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 .