ηAC + ηBC ≥ 0,
and
2 2 2 2 2 2 2ηABηAC ηBC − ηABηAC − ηABηBC − ηAC ηBC ≥ 0.
The proof is rather lengthy and can be found in the appendix of [98]. The resulting boundary is depicted in Fig. 3.6; the shape and its determining equation is recognized as the convex hull of the Roman surface (aka Steiner surface) [112, 128]. Comparing with Fig. 3.5(a), we see that, still, the quantum joinability limitation arising from from local-positivity is stricter than the corresponding classical one. However, it is closer to the classical limitations than the state/channel-positive limitations obtained in the previous section for d = 2. To shed light on the cause of the quantum bound- ary here, we can explicitly construct a product-state projector, whose probability would be negative if joinability outside of this shape were allowed. The family of joining states w that we need to consider (see appendix of [98] for details) may be parameterized in terms of the bipartite reduced state Werner parameters as 1 η η η w(η , η , η ) = I + AB (V − I/2) + AC (V − I/2) + BC (V − I/2). AB AC BC 8 4 AB 4 AC 4 BC 54 3.3 Three-party joinability settings with collective invariance
Consider, in particular, the following state on A-B-C:
1 cos 2π/3 cos 4π/3 |ψi = ⊗ ⊗ , (3.9) 0 sin 2π/3 sin 4π/3
which corresponds to the pure product state with the local Bloch vectors as anti- parallel with one another as possible. Computing its expectation with respect to w(ηAB = ηAC = ηBC ≡ η), the largest value of η that admits a non-negative value is η = 2/3. Hence, local-positivity limits the simultaneous joining of these Werner operators to a maximum of η = 2/3. The operational interpretation of this result deserves attention. Consider a local projective measurement made on each of three qubit systems. Furthermore, consider the three systems to have a collective unitary symmetry, in the sense that there are no preferred local bases. In our general picture, where local positivity is considered, the systems need not be three distinct systems – they may also be the same system at two different points in time. Local positivity enforces the rule that “all probabilities arising from such measurements must be non- negative”. In the example above (i.e. ηAB = ηAC = ηBC ), this rule implies that the three equal correlations (as measured by the ηs) can never exceed 2/3. As this example and Fig. 3.6 show, local-positivity enforces joinability limitations more strict than those of classical probability theory. Notwithstanding, these limitations reflect the same symmetry as the classical limitations do, namely, symmetry with respect to individually inverting two axes. The state-joining and channel-joining scenarios reflected a preference towards the negative axis (anticorrelation) and the positive axis (correlation) of the ηs, respectively.
Before concluding this section, we connect the above discussion to the relationship between local-positivity and separability. As mentioned earlier, the cone of local positive operators and the cone of separable operators are dual to one another. The operator subspace we are dealing with is spanned by the orthonormal operators √1 I, 8 q √1 (V − I/2), √1 (V − I/2), and √1 (V − I/2) with coordinates √1 , 3 η , 6 AB 6 AC 6 BC 8 8 AB q 3 q 3 8 ηAC , and 8 ηBC , respectively. In Theorem 3.3.4, we determined the algebraic surface bounding the local positive operators; hence, the dual to this surface will bound the separable operators within this space. The dual to the Roman surface is known as the Cayley’s cubic surface [129], which, for a given scale parameter w is characterized by
w x y
x w z = 0.
y z w
q 3 q 3 q 3 We first set x = 8 ηAB, y = 8 ηAC , and z = 8 ηBC . Then we must set w so that
55 Joinability of causal and acausal relationships the Cayley surface delimits the separable states. For each extremal separable state in our space, there is a corresponding local-positive operator acting as an entanglement witness; a state is separable if the inner product with its entanglement witness is nonnegative.
Consider the extremal local-positive operator ηAB = ηAC = ηBC = 2/3 that we made use of previously. This operator will act as an entanglement witness for another operator with ηAB = ηAC = ηBC = σ. We obtain σ by solving
√1 √1 q 8 q 8 3 2 3 8 3 8 σ q · q = 0, 3 2 3 8 3 8 σ q 3 2 q 3 8 3 8 σ
1 to arrive at σ = − 6 . With this, the only value of w allowing the Cayley surface to be solved by σ = − 1 is w = √1 . Setting the scaling value and evaluating the 6 24 determinant, we find that the separable states are bound by the surface
2 1 + 54ηABηAC ηBC − 9(ηAB + ηAC + ηBC )
+18(ηABηAC + ηABηBC + ηAC ηBC ) ≥ 0.
This inequality may also be obtained using Theorem 1 in [82]. The shape of the separable states is depicted in Figure 3.7. Several remarks may be made. First, the set of separable states exhibits the tetrahedral symmetry shared by the classical joinability boundary and the local-positive joinability boundary. Thus, among the various boundaries we have considered in this three dimensional Euclidean space, the state- and channel-positive boundaries are the only ones not obeying tetrahedral symmetry. However, both the convex hull and the intersection of the state- and channel-positive cones bound regions which recover this symmetry. It is a curious observation that the convex hull of these cones is “nearly” the local-positive region, while the intersection is “nearly” the set of separable states. Earlier we found, in the two-qudit case, that local-positivity coincides with the union of the state- and channel-positive regions, as well as that the separable region was their intersection. Here we consider the analog for three qubits. The result is that i) the convex hull of state- and channel- positive operators is strictly contained in the set of local-positive operators; and ii) the intersection of the state- and channel-positive operators is strictly contained in the set of separable states. We may further interpret the latter result in terms of PPT considerations. The operators which result from a homocorrelation-mapped channel necessarily have PPT. Corollary 1 in Ref. [82] states that the PPT and bi-separable Werner operators co-
56 3.4 Agreement bounds for quantum states and channels
(a) (b)
Figure 3.7: (a) Set of separable operators within the set of local-positive operators. (b) Intersection of the state- and channel-positive cones within the set of local-positive operators. While the separable operators are a subset of the intersection set, the two objects coincide (only) at their vertices. In panel (a) the closest point in the separable set to the (−1, −1, −1) corner of the figure is (−1/6, −1/6, −1/6), whereas in panel (b), the closest point in the intersection set is (−1/5, −1/5, −1/5). incide. Thus, any state-positive operator which is also a homocorrelation mapped channel is necessarily bi-separable. Hence, the intersection of the four cones will be the set of states which are bi-separable with respect to any of the three partitions. This set is clearly contained in the set of tri-separable states. These observations il- luminate the relationships among entanglement, quantum states, and quantum chan- nels. Specifically, the homocorrelation map allows us to place quantum channels in the same arena as quantum states, and hence to directly compare and contrast them. Finding that the tri-separable operators are a proper subset of the bi-separable ones, we wonder what features these strictly bi-separable operators possess, and what does bi-separability imply for the states or channels supporting such correlations.
3.4 Agreement bounds for quantum states and chan- nels
In what remains, we illustrate some crucial differences between channel- and state- positive operators. These differences inform the nature of their respective joinability limitations. In order to directly compare states to channels we restrict our consid- erations here to operators in B(Hd ⊗ Hd). Qualitatively, state-positive operators are restricted in the degree to which they can support agreeing outcomes, whereas
57 Joinability of causal and acausal relationships
channel-positive operators are restricted in the degree to which they can support disagreeing outcomes. We define the degree of agreement to be the likelihood of a certain POVM element. Specifically, consider a local projective measurement M = {|ijihij|}. We can coarse-grain this into a two-element projective measure- P ment with the bipartition into “agreeing” outcomes, EA = |iiihii|, and “disagree- P i ing” outcomes, ED = i6=j |ijihij|, respectively. Lastly, so that these outcomes are basis-independent, we can “twirl” EA and ED as follows: Z X † † EA = dµ(U)U ⊗ U |iiihii| U ⊗ U i Z X † † ED = dµ(U)U ⊗ U |ijihij| U ⊗ U , i6=j
where dµ(U) denotes integration with respect to the invariant (Haar) measure. It is simple to see that these two operators yield a resolution of identity and hence form a POVM. We can compute these two operators explicitly as follows. By the invariance of the Haar measure, we can rewrite EA as Z ⊗2 EA = d dµ(ψ)|ψihψ| ,
for which the above integral is proportional to the projector onto (or identity operator + ⊗2 ⊗2 I2 in) the totally symmetric subspace H+ ⊂ H [130]. Explicitly, we can write d + d I + V + + 2 + d − 1 EA = + I2 = + , d2 ≡ dim(H2 ) = , (3.10) d2 d2 2 2
ED = I − EA. (3.11)
We define the degree of agreement to be the likelihood of EA and, similarly, the degree of disagreement to be the likelihood of ED. Operationally, these values are the prob- ability that, for a randomly chosen local projective measurement made collectively, the local outcomes will agree or, respectively, disagree.
We now proceed to show how quantum channels differ from quantum states in their allowed range of agreement likelihood. In the case of a bipartite operator ρ ∈ B(H ⊗ H), we are familiar with computing this agreement probability as Tr(EAρ). To carry out the same computation for a channel operator, the homocorrelation map becomes expedient. Given a quantum channel M : B(H) → B(H), we wish to determine the probability that the outcome of a randomly chosen projective measurement (made on the completely mixed state) will agree with the outcome of the same measurement after the application of M. Assume the outcome was |ii from an orthogonal basis
58 3.4 Agreement bounds for quantum states and channels
{|ii}. Then the post-channel state is M(|iihi|), and the likelihood that the post- channel measurement will also be |ii is hi|M(|iihi|)|ii. Lastly, if we want to average this likelihood of agreement over all choices of basis we integrate, p(agree) = R dµ(U)Tr M(U|iihi|U †)U|iihi|U † = R dµ(ψ)Tr M(|ψihψ|)|ψihψ| .
If we wish to find the bounds on this value, the above form does not make transparent the fact that we are performing an optimization problem in a convex cone. But, recalling the namesake property of the homocorrelation map, Eq. (3.3), the above expression may be rewritten as Z p(agree) = Tr H(M)d dµ(ψ)|ψihψ| ⊗ |ψihψ| = Tr[H(M)EA].
Accordingly, the likelihood of agreement is calculated for channel operators in the homocorrelation representation in just the same way as it is for bipartite density operators. With the stage set, the desired bounds are described in the following theorem:
Theorem 3.4.1. Let w be an operator in B(Hd ⊗ Hd), and consider a POVM with operation elements {EA, ED} as in Eqs. (3.10)-(3.11). Then the degree of agreement for w ≥st 0 as calculated by the likelihood of EA is bounded by 2 0 ≤ Tr(wE ) ≤ , (3.12) A 2 + d − 1
while the degree of agreement for w ≥ch 0 is bounded by 1 ≤ Tr(wE ) ≤ 1. (3.13) 2 + d − 1 A
Proof. In the case of state-positive operators, the maximal value of Tr(wEA) is achieved by setting w = EA/Tr(EA), which results in Tr(wEA) = 2/(d + 1). For the lower bound, it is simple to see that choosing w to lie in the complement of the projector yields a value of zero. Hence, we have obtained the bound of Eq. (3.12).
In the case of channel-positive operators, the value of Tr(wEA), where w ≥ch 0, is unchanged by a partial transposition of both operators. Thus, we may seek bounds TA TA TA on the value of Tr(w EA ), where w is a density operator. By using Eq. (3.10),
59 Joinability of causal and acausal relationships
the partial transposition of EA is
TA TA d I + V EA = + . d2 2
TA TA Thus, the upper and lower bounds on Tr(wEA) are achieved by setting w = V /d and wTA = (I − V TA )/(d2 − d), respectively. Accordingly, the resulting bounds are d 1 d 1+d + 2 ≤ Tr(wEA) ≤ + 2 , which simplify to those of Eq. (3.13). d2 d2
By virtue of the homocorrelation map, the above result may be understood geo- metrically. The objects involved are the agreement/disagreement POVM operators EA and ED, and the state- and channel- positive cones Wst and Wch, respectively. Theorem 3.4.1 places an upper bound on the inner product between vectors in Wst and EA, and, similarly, on the inner product between vectors in Wch and ED. This geometric understanding is aided by the example of Werner operators shown in Fig. 3.2. Lastly, we proceed to show that general joinability limitations (though not strict ones) can be derived based solely on i) the above agreement bounds of channels and states; ii) joinability bounds of classical probabilities; and iii) the fact that the agreement likelihoods must obey rules of classical joinability. Ultimately, the reduced states must satisfy certain limitations arising from joining limitations of classical probability distributions. In the three-party joining scenario, the bipartite marginal distributions of three classical d-nary random variables must have probabilities of agreement αAB, αAC , and αBC satisfying the following inequalities [61]:
−αAB + αAC + αBC ≤ 1, (3.14)
αAB − αAC + αBC ≤ 1, (3.15)
αAB + αAC − αBC ≤ 1, (3.16)
and, in the case of d = 2, also
αAB + αAC + αBC ≥ 1. (3.17)
Since Tr(wEA) is a probability of agreement, it too is subject to the above con- straints. Hence, we identify Tr(ρiEA) ≡ α`, where ` = AB, AC, or BC. Consider the case where systems B-C are state-positive. Theorem 3.4.1 then sets the bound 2 Tr(ρBC EA) ≤ d+1 . Setting the parameter αBC = Tr(ρBC EA) to this upper limit of 2 d+1 , Eq. (3.16) becomes d + 3 α + α ≤ . AB AC d + 1
60 3.4 Agreement bounds for quantum states and channels
In the case of αAB = αAC ≡ α, this yields d + 3 α ≤ , 2(d + 1)
which corresponds precisely to the optimal bound for qudit cloning [131] (cf. Eq. (21) therein, where their F coincides with our α). We can similarly recover the exact bound for the 1-2 sharability of qubit Werner states determined in [61]. Again, we 2 set the B-C agreement to its extremal value Tr(ρBC EA) = d+1 , as given by Theorem 3.4.1. For d = 2, Eq. (3.17) applies, and substituting in the extremal value of αBC 1 1 we obtain αAB + αAC ≥ 3 . Again, in the case of αAB = αAC ≡ α, this yields α ≤ 6 , which is the exact condition for 1-2 sharability of Werner qubits. While obtaining a full generalization of Theorem 3.4.1 to multiparty systems would entail a detailed understanding of representation theory for Brauer algebras which is beyond our current purpose, we can nevertheless establish the following:
⊗N Theorem 3.4.2. Let w ∈ B(Hd ), and consider a POVM with operation elements d + {EA = + IN , ED = I − EA} (analogous to Eqs.(3.10)-(3.11)). Then the degree of dN agreement for w ≥st 0 as calculated by the likelihood of EA is bounded by d 0 ≤ Tr(wE ) ≤ . (3.18) A d−1+N N
Proof. The maximal and minimal values of Tr(wEA) are achieved by setting w = + + EA/Tr(EA) and w = (dI/dN − EA)/Tr(dI/dN − EA), respectively, which yields the desired bounds of Eq. (3.18). From the above multiparty bound, one may attempt to recover, for instance, the known bounds on 1-n sharability of Werner states [61]. However, thus far we have not been successful in this endeavor. In the tripartite qudit setting, such bounds were found to be sufficient, but this might be a special feature of this particular case. Therefore, it remains an open question to determine whether there exists a simple principle (or simple principles) which govern joinability limitations beyond the tripartite setting.
61
Chapter 4
Towards an alternative approach to joinability: enforcing positivity through purification
63 Towards an alternative approach to joinability: enforcing positivity through purification 4.1 Introduction
The aim of this chapter is to follow up on some of the ideas brought up in the previous two chapters and also to draw further connections using the concept of joinability. We develop a new approach to the joinability problem which allows us to formalize the “composition law” that was touched upon in the previous chapter, to easily incorporate the concept of measurement incompatibility into the joinability framework, and to find a simple principle that governs joinability failures in both the classical and quantum cases. The techniques used in this chapter are rather elementary. Yet, to our knowledge, they have not been applied to the problem of joinability. Surprisingly, with basic linear algebra, we unveil important connections among principles that govern the structure of quantum relationships. We emphasize that the ideas presented in this chapter are just sprouting and that the purpose of presenting them is to provide an environment in which they can grow. In the previous chapters, failures of joinability were due to a failure of positive- semidefiniteness or complete positivity. In this chapter we make positivity manifest by purifying the quantum state with an ancillary system or by “lifting” the quantum channel to an isometric description in a larger Hilbert space. In this way, failures of joinability are only due to failures of trace-normalization, such as the conclusion that the likelihood of every possible event is zero. We emphasize that the motivation for pursuing the ideas in this chapter are mostly theoretical, urged by the desire to better understand the fundamental concepts of monogamy of entanglement, no-cloning, and measurement incompatibility. Previously, we presented the following explanation for the fact that qubit pairs A-B and A-C could not both be in the singlet state. The singlet state exhibits per- fectly disagreeing measurement outcomes for the same spin measurement made on both qubits. Assuming a singlet state were shared between A-B and A-C, we can “compose” the disagreements to conclude that the same spin measurement made on B-C must produce perfectly agreeing outcomes. However, no valid quantum state (i.e. no positive-semidefinite trace-one operator) can produce such perfectly agreeing outcomes for all collective spin measurements. Thus, we have that the A-B disagree- ment and the A-C disagreement force the contradictory B-C agreement. A similar argument can explain the no-cloning result. The initial motivation for the ideas in this chapter had been to further explore the idea of “composing” two conditions, such as the A-B disagreement and the A-C disagreement, to obtain a third condition. Such an argument was made to work in the singlet state sharing example and to obtain the no-cloning principle. But, it is not clear how it could be properly generalized. Furthermore, could such an approach be extended beyond the three-body joinability scenario? In the spirit of generalized en- tanglement [132], could such an approach be applied to general consistency problems
64 4.1 Introduction
which do not admit a natural tensor product structure as in fermionic systems? Before discussing our new approach to joinability we establish the nature of the constraints that are considered. We consider constraints to be on the manifestly positive objects such as the purification of the quantum state or the isometry of the quantum channel. We briefly review the construction of these objects. Given a density operator ρ ∈ B(H ), there always exists a quantum state |ψi ∈ A √ H1 ⊗ H2 such that Tr (|ψihψ|) = ρ. This is easily verified. ρ ≥ 0 implies ρ = √A A 2 √ ρ† is well-defined. Then |ψi = ( ρ ⊗ I)|Ωi satisfies the partial trace condition, P where |Ωi = i |iii is the unnormalized maximally entangled state. The state |ψi is unique up to a norm-preserving linear map, or isometry, acting on the ancillary system. A similar argument can be used to show that, given a quantum channel 1 1 2 3 E : B(HA) → B(HB), there exists an isometry V : HA → HB ⊗ HB ⊗ HA such † that Tr23 V · V = E(·). Here, trace-preservation of E corresponds to the isometry condition on V . The isometry is defined up to the action of another isometry acting on the traced-out ancillary systems. We consider two types of constraints that we refer to as hard constraints and soft constraints. Hard constraints express knowledge of statistical impossibilities, while soft constraints bound the likelihoods of various outcomes. We represent a hard constraint on a quantum state as a set of linear equations satisfied by its purification:
(hφ|A ⊗ IB)|ψiAB = 0, ∀ |φi ∈ SA ⊆ HA, (4.1)
where SA is the set of states involved in the hard constraint. A hard constraint on a quantum channel is, similarly, represented by a set of linear equations satisfied by its isometry:
(hφ|AB ⊗ IC )(IA ⊗ VA→BC )|ΩiAA = 0, ∀ |φiAB ∈ SAB ⊆ HA ⊗ HB, (4.2) P where |ΩiAA = i |iii is the unnormalized maximally entangled state and SAB is, again, the set of vectors used to subject |ψi to the hard constraint.
As a simple example, consider a system of three qubits HA⊗HB⊗HC and constrain the density matrix of this system to have reduced state AB described by the (anti- symmetric) singlet state |ψ−i = √1 (|01i − |10i). Denote the space of symmetric 2 states on AB as S2(H). This is the space orthogonal to the singlet state. Hence, the condition that AB is described by the singlet state is expressed by the hard constraint
2 (hφ|AB ⊗ ICD)|ψiABCD = 0, ∀ |φi ∈ S (H) (4.3)
Standing alone, this means of expressing the AB-singlet condition is not very illu- minating. The value of this rephrasing is revealed when we consider multiple hard constraints in tandem. As we show in the following section, we can recover a number
65 Towards an alternative approach to joinability: enforcing positivity through purification of basic quantum “no-go” principles as stemming from an over-constraining of hard constraints.
Soft constraints bound the likelihoods of certain projective measurement out- comes. We represent a soft constraint on a quantum state as a set of inequalities satisfied by its purification:
2 2 2 k(hφ|A ⊗ IB)|ψiABk ≤ || |φi|| || |ψi|| , ∀ |φi ∈ SA ⊆ HA, (4.4) where represents the likelihood bound associated to the set of states SA. A soft constraint on a quantum channel is, similarly, represented by a set of inequalities satisfied by its isometry:
2 2 k(hφ|AB ⊗ IC )(IA ⊗ VA→BC )|Ωik ≤ || |φi|| || (IA ⊗ VA→BC )|Ωi|| ,
∀ |φiAB ∈ SAB ⊆ HA ⊗ HB. (4.5)
4.2 Joinability limitations from hard constraints
In this section we explore joinability where the partial descriptions of the system are given by hard constraints. We begin with a trivial example from classical probability theory. Consider the simplest classical system, the probabilistic bit. The probability distribution is given by the likelihoods p(0) and p(1). An example of inconsistent hard constraints is p(0) = 0 and p(1) = 0. No distribution on outcomes 0 and 1 can satisfy these constraints while also satisfying a non-trivial normalization condition.
This example seems too trivial to bear insight. However, we will argue that all inconsistent sets of hard constraints fail to be consistent for the same reason as the above example fails: the likelihood of any outcome is zero. Surprisingly, this same reasoning explains the non-sharability of maximally entangled states, the no-cloning principle, the incompatibility of certain observables, as well as the simple classical principle that three bits cannot disagree.
Consider three classical probabilistic bits A, B, and C with the constraints, A and B disagree, B and C disagree, and A and C disagree. These conditions are equivalent to the three hard constraints, A and B cannot agree, B and C cannot agree, and A
66 4.2 Joinability limitations from hard constraints and C cannot agree. These three conditions exclude all possible configurations:
0 0 0 0 0 0 × 0 0 0 × 0 0 0 × 0 0 1 0 0 1 × 0 0 1 × 0 0 1 × 0 1 0 0 1 0 0 1 0 0 1 0 × 0 1 1 0 1 1 0 1 1 × 0 1 1 × AB → BC → AC → 1 0 0 1 0 0 1 0 0 × 1 0 0 × 1 0 1 1 0 1 1 0 1 1 0 1 × 1 1 0 1 1 0 × 1 1 0 × 1 1 0 × 1 1 1 1 1 1 × 1 1 1 × 1 1 1 × (4.6) The constraints imply that the likelihood of any outcome is zero. Therefore, no distribution can satisfy these three conditions. Consider a single qubit S with the constraints “|0i is certain upon making a Pauli- Z measurement” and “|+i is certain upon making a Pauli-X measurement”. We can translate each into a hard constraint on the purification of the density operator to an ancillary system A: h1|S ⊗ IA|ψSAi = 0 and h−|S ⊗ IA|ψSAi = 0. Any state vector |φi ∈ HS can be written as some linear combination |φi = α|1i + β|−i. Hence, for any |φi ∈ HS , there exists α and β such that
∗ ∗ hφ| ⊗ IA|ψSAi = α h1| ⊗ IA|ψSAi + β h−| ⊗ IA|ψSAi = 0. (4.7)
Only |ψSAi = 0 satisfies hφ| ⊗ IA|ψSAi = 0 for all |φi. Thus, no non-trivial state can satisfy both constraints. It seems that we have taken a simple fact of quantum mechanics (i.e. uncertainty of Z and X measurements cannot both be zero) and given an unfamiliar or cumbersome argument for reaching it. However, a failure of the trace-one condition seems to be more direct or basic than one involving algebraic properties of observables, especially when the example is more complicated than the one above. Next, we use the concept of hard constraints to easily prove a fact used in the previous chapter: there is no bipartite quantum state that exhibits perfectly agreeing outcomes for all collective projective measurements. Considering the system to be described by the Hilbert space H = Cd ⊗ Cd we can rephrase the condition in terms of hard constraints as,
⊥ ⊥ ⊥ (hφ|A ⊗ hφ |B ⊗ IC )|ψiABC = 0, ∀ |φi, |φ i, s.t. hφ |φi = 0. (4.8)
Since the set of vectors |φi ⊗ |φ⊥i span Cd ⊗ Cd (this can be taken as an exercise for the enthusiastic reader), there is no non-trivial solution to the set of hard constraints, verifying the statement. Note that we did not have to make any reference to a positive-semidefinite constraint as we did while making this argument in the previous
67 Towards an alternative approach to joinability: enforcing positivity through purification
chapter. The key insight was to phrase the condition in terms of a set of negative statements, or impossibilities, as opposed to a number of positive expectation values. With a parallel argument we can show that no quantum channel can map every input pure state to an orthogonal pure state. Considering the quantum channel to d d be described by an isometry V : C → C ⊗ HA we rephrase the condition in terms of hard constraints as follows. The condition states that the output of the channel, † TrA V |φihφ|V , should be orthogonal to |φi, or, equivalently, hφ|V |φi = 0 for all |φi ∈ Cd. Defining hφT | ≡ T (|φi), and noting that |φi = (hφT | ⊗ I)|Ωi, we write the hard constraints as
T d (hφ | ⊗ hφ| ⊗ IA)(I ⊗ V )|Ωi = 0, ∀ |φi ∈ C . (4.9)
Then, because span{hφT | ⊗ hφ|} = (Cd ⊗ Cd)† (this, too, can be taken as an exercise), there is no non-trivial solution for V . These two previous observations address the quantum distinction between causal and acausal relationships. There is certainly a one-to-one correspondence between the vectors |φi ⊗ |φi and matrices |φi ⊗ hφ|. The former linear objects inhabit the space H ⊗ H, which corresponds to acausal quantum relationships, while the latter inhabit the space B(H) ∼= H⊗H∗, which corresponds to causal quantum relationships. Despite the one-to-one correspondence, the span of the |φi ⊗ |φi is a strict subspace of H ⊗ H, while the span of the |φi ⊗ hφ| is equal to H ⊗ H. In particular the vectors |φi ⊗ |φi ∈ Cd ⊗ Cd only span a d(d + 1)/2-dimensional space, while the set of |φihφ| ∈ Cd ⊗ Cd† spans Cd ⊗ Cd†. Next we consider the impossibility of having entanglement between both A-B and B-C. We return to the three-qubit system with the constraint that A-B and B-C are each described by the singlet state |ψ−i = √1 (|01i − |10i). We can rephrase this 2 in terms of a set of hard constraints on the purified state |ψiABCD,
2 (hφφ|AB ⊗ hτ|C ⊗ ID)|ψiABCD = 0 ∀ |φi, |τi ∈ C , (4.10)
and similarly for the BC system. Since span{|φφτi, |τφφi} = (C2)⊗3, again, there is no non-trivial solution. Thus, we can view the impossibility of singlet sharing to be due to the constraints having ensured zero likelihood for any measurement outcome. In this sense, the singlet sharing is impossible for the same reason that three bits disagreeing is impossible. For completeness, we prove the no-cloning theorem using the hard constraint approach. Consider a quantum channel from system A to BC, where HA = HB = HC ≡ H, represented by the isometry V : HA → HB ⊗ HC ⊗ HD. We can phrase the cloning condition as a negative statement by requiring that the output of the ⊥ ⊥ channel applied to |φi be orthogonal to span(|φ i)⊗HC and HB ⊗span(|φ i), where
68 4.2 Joinability limitations from hard constraints
hφ|φ⊥i = 0. As a set of hard constraints, we have
T (hφ φτ| ⊗ ID)(I ⊗ V )|Ωi = 0 ∀ |φi, |τi ∈ H, (4.11)
along with the analogous statement from switching the roles of B and C. Since span{hφT φτ|, hφT τφ|} = H† ⊗ H† ⊗ H†, no non-trivial solution exists for V . As with the no singlet sharing example, we can explain the impossibility of cloning as due to the set of constraints ruling out all possibilities. As a final example, we consider the incompatibility of measuring devices. A quantum measurement can be described by a quantum channel from the measured system A to a classical measuring device D. The measuring device is made “classical” by requiring that operators appearing on B(HD) are restricted to being diagonal in a particular basis. Equivalently, operators on D must be in the commutative algebra C⊕d, where d is the number of outcomes. As with any quantum channel, such measurement channels admit an isometric representation V : HA → HD ⊗ HE for d some ancillary system E. For a complete projective measurement on HA = C with outcomes corresponding to the basis of vectors |ji ∈ HA, an isometric representation P of the channel is V = j |jjihj|, where we have let the |ji also correspond to an orthonormal basis of vectors in HD and in HE. With this formalism in place, we consider the case of having a single system be measured by two different measurement devices simultaneously. Following [133], we say that the two measurements are compatible if there exists a single measurement device from which the two measurements can be obtained via partial trace of the outputs. Note that this is equivalent to the definition of channel joinability. Re- stricting to the qubit case, consider an X and a Z measurement device, from the quantum system A to the devices DX and DZ . We label the basis vectors of the output systems as {|+i, |−i} and {|0i, |1i}, respectively, such that the labels corre- spond to the states of the input system. We can ensure the X and Z measurement conditions by requiring that, with input |±i, the output on DX must be orthogonal to |∓i and that, with input |0/1i, the output on DZ must be orthogonal to |1/0i
(the “/” indicates “or”). Letting V : HA → HDX ⊗ HDZ ⊗ HE be the isometry of the quantum measurement, the hard constraints are (h∓| ⊗ hφ| ⊗ IE)V |±i = 0
for all |φi ∈ HDZ and (hφ| ⊗ h1/0| ⊗ IE)V |0/1i = 0 for all |φi ∈ HDX . Since span{h∓| ⊗ hφ| ⊗ |±i, hφ| ⊗ h1/0| ⊗ |0/1i} = H† ⊗ H† ⊗ H , no non-trivial solution DX DZ A exists. Thus, we have shown that many of the important quantum no-go principles, in- cluding the impossibility of singlet sharing, no-cloning, and measurement incompati- bility can be explained with the same argument used to show p(0) = 0 and p(1) = 0 are incompatible. The approach used is not restricted to needing a tensor product structure. Therefore, it would be interesting to explore these ideas for fermionic or
69 Towards an alternative approach to joinability: enforcing positivity through purification
bosonic systems, making better connection, for instance, with the N-representability problem [134, 28, 27].
4.3 Joinability limitations from soft constraints
We now turn to the case where, instead of ruling out any possibilities, we simply place bounds on their likelihoods. As mentioned, the motivation for the approach to joinability, developed in this chapter, was to elucidate the composition rule that we previously used to show the impossibility of singlet sharing. The first example returns to the singlet sharing scenario. Consider a tripartite system of qubits ABC. We aim to show that the correla- tions of AB in tandem with the correlations of BC ensure certain correlations on AC. For instance, we used the “perfect disagreement” of the singlet state to ensure that the singlet describing AB and BC ensures the correlations on AC to be perfectly agreeing. After much effort towards trying to develop and generalize this “compo- sition observation”, we have arrived at a simple idea which makes this possible. A key insight, which was conveyed in the previous section, is the utility of expressing constraints in terms of negative statements. There, instead of formalizing the com- position (disagree)AB + (disagree)BC ⇒ (agree)AC , we found it simpler to formalize the composition (not agree)AB + (not agree)BC ⇒ (not disagree)AC as represented by hard constraints. Then, we expect the corresponding composition rule in the soft con- straint case to be (low agreement)AB + (low agreement)BC ⇒ (low disagreement)AC . We enforce some degree of singlet correlation on a pair of subsystems by upper bounding the inner product of the state with the set of |φφi, which span the symmetric subspace. Assume that, for the purified state |ψi on ABCD, the soft constraint on AB is given by 2 2 k(hφφτ| ⊗ ID)|ψik ≤ αAB, ∀|φi, |τi ∈ C (4.12) while the soft constraint on BC is given by
2 2 k(hτφφ| ⊗ ID)|ψik ≤ αBC , ∀|φi, |τi ∈ C . (4.13)
We expect to be able to bound the likelihood of “disagreeing” outcomes on AC, such as |φi ⊗ |φ⊥i, where the two states are orthogonal. Consider, then, the expression for the likelihoods on AC,
⊥ 2 (hφτφ | ⊗ ID)|ψi . (4.14)
Since the Hilbert space of system B is two-dimensional, an arbitrary vector |τiB can be written as a linear combination |τi = λ|φi + µ|φ⊥i. Making this replacement in
70 4.3 Joinability limitations from soft constraints the likelihood expression, we obtain
⊥ ⊥ ⊥ 2 (λhφφφ | ⊗ ID + µhφφ φ | ⊗ ID)|ψi . (4.15)
We can bound the value of this expression using the triangle inequality,
⊥ 2 ⊥ 2 ⊥ ⊥ 2 (hφτφ | ⊗ ID)|ψi ≤ (λhφφφ | ⊗ ID)|ψi + (µhφφ φ | ⊗ ID)|ψi . (4.16)
Each term of the right-hand side of the inequality is the form of the constraints we ⊥ 2 place on AB and BC. Letting the maximum value of (hφτφ | ⊗ ID)|ψi be δAC and noting that |λ|, |µ| ≤ 1, we then obtain the monogamy-like inequality
δAC ≤ αAB + αBC . (4.17)
This inequality captures how an “agreement-bound” on AB and on BC (i.e. αAB and αBC ) enforce a “disagreement-bound” on AC (i.e. δAC ). Note that the hard-constraint case is recovered by setting αAB = αBC = 0, which forces δAC = 0 (an impossibility). Also, using the fact that span{|φφ⊥i} = C2 ⊗ C2, there is no vector which can be ⊥ orthogonal to all |φφ i. Thus, there is a lower bound to the value of δAC , which can 1 be computed to be 2 (show as an exercise). This expresses a “trade-off” in the values that αAB and αBC may take, 1 ≤ α + α . (4.18) 2 AB BC This inequality gives a necessary condition for determining if the soft constraints are consistent. It does not diagnose all inconsistent sets of constraints. This is, in part, due to the fact that the inequality is weakened by upper-bounding λ and µ. In future work, we hope to properly compare these observations to our previous Werner joinability findings. We only briefly make a few observations. First, applying the inequality in Eq. (4.18) to Werner states gives a linear trade-off for the Werner parameters (see Chapters 2 and 3 for various parameterizations). We found, however, that the exact trade-off between the Werner parameters for AB and BC is quadratic (see Eq. (2.22)). We can now understand another opportunity afforded by phrasing the constraints as negative statements (or upper bounds on likelihoods): the upper-bounds of the constraints allow us to use the triangle inequality to upper bound the derived con- straint. If instead, we had used lower-bounds, the direction of these inequalities would have opposed the direction of the triangle inequality. The singlet state is not particular with respect to monogamy constraints. We expect that other Bell states should lend themselves to similar composition laws. Sticking to the qubit case, each maximally entangled state is obtained by applying a particular unitary transformation to, without loss of generality, system B: |ψU i =
71 Towards an alternative approach to joinability: enforcing positivity through purification
(I ⊗ U)|ψ−i. With this, the hard constraint that ensures two qubits are described † 2 by |ψU i is (hφφ| ⊗ IC )(IA ⊗ UB ⊗ IC )|ψiABC = 0 for all |φi ∈ C . This is a simple modification of the singlet state hard constraints.
Accordingly, the soft constraints which ensure a state is “close to” the state |ψU i are stated as 2 † U 2 (hφφ| ⊗ IC )(IA ⊗ UB ⊗ IC )|ψiABC ≤ αAB, ∀ |φi ∈ C . (4.19) Consider the three-party joining scenario, let the above soft constraint apply to AB (where we purify the quantum state using an ancilla system D). Let the following soft constraint apply to BC,
2 † V 2 (hτφφ| ⊗ ID)(IAB ⊗ VC ⊗ ID)|ψiABCD ≤ αBC , ∀ |φi ∈ C . (4.20) We follow the reasoning of the singlet state example. Consider the following expression for certain likelihoods on AC
⊥ † † 2 (hφτφ | ⊗ ID)(IAB ⊗ (U V )C ⊗ ID)|ψi . (4.21)
Now, we can write |τi = λU|φi + µU|φ⊥i for any |τi, giving
2 ⊥ ⊥ ⊥ † † † (λhφφφ | ⊗ ID + µhφφ φ | ⊗ ID)(IA ⊗ UB ⊗ (U V )C ⊗ ID)|ψi . (4.22) We can bound the value of this expression using the triangle inequality,
⊥ † † 2 (hφτφ | ⊗ ID)(IAB ⊗ (U V )C ⊗ ID)|ψi (4.23) 2 ⊥ † † † ≤ (λhφφφ | ⊗ ID)(IA ⊗ UB ⊗ (U V )C ⊗ ID)|ψi 2 ⊥ ⊥ † † † + (µhφφ φ | ⊗ ID)(IA ⊗ UB ⊗ (U V )C ⊗ ID)|ψi . (4.24)
By setting |τ 0i = VU|φ⊥i and |φ0i = U|φ⊥i, we simplify the inequality
⊥ † † 2 (hφτφ | ⊗ ID)(IAB ⊗ (U V )C ⊗ ID)|ψi (4.25) 2 † ≤ (λhφφτ| ⊗ ID)(IA ⊗ UB ⊗ ICD)|ψi 2 0 0 † + (µhφφ φ | ⊗ ID)(IAB ⊗ VC ⊗ ID)|ψi . (4.26) Each term on the right-hand side of the inequality can be recognized as the expressions
72 4.3 Joinability limitations from soft constraints in the soft constraints on AB and BC, respectively. Define
VU ⊥ † † 2 δAC ≡ max (hφτφ | ⊗ ID)(IAB ⊗ (U V )C ⊗ ID)|ψi . (4.27) |φi,|τi
Using |λ|, |µ| ≤ 1, we obtain the inequality
VU U V δAC ≤ αAB + αBC . (4.28)
This generalized composition rule simplifies to the singlet state case by setting U = V = I. This inequality demonstrates how, by ensuring certain quantum correlations for AB and for BC, certain correlations are forced upon AC. We parameterized the maximally entangled qubit states with a unitary transformation. This shows that, if AB is close to the maximally entangled state |ψU i and BC is close to the maximally ⊥ entangled state |ψV i, then the likelihood of any measurement outcome |φi ⊗ VU|φ i is upper bounded. It would be interesting to explore further generalizations of these observations to qudits and to more than three systems. The above “composition rule” inequality does not, on its own, tell us when the initial soft constraints, themselves, are inconsistent with one another. As with the singlet example, we must further evaluate or place a lower bound on the disagreement parameter δAC , in order to obtain necessary conditions on the consistency of the soft constraints. We have yet to provide a general intuition behind joinability failures in the soft-constraint case. For instance, it would be useful to directly obtain an inequal- ity involving αAB and αBC , or involving a general set of soft constraint parameters i. We outline an approach to obtaining such inequalities and show that joinability fail- ures, in this case too, are on account of failures of the trace-normalization condition. Thus, we can view this cause of joinability failures as being a more general version of the “zero-total-probability” explanation in the hard-constraint case. As with the quantum case, for a classical probability distribution the sum of all likelihoods must be 1. Considering the classical soft constraints of p(0) ≤ and p(1) ≤ δ, the constraints are inconsistent unless 1 ≤ +δ. A simple quantum example obtains analogous inconsistency bounds. Consider a qubit system A (with ancilla B) 2 2 subject to the soft constraints k(h+| ⊗ I)|ψiABk ≤ δ √and k(h0| ⊗ I)|ψiABk ≤ . Writing the trace as h0| · |0i + h1| · |1i, we replace |1i = 2|+i − |0i to give √ Tr (ρ) = h0|ρ|0i + 2h+|ρ|+i − 2(h+|ρ|0i + h0|ρ|+i) + h0|ρ|0i. 2 2 ≤ 2k(h0| ⊗ I)|ψiABk + 2k(h+| ⊗ I)|ψiABk √ + 2 2k(h0| ⊗ I)|ψiABk · k(h+| ⊗ I)|ψiABk. (4.29) √ ≤ 2(δ + ) + 2 2δ, (4.30) where ρ = TrB (|ψihψ|AB). Thus, the trace-normalization condition is impossible
73 Towards an alternative approach to joinability: enforcing positivity through purification
√ 1 unless 2 ≤ δ + + 2δ. The inequality expresses the fact if δ and are both too small, the total probabilistic weight cannot amount to the proper normalization value. Intuitively, the constraints have “ruled out too much”. We can apply this same technique to the soft constraints in the singlet sharing example. Consider the following soft constraint that ensures proximity to the singlet state on AB
2 2 2 2 k(hφ|AB ⊗ hτ|C ⊗ ID)|ψik ≤ αAB, ∀ |φi ∈ S (C ), ∀ |τi ∈ C , (4.31)
2 2 2 2 2 where S (C ) denotes the symmetric subspace. Defining SAB ≡ S (C ) ⊗ C , we can equivalently express this soft constraint as
2 k(hν|ABC ⊗ ID)|ψik ≤ αAB, ∀ |νi ∈ SAB. (4.32)
2 2 2 Defining SBC ≡ C ⊗ S (C ), we can give the analogous soft constraint on BC as
2 k(hω|ABC ⊗ ID)|ψik ≤ αBC , ∀ |ωi ∈ SBC . (4.33)
2 ⊗3 Since span{SAB, SBC } = (C ) , there are no states |ψi which admit likelihood zero for all states in SAB and SBC . Furthermore, this ensures that, just like the previous qubit example, we can express the trace operation in terms of vectors chosen from SAB 2 ⊗3 and SBC . Let {|ji} be an arbitrary basis for (C ) . Each basis vector can be written as a linear combination |ji = xj|νji + yj|ωji, where |νji ∈ SAB and |ωji ∈ SBC . Letting ρ = TrD (|ψihψ|ABCD), we can write the trace of ρ as
8 X Tr (ρ) = hj|ρ|ji j=1 8 X 2 ∗ ∗ 2 = |xj| hνj|ρ|νji + xj yjhνj|ρ|ωji + xjyj hωj|ρ|νji + |yj| hωj|ρ|ωji. j=1 8 X 2 ∗ ∗ √ 2 ≤ |xj| αAB + (xj yj + xjyj ) αABαBC + |yj| αBC . (4.34) j=1 √ Thus, the agreement parameters are bound by 1 ≤ λαAB + η αABαBC + µαBC , where, the expressions for λ, η, and µ are given in the last line above. Crucially, the values of λ, η, and µ depend on the choice of basis {|ji} and the choice of decomposition |ji = xj|νji + yj|ωji. Some choices will lead to less-strict joinability limitations than other choices. Unfortunately, there is not a single choice of basis and decomposition which diagnoses all joinability failures. Rather, we expect there to be family of “optimal” decompositions. Nevertheless, any choice of decomposition
74 4.3 Joinability limitations from soft constraints gives finite values for λ, η, and µ and, therefore, leads to non-trivial constraints on √ the agreement parameters 1 ≤ λαAB + η αABαBC + µαBC . This approach can certainly be extended to more general settings. Furthermore, it would be valuable to investigate the structure of choosing the decompositions |ji = xj|νji+yj|ωji, and to understand the features of the optimal family of decompositions. We anticipate that convex geometry might play a role in determining such optimal families. If so, we will have returned to an undesirable vantage point, in that, one of the motivations of the work in this chapter was to avoid the use of convex geometry for understanding joinability. We concede that, it is possible (and probably likely), that any complete exploration of the concept of joinability must resolve to using the tools of convex geometry. Regardless, we have shown how the failure of the trace- normalization condition is responsible for certain failures of joinability. As we emphasized in the introduction to this chapter, the ideas presented here are very preliminary and represent a starting point for further investigation. While in the previous two chapters we have emphasized the role that positivity plays in limiting joinability, here, we have argued that an alternative explanation exists. In particular, we showed that failures of the consistency of hard or soft constraints can be explained by over-constraints on the total probabilistic weight of the quantum state or channel. We achieved this by enforcing the quantum state or channel to be manifestly positive-semidefinite or completely positive by means of purification or an isometric extension, respectively. When positive semi-definiteness is made manifest, joinability failures can be diagnosed with failures of the trace-normalization condition.
75
Chapter 5
Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics
77 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics
This chapter presents material that appeared in Quantum Information and Compu- tation, 7:0657, (2016), in an article titled “General fixed points of quasi-local frustration- free quantum semigroups: from invariance to stabilization”, which is joint work with Francesco Ticozzi and Lorenza Viola.
5.1 Introduction
Convergence of a dynamical system to a stable equilibrium point is a hallmark of dissipative, irreversible behavior. In particular, rigorously characterizing the na- ture and stability of equilibrium states of irreversible quantum evolutions is a long- standing problem central to both the mathematical theory of open quantum sys- tems and the foundations of quantum statistical mechanics [135, 136]. In recent years, renewed interest in these issues has been fueled by the growing theoreti- cal and experimental significance of techniques for quantum reservoir engineering [137] and dissipative quantum control [21] within Quantum Information Process- ing (QIP). Representative applications that benefit from engineered dissipation in- clude robust quantum state preparation, with implications for steady-state entan- glement [32, 138, 139, 140, 141], non-equilibrium topological phases of matter [142], and ground-state cooling [143, 144, 145]; as well as open-system quantum simulation [146, 147], steady-state dissipation-driven quantum computation [148, 149], dissipa- tive quantum gadgets and autonomous quantum error correction [150, 151], along with quantum-limited sensing and amplification [152, 153]. While applications are often developed by making reference to a specific physi- cal setting, a common theme is the key role played by constraints, that may restrict the allowed dynamical models and the extent of the available manipulations. This motivates seeking a rigorous system-theoretic framework for characterizing controlled open-quantum system dynamics subject to given resource constraints. In this work, we focus on dissipative multipartite quantum systems described by time-independent quasi-local (QL) semigroup dynamics, capturing the fact that, in many physically relevant scenarios, both the coherent (Hamiltonian) and irreversible (Lindblad) con- tributions to the semigroup generator must act non-trivially only on finite subsets of subsystems, determined for instance by spatial lattice geometry. The main question we address is to determine what properties an arbitrary target state of interest must satisfy in order to be the unique stationary (“fixed”) point for a given QL constraint, thereby making the state globally QL-stabilizable in principle, in an asymptotic sense. In previous work [31, 34], this question has been addressed under the assumption that the target state is pure, providing in particular a necessary and sufficient linear- algebraic condition for the latter to be stabilizable without requiring Hamiltonian dynamics. Such pure states are called purely Dissipatively Quasi-Locally Stabilizable (DQLS). While restricting to a pure target state is both a natural and adequate
78 5.2 Preliminaries
first step in the context of dissipatively preparing paradigmatic entangled states of relevance to QIP (such as W or GHZ states), allowing for a general mixed fixed- point is crucial for a number of reasons. On the one hand, since mixed quantum states represent the most general possibility, this is a prerequisite for mathematical completeness. On the other hand, from a practical standpoint, QL stabilization of a mixed state which is sufficiently close to an “unreachable” pure target may still be valuable for QIP purposes, a notable example being provided by thermal graph states at sufficiently low temperature [154]. Furthermore, as physical systems in thermal equilibrium are typically far from pure, characterizing mixed-state QL stabilization might offer insight into thermalization dynamics as occurring in Nature and on a quantum computer [155]. From this point of view, a stability analysis of thermal states of QL Hamiltonians is directly relevant to developing efficient simulation and sampling algorithms for the quantum canonical ensemble, so-called “quantum Gibbs samplers,” as analyzed in [156] for commuting Hamiltonians. In the mixed-state scenario, the problem of QL stabilization involves qualitatively different features and is substantially more complex. This is largely due to the fact that the analysis tools used in the pure-state setting do not lend themselves to a for- mulation where the invariance property of the globally defined target state translates directly at the level of QL generator components. We bypass this difficulty by restrict- ing to the important class of frustration-free (FF) semigroup dynamics [157, 156], for which global invariance of a state also implies its invariance under each QL compo- nent. Physically, the FF property is known to hold within standard derivations of Markovian semigroup dynamics, for instance based on Davies’ weak coupling limit or “heat-bath” approaches generalizing classical Glauber dynamics [136, 156].
5.2 Preliminaries
5.2.1 Notation and background Consider a finite-dimensional Hilbert space H, dim(H) = d, and let B(H) be the set of linear operators on H.X† shall denote the adjoint of X ∈ B(H), with self- adjoint operators X = X† representing physical observables. The adjoint operation corresponds to the transpose conjugate when applied to a matrix representation of X, with the simple transpose being denoted by XT and the entry-wise conjugation by X∗. To avoid confusion, we shall use I to indicate the identity operator on B(H), whereas I will indicate the identity map (or super-operator) from B(H) to itself. We shall use X ≡ Y to say that X is defined as Y . The convex subset D(H) ⊂ B(H) of trace-one, positive-semidefinite operators, called density operators, is associated to physical states. We are concerned with state changes in the Schrödinger picture between two arbitrary points in time, say 0 and
79 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics
t > 0, which are described by a completely-positive trace-preserving (CPTP) linear map (or quantum channel) on B(H) [136]. A map Tt is CP if and only if it admits an operator-sum representation,
X † ρ(t) = Tt(ρ0) = Mkρ0Mk , ρ0 ∈ D(H), (5.1) k
P † for some {Mk} ⊂ B(H), and is also TP if in addition k Mk Mk = I. The oper- ators Mk are referred to as (Hellwig-)Kraus operators or operation elements [158]. The operator-sum representation of a CPTP map is not unique, and new decompo- 1 sitions may be obtained from unitary changes of the operators Mk. Dual dynamics with respect to the Hilbert-Schmidt inner product on B(H) (Heisenberg picture) are associated to unital CP maps T †, that is, obeying the condition T †(I) = I. A continuous one-parameter semigroup of CPTP maps {Tt}t≥0, with T0 = I, characterized by the Markov composition property Tt ◦ Ts = Tt+s, for all t, s ≥ 0, will be referred to as a Quantum Dynamical Semigroup (QDS) [136]. We shall denote by Lt L the corresponding semigroup generator, Tt = e , with the corresponding dual QDS † † {Tt }t≥0 being described by the generator L . It is well known that L (also referred to as the “Liouvillian”) can be always expressed in Lindblad canonical form [159, 160], that is, in units where ~ = 1: X 1 ρ˙(t) = L (ρ(t)) ≡ −i[H, ρ(t)] + L ρ(t)L† − {L† L , ρ(t)} , t ≥ 0, (5.2) k k 2 k k k
where H = H† is a self-adjoint operator associated with the effective Hamiltonian (generally resulting from the bare system Hamiltonian plus a “Lamb shift” term), and the Lindblad (or noise) operators {Lk} specify the non-Hamiltonian component of the generator, resulting in non-unitary irreversible dynamics. Equivalently, L defines a valid QDS generator if and only if it may be expressed in the form (see e.g. Theorem 7.1 in [161]) 1 L(ρ) ≡ E(ρ) − (κρ + ρκ†), κ ≡ iH + E †(I), (5.3) 2 where E is a CP map and the anti-Hermitian part of κ identifies the Hamiltonian operator. We shall denote by L(H, {Lk}) the QDS generator associated to Hamiltonian H and noise operators {Lk}. Throughout this chapter, both H and all the Lk will be assumed to be time-independent, with (5.2) thus defining a linear time-invariant
1While from a probabilistic and operator-algebra viewpoint it would be more natural to consider the dynamics acting on the states as (pre-)dual, we follow here the standard quantum physics notation as it allows for a more direct connection with existing work as well as a more compact notation in our context.
80 5.2 Preliminaries dynamical system. It is important to recall that, as for CPTP maps, the Lind- blad representation is also not unique, namely, the same generator can be associ- ated to different Hamiltonian and noise operators (see e.g. Proposition 7.4 in [161]), and, further to that, the separation between the Hamiltonian and the noise oper- ators is not univocally defined [31, 161]. Specifically, the Liouvillian is unchanged, 0 0 L(H, {Lk}) = L(H , {Lk}), if the new operators may be obtained as (i) linear combi- 0 0 P ∗ † nations of H, {Lk} and the identity, Lk = Lk + ckI,H = H − (i/2) k(ckLk − ckLk), 0 P 0 with ck ∈ C; or (ii) unitary linear combinations, Lk = l uklLl, H = H, with U ≡ {ukl} a unitary matrix (and the smaller set “padded” with zeros if needed), corresponding to a change of operator-sum representation for E in Eq. (5.3). We will denote a †-closed associative subalgebra A ⊆ B(H) generated by a set of operators X1,...,Xk as A ≡ alg{X1,...,Xk}. If T ≡ T ({Mk}) and L ≡ L(H, {Lk}) are a CP map and a QDS generator, then we shall let alg{T } ≡ alg{Mk} and alg{L} ≡ alg{H,Lk}, respectively. These algebras are invariant with respect to the change of representation in the Kraus or, respectively, Hamiltonian and Lindblad operators since, as remarked, equivalent representations are linearly related to one another. Let ⊕ denote the orthogonal direct sum. It is well known that any †-closed associative subalgebra A of B(H) admits a block-diagonal Wedderburn decomposi- tion [162], namely, H may be decomposed in an orthogonal sum of tensor-product bipartite subspaces, possibly up to a summand: M M (A) (B) H ≡ H` ⊕ HR = H` ⊗ H` ⊕ HR, (5.4) ` ` in such a way that M (A) (B) A = B(H` ) ⊗ I` ⊕ OR, (5.5) ` (B) (B) where I` represents the identity operator on the factor H` and OR the zero operator 0 on HR, respectively. Relative to the same decomposition, the commutant A of A in B(H), given by A0 ≡ {Y | [Y,X] = 0, ∀X ∈ A}, has the dual structure 0 M (A) (B) A = I` ⊗ B(H` ) ⊕ B(HR). (5.6) `
Consider now a density operator ρ ∈ D(H) such that supp(A) ⊆ supp(ρ), where for a generic operator space W the support is henceforth defined as supp(W ) ≡ ∪O∈W supp(O). It then follows that
1 1 1 1 Aρ ≡ ρ 2 A ρ 2 = {Y | Y = ρ 2 Xρ 2 ,X ∈ A} ⊆ B(H) (5.7)
81 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics
is a †-closed subalgebra with respect to the modified operator product X1 ◦ρ X2 ≡ −1 X1ρ X2, where the inverse is in the sense of Moore-Penrose [163] if ρ is not invertible. In the QIP literature, an associative algebra like in Eq. (5.7), which may be thought of as arising from a standard associative algebra A upon replacing each identity factor (B) in Eq. (5.5) with a fixed matrix τ` in each factor has been termed a distorted algebra [164, 165]. In particular, we shall call Aρ a ρ-distorted algebra, and refer to the map 1 1 Φρ(X) ≡ ρ 2 Xρ 2 as a “distortion map”. The ρ-distorted algebra generated by a set of operators X1,...,Xk will be correspondingly denoted by Aρ ≡ algρ({X1,...,Xk}).
5.2.2 Fixed points of quantum dynamical semigroups States that are invariant (aka stationary or “fixed”) under the dissipative dynamics of interest will play a central role in our analysis. Let fix(T ) indicate the set of fixed Lt points of a CP map T ; when Tt = e for t ≥ 0, then clearly fix(Tt) = ker(L). In this section, we summarize relevant results on the structure of fixed-point sets for CPTP maps, and slightly extend them to continuous-time QDS evolutions. Recall that fixed points of unital CPTP maps form a †-closed algebra: this stems from the fact that alg{T } = alg{T †}, along with the following result (see e.g. Theo- rem 6.12 in [161]):
Lemma 5.2.1. Given a CPTP map T , the commutant alg{T }0 is contained in fix(T †). In particular, if there exists a positive-definite state ρ > 0 in fix(T ), then
alg{T }0 = fix(T †). (5.8)
If T is CPTP and unital, its dual map always admits the identity as a fixed point of full rank. It then follows that fix(T ) = alg{T }0 [166, 165]. A similar result can be established for QDS generators (Theorem 7.2, [161]):
Lemma 5.2.2. Given a QDS generator L, the commutant alg{L}0 is contained in the kernel of L†. In particular, if L(ρ) = 0 for some ρ > 0, then
alg{L}0 = ker(L†). (5.9)
A key result to our aim is that, in general, the set of fixed points of a QDS has the structure of a distorted algebra. The following characterization is known for arbitrary (non-unital) CPTP maps (see e.g. Corollary 6.7 in [161]):
Theorem 5.2.3. Given a CPTP map T and a full-rank fixed point ρ,
1 † 1 fix(T ) = ρ 2 fix(T ) ρ 2 , (5.10)
82 5.2 Preliminaries
† L (A) (B) Moreover, with respect to the decomposition fix(T ) = ` B(H` ) ⊗ I` , we have
M (A) (B) ρ = ρ` ⊗ τ` , (5.11) `
(A) (B) where ρ` and τ` are full-rank density operators of appropriate dimension. Building on the previous results, an analogous statement can be proved for QDS dynamics:
Theorem 5.2.4. (QDS fixed-point sets, full-rank case) Given a QDS generator L and a full-rank fixed point ρ,
1 † 1 ker(L) = ρ 2 ker(L ) ρ 2 . (5.12)
† L (A) (B) Moreover, with respect to the decomposition ker(L ) = ` B(H` ) ⊗ I` , we have
M (A) (B) ρ = ρ` ⊗ τ` , (5.13) `
(A) (B) where ρ` and τ` are full-rank density operators of appropriate dimension.
Lt Proof. In order for {e }t≥0 to be a QDS, and thus a semigroup of trace-norm con- tractions [136], L must have spectrum in the closed left-half of the complex plane and no purely imaginary eigenvalues with multiplicity. It is then easy to show, by resorting to its Jordan decomposition [161], that the following limit exists:
Z t 1 Lτ T∞ ≡ lim e dτ. t→∞ t 0 Being the limit of convex combination of CPTP maps, which form a closed convex set, T∞ it also CPTP. Furthermore, T∞ projects onto ker(L), namely, fix(T∞) = ker(L), and T∞ has only eigenvalues 0, 1 with simple Jordan blocks. Similarly, it follows that † † † the unital CP map T∞ ≡ (T∞) projects onto ker(L ). Using these facts along with Theorem 5.2.3, we then have:
1 1 1 1 2 † 2 2 † 2 ker(L) = fix(T∞) = ρ fix(T∞) ρ = ρ ker(L ) ρ .
The structure of the fixed point, Eq. (5.13), follows from Theorem 5.2.3 applied to T∞.
The above two theorems make it clear that, given discrete- or continuous-time CPTP dynamics admitting a full-rank invariant state ρ, the fixed-point sets fix(T )
83 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics
and ker(L) are a ρ-distorted algebra with structure
M (A) (B) Aρ = B(H` ) ⊗ τ` , (5.14) `
(B) where the states τ` are the same for every element in fix(T ) or ker(L). In addition, since ρ has a compatible block structure [Eq. (5.13)], it is immediate to see that fix(T ) and ker(L) are invariant with respect to the action of the linear map Mλ(X) ≡ λ −λ ρ Xρ for any λ ∈ C, and in particular for the modular group {Miφ} [167]. The same holds for the fixed points of the dual dynamics. In fact, we can show that modular invariance is also a sufficient condition for a distorted algebra to be a fixed-point set for a CPTP map that fixes ρ, as relevant to the problem of designing stabilizing dynamics for ρ. In order to do this, we need a result by Takesaki [168], which we give in its finite-dimensional formulation (adapted from [167], Theorem 9.2): Theorem 5.2.5. Let A be a †-closed subalgebra of B(H), and ρ a full-rank density operator. Then the following are equivalent: (i) There exists a unital CP map E † such that fix(E †) = A, (E †)2 = E † and E(ρ) = ρ. 1 − 1 (ii) A is invariant with respect to M 1 , that is, for every X ∈ A, ρ 2 Xρ 2 ∈ A. 2 These conditions are equivalent to saying that the map E † is a conditional expec- tation on A that preserves ρ. We can then prove the following: Theorem 5.2.6. (Existence of ρ-preserving dynamics) Let ρ be a full-rank den- sity operator. A distorted algebra Aρ admits a CPTP map T such that fix(T ) = Aρ if and only if it is invariant for M 1 . 2
Proof. First, notice that, if Aρ is a distorted algebra, then it is invariant for M 1 if 2 − 1 − 1 and only if the “undistorted” algebra A ≡ ρ 2 Aρρ 2 is invariant for M 1 . This follows 2 from the fact that M 1 commutes with both the distortion map and its inverse. In 2 particular, if Aρ is invariant for M 1 , we have: 2
− 1 − 1 − 1 − 1 − 1 − 1 M 1 (A) = M 1 (ρ 2 Aρρ 2 ) = ρ 2 M 1 (Aρ)ρ 2 ⊆ ρ 2 Aρρ 2 = A. 2 2 2
Thus, by Theorem 5.2.5, a unital CP projection E † onto A exists whose adjoint preserves ρ. By Theorem 5.2.3, the CPTP dual E ≡ T is such that fix(T ) = Aρ, as desired. To prove the other implication, it is sufficient to notice that Eq. (5.11) implies † that M 1 leaves A = fix(T ) invariant, and thus 2
1 1 1 1 1 1 M 1 (Aρ) = M 1 (ρ 2 Aρρ 2 ) = ρ 2 M 1 (A)ρ 2 ⊆ ρ 2 Aρ 2 = Aρ. 2 2 2
84 5.2 Preliminaries
If the dynamics admit no full-rank fixed state, we may restrict to the support of a given fixed point, which is an invariant subspace for the Schrödinger’s-picture evolution:
Theorem 5.2.7. (QDS fixed-point sets, general case) Given a finite-dimensional QDS generator L, and a maximal-rank fixed point ρ with H˜ ≡ supp(ρ), let L˜ denote the reduction of L to B(H˜). We then have
1 † 1 ker(L) = ρ 2 (ker(L˜ ) ⊕ O) ρ 2 . (5.15)
Proof. For any ρ ∈ ker(L), the subspace H˜ ≡ supp(ρ) is invariant for the dynam- ˜ ˜ ics [138]. Assume that L = L(H, {Lk}) and let Π: H → H denote the partial isometry onto H˜. Define the reduced (projected) operators ρ˜ ≡ Π˜ρΠ˜ †, H˜ ≡ Π˜HΠ˜ †, ˜ ˜ ˜ † ˜ and Lk ≡ ΠLkΠ . The dynamics inside H is then determined by the correspond- ˜ ˜ ˜ ing projected Liouvillian L(H, {Lk}) [169], and ρ˜ is, by construction, a full-rank state for this dynamics. Hence, the fixed-point set ker(L˜) is the distorted algebra 1 † 1 ker(L˜) = ρ 2 ker(L˜ )ρ 2 . Consider now a maximal-rank fixed point, satisfying supp(ρ) = H˜ = supp(ker(L)). It then follows from Theorem 9 in [138] that H˜ is not only invariant but also attractive for the dynamics. This means that ˜ ⊥ Lt lim Tr Π e (ρ0) = 0, ∀ρ0 ∈ D(H). t→∞
With H˜ being attractive, we have that ker(L) can have support only in H˜, and can thus be constructed by appending the zero operator on H˜⊥, so that, using Theorem 5.2.4:
1 † 1 ker(L) = ker(L˜)⊕O = ρ 2 (ker(L˜ )⊕O) ρ 2 .
In the above proof, we made the construction explicit in terms of a representation L = L(H, {Lk}) in order to make it clear that the result does not hold if we consider 1 † 1 † ρ 2 ker(L ) ρ 2 , since H˜ need not be invariant for L . Again, it follows that ρ admits a block decomposition as in Eq. (5.13), compatible with that of ker(L˜†) on its support.
85 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics
5.2.3 Quasi-local semigroup dynamics Throughout this chapter, the open quantum system of interest will consist of a finite number n of distinguishable subsystems, defined on a tensor-product Hilbert space
n O H = Ha, dim(Ha) = da, dim(H) = D. a=1 As in [31, 34], we shall introduce quasi-locality constraints on the system’s evolution by specifying a list of neighborhoods, namely, groups of subsystems on which operators are allowed to “act simultaneously”. Mathematically, neighborhoods {Nj} may be specified as subsets of the set of indexes labeling the subsystems, that is,
Nj ⊆ {1, . . . , n}, j = 1,...,M.
Each neighborhood induces a bipartite tensor-product structure of H as O O H = H ⊗ H , H ≡ H , H ≡ H . (5.16) Nj N j Nj a N j a a∈Nj a/∈Nj
Likewise, with a neighborhood structure N ≡ {Nj} in place, any state ρ ∈ D(H)
uniquely determines a list of reduced neighborhood states {ρNj }:
ρ ≡ Tr (ρ), ρ ∈ D(H ), j = 1,...,M, (5.17) Nj N j Nj Nj where Tr indicates the partial trace over H . Quasi-local dynamical constraints N j N j may be specified by requiring compatibility with the bipartitions in (5.16), in the following sense:
Definition 5.2.8. (Neighborhood operator) An operator X ∈ B(H) is a neigh- borhood operator relative to a given neighborhood structure N if there exists j such
that the action of X is non-trivial only on HNj , that is:
X = X ⊗ I , Nj N j where I is the identity operator on H . N j N j A similar definition may be given for neighborhood CPTP maps and generators. The relevant quasi-locality notion for QDS dynamics is then the following:
Definition 5.2.9. (QL semigroup) A QDS generator L is Quasi-Local (QL) rela- tive to a given neighborhood structure N if it may be expressed as a sum of neighbor- hood generators:
86 5.2 Preliminaries
X L = L , L ≡ L ⊗ I . (5.18) j j Nj N j j
Quasi-locality of a Liouvillian is well-defined, as the structural property in Eq. (5.18) is defined independently of a particular representation of the generator. In terms of an explicit representation, the above definition is equivalent to requiring that there exists some choice L ≡ L(H, {Lk}), such that each Lindblad operator Lk is a neigh- borhood operator and the Hamiltonian may be expressed as a sum of neighborhood Hamiltonians, namely: X L = L ⊗ I ,H = H ,H ≡ H ⊗ I . k k,Nj N j j j Nj N j j
A Hamiltonian H of the above form is called a QL Hamiltonian (often “few-body,” in the physics literature)2. Mathematically, this denomination is natural given that, for the limiting case of closed-system dynamics, a QL Hamiltonian so defined au- tomatically induces a QL (Lie-)group action consistent with Eq. (5.18), with Lj ≡ i adHj (·) = i [Hj, ·].
Remark 1. The above QL notion is appropriate to describe any locality constraint that may be associated with a spatial lattice geometry and finite interactions range (e.g., spins living on the vertices of a graph, subject to nearest-neighbor couplings). QL semigroup dynamics have also been considered under less restrictive assumptions on the spatial decay of interactions [171], and yet different QL notions may be po- tentially envisioned (e.g. based on locality in “momentum space” or relative to “error weight”). The present choice provides the simplest physically relevant setting that allows for a direct linear-algebraic analysis. We stress that, due to the freedom in the representation of the QDS generator, QL semigroup dynamics may still be induced by Lindblad operators that are not manifestly of neighborhood form. In principle, it is always possible to check the QL property by verifying whether a QDS generator L has components only in the (super-)operator subspace spanned by QL generators. While it may be interesting to determine more operational and efficient QL criteria in specific cases, in most practical scenarios (e.g. open-system simulators [147]) avail- able Lindblad operators are typically specified in a preferred neighborhood form from the outset.
2In particular, the notions of neighborhood Hamiltonian and QL Hamiltonian reduce to the standard uni-local and local ones for non-overlapping neighborhoods, see e.g. [170].
87 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics
5.2.4 Quasi-local stabilizability: prior pure-state results and frustration-free semigroup dynamics Our main focus will be on determining conditions under which a certain state of interest, ρ, is guaranteed to be invariant and the unique asymptotically stable state for some QL dynamics. Formally, an invariant state ρ ∈ D(H) for a QDS with generator L is said to be Globally Asymptotically Stable (GAS) if
Lt lim e (ρ0) = ρ, ∀ρ0 ∈ D(H). (5.19) t→+∞
For QDS dynamics not subject to QL constraints it is known that a state is GAS if and only if it is the unique fixed point [161, 30]. A definition of stabilizable states relevant to our constrained setting may be given as follows:
Definition 5.2.10. (QLS state) A state ρ ∈ D(H) is Quasi-Locally Stabilizable (QLS) relative to a neighborhood structure N if there exists a QL generator L for which ρ is GAS.
Existing work has so far focused on stabilizability of a pure state, with special emphasis on steady-state entanglement [31, 32, 34]. While even in this case, in general, a careful balancing of Hamiltonian and dissipative action is essential, a simple yet non- trivial stabilization setting arises by further requiring that the target can be made QLS by a generator without a Hamiltonian component, namely, by using dissipation alone. Given the freedom in the representation of a QDS generator, in order to formalize this additional constraint we introduced a standard representation for a generator L(H, {Lk}) that fixes a pure state ρ ≡ |ΨihΨ| ∈ D(H), as in the following result (Corollary 1 in [34]):
Proposition 5.2.11. If a generator L(H, {Lk}) makes ρ = |ΨihΨ| GAS, then the ˜ ˜ same generator can be represented in a standard form L(H, {Lk}), in such a way that ˜ ˜ H|Ψi = h|Ψi, h ∈ R and Lk|Ψi = 0, for all k. ˜ In the standard representation, the target |Ψi ∈ ker(Lk) may thus be seen as a common “dark state” for all the noise operators, borrowing from quantum-optics ter- minology. With this in mind, a pure state ρ = |ΨihΨ| may be defined as Dissipatively Quasi-Locally Stabilizable (DQLS) if it is QLS with H˜ ≡ 0 and QL noise operators ˜ {Lk} in standard form. Notice that such a definition implies that ρ is invariant for the ˜ dynamics relative to each neighborhood, namely, ρ ∈ ker(L({Lk})), for each k. Build- ing on this QL-invariance condition allows for proving the following characterization of DQLS states [31]:
88 5.2 Preliminaries
Theorem 5.2.12. A pure state ρ = |ΨihΨ| ∈ D(H) is DQLS relative to N if and only if \ supp(ρ) = supp(ρ ⊗ I ). (5.20) Nk N k k Remark 2. The proof of the above result includes the construction of a set of ˜ stabilizing Lindblad operators Lk that make ρ DQLS, also implying that one such operator per neighborhood always suffices. It is easy to show that any rescaled ˜0 ˜ version of the same operators, Lk ≡ rkLk, also yield a stabilizing QL generator 0 P 2 P L = k |rk| Lk ≡ k γkLk – incorporating “model (γ-)robustness,” in the terminol- ogy of [169]. However, the reasoning followed for QL stabilization of a pure state does not ex- tend naturally to a general, mixed target state. The main reason is that the standard form, hence the DQLS definition itself, do not have a consistent analogue for mixed states. A major simplification if ρ is pure stems from the fact that it is straightforward to check for invariance, directly in terms of the generator components (see Proposi- tion 1 in [34]); for general ρ, we seek a definition that extends the DQLS notion, and that similarly allows for explicitly studying what the invariance of ρ means at a QL level. A natural choice is to restrict to the class of frustration-free dynamics. That is, in addition to the QL constraint, we demand that each QL term in the generator leave the state of interest invariant. Formally, we define [156]: P Definition 5.2.13. (FF generator) A QL generator L = j Lj is Frustration Free (FF) relative to a neighborhood structure N = {Nj} if any invariant state ρ ∈ ker(L) also satisfies neighborhood-wise invariance, namely, ρ ∈ ker(Lj) for all j. Beside allowing for considerable simplification, FF dynamics are of practical interest because they are, similar to the DQLS setting, robust to certain perturbations. As in P Remark 5.2.4, given a QL generator L = j Lj, define a “neighborhood-perturbed” 0 P + QL generator L = j λjLj, with λj ∈ R . If L is FF, then L(ρ) = 0 implies 0 Lj(ρ) = 0 for each j; therefore, λjLj(ρ) = 0 for each j, and thus L (ρ) = 0 as well. Were L not FF, then the kernel of L would not be robust against such neighborhood- perturbations in general. With these motivations, we introduce the notion of QL stability that we analyze for the remainder of this chapter :
Definition 5.2.14. (FFQLS state) A state ρ ∈ D(H) is Frustration-Free Quasi- Locally Stabilizable (FFQLS) relative to a neighborhood structure N if it is QLS with a stabilizing generator L that is FF.
Remarkably, studying FFQLS states will allow us to recover the results for DQLS pure states as a special case. In fact, a pure state is FFQLS if and only if it is DQLS. This claim is proved in Appendix A of [172]. To summarize, the above definition consists of
89 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics
four distinct mathematical conditions that the generator L of the dissipative dynamics must obey for a given target invariant state ρ: • (QDS): L is a generator of a CPTP continuous semigroup; • (QL): L is QL, that is, L = P L , with L = L ⊗ I ; j j j Nj N j • (GAS): ρ is GAS, or equivalently ker(L) = span(ρ);
• (FF): L is FF, namely ker(Lj) ≥ span(ρ) for all Nj. The problem we are interested in is to determine necessary and sufficient conditions for a given state to be FFQLS and, if so, to design QL FF dynamics that achieves the task.
5.3 Frustration-Free Stabilizable States: Necessary Conditions
In this section, we derive necessary conditions for a target state to be FFQLS. Frustration-freeness requires such a state to be in the kernel of each neighborhood generator. We show that, if a neighborhood generator is to leave a global state in- variant, the size and structure of its kernel are constrained; in general, the kernel will be larger than the span of the reduced neighborhood state (as a vector in Hilbert- Schmidt space). However, if the target state is to be the unique fixed point of the QDS dynamics, then the intersection of all the neighborhood-generator kernels must coincide with the span of the target state. We shall show in Section 5.4.2 that this condition is also sufficient for a generic (full-rank) state to be FFQLS.
5.3.1 Linear-algebraic tools
Recall that, given a tensor product of two inner-product spaces V = VA ⊗ VB and a vector v ∈ V , a Schmidt decomposition of v is any decomposition X v = λiai ⊗ bi, i
where ai ∈ VA, bi ∈ VB, λi > 0, and {ai}, {bi} are each orthonormal sets of vectors. There are two instances of Schmidt decomposition which are relevant in our context, both well known within QIP [173]. The first is the Schmidt decomposition of a bipartite pure state |ψi ∈ HA ⊗ HB, namely, X |ψi = λi|aii ⊗ |bii. i
90 5.3 Frustration-Free Stabilizable States: Necessary Conditions
The second is the so-called operator-Schmidt decomposition, whereby a bipartite operator M ∈ B(HA ⊗ HB) = B(HA) ⊗ B(HB) is factorized in terms of elements in the vector spaces B(HA) and B(HB), relative to the Hilbert-Schmidt inner product. Specifically, X M = λiAi ⊗ Bi, i † † where Ai ∈ B(HA), Bi ∈ B(HB), λi > 0, and Tr(Ai Aj) = Tr(Bi Bj) = δij. Building on the concept of Schmidt decomposition, we introduce the Schmidt span:
Definition 5.3.1. (Schmidt span) Given a tensor product of two inner product P spaces V = VA ⊗ VB and a vector v ∈ V with Schmidt decomposition v = i λiai ⊗ bi, the Schmidt span of v relative to VA is the subspace n X o ΣA(v) = span ai ∈ VA | v = ai ⊗ bi, bi ∈ VB . (5.21) i
Without referring to a particular tensor-product decomposition, it is possible to show that the Schmidt span is the image of v under partial inner product:
† ΣA(v) = {a ∈ VA | a = (IA ⊗ b )v for some b ∈ VB}. (5.22)
dA dB † One example is when VA ⊗ VB is a matrix space, such as C ⊗ C (where the latter factor is meant as a space of row vectors). In this case, the Schmidt span of a matrix W ∈ CdA ⊗ CdB †, relative to the first factor CdA , is simply the range of the matrix W (namely, the set of all linear combinations of its column vectors), namely ΣA(W ) = range(W ). Similarly, the Schmidt span of W relative to the second factor CdB † is the orthogonal complement of the kernel of W or, in other words, the support: ΣB(W ) = supp(W ). Another example is when VA ⊗ VB is a bipartite operator space, such as B(HA)⊗B(HB). The Schmidt span of M ∈ B(HA)⊗B(HB) relative to B(HA), i.e. “on A”, is the operator subspace ΣA(M) = {TrB [(IA ⊗ B)M] ,B ∈ B(HB)}. The Schmidt span is a useful tool because conditions on how a neighborhood operator is to affect a global state constrains how such an operator must act on the entire operator-Schmidt span of that state. This intuition is formalized in the following Lemma:
Lemma 5.3.2. (Invariance of Schmidt span) Given a vector v ∈ VA ⊗ VB and 0 0 0 MA ∈ B(VA), if (MA ⊗ IB)v = λv, then (MA ⊗ IB)v = λv for all v ∈ ΣA(v) ⊗ VB. In particular:
span(v) ≤ ker(MA ⊗ IB) ⇒ ΣA(v) ⊗ VB ≤ ker(MA ⊗ IB), (5.23)
91 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics
and span(v) ≤ fix(MA ⊗ IB) ⇒ ΣA(v) ⊗ VB ≤ fix(MA ⊗ IB). (5.24)
P Proof. Consider the Schmidt decomposition of v, v = i γiai ⊗ bi, where ai ∈ VA, bi ∈ VB, γi > 0, and {ai}, {bi} are each orthonormal sets of vectors. Applying the eigenvalue equation for MA to this yields X X γiMAai ⊗ bi = λ γiai ⊗ bi. i i
† † Multiplying both sides by IA ⊗ bj, where bj is the dual vector of bj, selects out the VA-factor of the ith term, i.e., MAai = λai. This holds for each i and any linear combination of the ais. By definition, the Schmidt span of v is ΣA(v) = span{ai}. 0 0 P Denoting {βi} a basis for VB, we may write any v ∈ ΣA(v)⊗VB as v = ij µijai ⊗βj. Applying (MA ⊗ IB) to this we obtain
0 X 0 (MA ⊗ IB)v = µijMAai ⊗ βj = λv . ij
Thus, all elements in ΣA(v) ⊗ VB have eigenvalue λ with respect to MA ⊗ IB, as claimed. Eqs. (5.23) and (5.24) follow by specializing the above result to λ = 0 and λ = 1, respectively.
5.3.2 Invariance conditions for quasi-local generators As remarked, we require the global dynamics to be FF. This simplifies considerably the analysis, as global invariance of the target state is possible only if the latter is invariant for each neighborhood generator. Therefore, we examine the properties of a neighborhood generator that ensure the target state ρ to be in its kernel. Note that if ρ is factorizable relative to the neighborhood structure (i.e., a pure or mixed product state), ρ is invariant as long as each factor of ρ if fixed. Each such reduced neighborhood state can then be made not only invariant but also attractive by a neighborhood generator, if the reduced states are the only elements in the kernels of the corresponding LNj . This automatically makes the global factorized state also GAS. In other words, if ρ is factorizable, then QL stabilizability is guaranteed. If the target state is non-factorizable (in particular, entangled), the above scheme need not work; a non-factorizable state will have some operator Schmidt spans with dimension greater than one. The following Corollary, which follows from Lemma 5.3.2, illustrates the implication of quasi-locally fixing a state with non-trivial operator-Schmidt spans:
92 5.3 Frustration-Free Stabilizable States: Necessary Conditions
Corollary 5.3.3. Let L = L ⊗ I be a neighborhood Liouvillian. If ρ ∈ ker(L ), j Nj N j j then it must also be that
Σ (ρ) ⊗ B(H ) ≤ ker(L ). Nj N j j
Accordingly, if each neighborhood generator Lj is to fix a non-factorizable ρ (as is necessary for global invariance with FF dynamics), then each neighborhood generator must be constructed to leave invariant, in general, a larger space of operators – specifically, the corresponding neighborhood operator-Schmidt span of ρ. However, leaving only the Schmidt spans invariant is, in general, not possible if the dynamics are to be CPTP, since a Schmidt span need not be a distorted algebra (as required by Theorem 5.2.7). We show that, in order for ρ to be in the kernel of a valid QL generator, it is necessary that the dynamics leave certain “minimal fixed- point sets” generated by the Schmidt spans invariant as well. We give the following:
Definition 5.3.4. (Minimal modular-invariant distorted algebra) Let ρ ∈ D(H) be a density operator, and W ⊆ B(H). The minimal modular-invariant dis- torted algebra generated by W is the smallest ρ-distorted algebra generated by W 1 − 1 which is invariant with respect to M 1 (X) = ρ 2 Xρ 2 , where the inverse is in the 2 sense of Moore-Penrose if ρ is not full-rank.
In the finite-dimensional case that we consider, Fρ(W ) can be constructed by the 0 following iterative procedure: define F ≡ algρ(W ), and compute
k+1 k F = alg (M 1 (F )), ρ 2
k+1 k until F = F ≡ Fρ(W ). This particular distorted algebra is the smallest structure whose invariance is required if the dynamics are to be CPTP:
Lemma 5.3.5. (Minimal fixed-point sets) Let W ≤ B(H) be an operator subspace containing a positive-semidefinite operator ρ such that supp(ρ) = supp(W ). If W ≤ fix(T ) for a CPTP map T : B(H) → B(H), then
Fρ(W ) ≤ fix(T ). (5.25)
Proof. Given the iterative construction of Fρ(W ), it suffices to show that if some set W ⊆ fix(T ) includes a density operator with supp(ρ) = supp(W ), then both alg (W ) ⊆ fix(T ) and M 1 (W ) ⊆ fix(T ), and their support is still equal to supp(ρ). ρ 2 Since T is a CP linear map, fix(T ) is closed with respect to linear combinations and
93 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics
†-adjoint. We are left to show that fix(T ) is closed with respect to the ρ-modified product. Consider a partial isometry V : supp(ρ) → H, and define the reduced map
T˜ : B(supp(ρ)) → B(supp(ρ)), T˜(X) ≡ V †T (VXV †)V.
Since ρ is invariant, the set of operators with support contained in supp(ρ) = supp(W ) is an invariant subspace, and thus T (VV †XVV †) = VV †T (VV †XVV †)VV †. By construction, T˜ is CP, TP, and has a full-rank fixed point ρ˜ ≡ V †ρV . It follows from Theorems 5.2.3 and 5.2.6 that fix(T˜) is a ρ-distorted algebra; hence, it is closed with respect to the modified product, as well as modular-invariant. Now, if X,Y ∈ W are fixed points for T , so are X˜ = V †XV, Y˜ = V †YV † for T˜. Since their adjoint, linear combinations and ρ-distorted products are in fix(T˜), we have:
T (Xρ−1Y ) = T (VV †XVV †ρ−1VV †YVV †) = VV †T (V X˜ρ˜−1YV˜ †)VV † = V T˜(X˜ρ˜−1Y˜ )V † = V X˜ρ˜−1YV˜ † = V XV˜ †V ρ˜−1V †V YV˜ † = Xρ−1Y.
Hence, it must be algρ(W ) ≤ fix(T ), as desired, and we still have supp(algρ(W )) = supp(ρ). On the other hand, if X ∈ alg (W ), then supp(M 1 (X)) ∈ supp(W ), and we have: ρ 2
† 1 − 1 † † 1 − 1 † † T (M 1 (X)) = T (VV ρ 2 Xρ 2 VV ) = VV T (V ρ˜2 X˜ρ˜ 2 V )VV ρ 2 1 − 1 † 1 − 1 † = V T˜(˜ρ 2 X˜ρ˜ 2 )V = V ρ˜2 X˜ρ˜ 2 V 1 − 1 = ρ 2 Xρ 2 .
Accordingly, M 1 (W ) ∈ fix(T ) and supp(M 1 (X)) ⊆ supp(ρ) as well, as desired. 2 2
5.3.3 From invariance to necessary conditions for stabilizabil- ity In order to apply the above lemma to our case of interest, namely, finding necessary conditions for FFQLS, the first step is to show that the reduced neighborhood states of ρ may be used to generate the minimal ρ-distorted algebra containing the Schmidt span:
Proposition 5.3.6. Given a neighborhood Nj ∈ N , the support of the corresponding reduced state, ρ = Tr (ρ), is equal to the support of the operator-Schmidt span Nj N j
ΣNj (ρ).
94 5.3 Frustration-Free Stabilizable States: Necessary Conditions
Proof. Since ρNj ∈ ΣNj (ρ), supp(ρNj ) ≤ supp(ΣNj (ρ)). It remains to show the oppo- site inclusion, that is, by equivalently considering the complements, that ker(ρN ) ≤ j ker(ΣNj (ρ)). Let |ψi ∈ ker(ρNj ). Since ρNj ≥ 0, we then have Tr ρNj |ψihψ| = Tr (ρ(|ψihψ| ⊗ I)) = 0. Let {Ei} be a positive-operator valued measure (POVM) on HN which is informationally complete (that is, span{Ei} = B(HN )). The j P j POVM elements sum to I, giving i Tr (ρ(|ψihψ| ⊗ Ei)) = 0. Since each term is non-negative, Tr (ρ(|ψihψ| ⊗ E )) = 0 for all i. Letting ρ ≡ Tr (ρ(I ⊗ E )), we can i i N j i write 0 = Tr (ρ(|ψihψ| ⊗ Ei)) = hψ|ρi|ψi. Then, ρi ≥ 0 implies ρi|ψi = 0 for all i. Since the E span the operator space B(H ), by using Eq. (5.22), we have that the i N j
corresponding ρi span ΣNj (ρ). Hence, |ψi ∈ ker(ΣNj (ρ)).
The above Proposition, together with Lemma 5.3.2 and Lemma 5.3.5, imply the following:
Corollary 5.3.7. If a state ρ is in the kernel of a neighborhood generator Lj = L ⊗ I , then the minimal fixed-point set generated by the neighborhood Schmidt Nj N j span obeys F (Σ (ρ)) ⊗ B(H ) ≤ ker(L ). (5.26) ρNj Nj N j j
Proof. Assume that ρ ∈ ker(L ). By Lemma 5.3.2, we have Σ (ρ) ⊗ B(H ) ≤ j Nj N j
ker(Lj). By Proposition 5.3.6, we also know that the support of ρNj is equal to that of Σ (ρ), and hence supp(ρ ⊗ I ) = supp(Σ (ρ) ⊗ B(H )). With this and the Nj Nj N j Nj N j fact that ρ ⊗ I ∈ Σ (ρ) ⊗ B(H ) ≤ ker(L ), Nj N j Nj N j j Lemma 5.3.5 implies that FρN ⊗I ΣNj (ρ) ⊗ B(H ) ≤ ker(Lj), j N j N j
or, equivalently, F (Σ (ρ)) ⊗ B(H ) ≤ ker(L ), as desired. ρNj Nj N j j Summing up the results obtained on invariance so far, and recalling that unique- ness of the equilibrium state is necessary for GAS, we have the following necessary condition:
Theorem 5.3.8. (Necessary condition for FFQLS) A state ρ is FFQLS relative to the neighborhood structure N only if \ span(ρ) = F (Σ (ρ)) ⊗ B(H ). (5.27) ρNj Nj N j j
95 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics
Proof. Let ρ be FFQLS relative to N . Frustration-freeness of L implies that ker(L) = T ker(L ). From QLS, we have ker(L) = span(ρ). Thus, FFQLS implies Nj j \ ker(Lj) = span(ρ). j
Corollary 5.3.7 implies that for each neighborhood, F (Σ (ρ))⊗B(H ) ≤ ker(L ). ρNj Nj N j j Hence, \ \ F (Σ (ρ)) ⊗ B(H ) ≤ ker(L ). ρNj Nj N j j j j By construction, we also have \ span(ρ) ≤ F (Σ (ρ)) ⊗ B(H ). ρNj Nj N j j
Stringing together the three relationships above, we arrive at the desired result, Eq. (5.27).
5.4 Frustration-Free Stabilizable States: Sufficient Conditions
In this section, we move from necessary conditions for FFQL stabilization to sufficient ones, by providing in the process a constructive procedure to design stabilizing semi- group generators. A key step will be to establish a property that arbitrary (convex) sums of Liouvillians enjoy, namely the fact that, as long as the algebras associated with individual components of the generator are contained in the algebra associated to the full generator, the existence of a common full-rank fixed point suffices to prove frustration-freeness. Drawing on this result, we will prove that the necessary condi- tion of the previous section is also sufficient in the generic case where the target state is full-rank, and then separately address general target states.
5.4.1 A key result on frustration-free Markovian evolutions P Consider a QDS of the form L = k Lk, where individual terms need not, at this stage, correspond to neighborhood generators. The following general result holds:
Theorem 5.4.1. (Common fixed points of sums of Liouvillians) Let L = P k Lk be a sum of QDS generators, and assume that the following conditions hold: (i) alg{Lk} ≤ alg{L} for each k; (ii) there exists a positive definite ρ ∈ ker(L) such that ρ ∈ ker(Lk) for all k.
96 5.4 Frustration-Free Stabilizable States: Sufficient Conditions
0 Then ρ is invariant under L only if it is invariant under all Lk, that is:
0 0 ρ ∈ ker(L) =⇒ ρ ∈ ker(Lk) ∀ k.
T Proof. By linearity of L, we clearly have that ker(L) ≥ ker(Lk). We show that T k under the hypotheses, ker(L) ≤ ker(Lk), therefore effectively implying ker(L) = T k k ker(Lk). By (ii), ρ is a full-rank state in ker(L) and ρ ∈ ker(Lk) for all k. Theorem 5.2.4 implies that
1 1 1 † 1 2 † 2 2 2 ker(L) = ρ ker(L )ρ and ker(Lk) = ρ ker(Lk)ρ , ∀k. Then, by Lemma 5.2.2, we also have that
1 1 1 1 1 † 1 1 1 2 † 2 2 0 2 2 2 2 0 2 ρ ker(L )ρ = ρ alg{L} ρ and ρ ker(Lk)ρ = ρ alg{Lk} ρ . In view of condition (i), the relevant commutants satisfy
0 0 alg{L} ≤ alg{Lk} , ∀k.
The above inequality may then be used to bridge the previous equalities, yielding:
1 † 1 1 0 1 ker(L) = ρ 2 ker(L )ρ 2 = ρ 2 alg{L} ρ 2 ≤
1 † 1 1 1 2 2 2 0 2 ker(Lk) = ρ ker(Lk)ρ = ρ alg{Lk} ρ , T for all k. From this we obtain ker(L) ≤ k ker(Lk), which completes the proof.
Remark 3. We note that condition (i) above, namely alg{Lk} ≤ alg{L}, is only ever not satisfied due to the presence of Hamiltonian contributions in Lk. In fact, P if Lk ≡ Lk({Lj,k}) for each k in a given representation, then L = Lk also has a S k purely dissipative representation L( k{Lj,k}), and thus alg{Lk} ≤ alg{L}. On the other hand, suppose that Lk = Lk(Hk, {Lj,k}), with Hk 6= 0 in some representation. S P This implies that L = L(H, k{Lj,k}), with H = k Hk. In this case, since alg(H) need not contain alg(Hk), condition (i) does not hold in general. As a trivial example, consider two generators associated to H1 = M and H2 = −M, with M 6= I. Clearly, {O} = alg(H) does not contain alg(M). Likewise, if H = (C2)⊗3 and Z is a single- qubit Pauli operator, consider QL Hamiltonians H1 = ZZI and H2 = IZZ. Then alg(H1) alg(H1+H2). Intuitively, this stems from the fact that since noise operators enter “quadratically” (bilinearly) in the QDS, they cannot cancel each other’s action – unlike Hamiltonians, which by linearity may “interfere” with one another.
97 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics
Interestingly, the reasoning leading to Theorem 5.4.1 also applies to CPTP maps, with the simplification that, since no Hamiltonian is present, condition (i) is always satisfied. Formally:
Corollary 5.4.2. (Common fixed points of sums of CPTP maps) Let T = P P k pkTk be a sum of CPTP maps, with pk > 0 and k pk = 1. If there exists a 0 positive definite ρ ∈ fix(T ) such that ρ ∈ fix(Tk) for all k, then ρ is invariant under T only if it is invariant under all Tk, that is:
0 0 ρ ∈ fix(T ) =⇒ ρ ∈ fix(Tk).
The proof is a straightforward adaptation of the one above, and it can actually be extended to a positive semi-definite fixed state ρ, provided that some extra hypotheses on the support of ρ are satisfied. The precise statement and proof of this extended result are given in Appendix B of [172]. Another direct corollary of Theorem 5.4.1, which now specializes to locality- constrained dynamics, provides us with a useful tool to ensure that a QL generator be FF: the generator itself and all of its QL components must share a full-rank fixed state.
Corollary 5.4.3. (Frustration-freeness from full-rank fixed point) Let L = P j Lj be a QL generator, and assume that the following conditions hold: (i) alg{Lj} ≤ alg{L} for each j; (ii) there exists a positive-definite ρ ∈ ker(L) such that ρ ∈ ker(Lj) for all j. Then the QL generator L is FF.
5.4.2 Sufficient conditions for full-rank target states Theorem 5.4.4. (Sufficient condition for full-rank FFQLS) A full-rank state ρ is FFQLS relative to the neighborhood structure N if \ span(ρ) = F (Σ (ρ)) ⊗ B(H ). (5.28) ρNj Nj N j j
Proof. To show that this condition suffices for FFQLS, we must show that there exists some QL FF Liouvillian L for which span(ρ) = ker(L). Our strategy is to first construct a QL generator for which ρ is the unique state in the intersection of the QL-components’ kernels. Then, we use Thm. 5.4.3 to show that this generator is FF, yielding the desired equality.
Fix an arbitrary neighborhood Nj ∈ N , with associated bipartition H = HNj ⊗ H . We shall construct a neighborhood CPTP map E ≡ E ⊗ I , where E N j j Nj N j Nj
98 5.4 Frustration-Free Stabilizable States: Sufficient Conditions
projects onto the minimal fixed-point set containing the neighborhood-Schmidt span (that is, such projection maps are duals to a conditional expectation). Since, by construction, F (Σ (ρ)) is a modular-invariant distorted subalgebra of B(H ), ρNj Nj Nj
Theorem 5.2.6 ensures that there exists a CPTP map ENj such that
fix(E ) = F (Σ (ρ)). Nj ρNj Nj
In particular, we take E 2 = E , so that it projects onto its fixed points. Explicitly, Nj Nj its structure follows from the decomposition in Eq. (5.14):
M (A) (B) F (Σ (ρ)) = B(H ) ⊗ τ , ρNj Nj `,j `,j `
L L (A) (B) with a corresponding Hilbert space decomposition HNj ≡ ` H`,j = ` H`,j ⊗H`,j , (B) (B) and τ`,j a full-rank state on H`,j . Introducing partial isometries Π`,j : H`,j → HNj ,
the sought-after maps ENj can be constructed as:
M † (B) ENj (ρ) ≡ Tr (B) (Π`,jρΠ`,j) ⊗ τ`,j . (5.29) H`,j `
It is straightforward to verify that ENj (ρ) is CPTP. Recalling Eq. (5.3), we may then define a neighborhood QDS generator by taking κ = E † (I)/2 = I/2 and letting Nj
LNj ≡ ENj − INj , ∀j. (5.30)
Let now L ≡ P L = P L ⊗ I define the QL generator of the overall j j j Nj N j dynamics. We constructed each Lj in such a way that
ker(L ) = F (Σ (ρ)) ⊗ B(H ), ∀j. j ρNj Nj N j
Hence, by invoking the hypothesis (Eq. (5.28)), it follows that ρ is the unique state obeying \ span(ρ) = ker(Lj) ≤ ker(L). (5.31) j A priori, it is still possible that span(ρ) < ker(L). However, since we have chosen † κ = κ , the neighborhood generators Lj defined in Eq. (5.30) do not have any Hamiltonian contribution; recalling Remark 3, it follows that the algebra of the global generator contains the algebra of each neighborhood generator,
alg{L} ≥ alg{Lj}, ∀j.
99 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics
Thus, by Corollary 5.4.3, the generator L is FF. From L being FF, it follows in turn that \ ker(L) ≤ ker(Lj), j which, together with Eq. (5.31), implies span(ρ) = ker(L), as desired.
5.4.3 Sufficient conditions for general target states If the target state ρ is not full-rank, the necessary condition of Theorem 5.3.8 may still be shown to be sufficient for FFQLS if an additional condition (referred to as the “support condition” henceforth) is also obeyed:
Theorem 5.4.5. (Sufficient condition for general FFQLS) An arbitrary state ρ is FFQLS relative to the neighborhood structure N if \ span(ρ) = F (Σ (ρ)) ⊗ B(H ) (5.32) ρNj Nj N j j
and \ supp(ρ) = supp(ρNj ⊗ INj ). (5.33) j
Proof. Our strategy is to use the support condition of Eq. (5.33) to reduce the non-full-rank case to the full-rank one. As in the proof of the previous theorem, fix
an arbitrary neighborhood Nj, and consider the maps ENj , defined in Eq. (5.29). Let P ∈ B(H ) denote the Hermitian projector onto supp(F (Σ )(ρ)), and Nj Nj ρNj Nj P ⊥ = I − P the associated orthogonal projector. In this case, we compose each Nj Nj Nj
ENj with the corresponding map P E 0 (·) ≡ P (·)P + Nj Tr (P ⊥ ·), (5.34) Nj Nj Nj Nj Tr(PNj )
where E 0 is, like E , both CP and TP: Nj Nj Tr (E 0 (M)) = Tr M(P + P ⊥ ) = Tr(M), ∀M ∈ B(H ). Nj Nj Nj Nj
With this, consider new CPTP maps given by E ◦ E 0 , whereby it follows that new Nj Nj neighborhood generators may be constructed as
L ≡ E ◦ E 0 − I , L = L ⊗ I , (5.35) Nj Nj Nj Nj j Nj N j
100 5.4 Frustration-Free Stabilizable States: Sufficient Conditions
P with the global evolution being driven, as before, by the QL generator L = j Lj. Define now Π to be the projector onto supp(ρ), and consider the positive-semidefinite function V (τ) = 1 − Tr (Π τ) , τ ∈ B(H). The derivative of V along the trajectories of the generator we just constructed is
˙ X V (τ) = − Tr (Π Lj(τ)) . j
By LaSalle-Krasowskii theorem [174], the trajectories will converge to the largest in- variant set contained in the set of τ such that the above Lyapunov function V˙ (τ) = 0. We next show that this set must have support only on supp(ρ) = T supp(ρ ⊗I ). Nj Nj N j Since V is defined on global input operators, we first re-express each neighborhood generator Lj in Eq. (5.35) as
L = E ◦ E 0 − I, E ≡ E ⊗ I , E 0 ≡ E 0 ⊗ I , j j j j Nj N j j Nj N j
where we have used the property E ◦ E 0 = (E ◦ E 0 ) ⊗ I . Additionally, let j j Nj Nj N j P ≡ P ⊗ I denote the projector onto supp(F (Σ (ρ)) ⊗ I ). Assume now j Nj N j ρNj Nj N j that supp(τ) supp(ρ ⊗ I ) for some N ∈ N , that is, Tr(τP ⊥) > 0. By using * Nk N k k k the explicit form of the maps E 0 given in Eq. (5.34), we then have: Nj ˙ V (τ) ≤ −Tr (Π Lk(τ)) 0 = −Tr Π(Ek ◦ Ek )τ − Π τ
⊥ Tr (ΠEk(Pk)) = −Tr (ΠEk(PkτPk)) − Tr τPk + Tr (Πτ) . (5.36) Tr (Pk)
Since the target state ρ is invariant under Ek, its support is also invariant. This implies that Tr (ΠEk(PkτPk)) ≥ Tr (ΠEk(ΠτΠ)) = Tr (Πτ) . Hence, the sum of the first and the third term in Eq. (5.36) is less than or equal to zero. The second term, on the other hand, is strictly negative. This is because: (i) we ⊥ assumed that Tr(τPk ) > 0; (ii) with Π ≤ Pk, and Ek(Pk) having the same support of Pk by construction, it also follows that Tr (ΠEk(Pk)) > 0. We thus showed that no state τ with support outside of the support of ρ can be in the attractive set for the dynamics. Hence, the dynamics asymptotically converges onto the support of ρ, 0 which is invariant for all the Lj. By restricting to this set, the maps Ej have no effect and the same argument of Theorem 5.4.4 shows that the only invariant set in such a subspace is span(ρ), as desired.
Remark 4. We note that the support condition in Eq. (5.33) is indeed a natural can-
101 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics didate for a sufficient FFQLS condition, since if ρ is pure, it reduces to the necessary and sufficient condition of Theorem 5.2.12. However, it is provably not necessary in general, see Sec. 5.5.4. We conjecture that the necessary condition of Theorem 5.3.8 is in fact sufficient for general (non-full-rank) states. However, we currently lack a complete proof.
5.5 Illustrative Applications
In this section, we illustrate the general framework presented thus far through a num- ber of examples, with a twofold goal in mind: to both demonstrate the applicability and usefulness of the mathematical tools we have developed, and to gain insight into the problem of mixed-state QL stabilization, along with appreciating important dif- ferences from the pure-state setting. For simplicity, we shall focus in what follows on a multi-partite system consisting of n co-dimensional qudit subsystems, namely, n d ⊗n n H = ⊗a=1Ha = (C ) , with D = dim(H) = d . In the especially important case corresponding to qubit (or spin-1/2) subsystems, d = 2, we shall follow standard notation and denote by {|0i, |1i} an orthonormal (computational) basis in C2 and by {σα|α = 0, 1, 2, 3} ≡ {I,X,Y,Z} the set of single-qubit Pauli matrices, under the (a) natural extension to multi-qubit operators, e.g., σx ≡ Xa = I ⊗ ... I ⊗ X ⊗ ... I, with non-trivial action occurring only on the ath factor.
5.5.1 Some notable failures of quasi-local stabilizability Before exhibiting explicit classes of states which are provably FFQLS, it may be useful to appreciate some distinctive features that the mixed nature of the target state entails and, with that, the failure of some intuitively natural mechanisms to generate candidate FFQLS states. Recall that an arbitrary pure product (fully factorized) state is always DQLS (or equivalently, as shown, FFQLS) [31] thus, in other words, failure of a pure target state to be FFQLS always implies some entanglement in the state. In contrast to that, entanglement is not necessary for failures of FFQLS if the target is mixed. Consider n qubits arranged on a line, with neighborhood specified by nearest-neighbor (NN) pairs, Nj ≡ {j, j + 1}, j = 1, . . . , n − 1, and the manifestly separable target state 1 ρ ≡ ρ = (|0ih0|⊗n + |1ih1|⊗n). (5.37) sep 2 Because ρ is already in Schmidt decomposition form for all n, it easily follows that each Schmidt span has the form
ΣNj (ρ) = span{|00ih00|j,j+1, |11ih11|j,j+1},
102 5.5 Illustrative Applications
1 and is already a ρNj -distorted algebra invariant for M . Taking the intersection over 2 all neighborhoods then leaves the two-dimensional space, \ Σ (ρ) ⊗ B(H ) = span{|0ih0|⊗n, |1ih1|⊗n} > span(ρ), Nj N j j
which violates the necessary condition for FFQLS of Theorem 5.3.8. Likewise, mixing a pure FFQLS entangled state |ψi with a trivially FFQLS target such as the fully mixed state results in a “pseudo-pure” target state of the form
n ρ ≡ ρpp = (1 − )|ψihψ| + I/2 , (5.38)
which is not FFQLS in general: an explicit example may be constructed by taking |ψi to be the two-excitation Dicke state on n = 4 qubits, 1 D4,3 ≡ |(0011)i = √ (|0011i + |0101i + |0110i + |1001i + |1010i + |1100i), (5.39) 6
which was proved to be DQLS relative to the three-body neighborhoods N1 = {1, 2, 3}, N2 = {2, 3, 4} in [31] (see also Sec. 5.5.4 for explicit calculations).
5.5.2 Quasi-local stabilization of graph product states Multi-qubit pure graph states are an important resource across QIP, with applications ranging from measurement-based quantum computation [175] to stabilizer quantum error-correcting codes [176]. More recently, thermal graph states [177, 154] have been shown to both provide faithful approximations of pure graph states for sufficiently low temperatures and to support non-trivial multipartite bound entanglement over a temperature range [178]. In this section, we demonstrate that a broader class of mixed graph states on qudits, which we refer to as graph product states, are FFQLS. For the special case of qubits, both pure [32, 31] and thermal [154] graph states are known to be stabilizable with QL FF semigroup dynamics with respect to a natural locality notion induced by the graph. We recover and extend these results to d > 2 and a broader class of non-thermal graph states, without making reference, in the mixed-state case, to properties of the Davies QDS generator which is typically employed under weak-coupling-limit assumptions [136, 156, 154]. The key property of graph product states is that they can be transformed to a product form relative to a “logical subsystem factorization,” following a change of basis which is effected by a sequence of commuting neighborhood unitary transformations (a QL quantum circuit). Commutativity of the unitaries effectively reduces the problem of FFQLS to one of local stabilization of product states. While for graph states this observation allows to directly obtain QL FF stabilizing dynamics, they nevertheless serve as a
103 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics
first relevant example of FFQLS, in preparation for cases where the tools we propose become indispensable. We formally define general qudit graph states following [179]. Let G = (V,E) be a graph, where vertices j ∈ V are associated to qudits and edges (j, k) ∈ E label their allowed pairwise interactions. A natural neighborhood structure is derived from G, by letting the jth neighborhood Nj comprise vertex j along with the subset of vertices adjacent to it. Rather than associating the graph to a particular state, the graph is used to construct a set of commuting unitary edge-wise operators, say {U(j,k)}. The Q product of all such unitaries, UG ≡ (j,k)∈E U(j,k), constitutes the quantum circuit which is used to map an input product state into, in general, an entangled state3. Each U(j,k) is defined as the generalized controlled-Z transformation [179] associated to a symmetric qudit Hadamard matrix H. Hadamard matrices H are defined by † T the conditions H H = dI, H = H , and |hij| = 1, where hij ≡ [H]ij. To each Hadamard, there exists a corresponding generalized controlled-Z gate acting on two- H H qudits, defined by C |iji = hij|iji. With these in place, U(j,k) ≡ C(j,k) ⊗I(j,k), and the unitary transformation which transforms local operators to neighborhood operators on Nj is defined by Y H Uj ≡ C(j,k),
k∈Nj \j where each operator is defined on the global Hilbert space H, and acting non-trivially only on the subsystems by which it is indexed4. Standard pure qudit graph states ⊗n may be defined as [179] |ψGi ≡ UG |+i . Similarly, we define graph product states as
n † O † ρG ≡ UG ρprod UG = UG ρj UG , (5.40) j=1 where each ρj is an arbitrary qudit mixed state. We note that graph product states are distinct from (though overlapping with) so-called graph diagonal states [180], defined as those states obtained by applying the circuit UG to any state ρdiag diagonal in the eigenbasis of the Hadamard matrix. To construct QL Lindblad operators which stabilize a graph product state ρG, we may simply construct the local Lindblad operators which prepare each factor ρj of Eq. (5.40) in the un-rotated basis, and then transform these Lindblad operators with UG. P i † i Let each factor ρj be diagonalized by ρj = Vj( i γj |iihi|)Vj , where γj ≥ 0 are the
3In the cluster model of quantum computation, qubit graph states are constructed by applying this global unitary (as a sequence of commuting neighborhood-wise actions) on an initial product state |+i⊗n. 4 Since H is not uniquely defined, the above U(j,k) depend on the choice of H. For readability, our notation does not make this explicit. For standard qubit graph states, H is the discrete Fourier transform on C2.
104 5.5 Illustrative Applications
ordered (with i) eigenvalues of the qudit density operator ρj and Vj the diagonalizing unitary transformation. Stabilizing Lindblad operators may then be constructed as follows: q j j † † j q j † † Li,i+1 = γi UjVj|iihi + 1|Vj Uj ,Li+1,i = γi+1 UjVj|i + 1ihi|Vj Uj , where i, j = 1, . . . , n and each Lindblad operator is defined on the whole H, but by construction acts non-trivially only on the neighborhood Nj. That the resulting global P dynamics L = j Lj are FF follows from the commutativity of the neighborhood- Liouvillians Lj. It is interesting to note that, since any pure state ρdiag in Eq. (5.40) which is diagonal in the computational basis is necessarily a product, arbitrary pure qudit graph states are FFQLS. In general, however, since ρdiag may be separable but not necessarily of product form, mixed graph diagonal states need not be FFQLS (in line with similar conclusions for trivially separable states, as discussed in Sec. 5.5.1). Remark 4: Graph Hamiltonians. Pure graph states may be equivalently defined as a special class of stabilizer states, by assigning to each vertex in G a stabilizer generator, taken from the generalized Pauli group Gn for n qudits [176]. For qubits, for example, a graph state |ψGi may be seen to the the unique ground state of a QL graph Hamiltonian HG that is a sum of generators of Gn of the form:
n X X O † X HG ≡ HG,j = − Xj Zk = −UG Xj UG. (5.41)
j=1 j∈V k∈Nj \j j
By construction, HG is a sum of commuting terms, and may be easily seen to be FF (namely, such that |ψGi is also the ground state of each HG,j separately). Further to that, the last equality in Eq. (5.41) makes it clear how the graph Hamiltonian is mapped to a (strictly) local one in the “logical basis”, following application of the circuit UG. A feature that becomes evident from expressing graph-Hamiltonians in this form, and that is not shared by more general QL commuting Hamiltonians, is the large degeneracy of their eigenspaces – precisely 1/d of the global space dimension5. Thermal graph states [177, 154], relative to Hamiltonians as in Eq. (5.41), are a special case of graph product states, corresponding to each qudit being in a canonical † βj Xj Gibbs state, namely, ρj ∝ exp(−βjHG,j) in Eq. (5.40) (or, ρG ∝ UG(⊗je )UG), where βj denotes the inverse equilibrium temperature of the jth qubit. Thermal qubit graph states can thus provide a scalable class of mixed multiparty-entangled states. The construction leading to graph product states may be generalized to arbi- trary situations where a quantum circuit arising from commuting unitary neighbor-
5For d = 2, this feature is key in enabling graph-state preparation in finite time with discrete-time dynamics designed via splitting-subspace approaches [181].
105 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics
hood operators may be identified, not necessarily stemming from a graph. That is, Q say that U ≡ (j,k)∈E U(j,k), with each U(j,k), as above, being an edge-wise opera- 0 0 tor, with [U(j,k),U(j0,k0)] = 0 for all edges (j, k), (j , k ). Then we may define QL- transformed product states as resulting from the action of U on any product input † state: ρ ≡ U ρprod U . Note that ρprod is, clearly, FFQLS in the strongest sense, rela- tive to strictly local (single-site) neighborhoods, whereas ρ is FFQLS relative to the S structure {Nj = j∩(i,k)6=∅(i, k)}, which is imposed by the circuit. More generally, a QL commuting circuit may be used to extend neighborhoods of some input Nin into larger neighborhoods of some output Nout, associated to weaker QL constraints. FFQLS states maintain their property under this type of transformation, in the fol- lowing sense:
Proposition 5.5.1. (Circuit-transformed FFQLS) Let ρin be FFQLS relative to Q 0 N ≡ {N }, and let U = U , with U ≡ U ⊗ I and [U ,U 0 ] = 0 for all j, j . in in,i j j j Nj N j j j † Then the output state ρout = U(ρ) ≡ UρinU is FFQLS relative to Nout ≡ {Nout,k}, where [ Nout,k ≡ Nin,k ∪ Nj . (5.42)
Nj ∩Nin,i6=∅
Proof. This is easily verified by constructing FFQLS dynamics for U(ρ). If Lin = P i Lin,i is a FFQL stabilizing dynamics for ρin, construct a new Liouvillian by con- †. jugation, that is, Lout ≡ U ◦ Lin ◦ U . More explicitly,
Y X Y † Y Y † X Lout(ρ) = Uk Lin,i Uj ρ Uj0 Uk0 ≡ Lout,k(ρ), k i j j0 k k
where the neighborhood structure of the output Liouvillian relative to the (enlarged, in general) neighborhoods in Eq. (5.42) follows from the commutativity of the circuit unitaries Uj, as conjugation by all but those unitaries constrained by Eq. (5.42) has no net effect. Both the spectrum and the FF property are preserved as U is unitary, and ρout is stabilized by Lout because its kernel is U(ker(Lin)) = U(span(ρin)). Physically, the obvious way to construct a QL commuting circuit is via exponen- tiation of commuting QL Hamiltonians, namely, Uj ≡ exp(iHj), with [Hj,Hj0 ] = 0 for all j, j0. In fact, any QL commuting circuit arises in this way, in the sense that a family of QL commuting Hamiltonians may always be associated to U, for instance by letting Hj = −i log Uj in the basis which simultaneously diagonalizes all circuit unitaries. Remark 5: Rapid mixing. As mentioned, for both pure and thermal graph states on qubits, QL stabilizing dynamics have been thoroughly analyzed in the literature. In particular, rigorous upper bounds on the mixing time have been established, showing
106 5.5 Illustrative Applications
that such states may be efficiently prepared – that is, the (worst-case) convergence time scales only (poly-) logarithmically with the system size [182, 154]. Remarkably, rapid mixing has been shown to both lead to stability against QL perturbations of the generator [171] and to the emergence of effective area laws [183]. These results extend naturally to the broader classes of graph product states and encoded product FFQLS states considered here.
5.5.3 Quasi-local stabilization of commuting Gibbs states
In this subsection, we analyze FFQLS of another class of states derived from com- muting QL Hamiltonians. Consider a Gibbs state:
e−βH X ρ ≡ ,H ≡ H , β ∈ R+, (5.43) β Tr (e−βH ) j j
where each Hj is a neighborhood-operator relative to Nj ∈ N . If the neighborhood 0 Hamiltonians satisfy [Hj,Hj0 ] = 0 for all j, j , ρβ is also called a commuting Gibbs state [156]. Characterizing QL evolutions that have canonical Gibbs states as their unique fixed point has both implications for elucidating aspects of thermalization in naturally occurring dynamics and for quantum algorithms and simulation – most notably, in the context of quantum generalizations of Metropolis sampling [184]. Recent work [156] has shown that Gibbs states of arbitrary QL commuting Hamiltonians are FFQLS, the commutativity property being essential to ensure quasi-locality of either the weak- coupling (Davies) generator or the heat-bath QDS dynamics that dissipatively prepare them. The central result therein establishes an equivalence between the stabilizing dynamics being gapped and the correlations in the Gibbs state satisfying so-called “strong clustering”, implying rapid mixing for arbitrary one-dimensional (1D) lattice systems, or for arbitrary-dimensional lattice systems at high enough temperature. It is important to appreciate that in the derivation of such results, primitivity of the QDS generator is assumed from the outset, and verified, along with the QL and FF properties, by making explicit reference to the structure of the Davies or heat- bath generator (see respectively Lemma 9 and Theorem 10 in [156]). Conversely, the QL notion is not a priori imposed as a design constraint for the dynamics, but again emerges from the structure of the generator itself. In this sense, our framework may be seen to provide a complementary approach, providing in particular a necessary condition for thermal dynamics to be primitive relative to a specified neighborhood structure. Let us illustrate the potential of our approach by focusing on the simplest setting of commuting two-body NN Hamiltonians in 1D.
Proposition 5.5.2. A full-rank state ρ > 0 defined on a 1D lattice system is FFQLS
107 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics
relative to neighborhoods Nj = {j, j + 1} if and only if
O \ O span(ρ) = Fρj,j+1 {Σj,j+1(ρ)} Fρ1 {Σ1(ρ)} ⊗ Fρk,k+1 {Σk,k+1(ρ)} . j odd k even
Proof. The proof simply follows from noting that we can group the neighborhoods {j, j + 1} with odd and even j and, since neighborhoods in the same group are not overlapping, that the intersection of the minimal fixed-point sets corresponding to neighborhoods in the same group corresponds to their product.
One consequence of the above simplification is the following:
Proposition 5.5.3. Let ρ > 0 be FFQLS with respect to the above 1D NN neighbor- hood structure. If, for any neighborhood {l, l+1}, the minimal fixed-point set is the full algebra, that is, Fρl,l+1 {Σl,l+1(ρ)} = B(Hl,l+1), then ρ must factor as ρ = ρ1...l ⊗ρl+1...n.
Proof. Assume that ρ is FFQLS and Fρl,l+1 {Σl,l+1(ρ)} = B(Hl,l+1). Then, ρ satisfies the intersection condition, which is simplified to a tensor product of two intersections
span(ρ) = (int left)1,...,l ⊗ (int right)l+1,...,n, (5.44) where (int left)1,...,l stands for
l−2 l−1 O \ O Fρj,j+1 {Σj,j+1(ρ)} ⊗ B(Hl) Fρ1 {Σ1(ρ)} ⊗ Fρk,k+1 {Σk,k+1(ρ)} j odd k even
and, similarly, (int right)l+1,...,n is given by
n n O \ O B(Hl+1) ⊗ Fρj,j+1 {Σj,j+1(ρ)} Fρk,k+1 {Σk,k+1(ρ)} . j odd,j=l+2 k even,k=l+1
Eq. (5.44) can only be satisfied if ρ = ρ1...l ⊗ ρl+1...n.
The above proposition captures the fact that, for a state of a 1D NN-coupled chain to be FFQLS, there are limitations to the correlations that the state can exhibit. Similar restrictions are, generically, sufficiently strong to prevent Gibbs states of 1D
108 5.5 Illustrative Applications
commuting NN Hamiltonians to be FFQLS relative to NN neighborhoods. A simple example is a 1D Ising Hamiltonian:
n X H = − ZjZj+1, n > 3, (5.45) j=1
where periodic boundary conditions are assumed. The Schmidt span for each NN pair consists of the space of diagonal matrices, and is closed under generation of the distorted algebra. The intersection of all the Schmidt spans is thus the 2n-dimensional space of diagonal matrices, implying that, for all temperatures and size n, the Gibbs state is not FFQLS. In particular, no FF thermal dynamics subject to the NN QL constraint can stabilize (or be primitive with respect to) this state. The thermal dynamics of the Davies or heat-bath generators that stabilize com- muting Gibbs states are, in fact, both FF and QL, albeit relative to a different neighborhood structure than the one solely determined by the system’s Hamiltonian [156]. This is most transparent in the weak-coupling derivation of the QDS, whereby −itH the evolution induced by this Hamiltonian, Ut ≡ e , effectively “modulates” in time the bare system-bath neighborhood coupling operators, in turn determining the relevant Lindblad operators in frequency space [136]. The net effect is that Ut acts as a QL commuting circuit, resulting in a neighborhood structure which is expanded with respect to the one associated to H or to the coupling operators alone (recall Proposition 5.5.1). In the specific Ising example of Eq. (5.45), Davies generators are QL for three-body (next-to-NN, NNN for short) neighborhoods, Nj = {j − 1, j, j + 1}. Similarly, one can generalize the idea and define an enlarged “Davies QL notion”. With respect to this QL constraint, commuting Gibbs states may be shown to obey our necessary and sufficient condition for FFQLS, as expected on physical grounds:
Proposition 5.5.4. (FFQLS commuting Gibbs states) Gibbs states of 1D NN commuting Hamiltonians are FFQLS relative to the Davies (NNN) neighborhood structure.
Pn−1 Proof. Up to normalization and letting, for convenience, βH ≡ j=1 Hj,j+1, 0 < β < −H12 −H23 −Hn−1,n ∞, the Gibbs state of Eq. (5.43) may be written as ρβ ≡ e e . . . e = σ12σ23 . . . σn−1,n, where the σi,i+1 are pairwise-commuting, invertible matrices defined that are different from the identity only in NN sites. Our strategy is to first compute
the minimal fixed-point set Fρ234 (Σ234(ρ)) and its intersection with Fρ123 (Σ123(ρ)), and then, by iterating, to show that the resulting intersection is span(ρ). To compute
Fρ234 (Σ234(ρ)), we first obtain the corresponding Schmidt span. Using Eq. (5.22), Σ234(ρ) = span{Tr234 (ρM)}, for all M ∈ B(H1) ⊗ I234 ⊗ B(H5,...,n). Letting τ2 ≡
109 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics
Qn−1 j=5 σj,j+1, we can write
B(H1) ⊗ I234 ⊗ B(H5,...,n) = span{τ2(A1 ⊗ I234 ⊗ B5 ⊗ C6,...,n)}, where A, B, C range over all matrices acting on those sites. With this parameteriza- tion, the Schmidt span is simplified to
Σ234(ρ) = span{Tr1 ((A1 ⊗ I1)σ12) σ23σ34Tr5 ((B5 ⊗ I5)σ45)}
= σ23σ34[Σ2(σ12) ⊗ I3 ⊗ Σ4(σ45)], noting that σ23 and σ34 commute with all operators in this space. To calculate
Fρ234 (Σ234(ρ)), we first obtain the reduced state
ρ234 = σ23σ34Tr234 (σ12σ45 . . . σn−1,n) = σ23σ34(σ2 ⊗ I3 ⊗ σ4),
where σ2 = Tr2 (σ12) and σ4 = Tr4 (σ45). It follows that Fρ234 (Σ234(ρ)) has a simple structure:
Fρ234 (Σ234(ρ)) = σ23σ34[Fσ2 (Σ2(σ12)) ⊗ I3 ⊗ Fσ4 (Σ4(σ45))].
Direct calculation verifies that Fρ234 (Σ234(ρ)) obeys the required properties of closure under the distortion map Φρ234 and invariance under ρ234-modular action. Similarly, we have
Fρ123 (Σ123(ρ)) = σ12σ23[I12 ⊗ Fσ3 (Σ3(σ34))].
Finally, we compute the intersection of these two adjacent fixed-point sets. To highlight the necessary structure, we write F i,i+1 ≡ F (Σ (σ )), where k = i or σk σk k i,i+1 34 i + 1. The relevant intersection is then {σ12σ23[I12 ⊗ Fσ3 ⊗ B(H4)]} ∩ {σ23σ34[B(H1) ⊗ 12 45 Fσ2 ⊗I3 ⊗Fσ4 ]}. Factoring out the common invertible multiple of σ23, this intersection simplifies to
34 12 45 σ23[{(σ12) ⊗ (Fσ3 ⊗ B(H4))} ∩ {(B(H1) ⊗ Fσ2 ) ⊗ (σ34Fσ4 )]}] 12 34 45 = σ23[{(σ12) ∩ (B(H1) ⊗ Fσ2 )} ⊗ {(Fσ3 ⊗ B(H4)) ∩ (σ34Fσ4 )}. For each intersection, notice that one argument is contained in the other, giving
45 45 σ23[σ12 ⊗ σ34Fσ4 ] = σ12σ23σ34[I123 ⊗ Fσ4 ].
By iterating, we find that subsequent intersections simplify to σ12 . . . σj−1,j[I1,...,j−1 ⊗ F j,j+1]. For the final intersection, we may take σ = I ⊗ 1 , where we take the σj n,n+1 n n+1
(n+1)-th system to be trivial. This leads to Fσn (Σn(σn,n+1)) = span(In). Thus, after
110 5.5 Illustrative Applications taking the intersection over all fixed-point sets, we are left with \ Fρj,j+1,j+2 (Σj,j+1,j+2(ρ)) = σ12 . . . σn−1,n(I1,...,n−1 ⊗ span(In)) = span(ρ), (5.46) which, by Theorem 5.4.4, proves that these states are FFQLS.
5.5.4 Quasi-local stabilization beyond commuting Hamiltoni- ans So far, the identification of a commuting structure has played an important role in the verification of the FFQLS property. Thus, it is an important question to determine the extent to which “lack of commutativity” may hinder FFQLS. The issue is simpler and better (albeit still only partially) understood for pure target states, in which case families of QL stabilizable states not stemming from a commuting structure have been identified for arbitrary system size and complex multi-partite entanglement patterns. Notably, spin-1 AKLT states in 1D, which are the archetypal example of a valence-bond-solid state in condensed-matter physics [185], as well as a spin-3/2 (or higher) AKLT states in 2D, which provide a resource for universal quantum computation [186, 187], are unique ground states of FF anti-ferromagnet Hamiltonians. As such, they are FFQLS using NN, two-body dissipative dynamics [32]. Perhaps even more surprisingly, the FFQLS property still holds for long-range entangled states known as Motzkin states [188], which are also unique ground states of FF NN spin-1 Hamiltonians and have been proved to (logarithmically) violate the area law. In what follows, we first exhibit a family of “non-commuting” FFQLS multi-qudit generalized Dicke states, by also including a general result linking QL stabilizability of a pure state to its ability to be uniquely determined by its neighborhood marginals. Focusing then on mixed target states, we construct and analyze two explicit (non- scalable) examples showing that commutativity of the parent Hamiltonian or the generating QL circuit is, as for pure states, not necessary for FFQLS in general.
Pure Dicke states on qudits Since the main emphasis of this chapter is on mixed states, most of the technical proofs of the results in this section are omitted and can be found in Appendix C of [172]. Similar to qubit Dicke states from quantum optics [189], qudit Dicke states may be constructed by symmetrizing an n-qudit product state in which k of the n subsystems are “excited” to a given single-particle state, and the remaining (n − k) are in their “vacuum”. We may further naturally generalize by allowing multi-level
111 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics excitation. Specifically, let {|`i}, ` = 0,..., (d − 1), denote an orthonormal basis d in C and Sn ≡ {π} the symmetric (permutation) group on n objects. Then each generalized Dicke state is in one-to-one correspondence with a vector of integers, ~ Λ = (k0, . . . , kd−1), where each k` specifies the occupation number (multiplicity) of each single-qudit state. That is6,
~ 1 X |Λi ≡ |(0 ... 0, 1 ... 1, . . . , d − 1 . . . d − 1)i ≡ Vπ| 0 ... 0 1 ... 1 . . . d − 1 . . . d − 1i, | {z } | {z } | {z } n! | {z } | {z } | {z } k0 k1 kd−1 π∈Sn k0 k1 kd−1 (5.47) P where i ki = N and Vπ permutes the subsystems according to the permutation π. A useful fact to our purpose is that generalized Dicke states admit a simple Schmidt decomposition. Consider a partition of the system into two groups of nA and nB subsystems, respectively. It is then easy to show that the Schmidt decomposition of |Λ~ i is
s ~ 1 X ~ ~ nA nB |Λi = µ~ ~ |ΛAi ⊗ |ΛBi, µ~ ~ = . (5.48) q ΛA,ΛB ΛA,ΛB ~ ~ n ΛA ΛB ~ ~ ~ Λ~ ΛA+ΛB =Λ
In order to specify the relevant class of FFQLS generalized Dicke states, the choice of the neighborhood structure is crucial. The following definition captures the re- quired feature:
Definition 5.5.5. A neighborhood structure {Nk} is connected if for any bipartition of the subsystems, there is some neighborhood containing subsystems from both parts.
Our main result is then contained in the following: Proposition 5.5.6. (FFQLS Dicke states) Given n qudits and a connected neigh- borhood structure N , there exists a (non-factorized) FFQLS generalized Dicke state relative to N if d(m − 1) ≥ n, where m is the size of the largest neighborhood in N . The proof (found in Appendix C of [172]) is constructive, and yields, in particular, the state
Dn,m ≡ |(0 ... 0, 1 ... 1, . . . , d − 1 . . . d − 1)i, r = n − (d − 1)(m − 1), (5.49) | {z } | {z } | {z } m−1 m−1 r as a non-factorized (entangled) DQLS state for given, arbitrary system size. Nonethe- less, note that the product of the neighborhood size and qudit dimension must be
6States of this form have been recently analyzed in [190], where additionally non-uniform super- positions are also considered. We maintain permutation symmetry as a hallmark our generalization of standard Dicke states.
112 5.5 Illustrative Applications
scaled accordingly. For example, if N is fixed to be two-body NN (hence m = 2), then the qudit dimension itself must be at least d = n. In this sense, the resulting family of states is non-scalable. Remark 6. As a particular case of Propositions 5.5.6, we recover the fact (established in [31]) that the state |(0011)i = D4,3, also previously defined in Eq. (5.39), is FFQLS with respect to the neighborhood structure N1 = {1, 2, 3}, N2 = {2, 3, 4}. Generalized Dicke states are non-trivially entangled. Their multiparty correlations have the feature of being uniquely determined by a proper subset of all possible marginals [190], that is, of being “uniquely joined” [61]. Remarkably, an interesting connection may be made between the extent to which an arbitrary pure target state is uniquely determined by the set of its neighborhood-marginals and the FFQLS property. This is formalized in the following:
Proposition 5.5.7. (Unique joinability) If a pure state |ψi is FFQLS relative
to N , then |ψi is uniquely determined by its neighborhood reduced states {ρNj } = { Tr (|ψihψ|)}. N j
Proof. By contradiction, if |ψi is not uniquely joinable, then there exists a state τ, not necessarily pure, such that τ 6= |ψihψ| and Tr (τ) = ρ , for all N ∈ N . Clearly, N j Nj j supp(τ) ≤ supp(ρ ) ⊗ B(H ), for all j, hence by recalling Theorem 5.2.12 it also Nj N j follows that \ supp(τ) ≤ supp(ρ ) ⊗ B(H ) = span(|ψi). Nj N j j On the other hand, since by assumption supp(τ) 6= span(|ψi), the above inclusion must be strict, which is impossible since span(|ψi) is one-dimensional.
We now establish that the class of states of Eq. (5.49) constitute genuinely non- commuting examples of FFQLS. To do so, we show that, with respect to a certain class of neighborhood structures, these states are the unique ground states of non- commuting FF QL Hamiltonians, but they cannot be the unique ground states of any commuting FF QL Hamiltonians. First, since the ground-state space of a FF Hamiltonian is simply the intersection of the ground-state spaces of the individual Hamiltonian terms in the sum, replacing each such term with the projector onto its excited space preserves the ground space of the global FF Hamiltonian. Without loss of generality, we may then restrict atten- tion to QL Hamiltonians consisting of sums of neighborhood-acting projectors. An important consequence of this simplification is that for a given FFQL Hamiltonian, the only possible candidates for other FFQL Hamiltonians with the same ground- state space are sums of projectors with enlarged ground-state spaces, with respect to
113 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics
those of the given Hamiltonian. However, the following Lemma demonstrates that, in the two-neighborhood case, a non-commuting FFQL Hamiltonian does not admit a commuting enlargement with the same ground state space:
0 0 Lemma 5.5.8. (Commuting enlargements) Assume that Π1, Π2 are two non- commuting projections such that
0 0 n 0 lim (Π1 Π2) ≡ ΠGS 6= 0. n→∞
If Π1, Π2 are enlarged commuting projectors such that
0 0 0 Πk Πk = Πk Πk = Πk, k = 1, 2, (5.50)
0 then Π1Π2 ≡ ΠGS, with ΠGS ΠGS.
Proof. Using Eq. (5.50), we have that
0 0 0 0 0 0 0 0 Π1 Π2 = Π1 Π1 Π2 Π2, Π2 Π1 = Π2 Π2 Π1 Π1.
0 Towards proof by contradiction, assume that ΠGS = Π1Π2 = ΠGS. It follows that
0 0 0 0 0 0 0 0 0 0 Π1 Π2 = Π1 ΠGS Π2, Π2 Π1 = Π2 ΠGS Π1.
0 0 By using the defining property of Π1, Π2, however, the right hand-side in each of the 0 0 0 0 above equalities simplifies to Πk ΠGS Πk = ΠGS, for k = 1, 2. This in turn yields
0 0 0 2 1 Π1 Π2 = ΠGS = Π0 Π0,
which contradicts the non-commuting assumption.
With this Lemma in place, we now verify the genuine non-commutativity of these states.
Proposition 5.5.9. (Non-commutativity of FFQLS Dicke states) For each (non-factorized) Dicke state Dn,m of Eq. (5.49), there exists a neighborhood struc- ture for which Dn,m cannot be the unique ground state of any commuting FF QL Hamiltonian.
Proof. Consider the (non-factorized) Dicke state Dn,m. Correspondingly, we choose any connected neighborhood structure N with m-body neighborhoods, whereby there is at least one neighborhood (N1, say) containing a system that is not contained in any
114 5.5 Illustrative Applications
other neighborhood. Such a neighborhood structure always exists, and Proposition 5.5.6 ensures that Dn,m is FFQLS relative to that. From [31], any FFQLS state |ψi is the unique ground state of some FF QL Hamiltonian. In particular, letting |ψi ≡ Dn,m, one such parent Hamiltonian is X X H = H ≡ (I − Π ⊗ I ), (5.51) j Nj N j j j
where ΠNj is the projector onto the Schmidt span of Dn,m with respect to Nj. We first show that the Hj do not commute with one another. From the Schmidt decomposition P of Dn,m, given in Eq. (5.48), we have ΠNj = Λ |ΛihΛ|, where the sum extends over ~ all choices of m symbols from the symbols in Dn,m = |Λi. A direct calculation the
shows that the ΠNj , and therefore, the Hj, of overlapping neighborhoods do not commute with one another. In order to establish the desired result, note that enlarging the neighborhoods preserves FFQLS. Hence, Dn,m is also FFQLS with respect to the two-neighborhood neighborhood structure with one neighborhood being N1 and the other neighborhood, say, NU , being the union of the remaining neighborhoods of N . Furthermore, by building a parent Hamiltonian out of these two projectors as in Eq. (5.51), the reasoning above shows that the projectors of Dn,m with respect to N1 and NU do not commute. Hence, by Lemma 5.5.8, no commuting enlargement exists for {N1, NU }. Assume now that a commuting QL parent Hamiltonian exists for the original neighborhood structure. The projectors Πk onto the ground state spaces of these Hamiltonians also commute. Along with FF condition, this implies that
|ψihψ| = Π1Π2 ... Π|N | ≡ Π1ΠU ,
where ΠU ≡ Π2 ... Π|N |. Hence, the QL Hamiltonian H = (I − Π1) + (I − ΠU ) constitutes a commuting enlargement for {N1, NU }, which we showed cannot exist. This contradiction then implies that no commuting enlargement can exist for N .
Non-commuting Gibbs states
In order to demonstrate that genuinely mixed target states may also be FFQLS despite not being obviously associated to a commuting structure, specific examples may be constructed in 1D by considering a generalization of the NN Ising Hamiltonian
115 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics considered in Eq. (5.45), obtained by adding a transverse (magnetic) field. That is7:
n−1 n X X + H = − ZjZj+1 − g Xj, n ≥ 4, g ∈ R . (5.52) j=1 j=1
In particular, we consider (full-rank) Gibbs states, constructed as in Eq. (5.43), as well as variants inspired by non-equilibrium quench protocols, wherein an initial thermal state of a Hamiltonian with given g is evolved under a Hamiltonian with g0 6= g. In all cases, the relevant minimal distorted algebras and their intersection have been numerically constructed (in Matlab) for given QL constraints, and Theorem 5.4.4 used to determine FFQLS. The results are found to depend sensitively on the neighborhood structure: Gibbs (and generalized Gibbs) states on n = 4 qubits are found to be FFQLS for three-body neighborhoods (as in the corresponding commuting Ising case), however extending to n = 5, 6 qubits requires neighborhoods to be further enlarged to allow for four-body Liouvillians. While a direct (numerical) verification is beyond reach, this points to the possibil- ity that the (maximal) neighborhood size will have to scale extensively as n increases, thereby preventing scalable FFQLS. An intuitive argument in support of this is the observation that, as the size of the “neighborhood complements” increase, the dimen- sions of the extended Schmidt spans do as well; correspondingly, the uniqueness of their intersection, as is required for FFQLS, becomes less likely. Despite this limita- tion, these results show the general applicability of our framework.
Entangled mixed states As a final application, we analyze QL stabilizability of a one-parameter family of mixed entangled states on n = 4 qubits. Beside illustrating the full procedure needed to check if ρ is FFQLS and to construct the stabilizing maps, this example is useful for a number of reasons: first, it reinforces that genuinely multipartite entangled mixed states can be FFQLS; second, it explicitly shows that the support condition under which we proved sufficiency for general target states in Theorem 5.4.5 is not necessary in general; lastly, it shows how a non-FFQLS state may still admit arbitrarily close (in Hilbert-Schmidt space) states that are FFQLS, that is, in control-theoretic language, it may still in principle allow for “practical stabilization”. The family of mixed states we analyze may be parametrized as follows:
4 4 ρ ≡ (1 − ) |(0011)ih(0011)| + |GHZ2ihGHZ2|, ∈ (0, 1), (5.53) 7Hamiltonians such as in Eq. (5.52) are exactly solvable upon mapping to a free-fermion problem, and commuting once expressed in terms of appropriate quasi-particles. Indeed, thermal dynamics associated to free-fermion QDS is known to be hypercontractive [154], in the absence of a QL “real- space” constraint as we consider.
116 5.5 Illustrative Applications
4 with neighborhoods N1 = {1, 2, 3}, N2 = {2, 3, 4}. Here, GHZ√2 is the usual GHZ n ⊗n ⊗n state on qubits, that is, |GHZd i = (|0i + ... + |d − 1i ) / d. As established in [31], this state is not DQLS for any non-trivial neighborhood structure8, as one may verify by seeing that the d-dimensional space span{|0, 0,..., 0i,..., |(d − 1, d − 1, . . . , d−1i} is contained in each extended Schmidt span, and hence their intersection n is greater than just span(|GHZd i). Let us use the notation 123|4 to denote the partition of the index set {1, 2, 3, 4} in the neighborhood {1, 2, 3} and the remaining index {4}, and similarly for 1|234: the 1|234-Schmidt decomposition means the Schmidt decomposition with respect to such bipartition. In order to construct QL FF dynamics which render ρ GAS, the first step is to compute the operator Schmidt span for each neighborhood. Two properties aid our analysis. First, both the Dicke and GHZ components are permutation symmetric, so that the analysis of the 1|234 partition carries over to that of 123|4. Second, they have compatible Schmidt decompositions, in the sense that we can find a single operator basis in which to Schmidt-decompose both of them and their mixtures. Using that
|(0011)i = p1/2|0i|(011)i + p1/2|1i|(001)i, 4 p p |GHZ2i = 1/2|0i|000i + 1/2|1i|111i, the desired operator Schmidt decomposition is 1 ρ = |0ih0| ⊗ [(1 − )|(011)ih(011)| + |000ih000|] 2 1 + |0ih1| ⊗ [(1 − )|(011)ih(001)| + |000ih111|] 2 1 + |1ih0| ⊗ [(1 − )|(001)ih(011)| + |111ih000|] 2 1 + |1ih1| ⊗ [(1 − )|(001)ih(001)| + |111ih111|]. 2 Let us focus on the factors relative to subsystems {234} from each term above. To compute the minimal fixed-point set containing this Schmidt span, we first undo the distortion of the elements of this space by conjugating with respect to
− 1 1 1 ρ 2 = √ (|(011)ih(011)| + |(001)ih(001)|) + √ (|000ih000| + |111ih111|), 234 1 −
8 GHZ states are graph states, though with respect to a star graph (i.e., a central node connected to (n − 1) surrounding nodes). The neighborhood structure induced by this graph is trivial, in that it consists of a single neighborhood containing all n qubits. Hence, we treat GHZ states as separate from graph states, considering only non-trivial QL constraints.
117 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics where, as noted, the above inverses are taken as the Moore-Penrose inverse (with √1 √ 1 or 1− being replaced by 0 in the singular cases of = 0 or 1, respectively). 1 − 2 Conjugation of each Schmidt basis element with respect to this ρ234 removes the -dependence, namely,
|(011)ih(011)| + |000ih000|, |(011)ih(001)| + |000ih111|, |(001)ih(011)| + |111ih000|, |(001)ih(001)| + |111ih111|.
Via a unitary change of basis, we identify computational basis elements |0˜0˜0˜i, |0˜0˜1˜i, etc., with vectors
{|000i, |(011)i, |111i, |(001)i, |e1i, |e2i, |e3i, |e4i}, (5.54) where |e1i, |e2i, |e3i, |e4i are chosen to ensure orthonormality. This transformation reveals that the Schmidt-span operators share a common identity factor, as in this basis they read:
(|0˜ih0˜| ⊗ I) ⊕ O, (|0˜ih1˜| ⊗ I) ⊕ O, (|1˜ih0˜| ⊗ I) ⊕ O, (|1˜ih1˜| ⊗ I) ⊕ O, where the sector on which the zeros act is span{|e1i,..., |e4i}. The span of these operators is closed under ∗-algebra operations and constitutes a representation of the Pauli algebra. Thus, the distorted Schmidt spans of ρ are already *-closed algebras. A simple calculation verifies that each distorted-algebra basis element is also unchanged by M 1 . To find the minimal fixed-point sets, we need to apply 2 1 2 the distortion map again, by conjugating the generators with ρ234. We can write the Schmidt decomposition, with respect to the basis in Eq. (5.54), as
|0˜ih0˜| ⊗ τ ⊕ O, |0˜ih1˜| ⊗ τ ⊕ O, |1˜ih0˜| ⊗ τ ⊕ O, |1˜ih1˜| ⊗ τ ⊕ O, where we have defined τ ≡ |0˜ih0˜| + (1 − )|1˜ih1˜|. The last step is to construct QL Liouvillians for each neighborhood. The require- ment is that the kernels of each of these are the corresponding minimal fixed-point sets. This can be obtained by considering the operators:
1 1 L0 = |0˜ih1˜| ⊗ I ⊗ I,L+ = |0˜ih0˜| ⊗ I ⊗ τ 2 |0˜ih1˜|,L− = |0˜ih0˜| ⊗ I ⊗ τ 2 |1˜ih0˜|.
The first Lindblad operator L0 is responsible for asymptotically preparing the sub- space span{|000i, |(011)i, |111i, |(001)i}, while L+ and L− stabilize the τ factor. All three Lindblad operators must commute with the distorted algebra in order that it be pre- served. Using the standard definitions of ladder operators, σ+ ≡ |0ih1| = (σ−)†, we
118 5.5 Illustrative Applications
rewrite the τ-preparing Lindblad operators back in the original basis, in terms of standard Pauli matrices, as
√ + + + − − − − − − + + + L+ = [ σ2 σ3 σ4 (σ2 + σ3 + σ4 ) + σ2 σ3 σ4 (σ2 + σ3 + σ4 )], √ − − − + + + + + + − − − L− = 1 − [(σ2 + σ3 + σ4 ) σ2 σ3 σ4 + (σ2 + σ3 + σ4 ) σ2 σ3 σ4 ], √ q 1− † 2 where now L− = L+. Defining |(001)ωi ≡ (|001i + ω|010i + ω |100i)/ 3, ω ≡ e2πi/3, and similar terms to denote symmetric basis elements for the four dimensional space orthogonal to the symmetric subspace, the third Lindblad operator reads
L0 = |000ih(001)ω| + |(011)ih(001)ω2 | + |111ih(011)ω| + |(001)ih(011)ω2 |.
These Lindblad operators form the neighborhood Liouvillian on systems 234, namely, 1 L (ρ) = L ρL† + L ρL† + L ρL† − {L†L + L† L + L† L , ρ}, 234 0 0 + + − − 2 0 0 + + − −
The global generator L is obtained by constructing L123 in an analogous way, and by letting L = L234 + L123.
Using Matlab, we have verified that these dynamics are FF and stabilize ρ; the kernel of L is equal to span(ρ), as desired. In the limiting cases of = 0, 1, the above dynamics fail to have a unique fixed state. As we have already established, for the Dicke-state case of = 0, FFQL stabilizing dynamics can be constructed by a separate procedure, whereas for the GHZ case of = 1, no FFQL stabilizing dynamics exists relative to the given neighborhoods.
Remark 7: Failure of support condition. In Theorem 5.4.5, in order to obtain a general sufficient condition for FFQLS, we supplemented the necessary condition with the “support condition” of Eq. (5.33). We now show that ρ fails the support condition despite being FFQLS. The support of ρ is spanned by just |(0011)i and 4 |GHZ2i. On the other hand, we also have
supp(ρ123 ⊗ I4) = span{|0000i, |0001i, |(001)i|0i, |(001)i|1i, |(011)i|0i, |(011)i|1i, |1110i, |1111i},
supp(I1 ⊗ ρ234) = span{|0000i, |1000i, |0i|(001)i, |1i|(001)i, |0i|(011)i, |1i|(011)i, |0111i, |1111i}.
The intersection is found to be
span{|0000i, |(0001)i, |(0011)i, |(0111)i, |1111i}.
119 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics
Since this intersection properly contains the support of ρ, the claim follows.
Remark 8. Since stabilization of ρ is possible for arbitrarily close to the GHZ- value of one, the above example demonstrates that there exist FFQLS mixed states that are arbitrarily close to a non-FFQLS pure state. While this allows for practi- cal stabilization [174] or approximate stabilization in principle, we expect that the Liouvillian spectral gap will close as the non-FFQLS state is approached – making stabilization inefficient. If we normalize the Liouviallians to ||L||2 = 1, the corre- sponding gaps ∆ are found to behave as ∆ ≈ 0.049 (1 − ). It remains an interesting question for future investigation to determine whether similar conclusions about QL practical stabilization and associated efficiency trade-offs may be drawn for more gen- eral target states. In the following subsection we make these notions more precise and give a few examples.
5.5.5 Approximate FFQLS So far we have considered FFQL stabilizability as a binary property of quantum states. In practice, however, it is only ever possible (and suffices for all intents and purposes) to stabilize an approximate version of the target state. Furthermore, with time- independent continuous-time dynamics that we have considered, under even ideal conditions, for any finite wait time the target state is only reached approximately. Thus, as mentioned in the previous subsection, it is interesting to explore the notion of practical or approximate stabilization. There are various ways to define approximate stabilization. In the context of continuous-time stabilization that we have considered, it is useful to define a state to be approximately FFQLS if there is an FFQLS state in its proximity.
Definition 5.5.10. A state ρ is said to be -FFQLS with respect to neighborhood 0 1 0 structure N if there exists a state ρ which is FFQLS and satisfies 2 ||ρ − ρ ||1 ≤ , where || · ||1 denotes the trace-norm. We give two examples of states which are -FFQLS for arbitrarily small , despite not being FFQLS. Consider the 1D Ising Hamiltonian example from Sec. 5.5.3. The Hamiltonian is Pn H = − i=1 ZiZi+1 with Gibbs state ρβ = exp(−βH)/Tr (exp(−βH)). From either the results of [156] or of Sec. 5.5.3, this Gibbs state can be shown to be FFQLS with respect to the NNN neighborhood structure for any finite β. However, in the zero-temperature limit, the Gibbs state converges to the non-full rank state
1 ⊗N ⊗N ρ∞ ≡ lim ρβ = (|0ih0| + |1ih1| ). β→∞ 2
This classically correlated state coincides with that of Eq. (5.37), which was proven
120 5.5 Illustrative Applications
not to be FFQLS. As β increases, ρβ gets arbitrarily close to ρ∞, implying that ρ∞ is -FFQLS for all . To make this rigorous, we fix an arbitrary and show that there exists β such that ||ρ∞ − ρβ||1 ≤ .
To upper bound the trace distance, we use the inequality 1 ||ρ−σ|| ≤ p1 − F (ρ, σ)2, √ √ 2 1 where the fidelity of two states is F (ρ, σ) ≡ Tr p ρσ ρ. For commuting density P √ operators, as in our example, the fidelity simplifies to F (ρ, σ) = i piqi, where the pi and qi are corresponding eigenvalues in the common eigenbasis. Since ρ∞ is rank-2 1 with eigenvalues 2 , the fidelity sum involves just two terms
r1 r1 q F (ρ , ρ ) = h0|⊗N ρ |0i⊗N + h1|⊗N ρ |1i⊗N = 2h0|⊗N ρ |0i⊗N . (5.55) ∞ β 2 β 2 β β
where the second equality is a simplification due to the symmetry of ρβ. The par- tition function of the N-spin periodic boundary condition Ising model is calculated to be Z = Tr e−βH = 2N (coshN (β) + sinhN (β)). Then, after calculating that h0|⊗N ρ |0i⊗N = eβN , we can upper bound the trace-norm distance as β 2N (coshN (β)+sinhN (β)) s s 1 2eβN 2 ||ρ∞ −ρβ||1 ≤ 1 − = 1 − . 2 2N (coshN (β) + sinhN (β)) (1 + e−2β)N + (1 − e−2β)N (5.56) We must choose β such that is an upper bound for this expression. Set α = e−2β and rewrite the bound inequality as 1 1 ((1 + α)N + (1 − α)N ) ≤ , (5.57) 2 1 − 2 On the left-hand side, we find that the odd terms in the binomial expansions cancel, leaving bN/2c 1 X N ((1 + α)N + (1 − α)N ) = α2k. (5.58) 2 2k k=0 N 2k This can be upper bounded using 2k ≤ N , 1 ((1 + α)N + (1 − α)N ) ≤ 1 + (N 2α2) + ... + (N 2α2)bN/2c. (5.59) 2
1 2 By truncating the Taylor expansion of 1−2 , we obtain the lower bound 1 + + ... + 2 bN/2c 1 ( ) ≤ 1−2 . Comparing the two finite sums, we establish that choosing β such that Nα = Ne−2β = ensures the inequality of Eq. (5.57) to hold. Thus, for any 1 1 , setting β = 2 log(N/) ensures that 2 ||ρ∞ − ρβ||1 ≤ . For any , we can choose
121 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics
an FFQLS state ρβ that is within distance from ρ∞. This establishes that ρ∞ is -FFQLS for arbitrarily small . For the last example, we show that the n-qubit GHZ state is -FFQLS with re- spect to the NNN neighborhood structure for arbitrary . This example is somewhat surprising because, not only is GHZ not FFQLS, but it is not even QLS while allowing Hamiltonian control or frustrated dynamics, as shown in [34]. Towards this, we prove a result which connects FFQLS of pure states to a tensor network structure.
Proposition 5.5.11. If a state |ψi ∈ H⊗n can be written as the product of invertible, ⊗n mutually commuting NN-acting matrices Mi,i+1 applied to a product state |φi , then |ψi is FFQLS with respect to the NNN neighborhood structure.
⊗n Proof. By hypothesis, |ψi = M12M23 ...Mn−1,n|φi . We check the pure state FFQLS condition (i.e. the DQLS condition of [34]),
n−2 \ span(|ψi) = Σj,j+1,j+2(|ψi). (5.60) j=1
For any invertible matrix M, and vector subspaces U, V , and W , it is simple to prove that W = U ∩ V if and only if MW = (MU) ∩ (MV ). Let M ≡ M12 ...Mn−1,n. Then the above condition is true if and only if
n−2 −1 \ −1 span(M |ψi) = M Σj,j+1,j+2(|ψi). (5.61) j=1
Consider any intersection argument for 2 ≤ j ≤ n − 3 (i.e. excluding the boundary arguments). We can simplify as
−1 −1 −1 M Σj,j+1,j+2(|ψi) = H1...j−1 ⊗ Σj(Mj−1,j|φφi) ⊗ |φi ⊗ Σj(Mj+2,j+3|φφi) ⊗ Hj+3...n. (5.62) This space is contained in H1...j ⊗ |φij ⊗ Hj+2...n. Therefore, we have the containment
n−3 n−3 \ −1 \ ⊗n−2 M Σj,j+1,j+2(|ψi) ≤ H1...j ⊗|φij ⊗Hj+2...n = H12 ⊗|φi ⊗Hn−1,n. (5.63) j=2 j=2
For the j = 1 and j = n − 2 cases, the simplification is
−1 ⊗2 −1 M Σ123(|ψi) = |φi ⊗ Σj(M34 |φφi) ⊗ H4...n, (5.64)
⊗2 and similarly for j = n − 2. This space is contained in |φi ⊗ H3...n. Putting this all
122 5.5 Illustrative Applications
together, we find that
n−2 \ −1 ⊗n −1 M Σj,j+1,j+2(|ψi) ≤ span(|φi ) = span(M |ψi) (5.65) j=1
−1 −1 Since M |ψi ∈ M Σj,j+1,j+2(|ψi) for all j, the FFQLS intersection condition is satisfied by |ψi. We demonstrate that the n-qubit GHZ state can be δ-approximated by a state of the form above. Define the two-qubit invertible matrix
M = pcosh()(|00ih00| + |11ih11|) + psinh()(|01ih01| + |10ih10|).
Let Mi,i+1 = Mi,i+1⊗Ii,i+1. Since Mi,i+1 is diagonal in the computational basis, it com- 1 ⊗n mutes with all M 0 0 . The approximate GHZ state is |ψ i = √ M ...M |+i , i ,i +1 N 12 n−1,n cosh[(n−1)] where setting N = 2n−1 gives the proper normalization. The trace-norm distance between |ψi and |GHZi is 1 || |GHZihGHZ| − |ψ ihψ | || = p1 − |hGHZ|ψ i|2 2 1 " r 1 = 1 − h0|⊗nM ...M |ψ i 2N 12 n−1,n
!21/2 r 1 − h1|⊗nM ...M |ψ i 2N 12 n−1,n
1/2 r n!2 1 (n−1)/2 1 = 1 − 2 (cosh ) √ 2N 2 !1/2 cosh(n−1) = 1 − cosh[(n − 1)]
2 !1/2 1 + (n − 1) + ... = 1 − 2! 2 2 1 + (n − 1) 2! + ... r (n − 1)(n − 2) ≤ . (5.66) 2
q 2 Therefore, for any δ, by choosing = (n−1)(n−2) δ, we ensure that the FFQLS state
|ψihψ| is within trace-norm distance δ of the n-qubit GHZ state. This leads us to the
123 Asymptotic stabilization of quantum states with continuous-time quasi-local dynamics surprising conclusion that, even for a state which fails to be stabilizable in the weakest sense (i.e. GHZ is not even QLS with respect to frustrated QL dynamics), there may exist an arbitrarily close proxy state, which is stabilizable in a much stronger sense. As with the GHZ-Dicke mixture from the previous chapter, we expect the scaling of the time needed to stabilize such an approximate (with respect to the proximity) to be unfavorable.
124 Chapter 6
Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
125 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
This chapter presents material based on a paper titled “Exact stabilization of en- tangled states in finite time by dissipative quantum circuits” that is currently being finalized for submission and is joint work with Francesco Ticozzi and Lorenza Viola.
6.1 Introduction
Preparation of a target quantum state is a ubiquitous task in quantum information processing. In particular, measurement based quantum computing using graph states [191] or AKLT states [192] relies on initializing the system into a desired resource state. A more recent application of quantum state preparation is to quantum Gibbs samplers [156]. Determining the feasibility of preparing of a target state is only an interesting or relevant question when there are limitations to the control of the system. The particular control limitations from one implementation to the next can vary widely. Yet, a physically motivated and commonly assumed constraint is quasi-locality. Here, the dynamics are constrained to addressing only certain sets of subsystems. As an example, in an ion trap, one might only have control over nearest-neighbor couplings between adjacent ions, while being unable to directly couple distant ions. The standard approach to preparing a target pure state on a quantum computer begins with the initialization of the systems into a fiducial product state (e.g. |0i⊗N ) after which a circuit of unitary gates transforms the initial state into the desired state. Other approaches attempt to cool the quantum system to drive the system into an approximation of the ground state of the Hamiltonian [19]. An alternative approach, known as sequential generation [193], exploits a matrix product state (MPS) or pro- jected entangled pair state (PEPS) representation of the target state. The tensors in the matrix product determine a sequence of CPTP maps which act in a specified order to transform any unspecified input state into the target state. However, as emphasized in [194], such preparation schemes face a number of draw- backs. Most prominently, in these schemes, the target state is only available at the end of a “cycle”. The action of the intermediate steps alters the target state. The preparation scheme of target state stabilization addresses these drawbacks. With sta- bilizing dynamics, we demand that the CPTP maps which both drive the system to the target state with all-to-one dynamics and leave the target state invariant in each step. An important dichotomy among dynamical models is discrete vs continuous time. Much recent work has been devoted to analyzing how to engineer time-independent continuous-time Liouvillian dynamics for stabilizing a target state [31, 34, 172]. With such continuous-time dynamics, an arbitrary input state is driven asymptotically to- wards the target state; the target state is only prepared in infinite-time. In contrast, discrete-time dynamics, or a dissipative quantum circuit, have the po- tential for stabilizing a target state in finite time. In [194] we analyze the discrete-time
126 6.2 Preliminaries
analog of the above continuous-time cases; a fixed set of CPTP maps are alternatingly applied to drive the system towards the target state. Generally, the target state is ap- proached asymptotically, with stabilization in infinite-time. However, in cases when the CPTP maps commute with one another, the target state is stabilized in finite time. When the target state is stabilized independent of the order of the maps, we re- fer to the implementation as robust. Both the finite-time and robust implementation properties are favorable from a control-theoretic perspective. Two questions naturally arise:
(a) What ensures that a target state can be stabilized in finite time?
(b) Furthermore, what properties enable a robust implementation of the stabiliza- tion?
In this chapter we explore aspects of discrete-time stabilization of a target pure state in finite-time, providing some existential results as well as some results the provide a synthesis for the stabilizing dynamics. In Section 6.3 we develop both necessary and sufficient conditions for determining if a target state can, in principle, be finite-time stabilized. In the case that a state is verified to be finite-time stabilizable (FTS), we provide a scheme for its stabilization, although we lack an algorithm for synthesizing a part of the dynamics that is only shown to exist in principle. In Section 6.2 we provide the necessary background and mathematical tools used in this chapter. In Section 6.4 we turn to the notion of robust finite-times stabilization (RFTS), giving a number of examples and several necessary conditions which restrict the correlations on a target state if it is to be RFTS. In Section 6.5 we develop several sufficient conditions which ensure RFTS of a target state. Here, a useful structure that we rely on is a virtual subsystem factorization of the target state which facilitates robust finite-time stabilization. In Section 6.6 we explore the efficiency of the RFTS scheme for quasi-locality defined on a lattice and then make connections between RFTS and the existence of quasi-local continuous-time dynamics which “rapidly” prepare an equilibrium state. In Section 6.7 we extend a number of results to the case of a mixed target state. Finally, we have reserved all of the technical proofs to a final section at the end of the chapter.
6.2 Preliminaries
6.2.1 Quasi-local discrete-time dynamical semigroups We consider finite-dimensional multipartite qudit systems described by a Hilbert NN di space H' i=1 Hi, with each Hi ' C . B(H) denotes bounded operators on
127 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
Figure 6.1: 1-D nearest-neighbor (NN) neighborhood structure.
H. The state of the system at each time is a density operator in the space of positive- semidefinite, trace-one operators, denoted D(H). The time evolution of the system is modeled by non-homogeneous Markov dynamics. Such dynamics are represented by sequences of completely positive trace-preserving (CPTP) maps {Et}, whereby the evolution of the state ρt from step t to t + 1 is
ρt+1 = Et(ρt). (6.1)
For any t > s ≥ 0, we denote the evolution map from s to t as
Et,s = Et ◦ Et−1 ◦ ... ◦ Es. (6.2)
We assume that each step in the dynamics is engineered towards achieving stabiliza- tion of a target state. In a practical setting, such control of the quantum system can be limited by a variety of constraints. Without any restrictions, stabilization of an arbitrary target state becomes trivial. We thus assume that each map Et is chosen from some set of “available” maps reflecting the limited control of the physical system. In particular, we assume that each map must act quasi-locally. Following [31, 34, 172], the notion of quasi-locality we consider is described by a neighborhood structure on the multipartite Hilbert space. We define a neighborhood structure N to be specified by a list of subsets of subsystems, Nk ⊆ {1,...,N}, for k = 1,...,T . As an example, the 1-D nearest-neighbor (NN) neighborhood structure is depicted in Fig. 6.1. Definition 6.2.1. A CPTP map E is a neighborhood map with respect to a neigh- borhood Nj if
E = ENj ⊗ IN¯j , (6.3) where ENj is the restriction of E to operators on the subsystems in Nj and IN¯j is the identity map for operators on HN¯j . A QDS is quasi-local with respect to a neighbor- hood structure N if, for each map Et in the sequence, Et is a neighborhood map with respect to some Nj ∈ N . A useful tool for analyzing the neighborhood-wise features of a quantum state is the Schmidt span of a linear object (vector, operator, or tensor) [172]:
128 6.2 Preliminaries
Definition 6.2.2. Given the tensor product of two finite-dimensional inner-product P i i spaces W1 ⊗ W2 and v ∈ W1 ⊗ W2 with Schmidt decomposition v = i siv1 ⊗ v2, the i Schmidt span of v with respect to W1 is defined as Σ1(v) ≡ span{v1}. We will often use the notion of an extended Schmidt span, which is denoted and defined as Σ1(v) ≡ Σ1(v) ⊗ W2. We will mostly make use of the extended Schmidt
span of the target state |ψi with respect to neighborhood Hilbert spaces, ΣNk (|ψi) = Σ (|ψi) ⊗ H . Nk N k
6.2.2 Convergence notions The task that we analyze in this chapter is the design of dynamics which drive a system towards a target state. The following definitions are used to make rigorous the notion of target state stabilization as discussed in the introduction. A state ρ ∈ D(H) is invariant with respect to the dynamics {Es,t}t≥s≥0 if Et,s(ρ) = ρ for all τ ∈ S and all t ≥ s ≥ 0. With this, we define a notion of convergence towards the target state:
Definition 6.2.3. With respect to a dynamical semigroup {Es,t}t≥s≥0, an invariant state ρ is globally asymptotically stable (GAS) if, for any initial state σ,
lim |Et,s(σ) − ρ| = 0, ∀s ≥ 0. (6.4) t→∞ Following [194], we define a notion of GAS with respect to a quasi-local dynamical semigroup. Definition 6.2.4. A target pure state ρ = |ψihψ| is said to be quasi-locally stabiliz- able (QLS) with respect to neighborhood structure N if there exists a sequence {Et}t≥0 of N -acting maps rendering ρ GAS. A main result in [194] establishes the following necessary and sufficient condition for determining whether or not a target pure state is QLS. Adapting the notation to our purposes, we have: Theorem 6.2.5 ([194]). A target pure state ρ = |ψihψ| is QLS if and only if \ span(|ψi) = ΣNk (|ψi). (6.5) k For certain target states, the asymptotic scheme for QLS devised in [194] converges to the target state exactly in a finite number of steps. In this work, our aim is to build off of the QLS results in [194] to determine further conditions on the target state which enable such finite-time quasi-local stabilization.
129 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
Definition 6.2.6. A target pure state ρ = |ψihψ| is quasi-locally finite-time stabiliz- able (FTS) with respect to neighborhood structure N if there exists a finite sequence T {Et}t=1 of N -acting maps satisfying
(a) {Ei}(ρ) = ρ, for all i = 1,...,T , and
(b) ET ◦ ... ◦ E1(σ) = ρ for all σ ∈ D(H).
Furthermore, ρ is said to be robustly FTS (RFTS) if property 2 holds for any per- mutation of the T maps.
Although the focus of this chapter is on discrete-time dynamics, we briefly describe the analogous continuous-time dynamics in order to draw parallels between the two cases and to make contact with previous work. A continuous-time Markovian semi- group is generated by a Liouvillian master equation,
X 1 ρ˙ = L(ρ) ≡ L ρL† − {L†L , ρ}, (6.6) j j 2 j j j
where L is a linear superoperator and Lj are Lindblad operators which may vary in time. Following [172], a Liouvillian generator L is QL with respect to neighborhood N if L = P L ⊗ I , where I denotes the identity superoperator. Previous work k Nk N k has explored quasi-local stabilization of a target state with respect to such continuous- time dynamics [158, 31, 34, 172]. We emphasize that, in contrast to the finite-time discrete dynamics that we consider in this chapter, time-independent continuous-time dynamics can never stabilize a target state exactly for any finite wait time. One of the main motivations of this work, then, is to extend these investigations of continuous- time stabilization to the discrete-time case, where finite-time stabilization is possible.
6.2.3 Quasi-local parent Hamiltonians In this subsection we introduce a useful tool for analyzing QLS, FTS, and RFTS. Remarkably, QLS can be related to the existence of a particular Hamiltonian. A quasi-local Hamiltonian H ≡ P H = P H ⊗ I is frustration-free (FF) if the k k k Nk N k ground state space of H is contained in the ground state space of each neighborhood term Hk. In particular, if a state |ψi has minimal energy with respect to H, it has minimal energy with respect to each Hk. A corollary in [194] shows that a target pure state |ψi is QLS with respect to N if and only if |ψi is the unique ground state of some frustration-free (FF) quasi-local Hamiltonian H = P H ⊗I . This result, then, provides a physical interpretation k Nk N k for the QLS scheme: each neighborhood map serves to locally “cool” the system with
130 6.2 Preliminaries
respect to Hk, from which the FF condition on H ensures that these local coolings collectively achieve the global cooling to |ψi, the unique ground state of H. A Hamiltonian for which a state is a ground state is known as a parent Hamilto- nian. Among FF parent Hamiltonians of a given state, one is uniquely constructed from the state itself:
Definition 6.2.7. The canonical FF Hamiltonian is defined as X H = (I − Π ⊗ I ), (6.7) |ψi Nk N k k
where ΠNk is the projector onto the Schmidt span ΣNk (|ψi).
This Hamiltonian satisfies the following universal property: if there exists a FF Hamiltonian with |ψi as the unique ground state, then |ψi is the unique ground state of the canonical FF Hamiltonian H|ψi. Thus, a target pure state |ψi is QLS if and only if it is the unique ground state of its canonical FF Hamiltonian. As FTS and RFTS are special cases of QLS, a necessary condition for target state |ψi to be FTS or RFTS is that it satisfies Eq. (6.5). We seek further conditions which ensure FTS or RFTS of a target state. Especially in the RFTS case, we are led to investigate further properties of the canonical FF Hamiltonian. In [194], the states which were found to be RFTS had canonical FF Hamiltonians for which the neighborhood-acting terms commuted with one another, pairwise. Commutativity of a Hamiltonian is a useful property which can be related to the concept of hypercon- tractive dynamics [154] and rapid thermalization of a Gibbs state [156]. Surprisingly, we show that commutativity of the canonical FF Hamiltonian is not necessary for a state to be RFTS. Nevertheless, in developing sufficient conditions for RFTS in Section 6.5 we will use certain algebraic structures induced by the canonical FF Hamiltonian to diagnose RFTS. Furthermore, although the canonical FF Hamil- tonian need not commute for a state to be RFTS, we conjecture that if |ψi is RFTS, then there exists some commuting quasi-local Hamiltonian for which |ψi is the unique ground state. Unlike the property of QLS, we lack a general necessary and sufficient condition for FTS or RFTS. However, in Section 6.5.2, by restricting to certain neighborhood structures we establish that |ψi is RFTS if and only if the canonical FF Hamiltonian is commuting and |ψi is its unique ground state.
131 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
6.3 Finite-time stabilization
6.3.1 Conditions for finite-time stabilization We state a necessary condition for a target state to be finite time stabilizable. As already remarked, proofs are deferred to the final section.
Theorem 6.3.1. A pure state |ψi is FTS with respect to N only if |ψi satisfies Eq. (6.5) and there exists at least one neighborhood Nk ∈ N for which
dim(HNk ) ≥ 2 · dim(ΣNk (|ψi)). (6.8)
We refer to the above condition as the small Schmidt span condition. In principle, the small Schmidt span could be implied by satisfaction of Eq. (6.5), and, hence, be redundant. We give an example which shows that this condition is truly necessary. This verifies the existence of states which are only QLS in infinite-time.
Example 6.3.2 (2D AKLT state). Consider a system of six spin-3/2 systems. We define the spin-3/2 AKLT state on this system as described in Fig. 6.2. The neighborhood structure consists of pairs of systems that are connected by an edge in Fig. 6.2. AKLT states are the unique ground states of particular non-commuting two-body FF Hamiltonians, where the two-body terms act on the edges of the graph which define the state (i.e. the neighborhoods). Thus, the AKLT state satisfies Eq. (6.5) (which also follows from analysis in [158]). Nevertheless, as verified numerically in MATLAB, this state violates the small Schmidt span condition.
The necessity of the small Schmidt span condition can be understood intuitively. It must be the case that, for some step T in the sequence, a neighborhood map ET transforms some mixture of states into the target state. This mixture must include a contribution from a state |ψ⊥ihψ⊥| that is orthogonal to the target state so that ⊥ ⊥ ET (λ|ψ ihψ | + (1 − λ)σ) = |ψihψ| for some state σ. Hence, this neighborhood map must take |ψ⊥ihψ⊥| into |ψihψ|. This action can be viewed as correcting a neighborhood-acting error on |ψi. If the Schmidt span of |ψi on the neighborhood NT of ET is too large, no neighborhood-acting errors can transform |ψi into an orthogonal state. This highlights an important difference between finite-time and asymptotic stabilization. From the view of error correction, asymptotic dynamics are able to correct errors which cannot be exactly corrected by discrete dynamics in finite time. Next, we describe a checkable condition for determining if a state is FTS. To prove that any state satisfying this condition with respect to a neighborhood structure is FTS, we construct a sequence of CPTP maps which render the state FTS. A crucial component of the scheme that we present is the use of neighborhood-acting unitary maps. One might guess that unitary maps would not be useful for the task of state
132 6.3 Finite-time stabilization
Figure 6.2: Example of a non-FTS state: AKLT state on a bipartite cubic graph. The pairs of nodes connected by an edge are virtual spin-1/2 particles in a singlet state. The perforated circles contain the systems which are projected into the spin-3/2 subspace. The resulting system describes 6 spin-3/2 particles. As verified numerically in MATLAB, this AKLT state violates the small Schmidt span property since for each top-bottom pair of systems (i.e. each neighborhood), the Schmidt span dimension (= 9) exceeds half the Hilbert space dimension (= 16/2). stabilization, as they are not entropy-decreasing. However, in the scheme we develop, unitary maps play a crucial role in stabilizing a target state in finite time. Let U(H) to denote the group of unitaries acting on H, and u(H) the corresponding Lie algebra. The unitary subgroups (subalgebras) that we define below are subgroups (subalgebras) of U(H) (u(H)).
Definition 6.3.3. The unitary stabilizer group of a vector |ψi ∈ H is defined as
† U|ψi ≡ {U ∈ U(H) | U|ψihψ|U = |ψihψ|}.
This is a Lie subgroup of U(H). The Lie algebra associated to U|ψi is denoted u|ψi. The neighborhood unitary stabilizer group of a vector |ψi ∈ H with respect to Nk is defined as
† UNk,|ψi ≡ {U ∈ U(HNk ) ⊗ IN | U|ψihψ|U = |ψihψ|}.
The Lie algebras associated to these Lie groups are denoted uNk,|ψi.
An important component of our FTS scheme is the decomposition of elements of the global stabilizer group U|ψi into a finite product of elements from the neighborhood stabilizer groups UNk,|ψi. The following proposition describes a checkable condition for determining whether or not such a decomposition is possible.
133 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
Proposition 6.3.4 (Unitary generation property). Given a state |ψi and a neighborhood structure N , any element in U|ψi can be decomposed into a finite product
of elements in UNk,|ψi if and only if
huNk,|ψiik = u|ψi, (6.9)
where h·ik denotes the smallest Lie algebra which contains all Lie algebras from the set indexed by k.
The linear algebraic closure, h·ik, may be computed numerically. Hence, for a given state, we may determine whether or not the unitary generation property holds using software such as MATLAB. However, we note that determining such a decomposition might be difficult in practice. Some care must be taken when we apply the unitary generation property. Consider a neighborhood structure comprised of two disjoint sets of neighborhoods (i.e. no neighborhood from the first set and from the second set have non-trivial intersection), giving, say, H'HL ⊗ HR. Towards checking FTS, Eq. (6.5) can only be satisfied if the target state |ψi = |ψiL ⊗ |ψiR is factorized with respect to HL and HR. With this product state, the neighborhood unitary stabilizers can, at most, generate U ⊗ |ψiL U , which is strictly smaller than U . Disconnected neighborhood structures will |ψiR |ψi never allow the unitary generation property to hold. Trivially, any product state is FTS with respect to a disconnected neighborhood structure. What is needed then, is that the unitary generation property hold for each connected component of the neighborhood structure. For example, the unitary generation property should be checked for |ψiL with respect to the corresponding set of neighborhoods and similarly for |ψiR. We thus simplify our analysis by only considering neighborhood structures which are connected, as disconnected neighborhood structures trivially limit many- body entanglement. The following example demonstrates the general scheme that we will use in veri- fying that a state is FTS.
Example 6.3.5 (Dicke state). Consider a four qubit system with a connected neigh- borhood structure N = {N1, N2}, with N1 = {1, 2, 3} and N2 = {2, 3, 4}. The Dicke state 1 |(0011)i ≡ √ (|0011i + |0101i + |0110i 6 +|1001i + |1010i + |1100i) (6.10)
was shown to be stabilizable asymptotically in [194]. We show that this state is FTS with respect to the same 3-body neighborhood structure. The Schmidt span of |(0011)i
with respect to N1 is ΣN1 (|(0011)i) = span{|(001)i, |(011)i}. Thus, the small Schmidt
134 6.3 Finite-time stabilization
span condition is satisfied:
dim(H ) 8 Nk = = 4 ≥ 2. dim(ΣNk (|ψi)) 2
In our FTS scheme, entropy is removed from the system only by the action of a dissipative map W123 acting on neighborhood N1. This contrasts the QLS scheme of [194], wherein dissipative maps alternatingly act on the two neighborhoods to asymp- totically drive the system towards the target state. There, in stabilizing |(0011)i, the need for infinite-time convergence reflects some degree of “competition” between the two dissipative maps. For states satisfying certain sufficient conditions, we achieve finite-time convergence by designing “collaboration” among the CPTP maps.
In our FTS scheme, W123 is used just twice. Between these, in a finite sequence, unitaries alternatingly act on the two neighborhoods. As we show, W123 maps any density operator with support in a particular four-dimensional subspace into the target state. The unitaries serve to transform the state of the system into this space. Then, the final action of W123 maps this state to the target. The general scheme also employs this approach of interspersing a single-neighborhood dissipative map with sequences of unitaries, the maps collaborating to achieve FTS. However, in general, we will need to call on more than just two uses of the dissipative map and will need to design distinct sequences of unitaries for each step. P † The dissipative map W123 = i Ki · Ki is defined by its Kraus operators
K0 ≡ (|(001)ih(001)| + |(011)ih(011)|) ⊗ I,
K1 ≡ (|(001)ih000| + |(011)ih111|) ⊗ I,
K2 ≡ (|(001)ih(001)ω| + |(011)ih(011)ω|) ⊗ I,
K3 ≡ (|(001)ih(001)ω2 | + |(011)ih(011)ω2 |) ⊗ I,
1 2 2πi where |(abc) i ≡ √ (|abci + ν|bcai + ν |cabi) and ω ≡ e 3 . W maps the following ν 3 123 four orthogonal states (including the target state, itself) into |(0011)i:
|ψ0i ≡ |(0011)i, √ |ψ1i ≡ (|000i|1i + |111i|0i)/ 2, √ 2 |ψ i ≡ (|(001)ωi|1i + |(011)ωi|0i)/ 2, √ 3 |ψ i ≡ (|(001)ω2 i|1i + |(011)ω2 i|0i)/ 2.
The range of W123 is the set of operators with support on the Schmidt span ΣN1 (|(0011)i).
Thus, we design a sequence of neighborhood unitaries {Ui} which maps ΣN1 (|(0011)i)
135 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
into span{|ψii},
U = UT ...U1 = |ψ0ihψ0| + |ψ1ih(001)|h0| + |ψ2ih(011)|h1| √ 3 + |ψ i (h(001)|h1| − h(011)|h0|) / 2 + UR,
where UR is any matrix which ensures that U is unitary. The fact that U can be decomposed into such a product is ensured by the fact that |(0011)i satisfies the Lie algebraic generation property of Eq. (6.9). This condition was checked using MatLab. We do not determine the actual form of the Ui, but this check ensures us that such a finite sequence of unitaries exists. Finally, a simple calculation shows that
W123 ◦ UT ◦ ... ◦ U2 ◦ U1 ◦ W123(I/16) = |(0011)ih(0011)|.
Hence, |(0011)i is FTS, as desired. This Dicke state is provably not RFTS (see Prop. 6.4.6 in Sec. 6.5). With the simplicity of this neighborhood structure, in order for a state to be RFTS, the Schmidt span projectors must commute with one another. The lack of commutativity in the Dicke state example, then, renders this state not RFTS. This demonstrates that the condition of being RFTS is strictly stronger than the condition of being FTS, as expected. We now state our general sufficient condition for FTS. Theorem 6.3.6. A state |ψi is FTS relative to a connected neighborhood structure N if there exists at least one neighborhood satisfying the small Schmidt span condition and
huNk,|ψiik = u|ψi. (6.11) Satisfaction of these sufficient conditions certainly implies satisfaction of the nec- essary conditions in Thm. 6.3.1. However, it is interesting to directly prove a connec- tion between the linear intersection property of Eq. 6.5 and the Lie algebraic unitary generation property:
Proposition 6.3.7. If |ψi satisfies huNk,|ψiik = u|ψi with respect to neighborhood structure N , then |ψi satisfies Eq. (6.5) with respect to N . We conjecture that satisfaction of Eq. (6.5) along with the small Schmidt span condition (i.e. the necessary conditions in Thm. 6.3.1) is sufficient to ensure FTS. One avenue to proving this would be to establish the converse of Prop. 6.3.7. To conclude this section we describe an example of a state which is FTS with respect to a more complicated neighborhood structure.
136 6.3 Finite-time stabilization
Example 6.3.8 (1D AKLT state). Consider a chain of N spin-1 particles along with the NN neighborhood structure. The 1D AKLT state can be defined as
dN/2e Y − ⊗N−1 |AKLTN i ≡ P1 P2i,2i+1 PN |ψ i , (6.12) i=1
2 2 3 where P2i,2i+1 : C ⊗ C → C projects the two spin-1/2 systems into the spin-1 triplet space, P1 and PN are isometries mapping |0i → |m = 1i, |1i → |m = −1i, and |ψ−i = √1 (|01i − |10i). With respect to the “boundary neighborhoods” (1, 2) and 2 (N − 1,N), the Schmidt spans have dimension 2. With respect to the remaining, “bulk neighborhoods”, the Schmidt spans have dimension 4. The neighborhood Hilbert spaces have dimension 3 · 3 = 9, so the small Schmidt span condition is satisfied. It remains to show that the AKLT states satisfy the unitary generation property. For small values of N this may be checked numerically. Indeed, we have verified numerically that the unitary generation property is satisfied for N = 3 and N = 4. We conjecture that for all N, the 1-D AKLT state is FTS with respect to the NN neighborhood structure.
6.3.2 Efficiency of finite-time stabilization In the Dicke state example of Section 6.3.1, the sequence of stabilizing CPTP maps is of the form W ◦U ◦W, where W is a dissipative map acting on N1, and U is a unitary composed of a sequence of neighborhood unitaries. At most, W is able to reduce the rank of the input density operator by a factor of four. Acting on the completely mixed state, W outputs a density operator of rank = 16/4 = 4. Then, a proper choice for U shifts the support of this rank-four density operator in such a way that a second application of W reduces the rank to one (again, a reduction by a factor of four). In the general case, more than one iteration of the the unitary followed by the dissipative map is needed to achieve FTS. Two principles guide the general FTS scheme:
(i) Choose the dissipative map W to maximally reduce the rank of the completely mixed state.
(ii) Choose the unitaries Ui so that the subsequent action of W maximally reduces the rank of its input.
Consider a general target state |ψi and neighborhood structure N , where |ψi 0 satisfies the conditions of Thm. 6.3.6. For ease of notation, let Σ ≡ ΣN (|ψi). Ls−1 i i 0 Decompose HN ' i=0 Σ ⊕ R, where Σ are orthogonal isomorphic copies of Σ
137 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
Figure 6.3: Finite-time stabilization scheme for the N = 3 particle spin-1 AKLT state. In each numbered panel, each square represents one of the 27 dimensions in the C3 ⊗ C3 ⊗ C3 state space, while the dots represent the probabilistic weight of each basis vector for the current state. The task is to move all the probabilistic weight from the initial flat distribution (completely mixed state) into the box in the upper left-hand corner, corresponding to the target state. The Schmidt span on the first two systems is 2-dimensional, leading to the 6-dimensional extended Schmidt span Σ0 as represented by the first row of boxes. The remaining rows, labeled Σi, are isometric “copies” of this subspace, leaving the 3-dimensional remainder space R ⊗ HN . The dissipative map W acts only on the first two qutrits, cooling each Σi to Σ0. The unitaries U1 and U2 leave the target state invariant while preparing probabilistic weight to be cooled by the neighborhood map W. Since the AKLT state satisfies the unitary generation property with respect to this neighborhood structure, each Ui can be decomposed into a finite sequence of neighborhood-acting invariance-satisfying maps. and R is the remainder space of minimal dimension. Factoring out the index i, the global space decomposes as
s−1 M i s Σ ⊗ HN ⊕ R ⊗ HN ' C ⊗ (Σ ⊗ HN ) ⊕ R ⊗ HN . i=0
Now let the dissipative map W take each isomorphic copy Σi into Σ0. Using the above decomposition, the dissipative map is defined as
W ≡ (|0ih0|Tr) ⊗ I ⊕ I. (6.13)
Having fixed W, the interspersed unitary maps Ui are constructed so as to maximize the rank-reduction achieved by each W. This is done using the following algorithm:
0 0 0 (a) Choose an orthonormal basis {|ψαi} for Σ ⊗ HN where |ψ0i ≡ |ψi. This
138 6.3 Finite-time stabilization
i i determines isomorphic orthonormal bases {|ψαi} for the copies Σ ⊗ HN .
(b) Choose an orthonormal basis {|λβi} for R ⊗ HN . (c) Order the basis vectors as
0 1 s−1 0 1 s−1 |ψ0i, |ψ0i,..., |ψ0 i, |ψ1i, |ψ1i,..., |ψ1 i, . . 0 1 s−1 |ψl i, |ψl i,..., |ψl i, |λ0i,..., |λwi. (6.14)
(d) The choice of each Ui depends recursively on the input density matrix ρi = W(Ui−1(ρi−1)), beginning with ρ1 ≡ W(I).
(e) In each step, Ui is a permutation of the basis vectors, chosen so that the target state is fixed and, iteratively, each basis vector in the support of ρi is mapped to the first basis vector outside of the support of ρi according to the ordering of Eq. (6.14). The aim is for the output density matrix to have contiguous 0 support starting from |ψ0i. Note that, by taking the completely mixed state the initial state, the output of each map W and Ui is diagonal in the chosen ordered basis. Furthermore, the particular choice of each Ui ensures that, in the sequence of CPTP maps applied to the com- pletely mixed state, each W maximally reduces the rank of the input density matrix. Finally, since |ψi satisfies the unitary generation property with respect to N , each global stabilizer Ui can be decomposed into a finite number of neighborhood stabiliz- ers. The sequence of CPTP maps terminates after a finite number of steps because the rank of the input density matrix is necessarily reduced in each step (assuming the initial input is the completely mixed state). Having verified that the sequence converges to the target for the initial state being completely mixed, we are guaranteed that the sequence converges to the target for arbitrary input. The scheme is illustrated for the three spin-1/2 AKLT target state in Fig. 6.3. Each global unitary stabilizer map is decomposable into no more than 2 · 262 (twice the dimension of the Lie group, cf. Thm. 6.8.1) neighborhood stabilizer unitaries. The total number of neighborhood maps used in the sequence is thus at most 1 + 2 · 262 + 1 + 2 · 262 + 1 = 2707. More generally, consider a system of N qudits with target state |ψi and neighbor- hood structure N . We define the cooling rate of the neighborhood Nk with respect to
|ψi to be rk ≡ logd (bdim(HNk )/dim(ΣNk [|ψi])c). From this, we define the maximum cooling rate r = maxk rk. The small Schmidt span condition ensures that the max- imum cooling rate is greater than zero. A larger cooling rate affords the dissipative map W to more greatly reduce the rank of the input density matrix. In Fig. 6.3, the
139 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
0 0 cooling rate is log3(4), as W maps four such copies of Σ (including itself) into Σ . Take r to be the largest cooling rate of |ψi among the neighborhoods. For large d and N, the circuit size is dominated by the number of neighborhood unitaries, while the number of dissipative maps is negligible. The maximum number of neighborhood N 2 2N unitaries comprising each Ui is 2(d − 1) ∼ O(d ). The number of Ui needed de- pends on the rate at which W can reduce the rank of the input density matrix. By choosing Ui according to the above scheme, W is able to reduce the rank of the input density matrix by at least a factor of dr−1 ∼ O(dr). Thus, the number of W that must act to reduce the rank to 1 is at most O(N/r). Putting these together, the circuit size scales as O(d2N 2/r). For certain neighborhood structures, this circuit size scaling can be reduced. Con- sider a neighborhood structure, where the neighborhoods admit an L-layering. That is, the neighborhoods can be partitioned into L sets, such that all neighborhoods in a given set are mutually disjoint. The 1-D NN neighborhood structure admits a 2-layering. Assume that the cooling rate of all neighborhoods in a particular layer is r. In this case, instead of defining a single neighborhood-acting dissipative map W, we define a dissipative map Wi for each neighborhood in the particular layer, and Q then let W = i Wi. While a given Wi reduces the rank of the input density matrix by a factor of O(dr), W reduces the rank by a factor of O((dr)|N |/L), assuming that each layer contains about |N |/L neighborhoods. If we take |N | ∼ N, then the circuit size of the L-layered neighborhood case scales as O(d2LN/r). Hence, in the 1-D NN case, the circuit size scales as O(d4N/r). This exponential scaling with respect to the system size is quite unfavorable. We note however, that the exponential scaling is completely due to the compilation of ∼ d2N neighborhood stabilizer unitaries making up the global stabilizer unitaries. This is a worst-case scaling, which, in principle, could be drastically reduced for particular cases. The next section explores the far more efficient stabilization scheme of robust FTS. This scheme only uses dissipative maps, with a circuit size that is proportional to the number of neighborhoods. Furthermore, the stabilization does not depend on order of maps, leading to a favorable scaling of circuit depth. We will demonstrate that RFTS states are a proper subset of FTS states. In spite of this, we show that important classes of states, such as graph states used in measurement based quantum computing, are RFTS.
6.4 Robust finite-time stabilization: necessary con- ditions
We begin our analysis of RFTS by considering an obvious example:
140 6.4 Robust finite-time stabilization: necessary conditions
NN Example 6.4.1 (Product states). Given H' i=1 Hi, consider a strictly local neighborhood structure {Ni}, with Ni = {i}, and an arbitrary product state ρ = NN i=1 ρi. To each neighborhood let us associate the map Ei ≡ (ρiTri) ⊗ Ii. Then, any complete sequence of such maps gives
N N ! N ! O O O EN ◦ ... ◦ E2 ◦ E1 = (ρiTri) = ρi Tri i=1 i=1 i=1 = ρTr, (6.15) demonstrating that ρ is RFTS, as intuitively expected. Of course, by considering a neighborhood structure with enlarged neighborhoods (relative the the strictly local one above), a product state remains RFTS. Hence, any product state is RFTS with respect to any neighborhood structure which covers all systems.
Although the above example is trivial, its structure is important. In much of our analysis of RFTS, we seek to translate more complicated examples into the form of the product state example. Our next example demonstrates this translation and shows that RFT stabilizability is not limited to product states.
Example 6.4.2 (Graph states). Graph states are many-body entangled states which are known to be resources for universal measurement based quantum computing [191]. This example is valuable in that it demonstrates that the RFTS of a non-trivial target state can be translated into the task of stabilizing a product state with respect to “virtual degrees of freedom”. Following [179, 172], a graph state on N qudits is defined by a graph G = (V,E) with N vertices and a choice of Hadamard matrix H. The Hadamard matrix must † T H satisfy H H = dI, H = H , and |[H]ij| = 1 for all i, j. The edge-wise action C H is defined, according to the choice of Hadamard matrix, by C |iji = [H]ij|iji. The standard choice in the qubit case is that CH equals a controlled-Z transformation. Note that CH is diagonal in the computational basis and symmetric under swap of the two systems it acts on. We define the global graph unitary transformation as Q H † UG ≡ (i,j)∈E Ci,j, with the corresponding CPTP map UG(·) ≡ UG · UG. Then, the graph state associated to G is
⊗N |Gi ≡ UG|+i , |+i = H|0i. (6.16)
To each physical system i we associate a neighborhood Ni defined by that system along with the graph-adjacent systems (i.e. the set of j connected to i by some edge (i, j) ∈ E). For any given |Gi, we may construct a set of neighborhood maps which robustly stabilize |Gi relative to N . Let Eˆ : B(Cd) → B(Cd) be defined by Eˆ(·) = ˆ ˆ |+ih+|Tr (·). Let Ei indicate the map E acting on system i with trivial action on
141 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
ˆ −1 i. To each neighborhood Ni, we associate the map Ei = UG ◦ Ei ◦ UG . The Kraus ˆ α ˆ operators of Ei are of the form Xi ⊗ Ii. The unitary conjugation of Ei transforms α 0α 0α its Kraus operators into those of Ei as UG(Xi ⊗ Ii) = X i . Crucially, each X i acts non-trivially only on Ni. This is seen as follows:
0α α X k = UG(Xk ⊗ Ik) α † = UG(Xk ⊗ Ik)UG † Y H α † Y H = Ck,j (HXk H ⊗ Ik) Ck,j j∼k j∼k = (X0α) ⊗ I . k Nk N k
Hence, each Ei is a valid neighborhood map. Finally, we show that each Ei leaves |Gi invariant and that the composition of an arbitrary complete sequence of these maps ˆ † prepares |Gi. Invariance is demonstrated by Ei(|GihG|) = UG(Ei(UG(|GihG|))) = ⊗N UG(|+ih+| ) = |GihG|. Preparation is seen from,
ˆ ˆ ˆ −1 EN ◦ ... ◦ E2 ◦ E1 = UG ◦ EN ◦ ... ◦ E2 ◦ E1 ◦ UG ⊗N −1 = UG ◦ (|+ih+|Tr) ◦ UG ⊗N = UG(|+ih+| )Tr = |GihG|Tr.
Surprisingly, Example 6.4.2 shows that there exist “resourceful” many-body en- tangled states which are RFTS. Note that the scaling of the size of the dissipative circuit is O(N), a drastic improvement on that of the general FTS scheme. In a sense, this improvement comes along with a drawback. Although graph states are resourceful and many-body entangled, the correlation among their physical systems is limited in a particular sense. Namely, as with product states, graph states exhibit a finite correlation length with respect to the geometry imposed by the graph. Next, we describe a few necessary conditions for RFTS. These characterize the limits on the target state’s correlations with respect to the “geometry” that the QL constraint induced. For a given N , let the neighborhood expansion of A be defined as AN = S N . Ni∩A6=∅ i Intuitively, AN is the set of subsystems which are connected to A by some neighbor- hood. Using this notion, we show how, in a sense, RFTS states cannot support “long-range” correlations.
Proposition 6.4.3 (Finite correlation length). A target pure state ρ = |ψihψ| is RFTS with respect to N only if, for any two subsystems A and B having dis- N N joint neighborhood expansions (A ∩ B = ∅), arbitrary observables XA and YB are
142 6.4 Robust finite-time stabilization: necessary conditions
uncorrelated, that is, Tr (XAYBρ) = Tr (XAρ) Tr (YBρ).
The above result can be modified to address the case where the neighborhood expansions are overlapping. Towards this, we give a lemma which describes the “QL recoverability” property satisfied by RFTS states.
Lemma 6.4.4 (Recoverability property). Let target pure state ρ = |ψihψ| be ˜ RFTS with respect to N . If a map M acts on a subsystem A, M ≡ MA ⊗ IA, then N there exists a sequence of CPTP neighborhood maps EAN each acting only on A , such that ρ = El ◦ ... ◦ E1 ◦ M(ρ).
As a physical interpretation, consider M to be an error map such as a bit-flip. The above result shows that, for any RFTS state, such errors can be corrected with a recovery map that acts on a confined region of the system, determined by the neigh- borhood structure.
Proposition 6.4.5 (Zero CMI). A target pure state ρ = |ψihψ| is RFTS with respect to N only if, for any two regions A and B, with AN ∩ B = ∅, the quantum conditional mutual information (CMI)
I(A : B|C)ρ ≡ S(A, C) + S(B,C) − S(A, B, C) − S(C)
N satisfies I(A : B|C)ρ = 0, where C ≡ A \A.
A common feature of product states and graph states is that the neighborhood- acting terms of their canonical FF Hamiltonians commute with one another. While we will show that this commutativity is not necessary for RFTS, a weaker version of this property is necessary, nevertheless.
Proposition 6.4.6 (Commuting FF Hamiltonian). If a target pure state |ψi
is RFTS with respect to neighborhood structure N , then [Πk, Πk] = 0 for all neigh-
borhoods Nk, where Πk and Πk are the projectors onto ΣNk (|ψi) and ∩j6=kΣNj (|ψi), respectively.
With this proposition, we can verify that neither the Dicke nor the AKLT states are RFTS on account of the lack of commutativity among the terms in their canonical FF Hamiltonians. In the next section we build from the graph state example and the connection to commuting Hamiltonians to develop conditions for verifying RFTS of more general states.
143 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
6.5 Robust finite-time stabilization: sufficient con- ditions
In this section, we describe three distinct sufficient conditions for ensuring RFTS of a target state. The first condition incorporates all examples that we know of, although it is not a computable condition. The next condition is computable, applies to ar- bitrary neighborhood geometries, and incorporates a number of important examples, including graph states. However, there are some examples of RFTS states which it does not include. The final condition is applicable to a certain class of neighborhood structures, and target states satisfying this condition admit a simple tensor network description [195].
6.5.1 Non-operational sufficiency criteria To understand what features ensure RFTS of a general target state, we take a closer look at the graph states of Example 6.4.2. An important feature here is that the CPTP maps Ei rendering the graph state RFTS do not interfere with one another. Although the maps on adjacent neighborhoods may overlap, the degrees of freedom they transform are independent. Each map Ei prepares a single degree of freedom into the state |+i. However, these degrees of freedom do not correspond to physical qubits. Rather, they correspond to virtual subsystems [196, 197]. These virtual subsystems are analogous to the quasi-particles of condensed matter physics in that their observables are linear combinations of the physical observables. With respect to the virtual subsystems, the graph state is factorized. Using the language of [132], the graph state is generalized unentangled with respect to the virtual subsystem observables. However, this feature alone is not sufficient for RFTS. Additionally, each virtual subsystem must be “contained” in a neighborhood. This enables each neighborhood map to subject a contained virtual subsystem to any transformation, while leaving the remaining virtual subsystems unaffected. For graph states, each neighborhood map prepares a virtual subsystem into the state |+i. The graph state example is special with respect to the virtual subsystems. There is a 1-1 correspondence between physical subsystems and virtual subsystems. As we will find, this feature is not necessary for RFTS. But, in the case that there is such a correspondence, we are granted a unitary transformation which maps the physical subsystems into the virtual subsystems. In the graph state example, this transfor- mation was achieved by UG. This 1-1 correspondence makes possible the standard construction of graph states, whereby the physical product state |+i⊗N is transformed into the virtual product state |Gi by the unitary transformation UG. Accordingly, each local physical observable σi is transformed into the virtual subsystem observable † σˆi = UGσiUG, which, like the Kraus operators of Ei, acts only on the neighborhood
144 6.5 Robust finite-time stabilization: sufficient conditions
Ni. Finally, we emphasize that, when the maps Ei are represented in the appropriate virtual subsystem description, the sequence EN ◦ ... ◦ E1 is seen as a product of maps acting on distinct (virtual) subsystems analogous to the product state stabilization in Eq. (6.15). The essential feature of this structure is the identification of the physical subsystem Hilbert space with a virtual subsystem Hilbert space,
O O ˆ W : Hi → Hj, (6.17) i j
ˆ where, in general, we need not require any pair Hi and Hj to be isomorphic (e.g. the physical systems could be qubits, while the virtual subsystems are four-dimensional). N N ˆ We denote this identification, or isomorphism, by i Hi ' j Hj, where the “'” symbol serves to remind that the tensor product structure on the left and right-hand sides may not correspond. This relabeling of the degrees of freedom leads to two conditions which ensure RFTS of a target state. First, the target state should be factorized with respect to the virtual degrees of freedom,
O ˆ |ψi = |ψji. j
Second, the operators associated to any given virtual subsystem should, themselves, be neighborhood operators; that is, for every j, there exists k such that for any virtual ˆ ˆ subsystem operator Xj ∈ B(Hj),
Xˆ ⊗ I ∈ B(H ) ⊗ I . j j Nk N k We describe how these two features ensure RFTS. The logic closely parallels our previous demonstration of RFTS for graph states. Assume that the conditions above hold for some |ψi. We construct a finite se- quence of commuting QL CPTP maps which robustly stabilize |ψi. On the virtual systems, define the maps Ej such that ˆ ˆ ˆ ˆ Ej : B(Hj) ⊗ B(Hj) → B(Hj) ⊗ B(Hj), ˆ ˆ Ej(·) = (|ψjihψj|Tr)j ⊗ Ij. ˆ The Kraus operators of Ej are contained in B(Hj) ⊗ Ij. Hence, by the inclusion B(Hˆ ) ⊗ I ≤ B(H ) ⊗ I , the Kraus operators of E act non-trivially only on j j Nk Nk j neighborhood Nk. Thus, each map Ej is a valid neighborhood map. Finally, we must show that an arbitrary sequence of these maps prepares |ψi while leaving it invariant.
145 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
For invariance, we have
M ˆ O ˆ ˆ El(|ψihψ|) = El |ψjihψj| j=1 ˆ ˆ ˆ ˆ O ˆ ˆ = |ψlihψl|Tr |ψlihψl| ⊗ |ψjihψj| = |ψihψ|. (6.18) j6=l
To show that an arbitrary complete sequence of the neighborhood maps prepares |ψi, note that
T O ˆ ˆ ET ◦ ... ◦ E2 ◦ E1 = |ψjihψj|Trj j=1 T T O ˆ ˆ O = |ψjihψj| Trj j=1 j=1 = |ψihψ|Tr. (6.19)
The product state and graph state examples admit such a factorization structure. In the following example we demonstrate another class of RFTS states by establishing a similar factorization structure. Qubit graph states are an example of so-called stabilizer states; they can be written as the unique eigenvalue-1 state with respect to a commuting set of operators from the N-qubit Pauli group. We note that the class of states presented in the following example are not standard stabilizer states [198].
Example 6.5.1 (CCZ states). In [198] the authors introduce the class of so-called 3 (Z2) -states, which exhibit genuine two-dimensional symmetry-protected topological order and for which the construction parallels that of the graph states. While this class of states may be defined for any 3-uniform hypergraph (i.e. one with only 3-element edges), we restrict to the triangular lattice, which allows for scaling the system.√ As with graph states, each qubit in the lattice is initialized in |+i = (|0i+|1i)/ 2. Then, on each triangular cell a controlled-controlled-Z (CCZ) gate is applied, where
1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 CCZ = 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 −1
146 6.5 Robust finite-time stabilization: sufficient conditions
in the computational basis. Noting that all CCZ gates commute with one another, we Q define U∆ ≡ (i,j,k)∈T CCZijk, where T is the set of triangular cells on the lattice. The target CCZ-state is ⊗N |∆i ≡ U∆|+i , (6.20) with N the number of lattice sites. To each site we associate a neighborhood defined by that qubit along with the six adjacent qubits. We verify that |∆i is RFTS with respect to this neighborhood structure by identifying a virtual subsystem decomposition satisfying the needed properties. As with the graph state example, we can identify each physical subsystem to a virtual subsystem. The unitary transformation U∆ takes the physical subsystem observables into the virtual subsystem observables. easily verify that Then, each virtual subsystem algebra corresponds to a neighborhood-contained algebra thanks to the commutativity of the CCZ gates,
ˆ −1 B(Hi) ⊗ Ii = U∆(B(Hi) ⊗ Ii)U∆ h i = U (B(Hˆ ) ⊗ I )U −1 ⊗ I Ni i Ni\i Ni Ni ≤ B(H ) ⊗ I , Ni Ni where U ≡ Q CCZ acts only on the physical systems in neighborhood N . Ni k,l∈Ni\i ikl i Furthermore, by construction, the CCZ state is a virtual product state. Considering ⊗N |∆i = U∆|+i , U∆ maps each physical factor into a corresponding virtual subsystem ˆ ⊗N NN ˆ factor, giving |∆i ' |+i with respect to i=1 Hi. As the neighborhood containment property of the virtual subsystems and the factorization of |∆i are satisfied, the CCZ state is verified to be RFTS. To be concrete, we can construct commuting neighborhood acting CPTP maps which each prepare a factor of |∆i viewed as a virtual product state.
Next, we describe an important generalization of the above neighborhood factor- ization scheme. This generalization is motivated by the following example of a state which does not admit a simple neighborhood factorization, yet is RFTS.
Example 6.5.2 (Non-factorizable RFTS state). Consider the system H'HA ⊗ 2 5 2 HB ⊗ HC ' C ⊗ C ⊗ C . We decompose the middle system as
5 2 2 1 0 ˜ 0 HB ' C ' (C ⊗ C ) ⊕ C 'Hb ⊗ Hb0 ⊕ HB ' HB ⊕ HB,
147 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
by which we label the basis vectors as
|0i = | + +i, |1i = | + −i, |2i = | − +i, |3i = | − −i, |4i = |ei.
Define the target state as
|ψi = |000i + |011i + |120i + |131i.
With respect to the alternative decomposition above, we can write (see Fig. 6.4 for a schematic) |ψi = [(|0+i + |1−i) ⊗ (| + 0i + | − 1i)] ⊕ 0. 0 Note that |ψi is orthogonal to the space HA ⊗ HB ⊗ HC . Consider the neighborhoods to be N1 = {A, B} and N2 = {B,C}. We construct maps which render |ψi RFTS. 0 Define E : B(HB) → B(HB) to be 1 E 0(σ) ≡ (I − |eihe|)σ(I − |eihe|) + (I − |eihe|)he|σ|ei. 4 This CPTP map takes probabilistic weight from span(|ei) and maps it uniformly to ˆ 0 the complement. Also, define E1 : B([HA ⊗ Hb ⊗ Hb0 ] ⊕ [HA ⊗ HB]) → B([HA ⊗ Hb ⊗ 0 Hb0 ] ⊕ [HA ⊗ HB]) to be ˆ + + E1 ≡ [|φ ihφ |TrA,b ⊗ Ib0 ] ⊕ [I], where |φ+i = √1 (|0+i + |1−i). We define Eˆ acting on N similarly. With these, we 2 2 2 define the two neighborhood maps
ˆ 0 E1 ≡ (E1 ⊗ IC ) ◦ (IA ⊗ E ⊗ IC ), ˆ 0 E2 ≡ (IA ⊗ E2) ◦ (IA ⊗ E ⊗ IC ).
0 Crucially, since the outputs of both E1 and E2 cannot have support in HA ⊗ HB ⊗ HC , the support-moving action E 0 following either map is trivial,
0 (IA ⊗ E ⊗ IC ) ◦ E1 = E1,
148 6.5 Robust finite-time stabilization: sufficient conditions
Figure 6.4: Example of a non-factorizable RFTS state.
and similarly for E2. Hence, the product of either order of the maps is
ˆ 0 E1 ◦ E2 = (E1 ⊗ IC ) ◦ (IA ⊗ E ⊗ IC ) ◦ E2 ˆ = (E1 ⊗ IC ) ◦ E2 ˆ ˆ 0 = (E1 ⊗ IC ) ◦ (IA ⊗ E2) ◦ (IA ⊗ E ⊗ IC ) = |ψihψ|Tr.
The key feature of this state that enables it to be RFTS is that, once part of 0 the Hilbert space is removed locally (in this example, HB ≤ HB), there exists a neighborhood factorization of |ψi with respect to the remaining space, whereby the algebra of each factor is contained in a corresponding neighborhood. In the above example, the Hilbert space reduces into two parts; one containing a factorization of the target state and the other being orthogonal to the reduced states of the target state. In contrast to such cases where the Hilbert space can be reduced into sectors and following [199], we will refer to a factorization such as in Eq. (6.17) as an irreducible factorization. Next, we show that, in general, an irreducible factorization is not required for RFTS and describe the reducible factorizations which still ensure RFTS. Towards this, we introduce two useful concepts. First, we refer to a generalization of the reduction in Example 6.5.2 as a local restriction. N ˜ Definition 6.5.3. Given H' i Hi and a set of spaces Hi ≤ Hi, we define a locally ˜ N ˜ restricted Hilbert space as H' i Hi. Second, with respect to a given neighborhood structure, the physical locality of sub- systems may be more fine-grained than the locality determined by the neighborhood structure. For example, consider systems A, B, C, D, with neighborhoods {A, B, C} and {B,C,D}. The physical locality describes four subsystems. But, as far as the
149 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics neighborhood structure is concerned, the separation of system B from C is artificial. In such a scenario we are afforded a more-reductive local restriction if we reduce starting from the coarse-grained subsystem decomposition, HA ⊗ HBC ⊗ HD.
NN Definition 6.5.4. Given H' i=1 Hi and a neighborhood structure N , we define the coarse-grained subsystems to be the equivalence classes of the subsystems Hi with respect to the relation “is contained in the same set of neighborhoods as”.
Though usually stated explicitly, in the remainder of the chapter the decomposi- N tion of the Hilbert space H' i Hi denotes the coarse-grained subsystems, and N incorporates the coarse-graining.
Theorem 6.5.5 (Neighborhood factorization on local restriction). A state |ψi NN described on the coarse-grained subsystems H' i=1 Hi with respect to the neigh- ˜ NN ˜ borhood structure N is RFTS if there exists a locally restricted space H' i=1 Hi, 0 ˜⊥ ˜ NM ˆ with a complement H ' H , whereby a factorization H = j=1 Hj gives
M M O ˆ O ˆ 0 |ψi = |ψji ⊕ 0 ∈ Hj ⊕ H (6.21) j=1 j=1
ˆ and for each virtual subsystem Hj there exists a neighborhood Nk such that
B(Hˆ ) ⊗ I ⊕ I0 ≤ B(H ) ⊗ I . (6.22) j j Nk Nk
The following example details a construction whereby such a factorization arises.
Example 6.5.6 (Generalized Bravyi-Vyalyi states). We introduce a class of states inspired by the work of Bravyi and Vyalyi in [199]. These authors studied the complexity of the problem “Common Eigenspace”, whereby one is to determine whether or not there exists a state |ψi that is a common eigenstate of the commuting Hamil- tonians H1,...,Hr with respect to eigenvalues λ1, . . . , λr. An important example that they consider is the case where the given Hamiltonians are each 2-body operators NN with respect to H' i=1 Hi. Bravyi and Vyalyi show that the Hamiltonian terms being 2-body and commuting induces a tensor factor decomposition of each physical subsystem.
Consider a set of commuting two-body projectors {Πjk}, acting with respect to the edges of a graph G = (V,E), with vertices corresponding to the factors of the Hilbert NN space H' i=1 Hi. Adapting the notation from Lemma 8 of [199], the commuting
150 6.5 Robust finite-time stabilization: sufficient conditions
two-body projectors can be shown to induce a decomposition of H, M H' Hα α N N M O α O O α ' Hi ⊗ Hij , (6.23) α i=1 i=1 j|(i,j)∈E by which each projector Πij simplifies to the form
M α Πij = Irest ⊗ Πij, (6.24) α
α α α where Πij acts only on Hij ⊗Hji. This construction makes manifest the commutativity among the Πij since, within each sector Hα, the non-trivial parts of each projector (i.e. α the Πij) act on disjoint factors. This simplification in the structure of the projectors enables Bravyi and Vyalyi to show that the 2-local version of the Common Eigenspace problem is in NP. Bravyi and Vyalyi then describe the construction of states which are “simple” with respect to the induced virtual subsystem decomposition. Within a given sector Hα, they first define n O O |φi = |φii ⊗ |φiji , (6.25) i=1 (i,j)∈E which is a product of single-factor and bipartite states. Then, defining a set of isome- tries α O α Vi : Hi ⊗ Hij → Hi, j 0 they construct the state |φ i = (V1 ⊗ ... ⊗ Vn)|φi ∈ H. We call any state constructed as such a Bravyi-Vyalyi (BV) state. We remark that such a form can be recast as a tensor network state [195]. A BV state is RFTS with respect to the neighborhood structure determined by its interaction graph: Compare the factorization of Eq. (6.25) with that of Eq. (6.21), α α α α noting that the virtual subsystems are of the form Hi and Hj ⊗ Hij ⊗ Hji. 2) Each α α α virtual subsystem Hi or Hij ⊗ Hji is contained in Nk = (i, j):
B(Hα) ⊗ I ⊕ 0 ≤ B(H ) ⊗ I i rest Nk N k B(Hα ⊗ Hα ) ⊗ I ⊕ 0 ≤ B(H ) ⊗ I , (6.26) ij ji rest Nk N k L where the sector with zero is β6=α Hβ. Building off of this structure, we define a generalization of such states and show
151 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
that they are RFTS. Consider a neighborhood structure N and a virtual subsystem decomposition of each particle
fi 0 ˜ 0 O Hi ≡ Hi ⊕ Hi 'Hi ⊕ Hij, j=1
˜ N ˜ and, correspondingly, H ≡ i Hi. To each neighborhood Nk, we associate HNk ' N H ' N Nfi H . Then, for each neighborhood N , consider a subset S i∈Nk i i∈N j=1 ij k k of the ij (i.e. associated to H ) contained in N , defining Hˆ ≡ N H . Assume ij k k ij∈Sk ij that the Sk have been chosen such that each ij is accounted for exactly once. This ensures that N |N | ˜ O ˜ O ˆ H' Hi ' Hk, (6.27) i=1 k=1 where we emphasize that the tensor products over the i index are not equivalent to ˜ the tensor products over the k index. Let Vi : Hi → Hi be isometries from the virtual to the physical particles, with V ≡ (V1 ⊗ ... ⊗ VN ), and consider any product state N ˆ ˜ |ψi = k |ψki ∈ H. We define a generalized BV state to be any of the form 0 O ˆ |ψ i ≡ V |ψi = (V1 ⊗ ... ⊗ VN ) |ψki . (6.28) k
Again, the tensor product structure among the Vi is not the same as the tensor product ˆ structure among the |ψki. This mismatch of tensor product structures is precisely what allows for |ψ0i to exhibit entanglement among the physical particles. An example of a generalized BV state is depicted in Fig. 6.5 and the RFTS scheme for such states is sketched in the caption therein.
6.5.2 Operational sufficiency criteria Algebraic factorization
In the BV scheme of [199], the factorization is induced by a set of commuting Hamil- tonians. In this section we draw inspiration from their construction to investigate ways to induce a factorization of the Hilbert space, amenable to RFTS, using, as opposed to a set of commuting Hamiltonians, the target state and the neighborhood structure. In the BV decomposition, each virtual subsystem corresponds to, at most, two physical subsystems. Hence, this scheme can only induce a factorization when the neighborhoods are two-body. From Example 6.4.2, the graph state on a 2-D square
152 6.5 Robust finite-time stabilization: sufficient conditions
Figure 6.5: Example structure of a generalized BV state. Perforated circles denote physical particles, nodes correspond to virtual subsystems, and solid lines connect vir- tual subsystems which are entangled. The hatched semicircles indicate the subspaces 0 0 0 ˜ H2 and H5 (Hi ⊕ Hi = Hi) where the respective reduced states do not have support. Solid curves delineate neighborhoods. The tensor product of isometries V1 ⊗ ... ⊗ V7 ˆ ˆ ˆ ˆ is applied to the virtual product state |ψi = |ψ1i ⊗ |ψ2i ⊗ |ψ3i ⊗ |ψ4i to obtain the generalized BV state. Entanglement among the virtual subsystems is translated into many-body entanglement among the physical particles. Despite this many-body en- tanglement, the state is RFTS. In the RFTS scheme, the job of each neighborhood ˜ map Ek is to transfer probabilistic weight into the spaces Hi and then prepare the cor- ˆ ˆ responding virtual factor |ψki ∈ Hk, while acting trivially on the remaining degrees of freedom.
lattice is RFTS and admits a virtual subsystem factorization. Yet, because the neigh- borhoods are five-body, the BV scheme does not apply to this case. Drawing from the graph state example, we develop a scheme to induce a virtual subsystem decom- position as in Thm. 6.5.5 for an arbitrary state and neighborhood structure. An important feature of graph states is the fact that the physical representation ˆ of the virtual subsystems B(Hi) ⊗ Ii, acting on neighborhood Ni, commutes with all Schmidt span projectors Πk for k 6= i. Furthermore, this algebra of operators
is singled out as the algebra of operators acting on HNi which commute with the remaining neighborhood projectors (Πk, for k 6= i). Lastly, these neighborhood- acting algebras commute with one another and generate the full algebra B(H). This is seen by expressing these algebras in the virtual subsystem basis. In general, a set of associative algebras which is complete (has trivial commutant) and for which
153 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
the individual algebras mutually commute induces an irreducible factorization of the space they act on. This fact is detailed in the following proposition. Proposition 6.5.7 (Algebraically induced factorization). If a set of algebras {Aj}, Ai ≤ B(H), is complete and commuting, then each Aj has a trivial center NT ˆ and there exists a decomposition of the Hilbert space H' j=1 Hj for which Aj ' ˆ B(Hj) ⊗ Ij for each j. In the case of graph states, the factorization of these algebras is given. But, for an arbitrary state, the above proposition points to the possibility that, by constructing a set of commuting neighborhood-contained algebras, we may obtain a Hilbert space factorization which factorizes the target state. We will aim to induce the factoriza- tion from the neighborhood-acting terms of a QL Hamiltonian, much like the BV decomposition. One important difference, however, is that the Hamiltonians we use are derived from the target state itself. Recall that Prop. 6.5.5 describes a factorization of the target state on a locally restricted space. Accordingly, we will define neighborhood acting algebras with re- spect to a locally restricted space. Towards this, we provide a means of constructing a locally restricted space from a positive-semidefinite operator such as the target state. Definition 6.5.8. Given an operator M ≥ 0 acting on coarse-grained subsystems NN i=1 Hi with respect to the neighborhood structure N , the subsystem support and subsystem kernel of M on p are supp(Trp (M)) and ker(Trp (M)), respectively. The ˜ NN 0 ˜ local support of M is H ≡ i=1 supp(Tri (M)), with H'H ⊕ H. We note that, for a pure state |ψi, the subsystem support of M = |ψihψ| on p is simply the Schmidt span Σp(|ψi). Furthermore, the support of each neigh- borhood projector Πj is contained in the subsystem support of the target state ˜ NN ˜ H = i=1 supp(Tri (|ψihψ|)). This allows us to define projectors Πk ≡ Πk|H˜ re- stricted to the subsystem support of |ψihψ|. We also give a label for the sub- system support on a given neighborhood, H˜ ≡ N supp(Tr (|ψihψ|)). Also, Nj i∈Nj i H˜ ≡ N supp(Tr (|ψihψ|)) denotes the subsystem support restricted to the N j i/∈Nj i complement of Nj. With these, we consider the following candidate for the algebras that are to induce a factorization of the target state. Definition 6.5.9. Given a target state |ψi and a neighborhood structure N , for each neighborhood Nj, we first define
A˜ ≡ {X ∈ B(H˜ )|[X ⊗ I , Π˜ ] = 0, ∀ k 6= j}. (6.29) Nj Nj H˜N H˜ k j N j From this, the neighborhood algebra is defined as ˜ Aj ≡ (span(I) ⊕ ANj ) ⊗ I, (6.30)
154 6.5 Robust finite-time stabilization: sufficient conditions
with respect to the decomposition H' (H ⊕ H˜ ) ⊗ H , where H is the Nj ,0 Nj N j N ,0 ˜ complement of HNj in HNj .
Each neighborhood algebra Aj is verified to be an associative algebra by writing it as a commutant,
Aj = {Y ∈ B(H)|[Y, Πk] = 0 ∀ k 6= j and [Y,Z] = 0 ∀ Z ∈ I ⊗ B(H )}. (6.31) Nj N j If a set is closed under adjoint, as in our case, then its commutant is closed under ad- joint and therefore is a C*-algebra. Intuitively, each Aj is the largest C*-algebra of Nj neighborhood operators which commute with the remaining neighborhood projectors Πk. We now give the main result of this section which states how the structure of the neighborhood algebras can ensure a particular factorization of the target state and, hence, ensure RFTS.
Theorem 6.5.10. A state |ψi described on the coarse-grained subsystems H' NN i=1 Hi with respect to the neighborhood structure N admits a decomposition |ψi = N ˆ 0 0 ⊕ j |ψji induced by the neighborhood algebra-induced factorization H'H ⊕ N ˆ j Hj, and hence, is RFTS with respect to N if i) |ψi satisfies Eq. (6.5) with respect to N ; and ii) the neighborhood algebras Aj are commuting and complete on the local support space H˜.
The key feature of this sufficient condition for RFTS is that it is operationally checkable; satisfaction of Eq. (6.5) is determined by an intersection of vector spaces, and the neighborhood algebras, and their commutativity properties can, in principle, be computationally determined. This sufficient condition does not incorporate all examples of RFTS that we know of. However, a simple pre-processing of the neighborhood structure and local support space allows for the inclusion of RFTS states that are otherwise excluded. In some cases, the reduced states of the target state on a particular neighborhood will contain physical factors which are full rank: Tr (|ψihψ|) = ρ = ρ ⊗ρ , with ρ > 0. In N k Nk Nk\i i i such cases, invariance requires that any neighborhood map Ek act trivially on system i. Thus, if |ψi were RFTS with respect to the initial neighborhood structure, it will 0 be RFTS with respect to a neighborhood structure where Nk is replaced with Nk ≡ Nk\i. We have found cases in which the sufficient conditions of Thm. 6.5.10, while not initially satisfied, become satisfied after updating the neighborhood structure as above.
155 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
Matching overlap A simplification of the above condition is possible if the neighborhoods satisfy a certain geometry. Definition 6.5.11. A neighborhood structure is said to satisfy the matching over- lap condition if, for any set of neighborhoods that have a common intersection, this common intersection is the intersection of any pair of the neighborhoods in the set. Two-body neighborhoods necessarily satisfy the matching overlap condition, whereas neighborhood structures of graph states or those in Fig. 6.5 do not. The matching overlap condition ensures a simple relationship between the intersection of neigh- borhoods and the coarse-grained particles: the intersection of any two non-disjoint neighborhoods is a coarse-grained particle. This fact is used in the proof of the following proposition. NN Proposition 6.5.12 (Matching overlap RFTS). Let p=1 Hp be a Hilbert space of coarse-grained subsystems with respect to a neighborhood structure N that satisfies the matching overlap condition. If |ψi satisfies Eq. (6.5) with respect to N and [Πj, Πk] = 0 for all pairs of neighborhood projectors, then |ψi is RFTS. As noted, the BV decomposition uses commuting 2-body Hamiltonians to decom- pose the Hilbert space. The matching overlap condition, in a sense, ensures when neighborhood structures beyond 2-body are amenable to a similar type of decompo- sition. The above proposition then describes how, for a QLS state with commuting Schmidt span projectors, matching overlap ensures a factorization of the target state. We remark that, by further restricting the neighborhood structure, it is possible to establish that, for any state satisfying Eq. (6.5), commutativity of the neigh- borhood projectors is necessary and sufficient for RFTS. The restriction is that the neighborhood hypergraph should be connected and not have any cycles. This restric- tion includes the 1-D NN neighborhood structure, but excludes, for instance, the 2-D lattice NN neighborhood structure. It is tempting to conjecture that commuting Schmidt span projectors are necessary for a QLS state to, further, be RFTS. The following example provides a case where this is not true. Example 6.5.13 (RFTS ground state of non-commuting FF parent Hamil- tonian). Consider nine qubits labeled 1-9 described by the state
|ψiW ≡ |W i123 ⊗ |W i456 ⊗ |W i789,
where |W i = √1 (|001i + |010i + |100i). The neighborhood structure is depicted in 3 Figure 6.6. First, we demonstrate that |ψiW is RFTS, then we show that the neigh- borhood projectors Πk do not commute with one another. That |ψiW is RFTS follows
156 6.5 Robust finite-time stabilization: sufficient conditions
Figure 6.6: The ovals denote the neighborhoods of the neighborhood structure for which |ψiW is RFTS. The three neighborhoods are most easily described by their respective complements. Letting S ≡ {1,..., 9}, we define NA ≡ S\{6, 7}, NB ≡ S\{1, 9}, and NC ≡ S\{3, 4}.
simply from the fact that it can be factorized such that each factor is contained in a neighborhood. In this way, the three maps which compose to stabilize |ψiW are E123 ≡ (|W ihW |123Tr) ⊗ I123 and similary for E456 and E789. To show that the Πk do not commute, consider ΠA and ΠB. On systems 7, 8, and 9, these, respectively project onto supp(I7 ⊗ Tr7 (|W ihW |789)) and supp(Tr9 (|W ihW |789) ⊗ I9). A direct calculation shows that these two projections onto systems 7, 8, and 9 do not commute with one another. Hence, [ΠA, ΠB] 6= 0, and, by symmetry, this holds for any pair of Πk. Despite the fact that the canonical FF Hamiltonian does not have commuting terms, we can still construct a commuting FF Hamiltonian for which |ψiW is the unique ground state, namely ˜ H = (I − |W ihW |123 ⊗ I123) + (I − |W ihW |456 ⊗ I456)
+ (I − |W ihW |789 ⊗ I789).
We conjecture that, if a state is RFTS, there always exists some FF commuting parent Hamiltonian for which it is the unique ground state. P This example suggests that, in general, the canonical Hamiltonian k Πk is not necessarily a useful object for identifying whether a QLS state can be stabilized ro- bustly. We leave open the question of identifying a scheme for discovering neighbor-
157 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
Figure 6.7: Given 1-D lattice of nine systems and a NNN neighborhood structure, a state which is RFTS can be stabilized with a depth = 3 dissipative circuit by organizing the application of maps into layers as shown in the schematic above.
hood factorizations of a general target state.
6.6 Efficiency of robust finite-time stabilization
6.6.1 Circuit complexity In this section we analyze the circuit complexity of the RFTS scheme. We investigate how the number of CPTP neighborhood maps (circuit size) and degree of paralleliza- tion (circuit depth) scale with N, the number of systems. We focus on systems and neighborhood structures defined with respect to a finite d-dimensional lattice. By fixing a type of neighborhood structure (e.g. NNN as in Fig. 6.7), we show that the depth of the dissipative circuit rendering a state RFTS is upper-bounded by a constant. The lattice structure of the subsystems and neighborhoods affords a high degree of parallelization for the neighborhood maps. As demonstrated in Figure 6.7, this is achieved by partitioning the set of neighborhood maps into “layers” wherein the maps of a given layer are mutually disjoint. To appreciate the role played by the lattice structure, consider the following example of a scalable neighborhood structure for which constant depth is not achievable. Let the neighborhood structure N be given N N(N−1) by the set of all pairs of subsystems, giving |N | = 2 = 2 . The largest number of neighborhood maps which may act in parallel is bN/2c. Hence, the best possible parallelization will still require at least |N |/bN/2c = N − 1 layers of maps. We develop a scheme for parallelizing the RFT stabilizing neighborhood maps for a lattice system which ensures finite depth. First, consider the following example.
158 6.6 Efficiency of robust finite-time stabilization
Example 6.6.1 (CCZ states on Kagome lattice). In Example 6.5.1 we showed that the CCZ state defined on the triangular lattice is RFTS. The CCZ state can be defined similarly on the kagome lattice with CCZ gates acting on each triangle of systems. As depicted in Fig. 6.8, to each physical system we associate the five-body neighborhood made of that system along with its four nearest neighbors. By following the reasoning of Example 6.5.1, it is simple to show that the CCZ state defined on this lattice is RFTS with respect to such neighborhoods. Here we show that, for a lattice of any size, this stabilization can be achieved by a dissipative circuit with depth = 12. The unit cell of the kagome lattice consists of three physical systems, and, therefore, to three neighborhoods as shown in Fig. 6.8. By translating these three physical systems and three neighborhoods by the group of lattice translations (generated by unit lattice vectors eˆ1 and eˆ2), we generate the set of all systems and all neighborhoods. With an RFT stabilization scheme, the irrelevance of the map ordering allows us to organize the neighborhood maps into layers. To construct a layer, consider the set 0 0 0 0 of neighborhoods N in the unit cell labeled N1 , N2 , and N3 in Fig. 6.8. For each direction, translate this set until it becomes disjoint with respect to the un-translated set. The diameter of the set, the maximum number of such translations needed over all directions, is found to be two. By translating any neighborhood in the unit cell by this diameter, the resulting neighborhood is ensured to be disjoint from the former. We can generate a layer of disjoint neighborhoods by repeatedly translating a unit cell neighborhood by multiples of the diameter (i.e. an even number of translations) in each direction. Three of the layers will correspond to the three neighborhoods in the unit cell. We still need to account for the neighborhoods translated by an odd number of lattice vectors in either direction. These nine remaining layers are obtained by translating each of the previous three layers by lattice translations (0, 1), (1, 0) or (1, 1). Thus, we have partitioned the neighborhood maps into twelve layers. In each layer the neighborhood maps act in parallel ensuring that, for any lattice size, the CCZ state is RFTS with respect to a depth = 12 dissipative circuit.
We generalize this scheme to neighborhood structures defined on an arbitrary lattice. A lattice system is characterized by a unit cell containing an arrangement of c physical systems. The global system is generated by replicating this unit cell d z }| { by translations from the group Ld = Z × ... × Z = Zd. As in Example 6.6.1, we take the neighborhood structure N to be translationally invariant with respect to Ld. Hence, to each unit cell, there corresponds a set of neighborhoods N 0 from which the translation group generates the global neighborhood structure N . In defining a unit cell of neighborhoods, the neighborhoods may “spill over” into adjacent unit cells. As defined in Ex. 6.6.1, we denote the diameter of this set diam(N 0). In order to describe how circuit size and depth scales with system size, we consider
159 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
Figure 6.8: Definition and features of the Kagome lattice, its unit cell, and corre- sponding neighborhood structure for the CCZ-state. In constructing the CCZ-state, the system is initialized in |+i⊗N and a CCZ-gate is applied to each triangle of adja- cent systems (cf. Eq. (6.20)).
a sequence of finite-sized subsets of the infinite lattice. We take the system to be comprised of a width-L hypercube of Ld unit cells, totaling N = cLd subsystems and |N 0|Ld neighborhoods. For each N we denote the corresponding neighborhood structure as N N . Using a generalization of the scheme described in Example 6.6.1, we bound the circuit complexity of stabilizing an RFTS state with respect to a lattice neighborhood structure (see Sec. 6.8 for a proof detailing the scheme). Proposition 6.6.2 (Lattice circuit size scaling). Consider a scalable lattice neigh- borhood structure N N as defined above. If |ψi is RFTS with respect to this neigh- borhood structure, then |ψi can be stabilized by a dissipative circuit of size |N 0|(N/c) and depth D = |N 0|diam(N 0)d. We remark that the scheme we use is not guaranteed to give an optimal depth. It merely captures the essential features of the lattice for the purposes of ensuring finite depth. The following example gives a case in which a partition of neighborhoods deviates from the above scheme to give an improved circuit depth. Example 6.6.3 (Optimal depth for 2D graph states). Consider the 2D square lattice on which we define the 2D graph states. The group of lattice translations is
160 6.6 Efficiency of robust finite-time stabilization
isomorphic to G ' Z × Z. Define a single neighborhood on site (0, 0) as that site along with the four adjacent sites, (1, 0), (0, 1), (−1, 0), and (0, −1). We generate the neighborhood structure by translating this neighborhood with respect to G = Z × Z. Hence, there is one neighborhood per physical system and each neighborhood is labeled by an element of G. There is one neighborhood per unit cell, and the diameter of the neighborhoods in a unit cell is diam(N 0) = 3. Therefore, using the above scheme, we may stabilize the graph state with a circuit of depth D = |N 0| diam(N )d = 1 · 32 = 9. However, we can choose a different parallelizing scheme which results in a depth- five circuit. By translating the neighborhood on site (0, 0) with just the subgroup H ' h(1, 2), (2, −1)i ≤ G, the generated neighborhoods are disjoint. The size of the coset group is |G/H| = 5. Each coset gH corresponds to a layer of disjoint neighborhood maps which may act in parallel. The number of layers needed so that the resulting circuit includes all neighborhood maps is |G/H| = 5. This shows that the 2-D graph states on N systems can be stabilized with a circuit of size N and depth 5.
6.6.2 Connection to rapid mixing In this section we aim to relate RFTS to the existence of continuous-time QL dynamics which “efficiently” stabilize a target state. For continuous-time dynamics, if the target state is an equilibrium point of the dynamics, then, for any finite time, the state of the system can only approximate the target state. We show how RFT stabilizability of a scalable family of states ensures the existence of QL Liouvillian dynamics such that the time needed to approximate the target state scales favorably with system size. To make this connection rigorous, we will restrict our considerations to RFTS scenarios where the CPTP maps in the sequence commute with one another. Although we have not shown this feature to be necessary, each of the sufficient conditions in Sec. 6.5 ensure the existence of such commuting maps. Two special CPTP maps derived from the Liouvillian L are used in defining con- vergence to the target state [161]:
•E φ is the CPTP map projecting onto the operators for which L has eigenvalue Re(λ) = 0.
•E ∞ is the CPTP map projecting onto the operators for which L has eigenvalue λ = 0.
With these, the following definition provides a measure of how far the “worst-case” evolution is from an equilibrium state of the continuous-time dynamics.
Definition 6.6.4. Given a one-parameter semigroup of CPTP maps Et, define the
161 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
contraction of Et to be 1 η(Et) ≡ sup ||(I − Eφ)(Et(ρ))||1. (6.32) 2 ρ≥0,Tr(ρ)
In the case that the continuous-time dynamics is generated by a Liouvillian L, η characterizes the slowest decay into the space corresponding to the peripheral spec- trum of L. We note that, in the cases we are interested in, the Liouvillian has no purely imaginary eigenvalues, ensuring Eφ = E∞. From the contraction, we define the mixing time of Et to be the minimal time such 1 that η(Et) = 2 . Since we are interested in the scaling of the mixing time, the choice 1 of 2 as the “sufficient distance to the peripheral spectrum” is unimportant; the type of scaling of this time does not depend on this distance. The contraction η generated by a Liouvillian may be bounded using the spectral gap.
Definition 6.6.5. The spectral gap λ¯ of a Liouvillian L is defined as
λ¯ ≡ inf{abs(Re(λ))|Re(λ) < 0, λ ∈ spec(L)}. (6.33)
However, the gap of the Liouvillian alone is not always sufficient to ensure a favorable scaling of η as the system size is increased [182]. In light of this, we will make use of the gaps of neighborhood Liouvillians as opposed to the gap of the global Liouvillian. The authors of [182] proved the following two results.
Theorem 6.6.6. [182] (Contraction for commuting Liouvillians) Let {Lj} be a set of Liouvillians which commute with one another. Then,
P Lj t X Lj t η e j ≤ η e . (6.34) j
(Contraction theorem) Let L be a Liouvillian with gap λ¯. Then, there exists L > 0 and for any ν < λ¯ there exists R > 0 such that
Le−λt¯ ≤ η eLt ≤ Re−νt. (6.35)
The mixing time of a Liouvillian is inversely proportional to the scale of that Li- ouvillian. Thus, to make our results non-trivial, we fix our neighborhood Liouvillians to have a bounded norm: c > ||Lj|| for all j for some constant c. With these considerations, we can easily place an exponentially decaying upper bound on the contraction of sums of commuting Liouvillians which linearly scales with the number of Liouvillians.
162 6.7 Extension of results to mixed target states
Proposition 6.6.7 (Commuting Liouvillian contraction bound). Let {Lj} be a, possibly infinite, set of bounded norm Liouvillians each acting on a subsystem of uniformly-bound dimension. Assume that the spectral gaps λj are strictly bounded below by ν > 0. Then, there exists R > 0 such that for any subset S ⊆ {Lj} of P mutually commuting Liouvillians, defining L = S Lj, the contraction is bounded by η(eLt) ≤ |S|Re−νt. (6.36)
Such bounds are more generally related to the notion of rapid mixing.
(s) Definition 6.6.8. For a family of one-parameter semigroups of CPTP maps {Et }, (s) (s) indexed by s and corresponding to the finite dimensional Hilbert space H , {Et } satisfies rapid mixing if there exists constants c, γ, δ > 0 such that
(s) δ (s) −γt η(Et ) ≤ c log (dim H )e . (6.37)
We relate commuting RFTS scenarios to rapid mixing.
Proposition 6.6.9 (Rapid mixing for commuting RFTS). Consider a family of finite systems H(s) indexed by s such that for each s there is a neighborhood structure (s) N with dim(H (s) ) < D. Furthermore, assume the number of neighborhoods scales Nk (s) (s) (s) (s) no more than polynomially in system size, Nk ∈ N and |N | ≤ b log(dim(H )), (s) for some constant b > 0. For each s, let ρs be RFTS with respect to N by a set (s) of commuting neighborhood maps {Ek }, where there exists some ν > 0 such that for (s) any eigenvalue λ ∈ eig(Ek ), λ = 1 or λ < 1 − ν. Then, there exists a family of bounded norm QL Liouvillians satisfying rapid mixing with respect to ρs.
Such rapid mixing ensures that these dynamics are “stable” with respect to local perturbations. The main result in [171] was to show that rapid-mixing of a family of CPTP semigroups ensures such stability of the dynamics. Their result applies to the dynamics we’ve considered above since the QL Liouvillians defined with respect to a lattice neighborhood structure fit the definition of a “uniform family with finite range interaction”.
6.7 Extension of results to mixed target states
Although the focus of this chapter has been on target pure states, the notions of FTS and RFTS apply equally well to target mixed states. In this section we highlight a few finite-time stabilization results regarding target mixed states and also state a few conjectures.
163 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
In our analysis of FTS in Sec. 6.3 (as opposed to RFTS) the purity of the target state played a crucial role. Even the necessary condition of Thm. 6.3.1 involved criteria that only apply to target pure states. For this reason, analysis of (strict) FTS for target mixed state states remains unexplored and left to future work. In contrast, a number of the RFTS results of Sections 6.4 and 6.5 are directly applicable to, or admit analogs for, the mixed state case. Proposition 6.4.3, Lemma 6.4.4, and Proposition 6.4.5 each constrain the correlations of a state that is to be RFTS. The statements and proofs of these results generalize directly to the case of a target mixed state. In Sec. 6.5, we described how the existence of a particular virtual subsystem decomposition of the Hilbert space can ensure that a target state is RFTS. This result, too, extends directly to the case of a mixed target state. Here, instead of the N ˆ pure state being factorized with respect to Hj, the mixed state must be of the N j form ρ = j ρˆj. Accordingly, the RFTS scheme employs neighborhood maps which prepare the mixed state factors among the virtual subsystems Ej = (ˆρjTr)j ⊗ Ij. This observation can be applied to certain types of thermal states, as shown in the following example.
Example 6.7.1 (Gibbs states of factorizable QL Hamiltonians). Consider a P NN QL Hamiltonian H = k Hk acting on H' i=1 Hi for which there exists a virtual NM ˆ factorization H' j=1 Hj satisfying 1) for all k there exists a j = jk such that H ' Hˆ k ⊗ I and 2) for each j there exists a k such that B(Hˆ ) ⊗ I ≤ B(H ) ⊗ I . k j j j j Nk Nk Then, the Gibbs state
ρ(H) = exp(−βH)/Tr (exp(−βH)) (6.38)
is RFTS. This follows from the fact that each virtual subsystem algebra is contained in a neighborhood algebra and that the target state is factorized with respect to the virtual subsystems,
ρ(H) = exp(−βH)/Tr (exp(−βH)) X = exp(−β (Hˆ k) ⊗ I )/Tr (exp(−βH)) jk jk k M 1 O X = exp(−β Hˆ k). Tr (exp(−βH)) j=1 k s.t. jk=j
We expect that, in general, the algebraic dependence among the Hamiltonian terms should play a significant role in determining whether or not such a factorization exists. We leave this to be studied in future work. Theorem 6.5.5, involving a proper virtual subsystem decomposition, can also be
164 6.8 Proofs
generalized. Here, the local restriction is defined by the mixed state’s subsystem support, as in Def. 6.5.8. The construction, then, is completely analogous to that of the pure state case. The remaining results regarding RFTS involve the Schmidt span projectors derived from the target pure state. As we do not know of the analogous structure for the mixed state case, we do not yet know how these results could be extended. In Section 6.6.2 we showed that certain commuting RFTS scenarios implied the existence of rapidly mixing dynamics. Considering the converse, we show that there exist classes of mixed states which admit rapidly mixing dynamics, yet support cor- relations beyond what is allowed for RFTS.
Example 6.7.2 (Non-RFTS commuting Gibbs state). In [156] it is shown that for 1D lattice systems, the Davies generator derived from a commuting Hamiltonian constitutes rapidly mixing dynamics with respect to the corresponding Gibbs state. PN−1 i i+1 Consider the 1D Ising model HN = −J i=1 σz ⊗σz , with coupling strength J > 0. It is well known that, in the thermodynamic limit, for any finite temperature, the two-point correlations of this Gibbs state are exponentially decaying with distance: i i+L −ξL Tr σz ⊗ σz ρ ∼ e , see e.g. []. Therefore, spins with disjoint neighborhood expansions are correlated, which violates the necessary condition for RFTS given in Prop. 6.4.3.
6.8 Proofs
We present here complete proofs of all the technical results stated in the previous sections. • Necessary conditions for FTS.– Theorem 6.3.1 |ψi is FTS with respect to N only if |ψi satisfies Eq. (6.5) and there exists at least one neighborhood for which
dim(HN ) ≥ 2 · dim(ΣN (|ψi)).
Proof. We prove, by its contrapositive, that |ψi being FTS implies |ψi satisfies Eq. (6.5). Assume |ψi does not satisfy Eq. (6.5). Then there exists some |φi ∈/ span(|ψi) T for which |φi ∈ k ΣNk (|ψi). Any |ψi-preserving neighborhood map Ek must fix all states in ΣNk (|ψi). Any sequence of such maps fixes |φihφ| and, hence, cannot map |φihφ| into |ψihψ| as is required for FTS. We continue by showing that the remaining condition is also necessary. Assuming that |ψi is FTS, let ET ... E1(·) = |ψihψ|Tr (·) be a sequence of CPTP maps which stabilizes |ψi. Then, there must exist some map Ek for which Ek(ρ) = |ψihψ| for some ρ∈ / span(|ψi). Using the locality and |ψi-invariance of Ek, we show
165 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
that the equation Ek(ρ) = |ψihψ| places an upper bound on the Schmidt rank of |ψi with respect to the N |N bipartition. The analysis is made easier considering a purification of the equation Ek(ρ) = |ψihψ|, the purified equation being first-order in |ψi. In purifying, ancilla systems must be introduced for ρ and Ek, though not for |ψihψ|, as it is already pure. Letting HA be the ancilla purifying ρ, we have
ρ → |φi ∈ H ⊗ H ⊗ H , A N k Nk
s.t. TrA (|φihφ|) = ρ. (6.39)
Letting HB be the ancilla for Ek, we obtain an isometry representation,
E → V : H ⊗ H → H ⊗ H ⊗ H , k N k Nk N k Nk B V = I ⊗ V˜ , N k N →N B † s.t. TrB V · V = Ek(·). (6.40)
(Note: we drop the k in Nk and continue by using just N to label the neighborhood system.) With respect to the isometry representation, Ek(ρ) = |ψihψ| becomes
† |ψihψ|N N = TrB V ρV † = TrAB (IA ⊗ V )|φihφ|(IA ⊗ V ) . (6.41)
Hence, (IA ⊗ V )|φi is some pure state, which, upon tracing out AB, leaves the pure state |ψi. Therefore,
(IA ⊗ V )|φi = |λiAB|ψiN N , (6.42)
where |λiAB is some pure state on ancillary systems HA ⊗ HB.
The invariance condition, Ek(|ψihψ|) = |ψihψ|, constrains the form of V . Written in terms of V , invariance requires Tr (I ⊗ V˜ )|ψihψ|(I ⊗ V˜ )† = |ψihψ|. (6.43) B N k N →N B N k N →N B
Hence, (I ⊗ V˜ )|ψi is some pure state, which, upon tracing out B, leaves the N k N →N B pure state |ψi. Therefore, (I ⊗ V˜ )|ψi = |0i ⊗ |ψi, where |0i is some pure N k N →N B B B ˜ state on B. This equation ensures that VN →N B acts trivially on ΣN (|ψi), outputting ˜ ⊥ ⊥ |0i on B, when doing so. The action of VN →N B on ΣN (|ψi) , which we denote VN →N B ˜ is unconstrained as of yet. In summary, invariance ensures that VN →N B acts trivially on ΣN (|ψi), giving the decomposition
˜ ⊥ VN →N B = ΠN ⊗ |0iB + VN →N B, (6.44)
166 6.8 Proofs
⊥ ⊥ where ΠN is the projector onto ΣN (|ψi) and VN →N B satisfies VN →N BΠN = 0. ⊥ Trace-preservation of Ek constrains VN →N B. In terms of V , trace-preservation † † requires I = V V (= Ek(I)). Evaluating this in terms of the decomposition of Eq. (6.44), we have
⊥ IN N = [ΠN + (ΠN ⊗ h0|B)VN →N B + h.c. ⊥ † ⊥ + (VN →N B) VN →N B] ⊗ IN . (6.45)
The non-trivial part of this equation is on system N , where the equation may be block decomposed as
⊥ I 0 ΠN (ΠN ⊗ h0|B)V = ⊥ † ⊥ † ⊥ , (6.46) 0 I ((ΠN ⊗ h0|B)V ) (V ) V
⊥ showing that (ΠN ⊗ h0|B)VN →N B = 0. With these conditions on V we return to Eq. (6.42),
|λiAB|ψiNN = IA ⊗ V |φi
= (IA ⊗ IN ⊗ ΠN )|φi ⊗ |0iB (6.47) ⊥ + (IA ⊗ IN ⊗ VN →N B)|φi. (6.48)
Decompose this equation into three parts according to:
H' supp(IAN ⊗ ΠN ⊗ |0ih0|B)
⊕ supp(IAN ⊗ ΠN ⊗ (I − |0ih0|B))
⊕ supp(IAN ⊗ (I − ΠN ) ⊗ IB). (6.49)
The vector |λiAB|ψiNN lies entirely in the first two blocks. Later, we will need to use the fact that IA ⊗ (I − |0ih0|)B|λiAB 6= 0. This fact follows from |λiAB|ψiNN having a non-trivial part in the second block, which we now to follows from the assumption ρ∈ / span(|ψihψ|).
Towards reductio ad absurdum, assume that the norm-1 vector |λiAB|ψiNN lies 0 completely in the first block. Define |λ iA ≡ (IA ⊗ h0|B)|λiAB. Then, |λiAB|ψiNN = 0 0 (IA ⊗|0ih0|B ⊗IN ⊗ΠN )|λiAB|ψiNN = |λ iA|0iB|ψiNN , where k|λ ik = 1. Projecting the right-hand side of Eq. (6.47) into the first block,
0 |λ iA|ψiNN |0iB = (IAN ⊗ ΠN )|φiANN |0iB ⊥ + IAN ⊗ [(ΠN ⊗ h0|B)VN →N B]|φiANN |0iB. (6.50)
⊥ The last term is zero, as (ΠN ⊗h0|B)VN →N B = 0. Then, removing the common factor
167 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
of |0iB from the remaining terms, we have
0 |λ iA|ψiNN = (IAN ⊗ ΠN )|φi. (6.51)
0 The vector |λ iA|ψiNN is assumed to be norm-1, and hence (IAN ⊗ ΠN )|φi is as well. Since ΠN is a projector, k(IAN ⊗ ΠN )|φik = k|φik only if (IAN ⊗ ΠN )|φi = 0 |φi = |λ iA|ψiNN . Tracing out system A, this last equation becomes |ψihψ| = TrA (|φihφ|) = ρ. As this violates the assumption that ρ 6= span(|ψihψ|), it must be that |λiAB|ψiNN lies at least partly in the second block. From this, 0 6= (IANN ⊗ ⊥ ˆ (I − |0ih0|)B)|λiAB|ψiNN ≡ |λ iAB|ψiNN . Defining the matrix V ≡ ΠN ⊗ (IB − ⊥ |0ih0|B)VN →N B, the second block equation reads ⊥ ˆ |λ iAB|ψiNN = (IA ⊗ V ⊗ IN )|φi. (6.52)
Towards bounding the Schmidt rank of |ψi, it is useful to transform this vec- tor equation into a matrix equation by applying partial-transpose to the composite ⊥ Hilbert space HA ⊗ HN . Hence, the vectors |λ iAB, |ψiNN , and |φiANN are trans- formed into matrices:
⊥ λ : HA → HB
ψ : HN → HN
φ : HAN → HN (6.53)
Note that rank(ψ) = dim(ΣN (|ψi)). Eq. (6.52) is tranformed into the matrix equa- tion
λ⊥ ⊗ ψ = Vˆ φ. (6.54)
It follows that rank(λ⊥ ⊗ ψ) = rank(Vˆ φ). On the left hand side,
⊥ dim(ΣN (|ψi)) = rank(ψ) ≤ rank(λ ⊗ ψ), (6.55)
⊥ using the fact that |λ iAB 6= 0, as shown earlier. On the right hand side,
ˆ ˆ ⊥ rank(V φ) ≤ rank(V ) ≤ dim(ΣN (|ψi) ), (6.56)
ˆ ⊥ where the last inequality follows from ker(V ) ≥ ker(VN →N B) ≥ ΣN (|ψi). The ⊥ above two inequalities together imply dim(ΣN (|ψi)) ≤ dim(ΣN (|ψi) ). With H' ⊥ ΣN (|ψi) ⊕ ΣN (|ψi) , we obtain dim(H ) N ≥ 2. (6.57) dim(ΣN (|ψi))
168 6.8 Proofs
• Unitary generation property.– We develop a few results which build up to a proof of Prop. 6.3.4. The following is a repurposing of Prop. 1.2.2 in [200], specified to our own setting. N Q Corollary 6.8.1. Consider a quantum system H' i Hi of dimension D = i di,
a neighborhood structure N , and a quantum state |ψi ∈ H. Let UNk,|ψi be the neigh- borhood stabilizer groups of |ψi. Then, for any element U ∈ hUN ,|ψiik, there exists a 2 2 sequence of (D−1) elements Uj, drawn from the UNk,|ψi, such that U = U1 ...U(D−1) . Also, the group hUN ,|ψiik is connected. Lemma 6.8.2. Let a Lie subgroup H of a Lie group G be generated by connected Lie subgroups Hα, α ∈ A for set A. Then the Lie algebra h of H is generated by the Lie algebras hα of Hαs. ˆ ˆ Proof. Let h = hhαiα ⊆ h. Let H ⊆ H be the corresponding connected Lie subgroup. To show that two Lie algebras are equal, hˆ = h, it suffices to show that their Lie ˆ ˆ groups are equal, H = H. Thus, it remains to show that H ⊇ H. Since each Hα ˆ is connected, each is the exponential of its Lie algebra hα. Hence, h ⊇ hα for all α ˆ implies H ⊇ Hα for all α. H is the smallest Lie subgroup of G containing all Hα. ˆ ˆ ˆ ˆ Thus, H being a group and H ⊇ Hα for all α implies H ⊇ H. Finally, since H = H, we have hˆ = h.
Proposition 6.8.3 (Equality for stabilizer groups/algebras). huNk,|ψiik = u|ψi
if and only if hUNk,|ψiik = U|ψi. Proof. (⇐) Equal Lie groups have equal Lie algebras.
(⇒) Assume huNk,|ψiik = u|ψi. Let u˜ be the Lie algebra of the Lie group hUNk,|ψiik.
U|ψi and the UNk,|ψi are each connected Lie subgroups of U(H). Therefore, they are each equal to the exponential of their respective Lie algebras. Furthermore, Coro.
6.8.1 ensures that hUNk,|ψiik is connected due to the connectedness of the UNk,|ψi.
Hence, hUNk,|ψiik and U|ψi being connected implies that if hUNk,|ψiik = U|ψi, then
u˜ = u|ψi. The UNk,|ψi being connected ensures that, by Lemma 6.8.2, the Lie algebra
u˜ is generated by the Lie algebras uNk,|ψi. Thus, we have huNk,|ψiik = u|ψi. Proposition 6.3.4 (Unitary generation property) Given a state |ψi and a neigh- borhood structure N , any element in U|ψi can be decomposed into a finite product of
elements in UNk,|ψi if and only if
huNk,|ψiik = u|ψi, (6.58)
where h·ik denotes the smallest Lie algebra which contains all Lie algebras from the set indexed by k.
169 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
Proof. (⇐) First, we show that huNk,|ψiik = u|ψi implies the decomposition of elements in U|ψi. By Prop. 6.8.3, the Lie algebra generation implies the Lie group generation. Corollary 6.8.1, then, implies that the Lie group generation ensures that elements of
U|ψi decompose into finite products of elements of the UNk,|ψi.
(⇒) huNk,|ψiik ≤ u|ψi is true by construction. We show that huNk,|ψiik ≥ u|ψi under the decomposition assumption. Consider an arbitrary element X ∈ u|ψi. Then, by assumption, exp(X) = U ∈ U|ψi admits a decomposition, U = UT ...U1, with each Ui in a UNk,|ψi. As the UNk,|ψi are connected, each element in UNk,|ψi is the exponentiation of an element in uNk,|ψi: Ui = exp(Xi). Hence, U = exp(XT ) ... exp(X1). Iterating the Baker-Campbell-Hausdorff formula, we can write U = exp(Y ) for some element
Y ∈ huNk,|ψiik. Then, since U = exp(X) = exp(Y ), X and Y are proportional to one
another, ensuring λY = X ∈ huNk,|ψiik, for λ ∈ R.
• Sufficient conditions for FTS.– Theorem 6.3.6. A state |ψi is FTS relative to a connected neighborhood structure N if there exists at least one neighborhood for which
dim(HNk ) ≥ 2 · dim(ΣNk (|ψi)), (6.59)
and
huNk,|ψiik = u|ψi. (6.60)
Proof. We construct a finite sequence of CPTP maps which is ensured to stabilize the target state under the conditions given in the theorem. The Schmidt span dimension condition ensures that there exists some neighbor- ⊥ hood for which dim(ΣNk (|ψi)) ≤ dim(ΣNk (|ψi) ). Then, we can choose any subspace Σ1 ≤ dim(Σ (|ψi)⊥), such that dim(Σ1 ) = dim(Σ (|ψi)). We think of Σ1 as Nk Nk Nk Nk Nk ⊥ being a “copy” of ΣNk (|ψi) lying inside ΣNk (|ψi) . For convenience, we drop the 0 0 1 index k, and define ΣN ≡ ΣNk (|ψi). Then, HN = ΣN ⊕ ΣN ⊕ R, where R is the 0 1 remaining subspace of HN . Choosing an identification between ΣN and ΣN , we can write 0 1 2 ΣN ⊕ ΣN ' C ⊗ ΣN , (6.61) such that
0 ΣN ' |0i ⊗ ΣN , (6.62) 1 ΣN ' |1i ⊗ ΣN . (6.63)
The first map that we define to be used in our FTS sequence is
EN ≡ (|0ih0|Tr ⊗ I) ⊕ IR, (6.64)
170 6.8 Proofs
2 with respect to the decomposition HN = (C ⊗ ΣN ) ⊕ R. The corresponding global map is defined as W ≡ EN ⊗ IN . The global Hilbert space decomposes as
2 H'HN ⊗ HN = (C ⊗ ΣN ⊗ HN ) ⊕ (R ⊗ HN ), (6.65)
whereby the target state can be written as (|0i ⊗ |ψ˜i) ⊕ 0. From this decomposition and the definition of W, we can see that W(|ψihψ|) = |ψihψ|, and hence W satisfies the invariance condition. Furthermore, the only state orthogonal to |ψi whose density operator is mapped to |ψihψ| is |ψ0i ≡ (|1i ⊗ |ψ˜i) ⊕ 0. Hence, we can interpret W as correcting an arbitrary error U acting on the C2 qubit of |ψi. In brief, the main strategy we employ towards stabilizing |ψi is to iterate the following procedure: 1) apply a sequence of invariance-satisfying, neighborhood unitaries to map a state |αi to |ψ0i, 2) apply W to map |ψ0i to |ψi.
As mentioned, the remaining CPTP maps that we define for the FTS sequence are unitaries. For each vector in |αi ∈ ker(hψ|), we define
0 0 Uα ≡ |ψihψ| ⊕ (|ψ ihα| + |αihψ |) ⊕ I. (6.66)
0 0 This unitary has non-trivial action only on span{|αi, |ψ i}, acting as Uα|αi = |ψ i. Thus, we can see that the composition of Uα and W gives a map which takes |αi to † the target state |ψi. We label the corresponding CPTP map with Uα(·) ≡ Uα · Uα.
In Prop. 6.3.4, we show that the assumed property huNk,|ψiik = u|ψi ensures that any U ∈ U|ψi can be decomposed into a finite product of invariance-satisfying, neighborhood unitaries. Since any Uα, with hψ|αi = 0 is in U|ψi, such Uα can be composed from a finite sequence of |ψi-preserving neighborhood maps.
Finally, we construct the sequence of CPTP maps which renders |ψi FTS. Let {|αi} label an orthonormal basis set for ker(hψ|) with the following convention for the ordering:
|00i = |ψ0i 0 0 span{|1 i,..., |dki} = (|0i ⊗ ΣN (|ψi) ⊗ HN ) span(|ψi) 0 0 span{|(dk + 1) i,..., |T i} = R ⊗ HN . (6.67)
Then, we define the FTS sequence of CPTP as
E ≡ W ◦ UT 0 ◦ ... ◦ W ◦ U10 ◦ W ◦ U00 . (6.68)
The individual neighborhood maps manifestly satisfy invariance. Thus, it remains to show that E = |ψihψ|Tr. It suffices to show that E(I) = d|ψihψ|, where d is the global
171 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
Hilbert space dimension. In the first step,
WU00 (I) = W(I) = W([(|0ih0| + |1ih1|) ⊗ I] ⊕ 0 + (0 ⊗ 0) ⊕ I) = 2(|0ih0| ⊗ I) ⊕ 0 + (0 ⊗ 0) ⊕ I
= 2ΠN + ΠR, (6.69)
2 where the decomposition in the second and third lines is H' [C ⊗ (ΣN ⊗ HN )] ⊕ [R ⊗ HN ] and ΠR is the projector onto R ⊗ HN . In the next step,
WU10 WU00 (I) = W(2U10 (ΠN )) + U10 (ΠR)) 0 0 = W(2[U10 (ΠN − |1 ih1 |) 0 0 + U10 (|1 ih1 |)] + ΠR) 0 0 0 0 = W(2[(ΠN − |1 ih1 |) + |ψ ihψ | + ΠR) 0 0 = 2W(ΠN − |1 ih1 |) 0 0 + 2W(|ψ ihψ |) + W(ΠR) 0 0 = 2(ΠN − |1 ih1 |) + 2|ψihψ| + ΠR. (6.70)
The key property used above is the fact that all operators above have support in
(|0i ⊗ ΣN (|ψi) ⊕ R) ⊗ HN , where W acts trivially. Similarly, in the next step we have,
0 0 0 0 WU20 WU10 WU00 (I) = 2(ΠN − |1 ih1 | − |2 ih2 |)
+ 4|ψihψ| + ΠR. (6.71)
0 Continuing until |dki, we obtain
W ◦ U 0 ◦ ... ◦ W ◦ U 0 ◦ W ◦ U 0 (I) = 2d |ψihψ| + Π , (6.72) dk 1 0 k R where, again, dk is the dimension of ΣN (|ψi). At this point, we continue the sequence 0 with the unitaries transferring vectors from R ⊗ HN to |ψ i,
0 WU(dk+1) (2dk|ψihψ| + ΠR) = [dk + (dk + 1)]|ψihψ| 0 0 + ΠR − |(dk + 1) ih(dk + 1) |. (6.73)
Continuing in this way, the sequence terminates at
0 0 WUT WU(dk+1) (2dk|ψihψ| + ΠR) = (dk + T )|ψihψ| = d|ψihψ|. (6.74)
172 6.8 Proofs
Thus, we have shown that the following sequence renders |ψi FTS,
T Y (W ◦ Ui0 ) = |ψihψ|Tr. (6.75) i=0
Proposition 6.8.4. If |ψi satisfies huNk,|ψiik = u|ψi with respect to neighborhood structure N , then |ψi satisfies Eq. (6.5) with respect to N . T Proof. Assume that |ψi does not satisfy Eq. (6.5). Then k ΣNk (|ψi) = S >
span(|ψi). We show that huNk,|ψiik ≤ uS , where uS is the Lie algebra associated to the Lie group that stabilizes S, US . The definining property of US is that for all 0 † 0 0 U ∈ US , U|sihs |U = |sihs | for all |si, |s i ∈ S. Then, the defining property of the 0 0 corresponding Lie algebra is that for all X ∈ uS , [X, |sihs |] = 0 for all |si, |s i ∈ S.
Consider an arbitrary neighborhood Nk and an element Y ∈ uNk,|ψi. By definition, 0 0 Y satisfies [Y, |rihr |] = 0 for all |ri, |r i ∈ ΣNk (|ψi). Since S ≤ ΣNk (|ψi), we have 0 0 [Y, |sihs |] = 0 for all |si, |s i ∈ S. Thus, Y ∈ uS . As this inclusion holds for all
elements in uNk,|ψi for any k, and uS is closed with respect to linear combination and
Lie product, then any element in huNk,|ψiik is contained in uS . Since S > span(|ψi),
we have uS < u|ψi. Finally, since huNk,|ψiik ≤ uS , it follows that huNk,|ψiik < u|ψi, and in particular, |ψi does not satisfy
huNk,|ψiik 6= u|ψi.
• Necessary conditions for RFTS.– Proposition 6.4.3 (Finite correlation length) A quantum state ρ is RFTS with respect to N only if the following is satisfied: for any two subsystems A and B with N N disjoint neighborhood expansions (i.e. A ∩ B = ∅), arbitrary observables XA and YB are uncorrelated, that is, Tr (XAYBρ) = Tr (XAρ) Tr (YBρ). Proof. Assuming ρ is RFTS with respect to N , there exists a sequence of neigh- borhood maps such that ρTr (·) = ET ... E1(·). Let EAN be the composition of all such maps which act non-trivially on A, and similarly for EBN with B. By assumption, EAN and EBN act disjointly. Let Erest be the composition of the re- maining maps. By the robustness assumption, we may reorder the maps to write
ρ = ErestEAN EBN (σAN ⊗ τBN ⊗ ωAN BN ), with arbitrary input density operators. Let XA and YB be arbitrary observables acting on A and B. We have Tr (XAYBρ) = † Tr XAYBErestEAN EBN (σAN ⊗ τBN ⊗ ωAN BN ) . Since, Erest is unital and both XA
173 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
and YB are trivial where the map acts, the expression simplifies to Tr (XAYBρ) = Tr XAE AN (σAN ) ⊗ YBE BN (τBN ) , where E AN , E BN are defined to act on their re-
spective systems and we have traced out ω N N . The trace can be separated as A B Tr (XAYBρ) = Tr XAE AN (σAN ) Tr YBE BN (τBN ) . For the remaining steps, we first note that Tr XAE AN (σAN ) = Tr XAE AN (σAN ) ⊗ E BN (τBN ) ⊗ ωAN BN , (6.76) and Tr YBE BN (τBN ) = Tr E AN (σAN ) ⊗ YBE BN (τBN ) ⊗ ωAN BN . (6.77)
Finally, with Erest being trace-preserving, we may re-insert it into the trace to obtain
Tr (XAYBρ) = Tr XAE AN (σAN ) Tr YBE BN (τBN ) = Tr Erest[XAE AN (σAN ) ⊗ E BN (τBN ) ⊗ ωAN BN ] × Tr Erest[E AN (σAN ) ⊗ YBE BN (τBN ) ⊗ ωAN BN ] = Tr XAErestEAN EBN (σAN ⊗ τBN ⊗ ωAN BN ) × Tr YBErestEAN EBN (σAN ⊗ τBN ⊗ ωAN BN ) = Tr (XAρ) Tr (YBρ) . (6.78)
In the second to last step we have used the fact that Erest acts trivially on XA and YB.
Lemma 6.4.4 (Recoverability property) Let target pure state ρ = |ψihψ| be ˜ RFTS with respect to N . If a map M acts on a subsystem A, M ≡ MA ⊗ IA, then N there exists a sequence of CPTP neighborhood maps EAN each acting only on A , such that ρ = El ◦ ... ◦ E1 ◦ M(ρ).
0 Proof. Let Ek be the sequence of neighborhood maps which renders ρ RFTS. Define Q 0 the subsequence of maps E N ≡ E . Let E be the product of the remaining A Nk∩A6=∅ k R 0 Ek. We have that ER ◦ EAN (σ) = ρ for any density operator σ. We show that EAN
174 6.8 Proofs
acting on the transformed target state, M(ρ), recovers ρ:
EAN ◦ M(ρ) = EAN ◦ M ◦ ER ◦ EAN (σ)
= EAN ◦ ER ◦ M ◦ EAN (σ) 0 = EAN ◦ ER(ρ ) = ρ,
0 where σ is an arbitrary density operator and ρ = M ◦ EAN (σ).
Proposition 6.4.5 (Zero CMI) A quantum state ρ is RFTS with respect to N only if the following is satisfied: for any two regions A and B, with AN ∩ B = ∅, N I(A : B|C)ρ = 0, where C ≡ A \A.
Proof. To prove this result, we specify Lemma 6.4.4 to the case where M = (τATrA)⊗ AN IA, with τA the completely mixed state on A. Lemma 6.4.4 gives E M(ρ) = AN N (τATrA) ⊗ IA[ρ] = E (τA ⊗ ρA). Then, using the fact that A ∩ B = ∅, we trace N out all but A and B (i.e. all but systems ABC) to obtain ρABC = TrABC (ρ) = AN AN TrABC E (τA ⊗ ρA) = E (τA ⊗ ρBC ). Since ρ is written as a short quantum
Markov chain, we have I(A : B|C)ρ = 0.
We prove a necessary condition for RFTS regarding the commutativity of the canonical FF Hamiltonian projectors. Towards this, we prove a lemma which con- strains the form of stabilizing CPTP maps.
Lemma 6.8.5. If Ek acting on neighborhood Nk preserves |ψi, then, for arbitrary ρ, Ek satisfies ΠkEk(ρ)Πk = ΠkρΠk + ΠkσΠk, (6.79) ⊥ ⊥ where Πk is the orthogonal projector onto ΣNk (|ψi) and σ = Ek(Πk ρΠk ) ≥ 0.
Proof. If Ek is to preserve |ψi then the Kraus operators of Ek must act trivially on supp(Tr [|ψihψ|]). This requires the form, Nk
⊥ Ki = λiΠk + RiΠk . (6.80)
Trace preservation of Ek requires that
X 2 ∗ ⊥ ⊥ † ⊥ † ⊥ I = |λi| Πk + λi ΠkRiΠk + λiΠk Ri Πk + Πk Ri RiΠk . (6.81) i
175 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
P 2 ⊥ P † ⊥ ⊥ From this, it follows that i |λi| = 1, Πk i Ri RiΠk = Πk , and, most importantly,
X ∗ ⊥ Πk( λi Ri)Πk = 0. (6.82) i
Finally, applying these trace-preserving conditions to ΠkEk(ρ)Πk, we find
X 2 ⊥ † ΠkEk(ρ)Πk = Πk( |λi| ΠkρΠk + λiΠkρΠk Ri i ⊥ ⊥ † + h.c. + RiΠk ρΠk Ri )Πk ⊥ X † = ΠkρΠk + Πkρ(Πk λiRi Πk) i X ⊥ ⊥ † + h.c. + ΠkRiΠk ρΠk Ri Πk i
= ΠkρΠk + ΠkσΠk.
We will also make use of the following trace inequality:
Proposition 6.8.6 (Non-commuting penalty). Let Π1 and Π2 be projectors, with Π1∩2 the projector onto their intersection. Then 1 Tr (Π Π ) ≥ Tr (Π ) + Tr |[Π , Π ]|2 . (6.83) 1 2 1∩2 2 1 2 Proof. First, note that 1 1 |[Π , Π ]|2 = (Π Π Π + Π Π Π − (Π Π )2 − (Π Π )2). (6.84) 2 1 2 2 1 2 1 2 1 2 1 2 2 1 Taking the trace of both sides and rearranging terms, we have 1 Tr (Π Π ) = Tr (Π Π )2 + Tr |[Π , Π ]|2 . (6.85) 1 2 1 2 2 1 2
2 Note that, under conjugation, Π1∩2(Π1Π2) Π1∩2 = Π1∩2. Using that trace is non- 2 increasing under conjugation with respect to a projector, we obtain Tr ((Π1Π2) ) ≥ 2 Tr (Π1∩2(Π1Π2) Π1∩2) = Tr (Π1∩2). Making this replacement, we obtain the desired result. Combining the two results above, we obtain a necessary condition for robust sta- bilization. Proposition 6.4.6 (Commuting FF Hamiltonian) If |ψi is robust quasi-locally
176 6.8 Proofs
stabilizable with respect to neighborhood structure N , then [Πk, Πk] = 0 for all neigh-
borhoods Nk, where Πk and Πk are the projectors onto ΣNk (|ψi) and ∩j6=kΣNj (|ψi), respectively.
Proof. Assume |ψi is robust stabilizable with respect to the sequence of neighborhood
maps ET ◦ ... E1. Let Ek be the neighborhood map on Nk and El be the composition of the remaining neighborhood maps. Robust stabilizability, implies Ek ◦ Ek(·) =
|ψihψ|Tr (·). The invariance condition requires Ej(X) = X for any X ∈ ΣNj (|ψihψ|).
Since Πk ∈ ΣNj (|ψihψ|) for all j 6= k, each Ej with j 6= k must fix Πk. Hence, we have Y Ek(Πk) = ( Ej)(Πk) = Πk. (6.86) j6=k
Thus, applying the full sequence of CPTP maps to Πk, we have
|ψihψ|Tr(Πk) = Ek ◦ Ek(Πk) (6.87)
= Ek(Πk). (6.88)
Conjugating both sides of the equation with respect to Πk, we can apply Lemma 6.8.5 to obtain
|ψihψ|Tr(Πk) = ΠkΠkΠk + ΠkσΠk, (6.89)
where σ is some positive-semidefinite operator. Next, conjugating both sides of the ˜ equation with respect to the projector Πk ≡ Πk −|ψihψ| kills the left hand side, while leaving the sum of two positive semidefinite operators on the right hand side, ˜ ˜ ˜ ˜ 0 = ΠkΠkΠkΠkΠk + ΠkΠkσΠkΠk. (6.90)
The sum of two positive-semidefinite matrices is zero only if both matrices are zero. Taking the trace of the first zero matrix gives ˜ 0 = Tr(ΠkΠkΠk) (6.91)
= Tr((Πk − |ψihψ|)Πk) (6.92)
= Tr(ΠkΠk) − Tr(Πk∩k) (6.93) 2 ≥ Tr(|[Πk, Πk]| ). (6.94)
This only holds if [Πk, Πk] = 0. As the above arguments are made for an arbitrary neighborhood Nk, they must hold for all neighborhoods. Thus, we arrive at the stated consequence.
177 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
• Non-operational sufficient conditions for RFTS.– Theorem 6.5.5 (Neighborhood factorization on local restriction) A state NN |ψi described on the coarse-grained subsystems H' i=1 Hi with respect to the neighborhood structure N is robustly finite time stabilizable (RFTS) if there exists ˜ NN ˜ 0 ˜⊥ a locally restricted space H' i=1 Hi, with a complement H ' H , whereby a ˜ NM ˆ factorization H = j=1 Hj gives
M M O ˆ 0 O ˆ |ψi = 0 ⊕ |ψji ∈ H ⊕ Hj (6.95) j=1 j=1
ˆ and for each virtual subsystem Hj there exists a neighborhood Nk such that
I0 ⊕ B(Hˆ ) ⊗ I ≤ B(H ) ⊗ I . (6.96) j j Nk Nk
Proof. Assume the conditions above hold. We construct a finite sequence of com- muting QL CPTP maps which robustly stabilize ρ. First, we construct the maps 0 which prepare the locally restricted space. Define the map Ei : B(Hi) → B(Hi) to 0 † Pi ˜ be E (·) = Pi · P + Tr ((I − Pi)·), where Pi is the projector onto Hi. For each i i Tr(Pi) neighborhood N we construct a map E 0 ≡ N E 0, which prepares support on the k k i∈Nk i locally restricted space of all coarse-grained subsystems contained in that neighbor- hood. ˆ 0 ˆ ˆ 0 ˆ ˆ On the virtual systems, define the maps Ej : B(H ⊕Hj ⊗Hj) → B(H ⊕Hj ⊗Hj) as ˆ 0 Ej(·) = I ⊕ (ˆρjTr)j ⊗ Ij. (6.97)
Each virtual subsystem labeled j is associated to a neighborhood Nk on which its operators act non-trivially. Correspondingly, each neighborhood-acting map Ej is ˆ 0 constructed from Ej by pre-composing it with Ek ,
ˆ 0 Ej ≡ Ej ◦ Ek (·). (6.98)
ˆ 0 ˆ 0 ˆ The Kraus operators of Ej are contained in I ⊕B(Hj)⊗Ij. Hence, by I ⊕B(Hj)⊗Ij ≤ B(H ) ⊗ I , we have that the Kraus operators of E act non-trivially only on Nk Nk j neighborhood Nk. Thus, each map Ej is a valid neighborhood map. Finally, we must show that an arbitrary sequence of these maps prepares ρ while leaving it invariant. ˆ 0 ˆ NM NM For invariance, we have Ej(ρ) = EjEk (ρ) = Ej(0 ⊕ j=1 ρˆj) = 0 ⊕ j=1 ρˆj = ρ. 0 0 To demsonstrate preparation of ρ, we use the fact that Ei Ej = EjEi . Consider an
178 6.8 Proofs
arbitrary complete sequence of the neighborhood maps,
ˆ 0 EM ◦ ... ◦ E1 = (EM ◦ EM ) ◦ EM−1 ◦ ... ◦ E2 ◦ E1 ˆ 0 = EM ◦ EM−1 ◦ ... ◦ E2 ◦ E1 ◦ EM
We continue in this way, using the commutativity of the support projections with the Ej to move all of the support projections to act first. Since every coarse-grained particle will have been accounted for, we may combine the action of all of these 0 ` ˜ projections Ek into a single projection E which has the effect of projecting onto H, ˆ ˆ 0 0 EM ◦ ... ◦ E1 = (EM ◦ ... ◦ E1) ◦ (EM ◦ ... ◦ E1 ) ˆ ˆ ` = EM ◦ ... ◦ E1 ◦ E . (6.99)
Finally, we see that the composition of these maps constitutes a preparation of the target state,
M 0 O ` EM ◦ ... ◦ E1 = I ⊕ (ˆρjTr) ◦ E j=1 M O = (0 ⊕ ρˆj)Tr j=1 = ρTr. (6.100)
• Algebraic sufficiency for RFTS.–
Theorem 6.5.7 (Algebraically induced factorization) If a set of algebras {Aj}, Ai ∈ B(H), is complete and commuting, then each Aj has a trivial center and there NT ˆ ˆ exists a decomposition of the Hilbert space H' j=1 Hj for which Aj 'B(Hj) ⊗ Ij for each j.
Proof. First, assume that a neighborhood algebra Aj were reducible, so that there 0 exists some X ∈ Aj where X ∈ Aj, but X 6= c · I. As X ∈ Aj, it commutes with all elements of Ak for k 6= j, and hence the algebra generated by all the neighborhood algebras has a non-trivial commutant, which violates completeness. We obtain the Hilbert space factorization as follows. First, for any algebra Aj with trivial center ˆ ˆ acting on H, there exists a decomposition H' Hj ⊗ Hj for which Aj = B(Hj) ⊗ Ij. ˆ 0 ˆ Starting with A1, we have H' H1 ⊗ H1. From this, A1 = I1 ⊗ B(H1). As the 0 ˆ algebras are all commuting, A2 ≤ A1 = I1 ⊗ B(H1). Hence, A2 carries a natural action on H1, and A2 having a trivial center implies that there is a decomposition
179 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
ˆ ˆ ˆ H1 = H2 ⊗H1,2 for which A2 = I1 ⊗B(H2)⊗I1,2. So far we have H' H1 ⊗H2 ⊗H1,2. With the introduction of each additional algebra, we obtain another factor in the Hilbert space. Continuing in this way, completeness of the set of Aj ensures that once all Aj have been included, the global Hilbert space will have been decomposed N ˆ as H' j Hj. The following two lemmas will be used to formulate the decomposition of Prop. 6.8.9. This proposition will then be used for proving Prop. 6.5.10. N Lemma 6.8.7. Consider a Hilbert space H' i Hi, a neighborhood Nk containing Hp, and a state |ψi ∈ H. Then, the subsystem kernel of |ψihψ| on p coincides with the subsystem kernel of the neighborhood projector Πk on p.
Proof. We show this by direct calculation. With p ∈ Nk,
ker(Trp (|ψihψ|)) = ker(Trp (ρNk )). (6.101) P Using the spectral decomposition ρNk = j λj|jihj| along with properties of the kernel function, X ker(Trp (|ψihψ|)) = ker( λjTrp (|jihj|)) j X = ker( Trp (|jihj|)) j = ker(Tr Tr Π˜ ⊗ I ) p N k k N k
= ker(Trp (Πk)). (6.102)
Lemma 6.8.8. Given a positive-semidefinite operator P acting on HA ⊗ HB, let PA = TrB (P ). Then, ker(PA) = ker(ΣA(P )).
Proof. The direction ker(PA) ⊇ ker(ΣA(P )) is trivial since PA ∈ ΣA(P ). For ker(PA) ⊆ ker(ΣA(P )), assume PA|vi = 0. Since PA ≥ 0, this is equivalent to Tr (|vihv|PA) = 0. 2 dB In terms of P then, we have Tr (|vihv| ⊗ IP ) = 0. Let {Ei}i=1 constitute an informa- P tionally complete POVM on HB (i.e. span{Ei} = B(HB)). Then i Tr (|vihv| ⊗ EiP ) = 0. Since each term must be non-negative, we have Tr (|vihv| ⊗ EiP ) = 0 for all √ √ i. We may rewrite this as hv|TrB (I ⊗ Ei)P (I ⊗ Ei) |vi = 0, which implies √ √ TrB (I ⊗ Ei)P (I ⊗ Ei) |vi = 0 for all i. Since the POVM is informationally complete, 2 span{TrB ((I ⊗ Ei)P ) |i = 1, . . . , dB} = ΣA(P ).
Thus, |vi ∈ ker(ΣA(P )).
180 6.8 Proofs
NN Proposition 6.8.9. Let i=1 Hi be a Hilbert space on which we define a neighborhood structure N . Let |ψi be any state of this system. For any neighborhood Nk containing a system Hp, consider the reduced state ρp = Tr (|ψihψ|) and the decomposition Hp ' L 0 supp(ρp) ⊕ ker(ρp). There exists a decomposition Hp ' ( l Hl ⊗ Hl) ⊕ ker(ρp) such that ! M p alg{Σp(Πk)} = B(H ) ⊗ I 0p ⊕ span{I}. (6.103) l H l l
Proof. The above decomposition is ensured as long as alg{Σp(Πk)} commutes with
all of Isupp(ρp) ⊕ B(ker(ρp)). We show that an arbitrary basis element in Isupp(ρp) ⊕ B(ker(ρp)) commutes with all elements in Σp(Πk). Consider the non-orthonormal basis {I, |αihβ|}, where |αi, |βi are basis elements of ker(ρp). We need only verify that elements |αihβ| commute with Σp(Πk), as I does trivially. Since p ∈ Nk, we may apply Lemma 6.8.7 to obtain that ker(ρp) = ker(Trp (Πk)). From Lemma 6.8.8 we have ker(Trp (Πk)) = ker(Σp(Πk)). Thus, |αi, |βi ∈ ker(Σp(Πk)), ensuring that 0 |αihβ| ∈ Σp(Πk) .
Note that, by Lem. 6.8.7, since supp(ρp) = supp(Trp (Πk)), the positive semidefinite operator Trp (Πk) ∈ alg(Σp(Πk)) has maximal rank in the space B(supp(ρp)) ⊕ 0. NN Theorem 6.5.10 A state |ψi described on the coarse-grained subsystems H' i=1 Hi N ˆ with respect to the neighborhood structure N admits a decomposition |ψi = 0⊕ j |ψji 0 N ˆ induced by the neighborhood algebra-induced factorization H'H ⊕ j Hj, and hence, is robustly finite time stabilizable (RFTS) with respect to N if 1) |ψi satisfies Eq. (6.5) with respect to N and 2) the neighborhood algebras Aj are commuting and complete on the local support space H˜. ˜ Proof. Completeness and commutativity of the Aj induce the decomposition H' N ˆ ˆ j Hj. The decomposition ensures that each Aj is of the form I0 ⊕ B(Hj) ⊗ Ij. Each Πk commutes with all elements in Aj for j 6= k. This can only be the case if Πk acts ˆ as identity on each factor Hj with j 6= k,
ˆ O Πk = 0 ⊕ Πk ⊗ I, (6.104) j6=k where we have used the fact that the Πj do not have support on the local kernel space H0. Thus, the Πk are mutually commuting with one another. This commutativity along with satisfaction of Eq. (6.5) ensures that
O ˆ Π1Π2 ... ΠT = 0 ⊕ Πj = |ψihψ|. (6.105) j
181 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics
ˆ The trace of the left hand side is the product of ranks of projectors Πj and is equal to ˆ ˆ ˆ the trace of |ψihψ|, which is 1. Hence, each projector satisfies Πj = |ψjihψj|. Thus,
O ˆ |ψi = 0 ⊕ |ψji. (6.106) j
With this factorization of |ψi, as well as the fact that I ⊕ B(Hˆ ) ⊗ I ≤ B(H ) ⊗ I 0 j j Nj N j for each j (by construction), Thm. 6.5.5 ensures the |ψi is RFTS with respect to N .
• Matching overlap condition for RFTS.– NN Proposition 6.5.12 (Matching overlap RFTS) Let p=1 Hp be a Hilbert space of coarse-grained particles with respect to a neighborhood structure N that satisfies the matching overlap condition. If |ψi satisfies Eq. (6.5) with respect to N and [Πj, Πk] = 0 for all pairs of neighborhood projectors, then |ψi is RFTS.
Proof. We obtain a decomposition of each Hp that constitutes a global change of basis leading to a neighborhood factorization as in Prop. 6.5.5 which implies RFTS. Consider an arbitrary coarse-grained particle p with Hilbert space Hp. The decom-
position of Hp is induced by the algebra alg{Σp(Πk)}Nk3p. By Prop. 6.8.9, each
alg{Σp(Πk)} is contained in B(supp(ρp)) ⊕ span{Iker(ρp)}. We show that, furthermore,
alg{Σp(Πk)}Nk3p = B(supp(ρp)) ⊕ span{Iker(ρp)}, by establishing that its center is
equal to span{I, Isupp(ρp) ⊕ 0}. ˜ Assuming otherwise, there exists an X = X ⊕ 0 ∈/ span{I, Isupp(ρp) ⊕ 0} such 0 that X ∈ Σp(Πk) for each Nk 3 p. Then, [Ip ⊗ Xp, Πk] = 0 for all neighborhoods Nk (including Nk / ∈ p). Since X acts non-trivially on supp(ρp), we have Ip ⊗Xp|ψi = |τi ∈/ span(|ψi). Since |ψi satisfies Eq. (6.5), it is the only vector for which Πk|ψi = |ψi for all neighborhoods Nk. However, for |τi,
Πk|τi = Πk(Ip ⊗ Xp)|ψi
= (Ip ⊗ Xp)Πk|ψi
= (Ip ⊗ Xp)|ψi = |τi,
which is a contradiction. Hence, no such X can exist, implying that the center of
alg{Σp(Πk)}Nk3p is equal to span{I, Isupp(ρp) ⊕ 0}. This fact, together with the fact
that alg{Σp(Πk)}Nk3p is contained in B(Hsupp(ρp)) ⊕ span{Iker(ρp)}, ensures that
alg{Σp(Πk)}Nk3p = B(Hsupp(ρp)) ⊕ span{Iker(ρp)}. (6.107)
As described, the matching overlap condition ensures that the intersection of any
182 6.8 Proofs
non-disjoint neighborhoods Nj and Nk is some coarse-grained particle p. Thus, from [Πj, Πk] = 0, we have [Σp(Πj), Σp(Πk)] = 0, abusing notation. Hence, for any two 0 neighborhoods Nj and Nk containing p, we have alg{Σp(Πj)} ≤ alg{Σp(Πk)} . The al- gebra alg{Σp(Πk)}Nk3p, then, is seen to be generated by a finite number of mutually commuting algebras. Given the form of this algebra in Eq. 6.107, these generat- ing subalgebras alg{Σ (Π )} can only mutually commute if H = N Hˆk, p k supp(ρp) k|Nk3p p whereby ˆk alg{Σp(Πk)} = (B(H ) ⊗ I ˆk ) ⊕ span{Iker(ρ )}, p Hp p for each neighborhood Nk 3 p. We have obtained a decomposition for each coarse-grained particle Hilbert space H ' (N Hˆk) ⊕ H . Hence, the global Hilbert space decomposes as p k|Nk3p p ker(ρp) O O O ˆk H' Hp ' ( Hp) ⊕ Hker(ρp)
p p k|Nk3p O O ˆk O O ˆk ' Hp ⊕ H0 ' Hp ⊕ H0
p k|Nk3p k p∈Nk O ˆk ≡ H ⊕ H0. (6.108) k
By the way this decomposition was formed, the Πk act trivially on all but one of the ˆ N ˆ virtual factors, Πk = 0 ⊕ Πk j6=k Ij. Hence, |ψi satisfying Eq. (6.5) implies that
O ˆ Π1Π2 ... ΠT = 0 ⊕ Πj = |ψihψ|. (6.109) j
Similar to the proof of Thm. 6.5.10, the trace of the left hand side is the product of ˆ ranks of projectors Πj and is equal to the trace of |ψihψ|, which is 1. Hence, each ˆ ˆ ˆ projector satisfies Πj = |ψjihψj|. Thus,
O ˆ |ψi = 0 ⊕ |ψji. (6.110) j
With this factorization of |ψi, as well as the fact that I ⊕ B(Hˆ ) ⊗ I ≤ B(H ) ⊗ I 0 j j Nj N j for each j, Thm. 6.5.5 ensures the |ψi is RFTS with respect to N .
• Efficiency of RFTS.– Proposition 6.6.2 (Lattice circuit size scaling) Consider a scalable lattice neigh- borhood structure N N (see discussion above Prop. 6.6.2 in Sec. 6.6.1). If |ψi is RFTS
183 Finite-time stabilization of quantum states with discrete-time quasi-local dynamics with respect to this neighborhood structure, then |ψi can be stabilized by a dissipative circuit of size |N 0|(N/c) and depth D = |N 0|diam(N 0)d.
Proof. For any robustly finite-time stabilizable state, the circuit size is equal to the number of neighborhoods. From the unit cell definition, the number of neighborhoods is |N 0|(N/c), the number of neighborhoods per unit cell times the number of unit cells. To bound the depth of the circuit, we devise a scheme which parallelizes the circuit to one with constant depth. In particular, we show that there exists a partitioning of the neighborhoods of N , and hence N N , into |N 0|diam(N 0)d parts such that each part consists of a set of mutually disjoint neighborhoods. If the union of the unit cell neighborhoods N 0 is translated in any direction a distance D = diam(N 0), the resulting set is disjoint from N 0. In particular, if we select a single neighborhood 0 Nk ∈ N and construct the set of neighborhoods generated by linear combinations of Deˆi for each i = 1, . . . , d, the neighborhoods in this set are ensured to be disjoint from one another. Hence, the sequence of the corresponding neighborhood maps act in parallel and constitute a layer of the circuit. This set of neighborhoods is generated by a subgroup (DZ)d = DZ × ... × DZ of the translation group Zd = Z × ... × Z. Therefore, the translated copies of Nk for which this did not account each correspond d d d to a coset of (DZ) in Z with respect to elements ~m = (m1, . . . , md) ∈ Z . This d coset group is isomorphic to the finite group ZD = ZD × ... × ZD. The size of this d d group is |ZD| = D . Using group action notation, we denote the ~m-translated version of N0 as ~mN0. Each layer of neighborhood maps corresponds to a set of disjoint neighborhoods, d 0 d ~m(DZ) Nk, k = 1,..., |N |, ~m ∈ ZD. (6.111) Each neighborhood is accounted for and there are |N 0|Dd layers. With this scheme, we define, Y E~m,k ≡ E(~v+~m)Nk , (6.112) ~v∈(DZ)d
0 d for k = 1,..., |N |, ~m ∈ ZD. The sequence of neighoborhood maps which prepares Q|N 0| Q the target state can be parallelized as ρTr = EN ... E1 = d E~m,k, of which k=1 ~m∈ZD there are |N 0|Dd parallelized maps.
Proposition 6.6.7 (Commuting Liouvillian contraction bound) Let {Lj} be a, possibly infinite, set of bounded norm Liouvillians each acting on a subsystem of uniformly-bound dimension. Assume that the spectral gaps λj are strictly bounded below by ν > 0. Then, there exists R > 0 such that for any subset S ⊆ {Lj} of P mutually commuting Liouvillians, defining L = S Lj, the contraction is bounded by η(eLt) ≤ |S|Re−νt. (6.113)
184 6.8 Proofs
Lt P Lj t Proof. From Theorem 6.6.6.a, commutativity implies η(e ) ≤ S η(e ). With ν < λj, Theorem 6.6.6.b ensures that, for each Lj ∈ {Lj}, there exists Rj > 0 such Lj t −νt that η(e ) ≤ Rje . In [182], it is shown that, for fixed ν, Rj is upper bounded by 2 dj a function of order dj , where dj is the dimension of the system on which Lj acts. Let D ≥ dj be the uniform subsystem dimension bound. Then, we can find a constant R 2 D2 dj and c such that for all j, we have R > cD > cdj > Rj. With this,
Lt X Lj t X −νt η(e ) ≤ η(e ) ≤ Rje S S X ≤ Re−νt = |S|Re−νt. (6.114) S
Proposition 6.6.9 (Rapid mixing for commuting RFTS) Consider a family of finite systems Hα indexed by α such that for each α there is a neighborhood structure α N with dim(H α ) < D. Furthermore, assume the number of neighborhoods scales Nk α α α α no more than polynomially in system size, Nk ∈ N and |N | ≤ b log(dim(H )), α for some constant b > 0. For each α, let ρα be RFTS with respect to N by a set of α commuting neighborhood maps {Ek }, where there exists some ν > 0 such that for any α eigenvalue λ ∈ eig(Ek ), λ = 1 or λ < 1 − ν. Then, there exists a family of bounded norm QL Liouvillians satisfying rapid mixing with respect to ρα.
α α α Proof. For each α, define the neighborhood-acting Liouvillian operators Lk ≡ Ek −I . α These Liouvillians have bounded norm and the gap λk of each Lk is ensured to satisfy α λk > ν > 0. Take {Lk }k,α as a set of Liouvillians, and define the sequence of subsets α α S = {Lk }k, indexed by α. Then, for each α, the global Liouvillian is
α X α L = Lk . (6.115) k For each α, this Liouvillian is a sum of commuting terms with D and ν satisfying the conditions in Prop. 6.6.7 with some finite prefactor R. Thus,
η(eLαt) ≤ |Sα|Re−νt = |N α|Re−νt ≤ Rb log(dimHα)e−νt. (6.116)
Setting c = Rb, δ = 1, and λ = ν verifies rapid mixing.
185
Chapter 7
Towards finite-time dissipative quasi-local quantum encoders
187 Towards finite-time dissipative quasi-local quantum encoders
7.1 Preliminaries
We explore the notion of a finite-time dissipative encoder. In particular, we investigate the possibility of quasi-local implementations, giving motivations, rigorous definitions, and a few illustrative examples.
The traditional subspace encoding approach embeds a Hilbert space HQ into a larger Hilbert space H via an isometry. For example, to encode any qubit state |ψi = α|0i+β|1i, we could append a number of ancillary systems and apply a unitary transformation taking (α|0i + β|1i)|00 ...i into the encoded state α|0i + β|1i ∈ HQ ≤ H. As emphasized in [201], this subspace approach is, in general, inadequate because it does not account for the possibility that the encoding system H might couple to other degrees of freedom H0 in the larger space H˜ = H ⊗ H0. This becomes an issue when, for example, H interacts with H0. As anticipated in [202], and established in [196, 21], the more general notion of an encoding is a subsystem encoding. Ticozzi and Viola [203] give a natural operational interpretation for the subsystem principle, which clarifies connections among various notions of encoding. They posit that a quantum code is a linear map which preserves the distinguishability of quantum states.
Definition 7.1.1. A linear map on Hermitian operators, Φ: B(HQ) → B(HP ), defines a 1-isometric encoding if for all ρ1, ρ2 ∈ D(HQ) and p ∈ [0, 1],
||pΦ(ρ1) − (1 − p)Φ(ρ2)||1 = ||pρ1 − (1 − p)ρ2||1.
As shown in [203], this property is sufficient to ensure that such an encoding is a subsystem encoding: any encoding map Φ: B(HQ) → B(HP ) induces a subsystem decomposition H'HS ⊗ HF ⊕ HR, where HS 'HQ and Φ(ρ) = ρ ⊗ τ ⊕ 0R for some fixed τ ∈ B(HF ). A quantum code, then, is defined as CQ ≡ Φ(D(HQ)) and its elements take the form ρ ⊗ τ ⊕ 0R for fixed τ with respect to the subsystem decomposition. In the above notion of an encoder, the input system is an abstract logical quantum system, while the output system is a multi-qubit system. In a realistic scenario, we may desire a description of how to implement the quantum information transfer from locally-accessible degrees of freedom into the quantum code. It is useful, then, to decompose the encoder Φ above into two steps. In the first step the abstract logical quantum information is encoded by ΦL into locally-accessible upload qubits, such as a number of spatially local spin-1/2 particles. In the second step, the quantum information of the data qubits is carried into the code by the map ΦP . The local encoding and the physical encoding compose to give Φ = ΦP ◦ ΦL. The most familiar implementation of such an encoding is a unitary encoder ΦP = † UP · UP . The to-be-encoded quantum information |ψi is initialized in the upload
188 7.1 Preliminaries
qubits, while the remaining part of the system is initialized in some fixed pure state |φi. Then, the encoder ΦP is a global unitary transformation designed to map |ψi⊗|φi into the encoded state |ψi. In general, the success of the encoding requires the remaining part of the system to be sufficiently well-prepared in the fixed state |φi. The unitary encoding is not necessarily robust to errors in the initialization of |φi. For example, if the remaining system is afflicted by an error X which transforms |φi to an orthogonal state, hφ|X|φi = hφ|φ0i = 0, then the subsequent encoded state is sure to be orthogonal to the intended encoded state. With a dissipative encoder, as opposed to a standard unitary encoder, we are afforded some degree of robustness with respect to the initialization of the remaining system. N Definition 7.1.2. Consider a Hilbert space H = Hi, a quantum code CQ ⊆ B(H), N i and upload qubits HS = i∈S Hi with D(HS) 1-isometric to CQ.A dissipative encoder is a CPTP map ΦD : B(H) → B(H) which acts as
† ΦD(ρ ⊗ σ) = Uρ ⊗ τ ⊕ 0RU , (7.1)
where σ is from some set W ⊂ D(HS), while U and τ are those associated to the subsystem decomposition of the code. The set W is a linear space and we refer to it as the basin of attraction for the encoder ΦD. In [204] such a dissipative encoder was implemented via continuous-time quasi- local Liouvillian dynamics. As we will describe, this encoding is robust with respect to a non-trivial basin of attraction. Furthermore, the dynamics are autonomous in that the control procedure is constant in time, requiring no switching by the controller. A drawback is that, even under the assumption of ideal conditions, the encoding is sure to possess non-zero error for any finite time. We explore the construction of a dissipative encoder in terms of a finite sequence of quasi-local CPTP maps. Although the dynamics are not autonomous, we show some examples where encoding may be achieved exactly in finite time. Comparing to the continuous-time toric code example of [204], the finite-time dissipative encoder we construct for the toric code admits the same non-trivial basin of attraction. In practice, any encoding, discrete or continuous, is only achieved approximately. However, it is valuable to analyze a scenario in which the deviations from the exact encoding are due solely to undesirable errors, as opposed to additionally, a finite waiting time (as in the continuous-time case). N Definition 7.1.3. Consider a Hilbert space H = Hi, a quantum code CQ ⊆ B(H), N i upload qubits HS = i∈S Hi with B(HS) 1-isometric to CQ, and a neighborhood structure N . A finite-time quasi-local dissipative encoder with basin of attraction
W ⊆ B(HS) is any product of neighborhood maps Φk such that
ΦT ◦ ... ◦ Φ1 = ΦP (7.2)
189 Towards finite-time dissipative quasi-local quantum encoders
† maps any state ρS ⊗ σS ∈ B(HS) ⊗ W into UρS ⊗ τ ⊕ 0RU where τ and U are the fixed state and unitary transformation associated to the subsystem decomposition of the code CQ. The utility of an encoding depends on its ability to prevent, recover from, or pro- tect against errors subjected to the system. We establish some terminology regarding quantum error correction. The degree of error prevention of a code can be classified according to the following [205].
Definition 7.1.4. With respect to error model E : B(H) → B(H), a given code CQ = Φ(B(HQ)) is
• fixed if E(ρ) = ρ acts as identity on CQ.
P j • noiseless if for any probability distribution {pj}, j pjE acts as a 1-isometry on CQ.
• preserved if E acts as a 1-isometry on CQ. • correctable if there exists a recovery map R : B(H) → B(H) such that the code is noiseless for R ◦ E.
• protectable if there exists P : B(H) → B(H) such that the code is noiseless for E ◦ P.
These definitions are presented in order of decreasing strictness. In [203] it is shown that correctable 1-isometric encodings are equivalent to protectable 1-isometric en- codings. The quantum codes that we consider fall under the stabilizer formalism [20]. Here, the subspace code is specified as the common +1 eigenspace of a set of commuting operators {Sj}, which are all elements of the N-qubit Pauli group. The logical oper- ators Xα and Zα commute with all stabilizers and satisfy the commutation relations δαβ XαZβ + (−1) ZβXα = 0. Our strategy for constructing finite-time dissipative encoders employs correction maps that correspond to stabilizer generators. A correction map Φj corresponding to stabilizer operator Sj acts trivially on oper- ators with support in the code, has range in the +1 eigenspace of Sj, and coherently maps the −1 eigenspace of Sj into the +1 eigenspace. Often, the coherent mapping from the −1 to +1 eigenspace corresponds to undoing an error from the Pauli-group. An error E is detectable by measurement of Sj if {E,Sj} = 0; that is, if E exchanges the ± eigenspaces of Sj. Assuming E is from the Pauli-group, reapplying E undoes the detected error. In general, the correction map Kraus operators involve a Pauli- group correction operator Cj satisfying {Cj,Sj} = 0. The Kraus operators of the
190 7.2 Repetition code
correction map correspond to correcting an error conditional on the measurement outcome of Sj: 1 1 K(0) = (I + S ),K(1) = C (I − S ), (7.3) j 2 j j 2 j j We find that the following equivalent form will be more useful for our purposes, 1 1 K(0) = (I + S ),K(1) = (I + S )C . (7.4) j 2 j j 2 j j
For a given stabilizer Sj, there are many choices of correction operators Cj. The challenge in constructing a working finite-time dissipative encoder, then, lies in prop- erly choosing the correction operators along with the order of the correction maps. Through a number of examples, we will arrive at a list of principles which guide the construction of a finite-time dissipative encoder for general stabilizer codes. The finite-time dissipative encoders that we construct are closely related to the notion of discrete-time quasi-local stabilization [206, 194]. In particular, any finite- time dissipative encoder ΦD made from correction maps constitutes a conditional stabilization procedure for any state |ψi in the code. By construction, correction maps leave invariant any state in the code. Furthermore, if the system is conditioned on |ψihψ| ⊗ σ where σ is any state in the basin of attraction W, then ΦD maps the input into |ψihψ|. Our correction map approach is not the only means of constructing a finite-time dissipative encoder. As we will demonstrate at the end of Section 7.2, the technique of sequential generation can be implemented towards making a finite-time dissipative encoder for the repetition code with no constraints on the basin of attraction. It would be interesting to explore if such approaches could be applied to more general codes.
7.2 Repetition code
We demonstrate a finite-time quasi-local dissipative encoder for the quantum 3-qubit repetition code. This stabilizer code is correctable with respect to single bit flip errors. The code space is span{|000i, |111i} ≤ H⊗3. Sufficient stabilizer generators are {ZZI,IZZ}. We choose the logical operators to be X = XXX and Z = ZII. We show that the repetition code may be encoded from a localized data qubit using a sequence of two-body CPTP maps. The two correction maps are Φ12 and Φ23, which we consider to act in that order. (One can construct maps acting in the opposite order which will work as an encoder, but we choose to analyze one case for simplicity of presentation.) Each stabilizer generator admits the same set of
191 Towards finite-time dissipative quasi-local quantum encoders
neighborhood-acting correction operators on its respective neighborhood,
{XI,IX,XZ,ZX,YI,IY,YZ,ZY }.
To arrive at a principle for choosing the correction operators, consider any code with two stabilizer generators S1 and S2. Let Φ2 ◦ Φ1 be the composition of two (i1,i2) 1 correction maps. The Kraus operators of the composed map are K = 2 (I + i2 1 i1 S2)C2 2 (I + S1)C1 . In order for the range of these composed maps to be in the code, it is necessary and sufficient that C2 commutes with S1. For necessity, consider that C2 and S1 do not commute. Then, it must be that {C2,S1} = 0, since Pauli group elements either commute or anti-commute. The Kraus operator K(i1,1) would be of 1 1 i1 the form 2 (I+S2) 2 (I−S1)C2C1 , having a range which is orthogonal to the code. That commutativity is sufficient follows from the fact that the range of the composed map’s 1 Kraus operators is the range of 4 (I + S2)(I + S1), which is equal to the code. Note that this condition rules out the issue of choosing redundant correction operators; if a correction operator is used for two different stabilizers, then, in the composed map’s Kraus operators, one of the correction operators necessarily anti-commutes with a stabilizer to the right of it.
For the repetition code, the set of candidate correction operators for C23 is re- duced to {IIX,IZX,IIY,IZY }. We can narrow down further by the requirement that the logical operators of the code (i.e. X = XXX and Z = ZII) be left in- variant. This requires that correction operators commute with the logical operators, leaving C12 ∈ {IXI,ZYI} and C23 ∈ {IIX,IZY }. Furthermore, for general cor- rection maps, correction operators are only ever defined up to multiplication by the corresponding stabilizer. For instance, the Kraus operator of the first map can be written equivalently with respect to either correction operator, 1 1 1 (III + ZZI)ZYI = (ZZI + III)(ZZI)ZYI = (III + ZZI)IXI. (7.5) 2 2 2 Therefore, our requirements have singled out one possibility for the set of correction maps. The key property of these correction maps is that the range of their composition is in the code and the logical operator values are invariant. Having specified the correction maps, we determine the basin of attraction W ⊆ D(H23) which ensures that ΦD(ρ ⊗ σ) = ρ for σ ∈ W. For encoding to be achieved, the logical operator coordinates of the output density matrix must coincide with those of the data qubit:
i j i j Tr X Z ρ = Tr X Z ΦD(ρ ⊗ σ) , (7.6)
for all i, j ∈ {0, 1}. We have chosen ΦD to leave the values of the logical operators
192 7.2 Repetition code
invariant so that the above equation simplifies to
i j Tr XiZjρ = Tr X Z (ρ ⊗ σ) . (7.7)
Evaluating this for the repetition code, we obtain
Tr XiZjρ = Tr (XiZjρ) ⊗ [(XX)iσ] , (7.8)
giving Tr ((XX)iσ) = 1. The basin of attraction is therefore the set of density operators σ with support contained in the +1 eigenspace of IXX.
The above conclusions can be generalized to the N-qubit repetition code. This N−1 stabilizer error correcting code is correctable with respect to 2 uncorrelated bit flip errors. The code is defined as the subspace span{|0i⊗N , |1i⊗N } ≤ H⊗N . Sufficient N−1 stabilizer generators are {ZiZi+1}i=1 . The logical operators can be taken as X = X⊗N and Z = Z ⊗ I⊗N−1. As with the 3-qubit repetition code, we find that the correction operator for stabilizer ZiZi+1 is Xi+1. Applying Eq. (7.6) as before, we find that the basin of attraction W is the set of density operators σ with support in the +1 eigenspace of X⊗N−1. One state which can be locally prepared in the basin of attraction is |+i⊗N−1. Consider that the pairs of qubits in this system were subject to any degree of XX, YY , ZZ , YZ, or ZY interaction. The initialized state |+i⊗N−1 would be altered, but because it would remain in the +1 eigenspace of X⊗N−1, the subsequent encoding would still function properly.
Next we present a finite-time dissipative encoder that deviates from the above scheme, but admits a global basin of attraction. The scheme utilizes the concept of sequential generation [193]. Consider the two-body Kraus operators defined by
1 X K(j) = |iiihij|. (7.9) i=0 The corresponding superoperator is easily verified to be trace-preserving. Intuitively, this map traces out the second system, while classically copying the |0i or |1i state from the first qubit onto the second.
Consider a 1-D chain of N qubits. Let a sequence of N − 1 of these maps act on
193 Towards finite-time dissipative quasi-local quantum encoders subsequent nearest neighbor qubits. The Kraus operators of the composed map are
(jN ) (j2) X KN−1,N ...K1,2 = (|iN−1iN−1ihiN−1jN |)N−1,N ... (|i1i1ihi1j2|)1,2 iN−1,...,i1 X = δi1,i2 . . . δiN−2,iN−1 |i1i1 . . . iN−1ihi1j2 . . . jN |
iN−1,...,i1 X = |ii . . . iiihij2 . . . jN | (7.10) i
(~j) (jN ) (j2) Applying an arbitrary Kraus operator K ≡ KN−1,N ...K1,2 to the product of a data qubit state and an arbitrary state, we obtain
(~j) X K (α|0i + β|1i) ⊗ |φi2...N = |iihi|(α|0i + β|1i)|i . . . iiihj2 . . . jN ||φi i ⊗N ⊗N = (α|0i + β|1i )hj2 . . . jN ||φi (7.11)
Letting |ψi ≡ α|0i+β|1i and |ψi ≡ α|0i⊗N +β|1i⊗N , the corresponding superoperator action is therefore
† X (~j) (~j) X K |ψφihψφ|K = |ψihψ| hj2 . . . jN ||φihφ||j2 . . . jN i ~j ~j = |ψihψ|. (7.12)
Thus, the encoding is achieved independent of the state of the remaining system. In other words, the basin of attraction is global. The main distinction between this encoding and the previous one based of the stabilizer formalism is that the individual dissipative maps above do not leave the values of the logical operators invariant. This shows that, while the invariance condition may be useful for constructing dissipative encoders for stabilizer codes, it is not requisite for constructing a general finite-time dissipative encoder.
We have demonstrated that a single localized qubit can be dissipatively encoded into the repetition code using a finite sequence of nearest neighbor acting CPTP maps. We demonstrated the basin of attraction for this dissipative encoding as any state on the remaining qubits with support contained in the +1 eigenspace of X⊗N , allowing the encoding to be achieved with some degree of robustness in the initialization. In the remaining sections, we build off of this example to construct finite-time dissipative encoders for more compelling examples of stabilizer codes.
194 7.3 Toric code
7.3 Toric code
We now demonstrate a finite-time dissipative encoding of the toric code [207, 208]. The 2-D toric code employs a 2-D lattice of 2L2 qubits arranged on the edges of the squares of a grid. In this way, neighboring quartets of qubits either surround a square of the grid (plaquette) or surround the intersection of a vertical and a horizontal line of the grid (vertex). The name “toric code” is due to identifying the eastern and western border qubits as neighboring one another, while similarly for the northern and southern border qubits; hence, the geometry and topology induced by the adjacency of qubits is that of a flat 2-torus. The Hamiltonian of the system is constructed using this structure. To each pla- quette p and to each vertex v (see Fig. 7.1a) we assign a four-body Hamiltonian acting on the corresponding qubits defined by
⊗4 Hp ≡ (Z )p ⊗ Ip¯ (7.13) and ⊗4 Hv ≡ (X )v ⊗ Iv¯. (7.14) These operators establish the toric code as a stabilizer code as follows. Since each plaquette overlaps with an even number of systems in any vertex, we have [Hp,Hv] = 0 for all p and v. X X HT ≡ −( Hp + Hv). (7.15) p v A crucial consequence of the toroidal geometry is that the plaquette and vertex Hamil- tonians are not algebraically independent, Y Y Hp = I and Hv = I. (7.16) p v
This implies that HT is at least 4-fold degenerate. Since the above identities generate all algebraic dependence of the plaquette and vertex Hamiltonians, the ground space of HT is exactly 4-fold degenerate. This ground space constitutes the toric code and is equivalently defined as the space of vectors |ψi satisfying Hp|ψi = Hv|ψi = |ψi for all p and v. Later on, we will use the fact that it suffices to define the toric code as the +1 eigenspace of all but one Hp and all but one Hv. This follows from the algebraic redundancy expressed in Eq. (7.16). In [204], the authors construct a Liouvillian constructed as a sum of plaquette- and vertex-acting Liouvillian terms. This Liouvillian generates a continuous-time encoding from two localized physical qubits into the toric code. The labeling scheme of the qubits is borrowed adapted from [204] and depicted in Fig. 7.1b. The qubits
195 Towards finite-time dissipative quasi-local quantum encoders
(a) Plaquette and vertex stabilizer oper- (b) Qubit labeling scheme ators
Figure 7.1: a) The four-body stabilizer operators of the toric code act on vertices v as ⊗4 ⊗4 X or on plaquettes as Z . b) A1 and A2 are the two upload qubits whose initial state is mapped into the toric code. B1 and C1 are the systems (in addition to A1) on which the code’s logical operators for the first encoded qubit are defined to act. The same holds for B2 and C2 with respect to the second encoded qubit. D denotes the remaining qubits. The systems B1, C1, B1, and C2 must be properly initialized in order to achieve a faithful encoding.
196 7.3 Toric code
(a) Logical Xs of toric code (b) Logical Zs of toric code
Figure 7.2: A definition of logical operators for the toric code. Logical operators consist of topologically non-trivial loops of local bit-flip or local phase-flip errors.
A1 and A2 are chosen so as to share a plaquette and vertex which are labeled p∗ and v∗. Then, the qubits of the vertical and horizontal strips which pass through p∗ (except for qubits A1 and A2) are each prepared in |+i. These strips are labeled B1 and B2, respectively. Simliarly, the qubits of the bands passing through v∗ are each prepared in |0i. These strips are labeled C1 and C2, respectively. The logical operators are chosen as
X = X ⊗ X⊗L−1 ⊗ I 1 A1 B1 rest Z = Z ⊗ Z⊗L−1 ⊗ I 1 A1 C1 rest X = X ⊗ X⊗L−1 ⊗ I 2 A2 B2 rest Z = X ⊗ Z⊗L−1 ⊗ I . (7.17) 1 A2 C2 rest
The authors of [204] show that, with this initialization, the state of A1 ⊗ A2 is driven towards the corresponding state of the toric code by means of well-chosen Liouvillian dynamics. We review the correction maps used to construct the Liouvillian in [204]. Then we show that a judicious ordering of these maps constitutes a finite-time dissipative encoder. The stabilizer generators are {Hp,Hv}, where the set ranges over all pla- quettes and vertices except for p∗ and v∗. Correction maps Φp are Φv are associated to each of these stabilizer generators.
197 Towards finite-time dissipative quasi-local quantum encoders
(a) Vertex correction maps (b) Plaquette correction maps
Figure 7.3: Sequences of correction maps. The ordering of correction maps and location of correction operators are chosen so that 1) correction operators commute with all four logical operators and 2) subsequent correction operators are applied where no correction map has acted previously.
Plaquettes and vertices are labeled according to their lattice coordinates with respect to p∗ and v∗. The plaquette pα,β and vertex vα,β lie α sites north and β sites east of p∗ and v∗, respectively. We use a tensor product structure where, for each plaquette and vertex system, the north, east, south, and west qubits are N⊗E⊗S⊗W .
Cpα,0 ≡ (I ⊗ X ⊗ I ⊗ I)p ⊗ Ip¯
Cpα,β ≡ (X ⊗ I ⊗ I ⊗ I)p ⊗ Ip¯
Cv0,β ≡ (I ⊗ I ⊗ Z ⊗ I)v ⊗ Iv¯
Cvα,β ≡ (I ⊗ I ⊗ I ⊗ Z)v ⊗ Iv¯ (7.18)
(i) (i) The Kraus operators are labeled Kpα,β and Kvα,β for the plaquette and vertex correction maps, respectively. The scheme devised in [204] for “pushing” errors to- wards A1 and A2 suggests a choice of ordering for the correction maps as depicted in Fig. 7.3. From our analysis of the repetition code, we expect that the key feature of this choice of correction operators and ordering is that subsequent correction opera- tors commute with all previous stabilizer operators. This property is necessary and sufficient for the range of the composed map to be in the code. Each plaquette correction map commutes with each vertex correction map since exchange of their Kraus operators can, at most, accrue an irrelevant global phase (it is canceled in the superoperator). Without loss of generality, we consider the vertex
198 7.3 Toric code
maps to act first. As seen in Fig. 7.3, the ordering among the vertex (resp. plaquette) correction maps is chosen such that each subsequent correction operator acts where no previous correction map (and hence stabilizer operator) has acted. This verifies that subsequent correction operators commute with all previous stabilizer operators. Let ~i ∈ {0, 1}2(L2−1) indicate the i = 0, 1 of the plaquette and vertex Kraus operators. Then, moving all correction operators to the right of the stabilizers, each Kraus operator of the encoder is written as follows
(~i) i1 it K = (P1C1 ) ... (PtCt ) + i1 it = P (C1 ...Ct ) = P +C~i, (7.19)
1 + 2 where Pj = 2 (I + Hj), P is the projector into the toric code, and t = 2(L − 1) is the total number of correction maps. The encoder can be written as
† + X (~i) (~i) + ΦD(·) = P C · C P . ~i This verifies that the range of the encoder is contained in the code itself. Finally, we determine the initial conditions of the input state that ensure ρ on
A1 ⊗ A2 is encoded into ρ of the code. Let the initial state be ρA1A2 ⊗ σBCD. The basin of attraction for σ is determined by the analogue of Eq. (7.6),
i j k l Tr Xi Zj Xk Zl ρ = Tr X Z X Z Φ (ρ ⊗ σ ) , (7.20) A1 A1 A2 A2 1 1 2 2 D A1A2 BCD
for all i, j, k, l = {0, 1}. Since the correction operators and stabilizers commute with † i j k l i j k l the logical operators, we have ΦD(X1Z1X2Z2) = X1Z1X2Z2. With this, the above equation simplifies to
Tr Xi Zj Xk Zl ρ = Tr (Xi Zj Xk Zl ρ) ⊗ ((XiZjXkZl)⊗L−1σ ) A1 A1 A2 A2 A1 A1 A2 A2 BC BCD = Tr Xi Zj Xk Zl ρ Tr (XiZjXkZl)⊗L−1σ , (7.21) A1 A1 A2 A2 BC BC where in the last step, we have traced out D to obtain σBC = TrD (σBCD). Hence, the state of system D does not affect the encoding. This determines the basin of attraction i j k l ⊗L−1 to be states σBCD with reduced state on BC satisfying Tr (X Z X Z )BC σBC = 1 for all i, j, k, l. A sufficient choice of initial state, as given in [204], is the pure state |φi = |+i⊗L−1|0i⊗L−1|+i⊗L−1|0i⊗L−1. This demonstrates that an initialization can BC B1 C1 B2 C2 be implemented locally. The full basin of attraction can also be described as any density operator with support in the +1 eigenspace of each of the four commuting operators X⊗L−1, Z⊗L−1, X⊗L−1, and Z⊗L−1. B1 C1 B2 C2
199 Towards finite-time dissipative quasi-local quantum encoders
7.4 General stabilizer codes
We investigate finite-time dissipative encoders for several other quantum error cor- recting codes in the stabilizer formalism [20]. We use the following principles in con- structing a finite-time dissipative encoder for a given stabilizer code with stabilizers {Sj}, logical operators Xi, Zi, and data qubits A1 ⊗ ... ⊗ Ak:
(a) Choose correction operators which commute with the logical operators Xi, Zi. This ensures that the basin of attraction is easily determined. (b) Choose an order for the stabilizers and a corresponding set of correction opera- tors so that subsequent correction operators commute with all previous stabilizer operators. This ensures that the range of the correction maps is in the code. (c) Basin of attraction is calculated by requiring
i j i j Tr Xi1 Zj1 ...Xik Zjk ρ = Tr X 1 Z 1 ... X k Z k (ρ ⊗ σ ) (7.22) A1 A1 Ak Ak 1 1 k k A A
for all il, jl ∈ {0, 1}, where A and A denote A1 ⊗ ... ⊗ Ak and its complement, respectively. Note that Eq. (7.22) is the generalization of Eq. (7.6), where we have simplified using the invariance of the logical operators with respect to ΦD. We use these principles to design finite-time dissipative encoders for a few well-known stabilizer codes.
9-Qubit Shor code: The Shor code defined on 9-qubits has logical operators
X = XXXXXXXXX Z = ZZZZZZZZZ. (7.23)
For each stabilizer we choose a corresponding correction operator using the first prin- ciple above.
S1 = ZZIIIIIIIC1 = IXXIIIIII
S2 = ZIZIIIIIIC2 = XXIIIIIII
S3 = IIIZZIIIIC3 = IIIIXXIII
S4 = IIIZIZIIIC4 = IIIXXIIII
S5 = IIIIIIZZIC5 = IIIIIIIXX
S6 = IIIIIIZIZC6 = IIIIIIXXI
S7 = XXXXXXIIIC7 = ZIIIIIZII
S8 = XXXIIIXXXC8 = ZIIZIIIII. (7.24)
200 7.4 General stabilizer codes
(i) To each stabilizer we associate a correction map Φj with Kraus operators Kj = 1 i 0 1 2 (I + Sj)Cj, where i = 0, 1, giving Cj = I and Cj = Cj. Note that the quasi-locality of each Kraus operator is slightly increased from that of the corresponding stabilizer operator. This is necessarily the case if ever the form of a stabilizer operator and a logical operator are identical on qubits which the stabilizer acts non-trivially on. Otherwise, the correction operator could not anti-commute with the stabilizer while commuting with the logical operator as is needed. This increase in neighborhood size will occur whenever the logical operators act as collective unitaries and the stabilizers contain either only X or only Z operators.
The finite-time dissipative encoding map is defined as ΦD = Φ1 ... Φ7Φ8. Since the correction operators commute with all of the preceding stabilizers, the Kraus operators of ΦD can be written as
(i1) (i7) (i8) + i1 i7 i8 K1 ...K7 K8 = P C1 ...C7 C8 , (7.25)
where P + is the product of the stabilizer projectors and hence the projector into the code. This shows that the output of ΦD is necessarily in the code. We choose the first qubit to be the data qubit whose state is to be mapped into the code. Since the correction operators and stabilizers commute with the logical operators, the logical operators are preserved by ΦD. The initialized state is of the form ρ1 ⊗ σ2...9. The basin of attraction for σ is determined by
i j i j Tr X Z ρ = Tr X Z ΦD(ρ ⊗ σ) . (7.26)
† i j i j Using the fact that ΦD(X Z ) = X Z , we obtain
Tr XiZjρ = Tr XiZjρ ⊗ (XiZj)⊗8σ = Tr XiZjρ Tr (XiZj)⊗8σ , determining the basin of attraction to be states satisfying Tr ((XiZj)⊗8σ) = 1. There are no product states which satisfy this requirement. One pure state satisfying this condition is |φi = (|00i + |11i)⊗4. Although this state is not a product state, its entanglement is restricted to pairs of neighboring qubits and can be prepared from a product state by a circuit of depth one.
7-Qubit Steane code: The Steane code defined on 7-qubits has logical operators
X = XXXXXXX Z = ZZZZZZZ. (7.27)
201 Towards finite-time dissipative quasi-local quantum encoders
The stabilizers and sufficient correction operators are
S1 = IIIXXXXC1 = ZIIZIZZ
S2 = IXXIIXXC2 = ZIZIIII
S3 = XIXIXIXC3 = ZZIIIIIII
S4 = IIIZZZZC4 = XIIIXII
S5 = IZZIIZZC5 = XIXIIII
S6 = ZIZIZIZC6 = XXIIIII. (7.28)
As before, to each stabilizer we associate a correction map Φj with Kraus opera- (i) 1 i tors Kj = 2 (I + Sj)Cj. The finite-time dissipative encoding map is defined as ΦD = Φ1 ... Φ5Φ6. Since the correction operators commute with all of the preceding stabilizers, the Kraus operators of ΦD can be written as
(i1) (i5) (i6) + i1 i5 i6 K1 ...K5 K6 = P C1 ...C5 C6 , (7.29)
+ where P is the projector into the code, ensuring that the output of ΦD is necessarily in the code. We choose the first qubit to be the data qubit. Since the correction operators and stabilizers commute with the logical operators, the logical operators are preserved by ΦD. The initialized state is of the form ρ1 ⊗ σ2...7. The basin of attraction for σ is determined by
i j i j Tr X Z ρ = Tr X Z ΦD(ρ ⊗ σ) . (7.30)
† i j i j Using the fact that ΦD(X Z ) = X Z , we obtain
Tr XiZjρ = Tr XiZjρ ⊗ (XiZj)⊗6σ = Tr XiZjρ Tr (XiZj)⊗6σ , determining the basin of attraction to be states satisfying Tr ((XiZj)⊗6σ) = 1. There are no product states which satisfy this requirement, but, as before, a product of bipartite maximally entangled states serves the purpose: |φi = (|00i + |11i)⊗3.
5-Qubit code: The logical operators for the 5-qubit code are
X = XXXXX Z = ZZZZZ. (7.31)
202 7.5 Further questions
The stabilizer operators are
S1 = XZZXI
S2 = IXZZX
S3 = XIXZZ
S4 = ZXIXZ. (7.32)
The 5-qubit stabilizer operators are 4-body. We are unable to find corresponding 4-body correction operators which satisfy the sufficient properties for constructing a finite-time dissipative encoder. It is possible that the 5-qubit code is not amenable to constructing a finite-time dissipative encoder using the approach we have outlined.
7.5 Further questions
(a) Both the repetition code and toric code are examples of generalized surface codes. How can finite-time dissipative encoders be constructed for (generalized) surface codes? What about Bacon-Shor, Haah, or Bombìn’s codes?
(b) Given a finite-time dissipative encoder constructed from the stabilizer formalism as above, does the “Lindbladized” version constitute a continuous-time dissipa- tive encoder?
(c) Can the sequential generation approach be extended to the toric code using its tensor network representation?
203
Chapter 8
Summary and outlook
205 Summary and outlook
In this thesis we have described results on two fronts: quantum joinability and quasi-local stabilization. The part-whole relationship features as the unifying theme of the work. On the first front, we have analyzed the question of whether or not a description of the whole is consistent with constraints placed on the parts. On the second front, we have investigated preparing features of the whole using dynamics which are constrained to address specified parts. In both, we have confronted the phenomenon of multipartite quantum entanglement and its nuances which distinguish the quantum from the classical cases. Respectively, these results contribute to our understanding of the nature of quan- tum relationships among the parts of a quantum system and to our understanding of the limits to and opportunities for controlling multipartite entanglement using engineered dissipation. We have drawn from a diverse mathematical tool set encom- passing linear algebra, group theory, representation theory, Lie theory, operator alge- bra, among others. Additionally, we have developed several mathematical concepts that we have found to be useful, including homocorrelation map, degree of agreement, Schmidt span, and neighborhood algebra. In our exploration of quantum joinability, we began by analyzing the joinabil- ity and sharability of bipartite quantum states. By restricting to the special classes of Werner and isotropic states, we determined simple analytical expressions for 1-n sharability, namely, that the sum of the bipartite concurrences cannot exceed (d − 1) for Werner states and cannot exceed Cmax,d for isotropic states. Then, we determined analytical conditions for characterizing the joinability of these bipartite classes of states in the three-party setting. In other words, we determined what trios of Werner or isotropic states the pairs among Alice-Bob-Charlie are able to simultaneously pos- sess. Surprisingly, we found that the entanglement content of the joined bipartite states does not suffice to determine the resulting joinability properties. In this three- party setting, we investigated the role that the classical joining limitations play in restricting quantum joining. The comparison of the quantum and classical joinability settings illuminated an in- triguing distinction. In particular, the quantum joinability limitations do not obey the tetrahedral symmetry of the classical joinability limitations in the three-bit setting. We discovered that the tetrahedral symmetry could be regained in the quantum case by incorporating additional joinability settings. In particular, we were prompted to develop the notion of channel joinability, whereby one asks whether a set of subsystem quantum channels (causal relationships) are consistent with some quantum channel on the whole. This led us to develop the unifying framework of generalized joinability, which encompasses the quantum state joinability case, the quantum channel joinabil- ity case, as well as variants. Many problems regarding the part-whole relationship in multiparty quantum settings, such as the quantum marginal problem, the asym- metric cloning problem, and various quantum extension problems, are encapsulated
206 Summary and outlook by this framework. An important step was to introduce the homocorrelation map as a natural way to represent quantum channels with bipartite operators, making them geometrically comparable to quantum states. Using this tool, it is possible to directly contrast the joinability properties of quantum states with those of quantum channels. As shown in Fig. 3.5, by directly comparing the joinability limitations for Werner states and depolarizing channels, the union of their joinability bounds regains the tetrahedral symmetry of the classical joinability scenario. Taking quantum channels to express causal relationships and bipartite states to express acausal relationships, we find it intruiging that quantum mechanics distinguishes between the two in a way that the classical probability theory does not. From these joinability findings, it is tempting to view quantum causal and quantum acausal relationships as two puzzle pieces, neither of which completes the picture alone. This distinction between the joinability properties of quantum channels and quan- tum states led us to further investigate the quantum causal-acausal asymmetry. We identified a difference between quantum states and quantum channels in the corre- lations that are obtainable from each. Namely, bipartite quantum states are lim- ited in their allowed degree of agreement, whereas quantum channels are limited in their allowed degree of disagreement. This difference is made explicit by representing quantum channels with the homocorrelation map. We showed how these differences, expressed in terms of agreement bounds, in turn inform the joinability properties of channels vs states. We view the quantum joinability results that we have presented as a starting point and a framework for further exploration. Throughout our analysis of quan- tum joinability, we have only considered scenarios with a pre-defined tensor prod- uct structure, and consequently all operator reductions are obtained via the usual partial-trace construction. However, it is important to appreciate that this was not a necessary restriction. Following [170], one may also consider a more general notion of a reduced state, which results from appropriately restricting the global state to a dis- tinguished operator subspace. Such a notion of reduction is operationally motivated in situations where a tensor product structure is not uniquely or naturally afforded on physical grounds (notably, systems of indistinguishable particles or operational quantum theory, see e.g. [132]). This points to a further extension of the present joinability framework “beyond subsystems”. This perspective also highlights another difference between quantum causal and acausal relationships. In the acausal case, we are free to consider various factorizations of the multipartite Hilbert space, or even forgo mention of the factorization, viewing all states on equal footing. It seems that the same does not hold for quantum causal relationships expressed by quantum chan- nels. Here, it seems that the input-output distinction is critical. Yet, do we need to make such a sharp distinction between these subsystems? What operational grounds lead to the need for a fixed subsystem distinction in the causal case, while allowing
207 Summary and outlook for some freedom in the acausal case? We expect that it would be fruitful to connect the insights from our joinability work to the recent, intriguing finding regarding the indefiniteness of causal order in quantum theory [209]. In the second half of the thesis, we investigated the potential of using quasi-local controlled dissipation for achieving the tasks of quantum state stabilization and en- coding quantum information into a quantum error-correcting code. We began our exploration of quasi-local stabilization in the setting of continuous-time control dy- namics. We discovered conditions determining whether a general mixed state of a finite-dimensional multi-partite quantum system may be the unique fixed point for a natural class of quasi-local frustration-free Markovian dynamics – for given locality constraints. In any case that a target state had been diagnosed as stabilizable, we have provided a constructive procedure to synthesize stabilizing dynamics. We have presented a number of quantum information processing and physically motivated examples demonstrating how our tools can naturally complement and gen- uinely extend available techniques for fixed-point convergence and stability analysis – including quasi-local stabilization of Gibbs states of non-commuting Hamiltonians (albeit for finite system size). Altogether, beside filling a major gap in the existing pure-state quasi-local stabilizability analysis, we believe that our results will have direct relevance to dissipative quantum information processing and quantum engi- neering, notably, open-system quantum simulators. We then moved on to explore quasi-local stabilization in the setting of discrete- time control dynamics. Specifically, we investigated the task of stabilizing a target pure state in with a finite sequence of quasi-local-acting dissipative maps. In parallel, ongoing work [194] (not presented in this thesis), we have been developing a method of alternating completely-positive trace-preserving map projectors to drive the sys- tem asymptotically towards a target state. With this scheme, many quasi-locally stabilizable states are only stabilized asymptotically, requiring infinite-time for exact stabilization. For a subset of states, however, we found that the sequence of stabi- lizing maps converges in a finite number of steps, exhibiting finite-time stabilization. Exact stabilization in finite-time is desirable from a control-theoretic, as well as prac- tical, perspective. Thus, the aim of our work was to develop control strategies and understand the features of target states which enable finite-time stabilization. We show that the set of finite-time stabilizable states is strict subset of the asymptoti- cally stabilizable states. For this subset of states, finite-time stabilization is achieved by designing the completely-positive trace-preserving maps to act collaboratively, by- passing the frustrated interactions of the alternating projection approach that led to infinite-time stabilization. With the scheme that we developed, we showed that certain asymptotically stabilizable states, such as certain AKLT states [185], can, in principle, be quasi-locally stabilized exactly in finite time. Beyond finite-time stabilization, we have investigated the notion of robust finite-
208 Summary and outlook
time stabilization, whereby the stabilization is achieved independently of the order of the dissipative maps. While exact robust finite-time stabilization is very desir- able from an implementation perspective, we found that the set states amenable to this task is strictly contained in the set of finite-time stabilizable states, as expected. Nevertheless, we showed that relevant multipartite entangled states, such as the re- sourceful graph states of measurement-based quantum computing, are still robustly finite-time stabilizable. Our general approach to establishing robust finite-time sta- bilization of a target state was to discover a virtual subsystem factorization of the target state, whereby the role of each neighborhood-acting map simplifies to preparing a pure state factor of the virtual subsystems. In developing the use of a virtual subsystem decomposition for robust finite-time stabilization, we have clarified the role played by “commuting structures” towards efficiently stabilizing a target state. We believe that the techniques used for “discov- ering” such commuting structures (e.g. the neighborhood algebra in Sec. 6.5.9) may be of general interest to researchers working with virtual subsystem decompositions. As robust finite-time stabilization constitutes an efficient means of preparing a target state, we have sought to connect it to notions of efficient preparation in the continuous-time setting, such as rapid mixing. We showed that, for scalable cases of robust finite-time stabilization where the dissipative maps mutually commute, there, indeed, exist rapid mixing continuous-time quasi-local dynamics which stabilize the target state. Lastly, we have highlighted the stabilization results which may be ex- tended to the case of a target mixed state. We leave a number of directions here to future work. Beyond quasi-local stabilization of a target state, we have also started to inves- tigate the task of encoding quantum information into a quantum error-correcting code using a finite sequence of quasi-local dissipative maps. By focusing on quantum error-correcting codes within the stabilizer formalism, we develop several principles that aid the design of finite-time “dissipative encoders” and give a number of relevant examples. Of particular interest is the dissipative encoder we construct for the toric code. A number of research questions are prompted by the present analysis and call for further investigation. Determining whether our necessary condition for frustration- free quasi-local stabilization is, as we conjecture, always sufficient on its own even for non-full-rank states is a first obvious issue to address. From a physical standpoint, in order to both understand the role of Hamiltonan control and to make contact with naturally occurring dissipative dynamics, it is important to scrutinize the extent to which the stabilizing dynamics obtained in our framework may be compatible with rigorous derivations of the QDS – in particular in the weak-coupling-limit, where the interplay between Hamiltonian and dissipative components is crucial and demands to be carefully accounted for [136, 210].
209 Summary and outlook
Still within the present quasi-local stabilization setting, an interesting mathemat- ical question is to obtain a global characterization of the geometrical and topologi- cal properties of the frustration-free quasi-local stabilizable set, beginning from pure states. As alluded to in the last remark above, answering these questions may also have practical implications, in terms of approximate quasi-local stabilization, which we briefly discussed in Section 5.5.5. Related to that, while our main focus has indeed been on exact asymptotic stabilization, it may also be beneficial (possibly necessary) to tailor approximate methods for analysis and/or synthesis to specific classes of states. A natural starting point could be provided here by graph states, which have recently been shown to arise as arbitrarily accurate approximations of ground states of two-body frustration-free Hamiltonians [211]. From a more general perspective, the present analysis fits within our broader program of understanding controlled open-quantum system dynamics subject to a re- source constraint. In that respect, an important next step will be to tackle different kinds of constraints – including, for instance, less restrictive notions of quasi-locality, which allow for exponentially decaying interactions in space, as in [156]; or possibly reformulating the quasi-local constraint away from “real” space and the associated tensor-product-decomposition, but rather relative to a preferred operator subspace, in the spirit of “generalized entanglement” [170]. Lastly, and perhaps somewhat counter-intuitively, as our results on separable non-FFQLS evidence, it is interest- ing to acknowledge that the quasi-local stabilization problem need not be trivial for classical probability distributions. In work not presented in this thesis, we have be- gun to investigate further the connections between quasi-local stabilization and the quantum marginal problem. A practical motivation for making this connection is to- wards simplifying our characterizations of “naturally-occuring” many-body quantum states. Recent work has investigated the use of ideas from the quantum marginal problem towards efficient quantum state tomography [212]. They achieve quantum tomography using just quasi-local expectation values, for example, depending only on just the two-body reduced density matrices. The extent to which naturally occurring quantum states can be characterized by their marginals is not well understood. But, if naturally occurring states are taken to be steady states of quasi-local dynamics, and we can relate steady states of quasi-local dynamics to states that are uniquely determined by their marginals, then there is hope for “efficient” characterization of such states. Several open questions also remain in our pursuit of understanding finite-time stabilization. One clear, outstanding problem is to obtain constructive procedures for synthesizing the sequences of stabilizing neighborhood unitary maps for finite- time stabilization. Furthermore, we would like to understand if, as conjectured, the “unitary generation property” is truly a necessary condition for finite-time stabiliza- tion. Towards robust finite-time stabilization, we have made significant progress in
210 Summary and outlook understanding the role of “commuting structures”. While we have shown that the commuting structure inherent in the virtual subsystem factorization ensures robust finite-time stabilization, we would like to understand the extent to which it is neces- sary. In thinking towards approximate robust finite-time stabilization, analogous to approximate frustration-free quasi-local stabilization of Sec. 5.5.5, we expect that it would be fruitful to explore the notion of “approximate factorization” with respect to virtual subsystems. More generally, we would like to explore the use of virtual sub- system factorization for analyzing the efficiency of approximate preparation states using either continuous-time or discrete-time dynamics. A last, important direction to pursue would be to assess the robustness of these finite-time stabilization schemes against various implementation errors. Intuitively, we expect the robust finite-time stabilization schemes to exhibit some degree of resilience to errors.
211
Appendix A
List of abbreviations
CMI: conditional mutual information, pg. 209 CPTP: completely positive trace-preserving, pg. 114 DQLS: dissipatively quasi-locally stabilizable, pg. 127 FF: frustration free, pg. 128 FFQLS: frustration-free quasi-locally stabilizable, pg. 129 FTS: finite-time stabilizable, pg. 189 GAS: globally asymptotically stable, pg. 126 LOCC: local operations and classical communication, pg. 26 MPS: matrix product state, pg. 183 NN: nearest-neighbor, pg. 149 and Fig. 6.1 PEPS: projected entangled pair state, pg. 183 POVM: positive-operator valued measure, pg. 137 QDS: quantum dynamical semigroup, pg. 114 QIP: quantum information processing, pg. 13 QL: quasi-local, pg. 124 QLS: quasi-locally stabilizable, pg. 126 RFTS: robust finite-time stabilizable, pg. 189
213
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