PHYSICAL REVIEW X 6, 041031 (2016)
Geometry and Response of Lindbladians
Victor V. Albert,1,* Barry Bradlyn,2 Martin Fraas,3 and Liang Jiang1 1Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06511, USA 2Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08540, USA 3Mathematisches Institut der Universität München, University of Munich, Munich 80333, Germany (Received 31 January 2016; revised manuscript received 18 August 2016; published 16 November 2016) Markovian reservoir engineering, in which time evolution of a quantum system is governed by a Lindblad master equation, is a powerful technique in studies of quantum phases of matter and quantum information. It can be used to drive a quantum system to a desired (unique) steady state, which can be an exotic phase of matter difficult to stabilize in nature. It can also be used to drive a system to a unitarily evolving subspace, which can be used to store, protect, and process quantum information. In this paper, we derive a formula for the map corresponding to asymptotic (infinite-time) Lindbladian evolution and use it to study several important features of the unique state and subspace cases. We quantify how subspaces retain information about initial states and show how to use Lindbladians to simulate any quantum channels. We show that the quantum information in all subspaces can be successfully manipulated by small Hamiltonian perturbations, jump operator perturbations, or adiabatic deformations. We provide a Lindblad-induced notion of distance between adiabatically connected subspaces. We derive a Kubo formula governing linear response of subspaces to time-dependent Hamiltonian perturbations and determine cases in which this formula reduces to a Hamiltonian-based Kubo formula. As an application, we show that (for gapped systems) the zero- frequency Hall conductivity is unaffected by many types of Markovian dissipation. Finally, we show that the energy scale governing leakage out of the subspaces, resulting from either Hamiltonian or jump-operator perturbations or corrections to adiabatic evolution, is different from the conventional Lindbladian dissipative gap and, in certain cases, is equivalent to the excitation gap of a related Hamiltonian.
DOI: 10.1103/PhysRevX.6.041031 Subject Areas: Condensed Matter Physics, Quantum Information, Statistical Physics
I. MOTIVATION AND OUTLINE On one hand, AsðHÞ that support quantum information – Consider coupling a quantum mechanical system to a [7 10] are promising candidates for storing, preserving, Markovian reservoir which evolves initial states of the and manipulating such information, particularly when their system into multiple nonequilibrium (i.e., nonthermal) states can be engineered to possess favorable features (e.g., – asymptotic states in the limit of infinite time. After tracing topological protection [11 13]). They have been subject out the degrees of freedom of the reservoir, the time to intense experimental investigation in quantum optics – evolution of the system is governed by a Lindbladian L [14,15], liquid-state NMR [16,17], trapped ions [18 23], [1,2] (see also Refs. [3–6]), and its various asymptotic and (most recently) circuit QED [24]. With many current states ρ∞ are elements of an asymptotic subspace AsðHÞ— experimental efforts aimed at engineering Markovian envi- a subspace of OpðHÞ, the space of operators on the system ronments admitting asymptotic subspaces, it is important to Hilbert space H. The asymptotic subspace attracts all initial gain a comprehensive understanding of any differences ρ ∈ H between the properties of these subspaces and analogous states in Opð Þ, is free from the decoherence effects of L, and any remaining time evolution within AsðHÞ is subspaces of Hamiltonian systems (e.g., subspaces spanned exclusively unitary. If AsðHÞ has no time evolution, all ρ∞ by degenerate energy eigenstates). are stationary or steady. This work provides a thorough On the other hand, response properties of AsðHÞ, which investigation into the response and geometrical properties do not necessarily support quantum information, can help of the various asymptotic subspaces. model experimental probes into exotic nonequilibrium phases of matter resulting from engineered Markovian reservoirs [25–35] (realized in, e.g., optical lattices [36–46] or microwave cavity arrays [47–49]). Because of, e.g., sym- *[email protected] metry [50,51] or topology [11], the asymptotic subspace can be degenerate yet not support a qubit [e.g., an AsðHÞ spanned by Published by the American Physical Society under the terms of ψ ψ ψ 0 ψ 0 the Creative Commons Attribution 3.0 License. Further distri- two projections j ih j and j ih j]. For these and similar bution of this work must maintain attribution to the author(s) and cases, standard thermodynamical concepts [52–55] may not the published article’s title, journal citation, and DOI. apply and steady states may no longer be thermal or even full
2160-3308=16=6(4)=041031(36) 041031-1 Published by the American Physical Society ALBERT, BRADLYN, FRAAS, and JIANG PHYS. REV. X 6, 041031 (2016) rank. (We remind the reader that the rank of a diagonalizable (4) What is the natural metric governing distances matrix is the number of its not necessarily distinct nonzero between various ρ∞? We introduce in Sec. VI a eigenvalues.) Our approach is directly tailored to such systems, Lindbladian version of the quantum geometric i.e., those possessing one or more nonequilibrium steady states tensor (QGT) [74,75], which encodes both the whose rank is less than the dimension of the system curvature associated with adiabatic deformations Hilbert space. and a metric associated with distances between Unlike Hamiltonians, Lindbladians have the capacity to adiabatically connected steady states. model decay. As a result, Lindbladians are often used to (5) What is the energy scale governing leakage out of describe commonplace non-Hamiltonianprocesses (e.g., cool- the asymptotic subspace? Extending Ref. [69],in ing to a ground state). In general Lindbladian-based time Secs. IV C and VC we determine the energy scale ρ H evolution, all parts of an initial state in that are outside of governing leakage out of Asð Þ due to both Ham- H ρ Asð Þ will decay as in evolves toward an asymptotic state iltonian perturbations and adiabatic evolution. Con- ρ∞ ∈ AsðHÞ.SinceAsðHÞ may be multidimensional, the trary to popular belief, this scale is not always the ρ L resulting asymptotic state may depend on in. The decay of dissipative gap of (the nonzero eigenvalue with the ρ ρ ρ parts of in and the nontrivialdependenceof ∞ on in stand out smallest real part). We demonstrate this with an as two distinct features of Lindbladian-based evolution. example from coherent state quantum information Nonetheless, ρ∞ is a collection of states whose behavior is processing [61]. otherwise familiar from Hamiltonian-based quantum mechan- (6) What is the linear response of ρ∞ to time-dependent ics. An asymptotic subspace can thus be thought of as a Hamiltonian perturbations? In. Sec. IV, we derive a Hamiltonian-evolving subspace embedded in a larger Lindbladian-based Kubo formula for response of ρ∞ Lindbladian-evolving space. The aim of this paper is to and determine when it reduces to the familiar determine the effects of Lindbladian evolution on the proper- Hamiltonian-based Kubo formula [76]. As an ap- ties of ρ∞. Namely, we prove a formula for the effect of L in the plication, we show that the zero-frequency Hall limit of infinite time (Proposition 2 in Sec. III) and apply it conductivity [77] remains quantized under various to the following physically motivated questions, noting that kinds of Markovian dissipation. (4)–(6) contain results relevant also to L with a unique steady state. ρ ρ II. STATEMENT OF KEY RESULTS (1) What is the dependence of ∞ on in? Building on previous results [50], in Sec. III we show that ρ∞ In this section, we introduce necessary notation, state our does not depend on any initial coherences between key result, summarize its ramifications in the form of two AsðHÞ and subspaces outside of AsðHÞ and that the properties, the no-leak and clean-leak properties Eqs. (2.9) presence of unitary evolution within AsðHÞ can and (2.11), and apply it to various types of AsðHÞ.We actually suppress the purity of ρ∞. We provide a conclude with a summary of earlier work and outline the recipe for using infinite-time Lindbladian evolution rest of the paper. Readers unfamiliar with Lindbladian to implement arbitrary quantum channels, i.e., com- evolution are welcome to browse Appendix A. pletely positive trace-preserving maps [56]. This recipe should prove useful in experimental quantum A. Four-corners decomposition channel simulation [57] and autonomous or passive quantum error correction [58]. Since decay of states is an unavoidable feature of (2) What is the effect of time-independent Hamiltonian Lindbladian evolution, it is important to make a clear perturbations on ρ∞ within AsðHÞ? It was recently distinction between the decaying and nondecaying parts of shown [59,60] that Hamiltonian perturbations and the N-dimensional system Hilbert space H. Let us group all perturbations to the jump operators of L generate nondecaying parts of OpðHÞ into the upper left corner [of unitary evolution within some AsðHÞ to linear order. the matrix representation of OpðHÞ] and denote them by the In Sec. IV, we prove that such perturbations induce “upper-left” block . Thereby, any completely decaying unitary evolution within all AsðHÞ to linear order, parts will be in the complementary block, and coher- extending the capabilities of environment-assisted ences between the two will be in the “off-diagonal” blocks quantum computation and quantum Zeno dynamics . We can discuss such a decomposition in the familiar [61–66]. language of NMR: the block consists of a degenerate (3) What is the geometric “phase” acquired by ρ∞ after ground state subspace immune to decay and dephasing, the cyclic adiabatic deformations of L?InSec.V,we block contains the set of populations decaying with the extend previous results [67–71] to show that cyclic well-known rate 1=T1, and the block is the set of Lindbladian-based [72] adiabatic evolution of states in coherences dephasing with rate 1=T2. More generally, there AsðHÞ is always unitary, extending the capabilities of can be further dephasing within without population holonomic quantum computation [73] via reservoir decay, so AsðHÞ [gray region in Fig. 1(a)] is in general a engineering. subspace of .
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The subspace ≡ POpðHÞ consists of all coherences between PH and QH, and the “diagonal” subspace ≡ POpðHÞ consists of all operators that do not contain any such coherences. Nontrivial decaying subspaces are ubiquitous in actively researched quantum information schemes (see, e.g., Refs. [61,62]). For instance, consider a bosonic Lindbladian whose steady states are the two coherent states jαi and j−αi (recently realized experimentally [24] FIG. 1. Decompositions of the space of matrices OpðHÞ acting and discussed in more detail in Sec. IV C 1). All states H on a Hilbert space using the projections fP; Qg defined in orthogonal to j αi constitute the decaying subspace, Eq. (2.1) and their corresponding superoperator projections and our results apply. We thoroughly discuss how our fP; P; P; Pg defined in Eq. (2.2). Panel (a) depicts the work applies to various AsðHÞ in Sec. II C. Here, before block diagonal structure of the asymptotic subspace AsðHÞ, which is located in and spanned by steady-state basis elements summarizing our key results, we mention two cases without decaying subspaces for which our work reduces Ψμ. Panel (b) depicts the subspace of OpðHÞ, spanned by conserved quantities Jμ, that may leave a footprint on states in to known results. — L − AsðHÞ in cases when there are multiple steady states. Hamiltonian case. If ¼ i½H;· for some Hamiltonian, any state written in terms of the N eigenstate Let us now define the superoperator projections on the projections jEkihEkj of H (HjEki¼EkjEki) is a steady blocks. Let P be the orthogonal operator projection state. Therefore, there is no decaying subspace in (P ¼ P2 ¼ P†)onand only on the nondecaying subspace Hamiltonian evolution (P ¼ I). — of H. This projection is uniquely defined by the following Unique state case (full rank). In the case of a one-dimensional AsðHÞ, P is the projection on the rank conditions: for all ρ∞ ∈ AsðHÞ, of the unique steady state ρ∞ ≡ ϱ. If the state’s spectral P −1 ρ ρ ϱ dϱ λ ψ ψ ∞ ¼ P ∞P; decomposition is ¼ k¼0 kj kih kj (with dϱ being λ ϱ ρ the numberP of nonzero eigenvalues k of ), then TrfPg¼maxfrankð ∞Þg: ð2:1Þ dϱ−1 ρ∞ ψ ψ P ¼ k¼0 j kih kj. If all N eigenvalues are nonzero, then ϱ is full rank (e.g., in a Gibbs state) and there is no The first condition makes sure that P projects onto all decaying subspace (P ¼ I). nondecaying subspaces, while the second guarantees that P does not project onto any decaying subspace. Naturally, the B. Key results orthogonal projection onto the maximal decaying subspace of H is Q ≡ I − P [with PQ ¼ QP ¼ 0 and QρðtÞQ → 0 States undergoing Lindbladian evolution evolve into as t → ∞]. asymptotic states for sufficiently long times [78]: We define the four-corners projections acting on A ∈ t→∞ H ρ ⟶ ρ ≡ −iH∞tP ρ iH∞t Opð Þ as follows: in ∞ e ∞ð inÞe : ð2:5Þ
A ≡ PðAÞ ≡ PAP; The nonunitary effect of Lindbladian time evolution is ≡ P ≡ encapsulated in the asymptotic projection superoperator A ðAÞ PAQ; 2 P∞ (with P∞ ¼ P∞). The extra Hamiltonian H∞ quanti- A ≡ PðAÞ ≡ QAP; fies any further unitary evolution within AsðHÞ, which of course does not cause any decoherence. For simplicity, we A ≡ PðAÞ ≡ QAQ: ð2:2Þ state our result for the H∞ ¼ 0 case and outline the ≠ 0 By our convention, taking the conjugate transpose of nontrivial consequences of H∞ later. The asymptotic the upper-right part places it in the lower-left subspace projection is a trace-preserving quantum process taking a ρ ∈ H † † † density matrix in Opð Þ into an asymptotic density (projection acts before adjoint): A ≡ ðAÞ ¼ðA Þ. matrix in ρ∞ ∈ AsðHÞ. We determine the following for- The operators P⊞ (with ⊞ ∈ f; ; ; g) are projec- P 2 mula for ∞ (Proposition 2): tions (P⊞ ¼ P⊞), which partition the identity I on OpðHÞ, −1 P∞ ¼ PΨðP − PLPL PÞ; ð2:6Þ P þ P þ P þ P ¼ I; ð2:3Þ
where the minimal projection PΨ further maps onto analogous to P Q I. They conveniently add, e.g., þ ¼ AsðHÞ. The form of PΨ, a projection onto AsðHÞ of L which do not possess a decaying subspace, depends on the P ≡ P P P ≡ P P þ and þ : ð2:4Þ details of AsðHÞ and is already known [78,79]. Therefore,
041031-3 ALBERT, BRADLYN, FRAAS, and JIANG PHYS. REV. X 6, 041031 (2016) our work extends previous Lindbladian results to cases where P ≡ I − P. Moreover, despite two actions of O when a decaying subspace is present. connecting to , Eq. (2.9) still provides some insight The above formula allows us to determine which parts of into second-order effects within AsðHÞ (Sec. IV C). ρ in are preserved in the large-time limit [question (1); see The no-leak property (2.9) is important in determining Sec. III C]. For example, since the projection P is not the energy scale governing leakage out of AsðHÞ [question present in the above formula, we can immediately read off (5); see Secs. IV C and VC]. Let us apply this property to that no coherences between the nondecaying subspace and the second term in Eq. (2.8): its counterpart are preserved. Moreover, the piece P∞P can be used to simulate an arbitrary quantum channel −1 −1 −1 L Oðρ∞Þ¼L POðρ∞Þ¼L Oðρ∞Þ; ð2:10Þ (Sec. III D). Finally, the formula describes how states that are already in AsðHÞ respond to perturbations. We now −1 −1 apply the formula to show why PΨ is the only part relevant where L⊞ ≡ ðP⊞LP⊞Þ and ⊞ is any block. Note that to answering questions (2)–(4). the last step in Eq. (2.10) also uses a property of L, We sketch the effect of small perturbations O on a state PLP ¼ 0 [see Eq. (3.3)], which can be understood by ρ∞ already in AsðHÞ. The perturbations of interest are remembering that evolution under L draws states to . either Hamiltonian perturbations V ≡ −i½V;· (with Since the restriction to studying L on in linear response Hamiltonian V and small parameter ϵ) or derivatives ∂α ≡ has previously gone unnoticed, it is conventionally believed ∂=∂xα (with parameters xα and adiabatic evolution time T) that the leakage energy scale is determined by the dis- Δ — of the now parameter-dependent ρ∞ðxαÞ and LðxαÞ: sipative (also, dissipation or damping) gap dg the non- zero eigenvalue of L with smallest real part. As shown in 1 Eq. (2.10), that energy scale is actually governed by the O ∈ ϵV; ∂α : ð2:7Þ Δ ≥ Δ — T effective dissipative gap edg dg the nonzero eigen- value of L with smallest real part. In Hamiltonian systems We show that both of these can be used to induce unitary (L ¼ −i½H; · ), a special case of the no-leak property states operations on AsðHÞ. We show later in the paper that this that the energy denominator in the first-order perturbative analysis holds for jump operator perturbations as well, but correction to the kth eigenstate of H contains only energy omit discussing those perturbations for now to keep things differences involving the energy Ek of that eigenstate (and − simple. The perturbations ∂α determine adiabatic connec- not, e.g., Ek−1 Ekþ1). tion(s) and thus help with defining parallel transport [i.e., We now project Oðρ∞Þ back to AsðHÞ to examine the adiabatic evolution of AsðHÞ]. Within first order for the first term in Eq. (2.8). Applying P∞ to Eq. (2.9) and using case of perturbation theory (ϵ → 0) and approaching the P∞P ¼ 0 from Eq. (2.6) removes two more squares: adiabatic limit for the case of parallel transport (T → ∞), two relevant perturbative processes after the action of O on P∞Oðρ∞Þ¼P∞POðρ∞Þ¼PΨOPΨðρ∞Þ: ð2:11Þ an asymptotic state are subsequent projection onto AsðHÞ and leakage out of AsðHÞ via the perturbation and L−1: The clean-leak property shows that any leakage of the −1 perturbed ρ∞ into does not contribute to the first-order ρ∞ → P∞Oðρ∞Þ − L Oðρ∞Þ: ð2:8Þ effect of O within AsðHÞ. Essentially, the clean-leak property (2.11) makes AsðHÞ resistant to the nonunitary We show below that these two terms occur both in the Kubo effects of Lindbladian evolution and allows for a closer formula and in adiabatic response. analogue between AsðHÞ and subspaces of unitary systems. We first observe that O is limited in its effect on ρ∞. The clean-leak property simplifies calculations of both Acting with O once does not connect with because O Hamiltonian perturbations [question (2); see Sec. IV] and does not act nontrivially on ρ∞ from both sides simulta- adiabatic or Berry connections [question (3); see Sec. V]. It neously. This no-leak property can be understood if one can be used to show that PΨ (instead of the full P∞) fully observes that Hamiltonian superoperator perturbations V governs adiabatic evolution, so the Lindbladian generali- act nontrivially on ρ∞ only from one side at a time due to zation of the QGT [question (4); see Sec. VI]is their commutator form. Likewise, derivatives ∂α act non- trivially on either the “ket” or “bra” parts of all basis elements used to write ρ∞ due to the product rule. Qαβ ≡ PΨ∂αPΨ∂βPΨPΨ: ð2:12Þ Therefore, acting with O once only connects to itself and nearest-neighbor squares ( ) and does not cause The part of the QGT antisymmetric in α, β corresponds to “transitions” into : the adiabatic curvature F αβ (determined from the Berry connections). The part of the QGT symmetric in α, β O ρ∞ PO ρ∞ ; : ð Þ¼ ð Þ ð2 9Þ corresponds to a metric Mαβ on the parameter space.
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C. Examples corresponding metric also reduce to that of the DFS states. We now apply the four-corners decomposition and leak In other words, all such results are the same regardless of conditions to various types of AsðHÞ and summarize some whether the states form a DFS of a Lindbladian or a of our main results. degenerate subspace of a Hamiltonian. 3. Noiseless subsystem (NS) case 1. Unique state case This important case is a combination of the DFS and In this case, AsðHÞ is one-dimensional (with unique unique steady-state cases. In this case, the nondecaying steady state ϱ) and the asymptotic projection preserves only portion of the system Hilbert space (PH) factors into a the trace of the initial state: H d-dimensional subspace DFS spanned by DFS states and a H P∞ðρ Þ¼ϱTrfρ g and PΨðρ Þ¼ϱTrfPρ g: ð2:13Þ dax-dimensional auxiliary subspace ax, which is the range in in in in ϱ ϱ of some unique steady state ax [dax ¼ rankð axÞ]. This ϱ ϱ Note that we use for states which are determined only by combination of a DFS tensored with the auxiliary state ax L ρ (meaning they are independent of in). Since there is only is called a noiseless subsystem (NS) [9]. For one NS block, one steady state, there is nowhere to move within AsðHÞ. H decomposes as Indeed, it is easy to show that H H ⊕ H H ⊗ H ⊕ H ¼ P Q ¼ð DFS axÞ Q : ð2:19Þ P∞OP∞ ¼ TrfOðϱÞgP∞ ¼ 0 ð2:14Þ A NS block is possible if L respects this decomposition and for both types of perturbations O (2.7). Thus, the only does not cause any decoherence within the DFS part. The novel application of our results to this case is the metric DFS basis elements jψ kihψ lj from Eq. (2.17) generalize to arising from the QGT, ψ ψ ⊗ ϱ j kih lj ax. For this case, states in are not perfectly preserved, but are instead partially traced over the auxiliary Mαβ ¼ Trf∂ αP∂β ϱg; ð2:15Þ ð Þ subspace: where A αBβ ¼ AαBβ þ AβBα. This metric is distinct from ð Þ P ρ ρ ⊗ ϱ ∂ ϱ∂ ϱ ϱ Ψð inÞ¼TraxfP inPg ax; ð2:20Þ the Hilbert-Schmidt metric Trf ðα βÞ g for mixed and is nonzero only when ϱ is not full rank. For pure steady ⊗ where P ¼ PDFS Pax and PDFS (Pax) is the identity on states, both metrics reduce to the Fubini-Study metric [74]. H H H DFS ( ax) and Trax is a trace over ax. ϱ Note that the auxiliary factor becomes trivial when ax is 2. Decoherence-free subspace (DFS) case 1 a pure state (dax ¼ ), reducing the NS to a DFS. This The simplest multidimensional AsðHÞ which stores means that the NS case is distinct from the DFS case only ϱ ≠ 1 quantum information is a decoherence-free subspace when ax is mixed (dax ). Similarly, if the dimension of 2 (DFS) [8].Ad2-dimensional DFS block, the DFS d ¼ d ¼ 1, the NS reduces to the unique steady- state case. The NS case thus encapsulates both the DFS and ¼ AsðHÞ; ð2:16Þ unique state cases. For this case, the effect of perturbations V on AsðHÞ is ψ ψ d−1 ψ d−1 is spanned by matrices fj kih ljgk;l¼0, where fj kigk¼0 is more subtle due to the auxiliary factor, but the induced time a basis for a subspace of the d ≤ N-dimensional system evolution on the DFS is nevertheless still unitary. The space. The decaying block is then spanned by effective DFS Hamiltonian is N−1 fjψ kihψ ljg . Evolution of the DFS under L is exclu- k;l¼d ϱ sively unitary, W ¼ Traxf axVg: ð2:21Þ x ∂tðjψ kihψ ljÞ ¼ Lðjψ kihψ ljÞ ¼ −i½H∞; jψ kihψ lj ; ð2:17Þ Similarly, if we define generators of motion Gα in the α direction in parameter space (i.e., such that ∂αρ∞ ¼ ≤ − 1 where H∞ is the asymptotic Hamiltonian and k; l d . −i½Gα; ρ∞ ), then the corresponding holonomy (Berry Since the entire upper-left block is preserved, phase) after a closed path is the path-ordered integral of the various DFS adiabatic connections P ρ P ρ ρ Ψð inÞ¼ ð inÞ¼P inP ð2:18Þ ~ DFS ϱ Aα ¼ Traxf axðGαÞg: ð2:22Þ for a DFS. We can thus deduce from Eq. (2.11) that the effect of Hamiltonian perturbations V within AsðHÞ is In both cases, the effect of the perturbation on the DFS part — ϱ ϱ V ¼ PVP the Hamiltonian projected onto the DFS. depends on ax, meaning that ax can be used to modulate Likewise, if O ¼ ∂α, then the Lindbladian adiabatic con- both Hamiltonian-based and holonomic quantum gates. ∂ ϱ nection can be shown to reduce to αP · P, the adiabatic The QGT for this case is rather complicated due to the ax- connection of the DFS. Naturally, the QGT and its assisted adiabatic evolution, but we show that the QGT
041031-5 ALBERT, BRADLYN, FRAAS, and JIANG PHYS. REV. X 6, 041031 (2016) does endow us with a metric on the parameter space for a evolution within AsðHÞ or causing leakage out of it? and NS block. (b) does the term acting within AsðHÞ generate unitary evolution? Regarding the first question, it has been widely 4. Multiblock case believed (and often numerically verified, e.g., in Ref. [61]) that the term governing evolution within AsðHÞ, P∞VP∞, The noiseless subsystem is the most general form of one H L dominates over the term governing leakage out of Asð Þ block of asymptotic states of , and the most general (provided that V is turned on for some finite time). Several As H is a direct sum of such NS blocks [78,80,81] [see ð Þ works [59,62,96] have formally justified this claim and Fig. 1(a)] with corresponding minimal projection PΨ. This provided the necessary constraints on the time scale of the important result applies to both Lindbladians and more perturbation, interpreting AsðHÞ as a quantum Zeno sub- general quantum channels [79,82–85] (see Ref. [86] for a space [63,64,66] (see also Refs. [97,98]). Regarding the technical introduction). Throughout the paper, we explicitly d−1 second question, Zanardi and Campos Venuti [59] recently calculate properties of one NS block fjψ ihψ j ⊗ ϱ g 0 k l ax k;l¼ proved that if P∞P ¼ 0, then P∞VP∞ generates unitary and sketch any straightforward generalizations to the evolution for the DFS case. They also showed [60] that multiblock case. Lindbladian jump operator perturbations induce unitary Both Eqs. (2.21) and (2.22) extend straightforwardly to evolution on Lindbladians without decaying subspaces. We the multiblock case, provided that the blocks maintain their generalize both of these results (by showing that P∞P is shape during adiabatic evolution. We do not derive a metric always zero) to all AsðHÞ. for this case, so taking into account any potential inter- Regarding reservoir-engineered holonomic quantum action of the blocks during adiabatic evolution remains an computation [73] on AsðHÞ [question (3)], we are faced open problem. again with two similar questions: (a) Is there an adiabatic limit for open systems? and (b) is the holonomy after a D. Earlier work closed adiabatic deformation unitary? Regarding the first We review efforts related to our work, including studies question, the adiabatic theorem has indeed been general- of the structure, stability, and control of Lindbladian ized to Lindblad master equations [70,72,99–102] and all steady-state subspaces. orders of corrections to adiabatic evolution have been Regarding the formula for P∞ (Proposition 2), we have derived (see, e.g., Ref. [72], Theorem 6). This is the mentioned that the piece P∞P has already been deter- adiabatic limit dominated by steady states of L. Another mined in two seminal works, Baumgartner and Narnhofer adiabatic limit exists which is dominated by eigenstates of [78]andBlume-Kohoutetal.[79](see also TicozziandViola the Hamiltonian part of L [103–105], which we do not [80]). Our four-corners partition of L produces constraints address further here. Regarding question (b), Sarandy and on the Hamiltonian and jump operators of L (Proposition 1), Lidar [68] were the first to make contact between adiabatic which are already known from Refs. [78,80,87]. There exist or Berry connections and Lindbladians. Avron et al. related formulas for the parts of P∞P corresponding to (Ref. [72], Proposition 3) showed that the corresponding fixed points of discrete-time quantum channels in Lemma holonomy is trace preserving and completely positive. 5.8 of Ref. [79] and Proposition 7 of Ref. [88] and of Markov Carollo, Santos, and Vedral [67] showed that the holonomy chains in Theorem 3.3 of Ref. [89]. In addition, previous is unitary for Lindbladians possessing one DFS block. results assume no residual unitary evolution within AsðHÞ Oreshkov and Calsamiglia [69] proposed a theory of (i.e., H∞ ¼ 0). adiabaticity which extended that result to the multiblock case and arrived at Eq. (2.22). They showed that corrections Regarding question (1), Jakob and Stenholm [90] men- pffiffiffiffi tioned the importance of conserved quantities in determin- to their result were Oð1= TÞ (with T being the traversal ρ ρ H 1 ing ∞ from in, but did not generalize to all Asð Þ. This time), as opposed to Oð =TÞ as in a proper adiabatic limit. generalization was done by two of us [50], showing that ρ∞ By explicitly calculating the adiabatic connections, we does not depend on dynamics at any intermediate times. connect the result of Ref. [69] with the formulation of Here, we provide an analytical formula for the conserved Ref. [68], showing that nonadiabatic corrections are quantities for multidimensional AsðHÞ. In contrast, current actually Oð1=TÞ. We also extend Ref. [69] to NS cases applications of the Keldysh formalism to Lindbladians [91] where the dimension of the auxiliary subspace (i.e., the ϱ do not tackle such cases. Regarding channel simulation, rank of ax) can change. Finally, Zanardi and Campos theoretical efforts have focused on minimizing the ancillary Venuti (Ref. [60], Proposition 1) showed that first-order resources required to simulate channels on a system Hamiltonian evolution within AsðHÞ can be thought of as a [92–95]. To our knowledge, previous efforts did not holonomy. We develop this connection further by showing consider constructing a more general quantum channel that, for both processes, evolution within AsðHÞ is gen- out of less general Markovian ones. erated by the same type of effective Hamiltonian Regarding Hamiltonian control of AsðHÞ [question (2)], [Eqs. (2.21) and (2.22)], and leakage out of AsðHÞ is there are two questions: (a) Is the dominant term generating governed by the same energy scale. We make the same
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Δ connection between ordinary and adiabatic perturbations effective dissipative gap edg, and touch upon second- to jump operators of L; the latter were first studied in Avron order perturbative effects. In a similar fashion, we study the et al. [70]. adiabatic response formula for Lindbladians in Sec. V. Next, we review the QGT, introduced for Hamiltonian There, we prove that adiabatic evolution within AsðHÞ is “ ” Δ systems in Ref. [74] (the term QGT was coined by Berry unitary and link edg to nonadiabatic corrections. In [75]). It encodes both a metric for measuring distances Sec. VI, we introduce the Lindbladian QGT and calculate [106] and the adiabatic curvature. The QGT is experimen- it for most of the examples discussed above. We discuss tally probable (e.g., via current noise measurements [107]). future directions in Sec. VII. Examples and links to the The Berry curvature can be obtained from adiabatic trans- appendixes are placed throughout the paper when physical port in Hamiltonian [108–110] and Lindbladian [70,111] concreteness or extra pedagogy are desired. systems and even ordinary linear response ([112] and Appxendix C of [113]). Singularities and scaling behavior of the metric are in correspondence with quantum phase III. ASYMPTOTIC PROJECTION transitions [114–116]. Conversely, flatness of the metric In this section, we apply the four-corners partition to and curvature may be used to quantify stability of a given Lindbladian superoperators and derive a formula for the – phase [117 120], a topic of particular interest due to its asymptotic projection P∞ for nonsteady AsðHÞ (H∞ ≠ 0). applications in engineering exotic topological phases. We also show how the presence of H∞ can influence the Regarding generalization of the QGT [question (4)], to ρ ρ dependence of ∞ on in and demonstrate how one can our knowledge there has been no introduction of a tensor embed any quantum channel in P∞. including both the adiabatic curvature and a metric for AsðHÞ. However, Refs. [121,122] did apply various known metrics to study distinguishability within families A. Four-corners partition of Lindbladians of Gaussian fermionic and spin-chain steady states, As we introduce in Sec. II, the four-corners projections respectively. Eq. (2.2) partition every operator A ∈ OpðHÞ into four Regarding leakage out of AsðHÞ [question (5)], the idea independent parts. Combining this notation with the that is not relevant to first-order nonadiabatic corrections vectorized or double-ket notation for matrices in OpðHÞ was mentioned in the Supplemental Material of Ref. [69]. (see Appendix A), we can express any A as a vector whose We extend that result to ordinary first-order perturbation components are the respective parts. The following are, theory. Regarding response (6), both ordinary [123–125] therefore, equivalent, and adiabatic [72,126] time-dependent perturbation theory 2 3 for Lindbladians have been developed earlier. In parallel jA⟫ 6 7 to this work, Campos Venuti and Zanardi [127] further A A 6 7 A ¼ ⟷ jA⟫ ¼ 4 jA⟫ 5; ð3:1Þ developed the Kubo formula for response to Lindladian A A perturbations to specific Lindbladians, most of which do jA⟫ not possess a decaying subspace. Lastly, regarding Hall conductivity, Avron et al. [70] and A ¼ A þ A. With A written as a block vector, used adiabatic perturbation theory to show that the zero- superoperators can now be represented as 3-by-3 block frequency Hall conductivity is unaffected by a Lindbladian matrices acting on said vector. Note that we use square whose jump operators are the Landau level lowering † brackets for partitioning superoperators and parentheses for (raising) operators b (b ). We confirm their result using operators in OpðHÞ [as in Fig. 1 and Eq. (3.1)]. We do so as linear response (calculated for all frequencies) and extend it well with the Lindbladian L. Recall that to jump operators that are powers of b. Still other jump operators are considered in Refs. [111,124]. 1X L ρ − ρ κ 2 lρ l† − l† lρ−ρ l† l ð Þ¼ i½H; þ2 lð F F F F F F Þ; l E. Structure of the paper ð3:2Þ In Sec. III, we prove Eq. (2.6) for P∞ by applying the L four-corners decomposition to . We also study the l ∈ H ρ ρ P with Hamiltonian H, jump operators F Opð Þ, and dependence of ∞ on in and show how ∞ can be used positive rates κl. By writing L ILI using Eqs. (2.3) to generate any quantum channel. The strategy of the rest of ¼ the paper is to apply the four-corners decomposition to and (2.4) (see Appendixes B and C), we find that leading-order response formulas from ordinary and adia- 2 3 L P LP P LP batic perturbation theory. In Sec. IV, we study the Kubo 6 7 6 7 formula for Lindbladians and state conditions under which L ¼ 4 0 L PLP 5; ð3:3Þ it reduces to a Hamiltonian-based formula. We also prove 00 L that the evolution within AsðHÞ is unitary, study the
041031-7 ALBERT, BRADLYN, FRAAS, and JIANG PHYS. REV. X 6, 041031 (2016) where L⊞ ≡ P⊞LP⊞. Note that L is a bona fide eigenvalue iΔ (used here as an index) and degeneracy index Lindbladian governing evolution within , and the minimal μ (that depends on Δ). By definition, jΨΔμ⟫ ∈ and the projection PΨ is exactly the asymptotic projection of L. eigenvalue equation is The reason for the zeros in the first column is the inability of L to take anything out of (stemming from the definition of LjΨΔμ⟫ ¼ iΔjΨΔμ⟫: ð3:8Þ the four-corners projections). This turns out to be sufficient P LP Since L is not always diagonalizable, any degeneracy may for to also be zero, leading to the block upper- Δ triangular form above. These constraints on L translate to induce a nontrivial Jordan block structure for a given . well-known [78,80,87] constraints on the Hamiltonian and However, it can be shown (see, e.g., Ref. [50], Appendix C) jump operators as follows (see Appendix B). that all Jordan blocks corresponding to asymptotic eigen- matrices are diagonal. Therefore, there exists a dual set of Proposition 1.—Let fP; Qg be projections on H and left asymptotic eigenmatrices ⟪JΔμj such that fP; P; P; Pg be their corresponding projections on Op H . Then Δμ Δμ ð Þ ⟪J jL ¼ iΔ⟪J j: ð3:9Þ ∀l∶ l 0 F ¼ ; ð3:4Þ The J are either conserved or oscillating indefinitely: i X − κ l† l ⟪ Δμ ρ ⟫ ⟪ Δμ tL ρ ⟫ iΔt⟪ Δμ ρ ⟫ H ¼ 2 lF F: ð3:5Þ J j ðtÞ ¼ J je j in ¼ e J j in ð3:10Þ l
l by trivial integration of the equations of motion [Eq. (2.17)]. These constraints on H and F (due to Hermiticity, For Δ ¼ 0, such J are conserved quantities, so a natural † H ¼ H) leave only their complements as degrees of question is whether they always commute with the freedom. The four-corners decomposition provides simple Hamiltonian and the jump operators. It turns out that they expressions for the surviving matrix elements of Eq. (3.3) do not always commute [50,78], and so various generaliza- l ’ in terms of H, F; these are shown in Appendix C. tions of Noether s theorem have to be considered [70,128]. Using the following analysis, we can say that J’s always DFSP case.—Recall that, in this case, AsðHÞ¼ and P ¼ d−1 jψ ihψ j is the DFS projection. In the case of a commute with both the Hamiltonian and jump operators k¼0 k k L nonsteady DFS, evolution within is exclusively unitary of when there is no decaying subspace (P ¼ I). If there is for all times and generated by a Hamiltonian superoperator decay, then conserved quantities still commute with jump operators and the Hamiltonian in the non-decaying subspace H∞ ≡ L. The jump operators in L, Eq. (C1), must then l 0 act trivially: (½J;F ¼ ; see Appendix B), but no longer have to commute in general (½J; Fl ≠ 0). l F ¼ alP ð3:6Þ The left and right eigenmatrices are dual in the sense that they can be made biorthogonal (while still maintaining the for some complex constants al. This implies that PLP orthonormality of the right ones): from Eq. (C5) is zero and the partition Eq. (3.3) becomes ⟪ Δμ Ψ ⟫ δ δ 2 3 J j Θν ¼ ΔΘ μν; H 0 P LP ∞ ⟪ΨΔμ ΨΘν⟫ δΔΘδμν: : 6 7 j ¼ ð3 11Þ L ¼ 4 0 L PLP 5: ð3:7Þ 00 L Outer products of such eigenmatrices can then be used to express the asymptotic projection X If we assume that jψ ki are eigenstates of H∞ (with Δμ P∞ ¼ jΨΔμ⟫⟪J j: ð3:12Þ H∞ ≡ −i½H∞; · ) and remember condition Eq. (3.5), Δ;μ we reduce to well-known conditions guaranteeing L ψ ψ 0 2 ðj kih kjÞ ¼ (Ref. [25], Theorem 1). This is indeed a projection (P∞ ¼ P∞) due to Eq. (3.11). Since it was shown that evolution of asymptotic states is B. Nonsteady asymptotic subspaces exclusively unitary (Ref. [78], Theorem 2), it must be that Armed with the partition of L from Eq. (3.3), we study the eigenvalue set fΔg is that of a Hamiltonian super- cases where AsðHÞ contains unitarily evolving states operator, which we define to be H∞ ≡ −i½H∞; · . In other [H∞ ≠ 0 from Eq. (2.5)]. The basis for AsðHÞ consists words, we use the set fΔg to construct a Hamiltonian of right eigenmatrices of L with pure imaginary eigenval- H∞ ∈ POpðHÞ (defined up to a constant energy shift) ues. By definition, we can expand jρ∞⟫ in such a basis such that each Δ is a difference of the energies of H∞ and L since all other eigenmatrices will decay to zero under et jΨΔμ⟫ are eigenmatrices of H∞. (Note that H∞ shares the for sufficiently large t. We call such eigenmatrices right same eigenvalues as P∞LP∞,butH∞ ≠ P∞LP∞ because asymptotic eigenmatrices jΨΔμ⟫ with purely imaginary the latter is not anti-Hermitian.) Because of this, the
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Ψ Δμ eigenmatrices f ;Jg must come in complex conjugate quantities of L ¼ H∞ are equal: jJ ⟫ ¼jΨΔμ⟫. † pairs: Ψ−Δμ ¼ ΨΔμ (which obstructs us from constructing a Splitting the degeneracy index μ into two indices k, l for Δμ Hermitian basis for fΨΔ≠0;μg) and the same for J . The convenience, one can express the right asymptotic eigen- Ψ explicit form of H∞ depends on the block diagonal vectors as Δ;kl ¼jψ kihψ lj, where fjψ kig is a basis for the structure of P∞. Combining P∞ with the definition of DFS consisting of eigenstates of H∞ with energies fEkg. H∞ yields The eigenvalue equation for ΨΔ;kl becomes X tL iΔt Δμ tH∞ lime ¼ e jΨΔμ⟫⟪J j ≡ e P∞: ð3:13Þ H∞ðΨΔ;klÞ¼−i½H∞; ΨΔ;kl ¼iðEl − EkÞΨΔ;kl; ð3:18Þ t→∞ Δ;μ where Δ ≡ Ek − El is a difference of the energies of H∞. The asymptotic state is then expressible as NS case.—Let us now focus on a stationary AsðHÞ (H∞ ¼ 0), meaning that all Δ ¼ 0, and we denote the ρ ⟫ tH∞ P ρ ⟫ j ∞ðtÞ ¼ e ∞j in ð3:14aÞ respective AsðHÞ basis elements and conserved quantities X Ψ ⟫ ≡ Ψ ⟫ μ⟫ ≡ Δ¼0;μ⟫ H as j μ j Δ¼0;μ and jJ jJ . Since Asð Þ is iΔt Ψ ⟫ ¼ cΔμe j Δμ ; ð3:14bÞ stationary, we can construct a Hermitian matrix basis for Δ;μ both AsðHÞ and the corresponding conserved quantities that uses one index and is orthonormal (under the trace). with complex coefficients For the DFS part of the NS, we define the matrix basis DFS d2−1 DFS ≡ ⟪ Δμ ρ ⟫ Δμ †ρ fjΨμ ⟫gμ 0 . Each Ψμ consists of Hermitian linear cΔμ J j in ¼ TrfðJ Þ ing: ð3:15Þ ¼ superpositions of the outer products jψ kihψ lj and is not ρ These coefficients determine the footprint that in leaves on a density matrix. In this new notation, the basis elements for ρ ρ ⟫ P ∞. In general, any part of j in not in the kernel of ∞ one NS block are then imprints on the asymptotic state since, by definition, that part Δμ 1 ΨDFS⟫ ⊗ ϱ ⟫ 0 overlaps with some J . Ψ ⟫ j μ j ax Δμ⟫ j μ ¼ : ð3:19Þ We proceed to determine jJ by plugging in the nax 00 partition of L from Eq. (3.3) into the eigenvalue equa- tion (3.9). The block upper-triangular structure of L readily We normalize the states using the auxiliary state norm Δμ implies that jJ ⟫ are left eigenmatrices of L: (purity), pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Δμ Δμ ≡ ⟪ϱ ϱ ⟫ ϱ2 ⟪J jL ¼ iΔ⟪J j: ð3:16Þ nax axj ax ¼ Trf axg; ð3:20Þ ⟪Ψ Ψ ⟫ δ Writing out the conditions on the remaining components to ensure that μj ν ¼ μν. Since a NS block is a Δμ Δμ combination of the unique and DFS cases, the conserved jJ ⟫ yields an analytic expression for jJ ⟫. We state this quantities of (i.e., of L) are direct products of the DFS formula below, noting that ½L; P ¼0; the proof is and auxiliary conserved quantities [78,79]. The unique given in Appendix B. — L auxiliary conserved quantity is Pax, the identity on the Proposition 2. The left eigenmatrices of correspond- H auxiliary subspace ax. Combining this with the result ing to pure imaginary eigenvalues iΔ are μ n Ψμ J above and multiplying by ax so that and are P biorthogonal [see Eq. (3.11)], we see that ⟪ Δμ ⟪ Δμ P − L J j¼ J j ; ð3:17Þ L − iΔP ⟪ΨDFS ⊗ ⟪ 0 μ μ j Paxj ⟪J j¼n μ : ð3:21Þ Δμ ax 0 ⟪J where ⟪J j are left eigenmatrices of L. j Δ 0 Plugging this result into Eq. (3.12) and setting ¼ We use the NS block basis of the above form throughout the yields the formula for P∞ from Sec. II for the case when paper. The asymptotic projection P∞ is then 0 H∞ ¼ . We now go through the relevant special cases, X introducing notation used throughout the rest of the paper. μ −1 P∞ ≡ jΨμ⟫⟪J j¼PΨ − PΨLL ; ð3:22Þ Unique state case.—Here, AsðHÞ is stationary because μ there is only one state ϱ. The corresponding conserved L where the minimal projection is quantity is the identity I (since et preserves the trace). In X the double-ket notation, the asymptotic projection P ≡ Ψ ⟫⟪ μ ≡ P ⊗ ϱ ⟫⟪ P ϱ⟫⟪ Ψ j μ Jj DFS j ax Paxjð3:23Þ Eq. (2.13) can be written as ∞ ¼j Ij. Note that P μ is the conserved quantity of L. P P ΨDFS⟫⟪ΨDFS ⟫ DFS case.—In this case, all states in are asymptotic. and DFSð·Þ¼ μj μ μ j · ¼ PDFS · PDFS is the Therefore, steady-state basis elements and conserved superoperator projection onto the DFS part. Applying this
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⊗ 1. Example: Four-level system to a state and remembering that P ¼ PDFS Pax yields the NS projection formula (2.20). Let H be four dimensional, with the first two levels Multiblock case.—If there are two NS blocks (charac- ⊥ ⊥ fjψ 0i; jψ 1ig being a DFS and the latter two fjψ 0 i; jψ 1 ig ðϰÞ ⊗ ðϰÞ ϰ ∈ 1 2 terized by projections PDFS Pax with f ; g) and decaying into the DFS. Let H ¼ 0 and no decaying subspace, then the conserved quantities ϰ;μ DFS ðϰÞ X1 X1 J ¼ Ψϰ μ ⊗ P do not have presence in the subspace ; ax F ¼ jψ ihψ ⊥jþα ð−1Þkjψ ⊥ihψ ⊥j; ð3:27Þ of coherences between the blocks. Since the most general k k k k k¼0 k¼0 AsðHÞ is a direct sum of such NS blocks [78,80,81],we μ can shade gray the blocks in which J may not be zero where the first term F makes sure that everything flows Ψ [Fig. 1(b)], dual to μ [Fig. 1(a)]. into the DFS and the last term F dephases the non-DFS Bloch vector (with α ∈ R). The steady-state basis elements C. Dependence of ρ∞ on ρin and H∞ are Ψkl ¼jψ kihψ lj since F ¼ 0. We can then use Eq. (3.17): acting on Ψ with the adjoint of L (see Here, we examine how ρ∞ depends on ρ , showing how kl in L−1 H∞ can suppress purity of ρ∞. The coefficients Eq. (3.15) Appendix A) and then the adjoint of [Eq. (C4)] yields ρ ⟫ ρ ⟫ the corresponding conserved quantities determining the dependence of j ∞ on j in can be split into two parts, jψ ⊥ihψ ⊥j Jkl ψ ψ k l : 3:28 ⟪ Δμ ρ ⟫ ⟪ Δμ ρ ⟫ ¼j kih ljþ1 2α2 1 − δ ð Þ cΔμ ¼ J j in þ J j in ; ð3:24Þ þ ð klÞ
P ρ ⟫ kl with each part representing the footprint left by j in One can see that J ¼ Ψkl, a feature of the DFS case, and P ρ ⟫ ⊥ kl and j in , respectively. We can readily see that coher- the absence of jψ kihψ j terms in J , a key result of the P ρ ⟫ ρ ⟫ l ences j in decay and cannot imprint in j ∞ . The paper. The only nontrivial feature of the steady state is −1 second term can be expressed using Proposition 2: due to F and the resulting “scrambling term” L in
Δμ Δμ −1 Eq. (3.25). Namely, an initial nonzero coherence ⟪J jρ ⟫ ¼ −⟪J jPLPðL − iΔÞ jρ ⟫: ð3:25Þ ψ ⊥ ρ ψ ⊥ in in h 0 j inj 1 i leads necessarily to a mixed steady state due to coherence suppression of order Oðα−2Þ. Reading from right to left, this part first “scrambles” α 0 P ρ ⟫ “ ” Letting ¼ , a similar effect can be achieved by j in via the inverse term, then transfers the result 1 adding the Hamiltonian H¼ βðjψ 0ihψ 0j−jψ 1ihψ 1jÞ (with ρ⋆ into via the channel [Eq. (C7)] 2 β ∈ R). Now the DFS is nonstationary (with H∞ ¼ H) and X Ψ P LP ρ⋆ κ l ρ⋆ l† the off-diagonal DFS elements k≠l rotate. Abusing nota- ð Þ¼ lF F ; ð3:26Þ Δ β l tion by omitting the corresponding eigenvalue ¼ , the Δμ left asymptotic eigenvectors become and, finally, “catches” that result with ⟪J j. The footprint thus depends on all three actions. The transfer channel in ψ ⊥ ψ ⊥ kl ψ ψ j k ih l j Eq. (3.26) is completely positive (Ref. [56], Theorem 8.1). J ¼j kih ljþ l : ð3:29Þ 1 þ iβð−Þ ð1 − δklÞ One can see that this map has to be nonzero for J ≠ 0, i.e., for any footprint to be left at all. This is indeed true when Despite F ¼ 0, the scrambling term still inflicts damage one remembers that all populations in are transferred to the initial state due to H∞ (for nonzero β), but now the since Lindbladian evolution is trace preserving (see coherence suppression is of order Oðβ−1Þ. Appendix C). L − Δ −1 Δ Now observe the scrambling term ð i Þ . Since is D. Quantum channel simulation an energy difference from H∞, this tells us that unitary Here, we show how to embed any quantum channel into evolution in AsðHÞ affects the dependence of jρ∞⟫ on P∞. Recall that a quantum channel E taking a state ρ from a Pjρ ⟫. This effect cannot be removed by transforming in H tH∞ d -dimensional input space to a d -dimensional into a rotating frame via e . In such a frame, jρ∞⟫ in in out Δμ output space H acts as becomes a steady state, but the Δ dependence of J (and out X therefore the expression for cΔμ) remains. This is because † EðρÞ ≡ ElρEl; ð3:30Þ tH∞ the evolution caused by e is happening in conjunction l P ρ ⟫ with the nonunitary decay of j in , which can be interpreted as H∞ affecting the “flow” of parts of whereP El are dout-by-din-dimensional matrices and P ρ ⟫ H † j in into Asð Þ. One can thus see that the energy lElEl is the identity on din. We construct a correspond- denominator (due to H∞ ≠ 0) may dampen the purity of the ing L such that E ¼ P∞P, with the input space matched asymptotic state. We highlight this with a specific example. to and output space to . First, set all rates κl of the
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κ Lindbladian equal to one rate eff , which quantifies con- evolving from the right side of the expression to the left. vergence to AsðHÞ. Let H ¼ 0 and pad El with zeros to Reading the integrand from right to left, the steady state is obtain jump operators of dimension din þ dout, perturbed by δL at a time τ, then evolved under the unperturbed Lindbladian L, and finally evaluated using 0 El the observable A at a time t ≥ τ. The integral represents a Fl ¼ Fl ¼ : ð3:31Þ 00 sum over different times τ of the perturbation acting on the steady state. Removing ⟪Aj produces the first-order term in This DFS case greatly simplifies the matrix elements of L the Dyson series for jρðtÞ⟫. in Appendix C. The decay-generating terms reduce to We dissect Hamiltonian and jump operator perturbations L − 1 κ P L −κ P L ¼ 2 eff and ¼ eff , so one can think of of [Eq. (3.2)], respectively, κ as the inverse of a relaxation time T1 for . Using the eff → Kraus form for the transfer term of P∞ from Eq. (3.26) and H H þ gðtÞV; ð4:3aÞ simplifying yields Fl → Fl þ gðtÞfl ð4:3bÞ 1 P P −P LL−1 P LP E ∞ ¼ ¼ κ ¼ : ð3:32Þ [for V, fl ∈ Op H and V† V],showingthatbothgenerate eff ð Þ ¼ unitary evolution within all AsðHÞ and leakage caused by both In other words, while not all quantum channels can be does not take states into . We handle the Hamiltonian case expressed as etL for any finite t, all can be embedded in first for simplicity, tL some P∞ ¼ limt→∞e . δL ¼ −i½V;· ≡ V; ð4:4Þ
IV. LINEAR RESPONSE returning to the general case in Sec. IV D. In this section, we apply the four-corners decomposition Hamiltonian case.—As a sanity check, we let L ¼ H ¼ to the Kubo formula. For both Hamiltonian and jump −i½H; · and massage Eq. (4.2) into standard form. operator perturbations, we show that evolution within For that, let OðtÞ ≡ eiHtOe−iHt ¼ e−tHðOÞ and recall that AsðHÞ is of Hamiltonian form and that leakage out of ½H; ρ∞ ¼0, since ρ∞ is generically a superposition of AsðHÞ is governed by the effective dissipative gap. projections on eigenstates of H. We can then commute eiHt with ρ∞ and cyclically permute under the trace to obtain A. Decomposing the Kubo formula Z 1 t Let us assume that time evolution is governed by a ⟪δAðtÞ⟫ ¼ dτgðτÞTrf½Aðt − τÞ;V ρ∞g: ð4:5Þ i −∞ Lindbladian L and the initial state ρ∞ is steady; i.e., Lðρ∞Þ¼0. The system is then perturbed as We now use four-corners projections P⊞ to partition Eq. (4.2). Because of the no-leak property [Eq. (2.9)], L → L þ gðtÞδL; ð4:1Þ PVP ¼ 0. Remembering that the Lindbladian is block upper-triangular in the four-corners partition [see where the perturbation superoperator δL is multiplied by a Eq. (3.3)], it follows that eLt is also block upper- time-dependent factor gðtÞ. The Lindbladian-based Kubo triangular. We do not make any assumptions on A: formula [123,125,127,129,130] is derived analogously to ⟪A ⟪A ⟪A ⟪A . Further decomposing the first the Hamiltonian formula; i.e., it is a leading-order Dyson j¼½ j j j term using the asymptotic projection P∞ from Eq. (3.22) expansion of the full evolution. The main difference is that Q ≡ I − P the derivation is performed in the superoperator formalism. and its complement ∞ ∞ yields Z We study the difference between the perturbed and unper- t ðt−τÞH∞ turbed expectation values, ⟪δAðtÞ⟫ ≡ ⟪AjρðtÞ − ρ∞⟫ for ⟪δAðtÞ⟫ ¼ dτgðτÞ⟪Aje Wjρ∞⟫ ð4:6aÞ some observable A. We remind the reader that we use −∞ vectorized notation for matrices and the Hilbert-Schmidt Z † t ⟪ ρ ⟫ ≡ ρ ðt−τÞL inner product Aj ðtÞ TrfA ðtÞg (see Appendix A). þ dτgðτÞ⟪Aje Q∞PVjρ∞⟫ ð4:6bÞ Within first order in g, the Kubo formula is −∞ Z Z t t ðt−τÞL ðt−τÞL ⟪δAðtÞ⟫ ¼ dτgðτÞ⟪Aje δLjρ∞⟫: ð4:2Þ þ dτgðτÞ⟪Aje PVjρ∞⟫: ð4:6cÞ −∞ −∞
While this superoperator form looks very different from the The terms differ by which parts of V perturb ρ∞ and also usual time-ordered commutator expression, it offers an which parts of A “capture” the evolved result. The three intuitive interpretation if one thinks of the system as relevant parts of A correspond to the three labels in Fig. 2.One
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above decomposition is useful with an important example.
1. Example: Hall conductivity with dissipation As an application of the Lindblad Kubo formula, let us consider a quantum Hall system with Markovian dissipa- tion. We do not aim to represent physically sensible environments of electronic systems; such environments have already been thoroughly studied (see, e.g., FIG. 2. Sketch of regions of linear response of the asymptotic H Ref. [131]). Rather, we aim to describe artificial quantum subspace Asð Þ (gray) to a Hamiltonian perturbation. Each of Hall systems induced by light-matter interactions and/or three regions A, B, and C corresponds to the respective response term Eqs. (4.6a), (4.6b), and (4.6c) in the text. photonic reservoir engineering. Such systems are being extensively studied both theoretically [38–41] and exper- imentally [43–45,132]. can readily see that A is irrelevant to this order due to Consider a two-dimensional system of N particles of Eq. (2.9).Equation(4.6a) consists of perturbing and evolving mass m, charge e ¼þ1, position r, and momentum p,inan A 2 within the asymptotic subspace , shaded gray in the Fig 2. area A ¼ L and external magnetic field B (with ℏ ¼ 1). The effect of the perturbation within AsðHÞ is W ≡ P∞VP∞ The Hamiltonian is (shown in Sec. IV B to be of Hamiltonian form), and H∞ is the part of the unperturbed L that generates unitary evolution 1 X 1 X πi πi within AsðHÞ. Equation (4.6a) therefore most closely resem- H ¼ 2 ς ς þ 2 Uij m ≠ bles the traditional Hamiltonian-based Kubo formula. The i i j H X 1 1 X remaining two terms quantify leakage out of Asð Þ and ω † contain non-Hamiltonian contributions. Equation (4.6b) con- ¼ c bi bi þ 2 þ 2 Uij; ð4:8Þ ≠ sists of perturbing into regions B and C in Fig. 2,butthen i i j P tLP B evolving under e strictly into region (since where i; j ∈ f1; …;Ng are particle indices and ς, τ ∈ P tLQ 0 ∞e ∞ ¼ ). Equation (4.6c) consists of perturbing into fx; yg index the spatial direction (with repeated indices tL region C and remaining there after evolution due to Pe P. summed). Above, we define the kinetic momentum 0 This term is eliminated if A ¼ , i.e., if the observable is πi ¼ pi − A (with A the magnetic vector potential, strictly in . i j ½πς; πτ ¼iBϵςτδ , and ϵςτ the antisymmetric Levi-Cività DFS case.—Recall that in this case is a DFS ij symbol), the Landau level lowering operators (P∞P ¼ P), and we do not assume it is stationary (H∞ ≠ 0). From Eq. (3.7), we can see that L cannot take 1 ffiffiffiffiffiffi πj πj any coherences in back into the DFS (PLP ¼ 0). bj ¼ p ð x þ i yÞ; ð4:9Þ 2B Therefore, the interference term [Eq. (4.6b)] is eliminated and the response formula reduces to a two-electron interaction potential Uij, and the cyclotron Z frequency ωc ¼ B=m. For simplicity, we assume t ⟪δ ⟫ τ τ ⟪ ðt−τÞH∞ W ρ ⟫ AðtÞ ¼ d gð Þ Aje j ∞ ð4:7aÞ † −∞ 0 ½Uij;bk ¼½Uij;bk ¼ : ð4:10Þ Z t ðt−τÞL Let us take the number of electrons N to satisfy ν ≡ þ dτgðτÞ⟪Aje PVjρ∞⟫: ð4:7bÞ −∞ 2πN=BA ¼ p=q ≤ 1 for p; q ∈ Z, and let us assume the interaction potential Uij is chosen such that there is a gap If, furthermore, A ¼ 0, there are no interference terms above the ground state j0i [133] in the absence of coming from outside of the DFS and the Lindbladian linear dissipation. We take for our perturbation the electric response reduces to the purely Hamiltonian-based term potential corresponding to a uniform electric field, [Eq. (4.7a)]. Such a simplification can also be achieved when X 0 i V ¼ , which implies that the Hamiltonian perturbation does Vτ ≡ −i½Vτ; · with Vτ ≡ −Eτ rτ; ð4:11Þ not take ρ∞ out of the DFS to begin with (PVP ¼ 0). i In the next section, we use the no-leak and clean-leak properties to determine that evolution within AsðHÞ is of and we measure the total current in the ς direction. The 0 Hamiltonian form and to quantify the leakage scale of the frequency-dependent conductivity tensor σςτ for the remaining two terms [Eqs. (4.6b) and (4.6c)]. Before Hamiltonian system can be extracted from Eq. (4.2) and doing that, however, let us first show how and when the is given by [77]
041031-12 GEOMETRY AND RESPONSE OF LINDBLADIANS PHYS. REV. X 6, 041031 (2016) Z 1 ∞ 1 X γ 0 iðωþiϵÞt 0 tH 0 i σςτðωÞ¼ lim dte ⟪Jςje Vτjρ∞⟫ ð4:12aÞ Jς ¼ Jς þ ϵςτ πτ: ð4:20Þ ϵ↓0 ω A 0 m i c νω c ωδ − ω ϵ ¼ 2 2 ði ςτ c ςτÞ; ð4:12bÞ With this form of the current operator and our choice of Fi 2πðω − ωcÞ and Uij, Eq. (4.19) can be evaluated for all frequencies: where ρ∞ ¼j0ih0j is the ground state, the total current γ σ ω δ ϵ σ0 ω~ 1 X ςτð Þ¼ ςλ þω ςλ λτð Þ 0 i c Jς ¼ πς ð4:13Þ m νω γω~ i c ωδ −ω 1− ϵ ¼ ~ 2 2 i ςτ c i 2 ςτ ; ð4:21Þ 2πðω −ωcÞ ωc is obtained from the Hamiltonian-based continuity equation, H − and ¼ i½H; · . We can further extract the quantized zero- with complex frequency ω~ ≡ ω þ iγ. Quite surprisingly, frequency Hall conductivity: the Hall conductivity at zero frequency is still given by its quantized value, 0 ν σH ≡ σxyðω → 0Þ¼ : ð4:14Þ 2π ν σxyðω → 0Þ¼σH ¼ ; ð4:22Þ We now examine the fate of the conductivity in the 2π presence of dissipation. Let us subject the system to due to an interesting interplay between the Lindbladian Lindblad evolution [Eq. (A3)] with rates κi ¼ 1 and single-particle jump operators time evolution and the modification to the current operator. This effect can also be observed when calculating the pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 quantized Hall conductivity using adiabatic perturbation F ¼ 2γ1b þ 2γ2b þ : ð4:15Þ i i i theory (Ref. [70], Sec. 7). It is even present when we extend this case to the case of a low (but nonzero) temperature Note that the coefficients γȷ must be independent of particle thermal bath, up to exponential corrections due to leakage index i for identical particles. One has to be careful about out of the lowest Landau level (see Appendix D). defining the current operator Jς. The current density jς r ð Þ Additionally, we see that the usual cyclotron pole at now obeys the Lindbladian-based continuity equation, ω ¼ ωc—guaranteed to be present in the Hamiltonian case by Kohn’s theorem [134]—is broadened into a Lorentzian ∂ ∂ L‡ ∂ 0 tn þ ςjς ¼ ðnÞþ ςjς ¼ ; ð4:16Þ due to the presence of dissipation. This shows that while the P δ r − r cyclotron resonance is independent of the details of where n ¼ i ð iÞ is the particle density operator (see interactions, it is in fact sensitive to dissipation. Appendix A for a formal definition of ‡). The total current DFS case.—Here, we look at the case when γ1 ¼ 0 and is then expressed as γȷ>1 ≠ 0. Now the asymptotic subspace consists of all states Z in the lowest two Landau levels—a DFS case. Therefore, it 1 2 Jς ¼ d rjςðrÞ: ð4:17Þ is useful to consider the DFS Kubo formula [Eq. (4.7)]. The L key point now is that the perturbation Vτ leaves ρ∞ in the This is the sensible and measurable definition of current in steady-state subspace, and, hence, the second term a dissipative system (Ref. [70], Sec. 5.2) since it represents [Eq. (4.7b)] in the Kubo formula vanishes. Although the the time-rate change of charge density in a region. Taking current operators Jς, determined by Eq. (4.18), depend on the Fourier transform of Eq. (4.16) and expanding to lowest the jump operators Fi, the projection ðJςÞ, which appears order in wave vector yields in the first term [Eq. (4.7a)], is independent of Fi and equivalent to the Hamiltonian-based current Eq. (4.13): X X 0 i ‡ i ðJςÞ ¼ Jς. These two observations conspire to ensure that Jς ¼ ∂trς ¼ L ðrςÞ; ð4:18Þ i i the conductivity at all frequencies is unaffected by dis- 0 sipation and is still given by σςτðωÞ from Eq. (4.12b). and the Kubo formula (4.12a) generalizes to Z 1 ∞ B. Evolution within AsðHÞ iðωþiϵÞt tL σςτðωÞ¼ lim dte ⟪Jςje Vτjρ∞⟫: ð4:19Þ A ϵ↓0 0 Let us now focus on the term W ≡ P∞VP∞ [Eq. (4.6a)] quantifying the effect of the perturbation within AsðHÞ. Unique state case.—We first consider the case when Becayse of a lack of a formula for P∞, it was previously γ ≡ γ1 ≠ 0, γȷ>1 ¼ 0, so that Fi ∝ bi. The key observation unclear whether W is capable of causing any decoherence is that the current operator is given by within AsðHÞ. We now show that it is not. Therefore, the
041031-13 ALBERT, BRADLYN, FRAAS, and JIANG PHYS. REV. X 6, 041031 (2016) first-order effect of the perturbation within AsðHÞ will weighed by the expectation value of the corresponding ι ϱ always be of Hamiltonian form. auxiliary operator Vax in the state ax. A swift application of the no-leak and clean-leak proper- ties, Eqs. (2.9) and (2.11), allows us to substitute PΨ ≡ C. Leakage out of AsðHÞ P∞P for P∞. Recall that PVP ¼ 0 and that W is Now, let us set H∞ ¼ 0 and focus on the two leakage acting on a steady state ρ∞ ∈ As H , yielding ð Þ terms [Eqs. (4.6b) and (4.6c)] from the Kubo formula. For V W ρ ⟫ P P VP ρ ⟫ P VP ρ ⟫ simplicity, let us slowly ramp up the perturbation gðtÞ to a j ∞ ¼ ∞ ∞j ∞ ¼ Ψ Ψj ∞ : ð4:23Þ ηtΘð−tÞ constant, so gðtÞ¼limη→0e , with ΘðtÞ the Heaviside step function. This simplifies the leakage part of the Kubo As seen from the full Kubo formula, this term is of the same L order in the perturbation as the two leakage terms formula using the Drazin inverse of : 0 Z Z [Eqs. (4.6b) and (4.6c)]. However, if H∞ ¼ and if the t ∞ ðt−τÞL tL −1 perturbation is turned on for a finite time T and rescaled by dτgðτÞe Q∞ ¼ dte Q∞ ≡ −L : ð4:29Þ 1=T, it can be shown [59,62,96] that W is the only leading- −∞ 0 order effect. Therefore, the entire state undergoes quantum −1 −1 This pseudoinverse (L L ¼ LL ¼ Q∞) is also the Zeno dynamics according to W (Refs. [63,64,66]; see also inverse of all invertible parts in the Jordan normal form Refs. [97,98]). We show below that such dynamics is of L (Ref. [59], Appendix D). Plugging this in and unitary for all As H . ð Þ omitting ⟪Aj, the leakage terms Eqs. (4.6b) and (4.6c) DFS case — . We immediately read off the effective reduce to Hamiltonian for the DFS case. Since PΨ ¼ P, −1 Q∞jρðtÞ⟫ ¼ −L Vjρ∞⟫: ð4:30Þ W ¼ −i½V; · ; ð4:24Þ Now we can apply the clean-leak property Eq. (2.11) to with V the perturbation projected onto the DFS. narrow down those eigenvalues of L that are relevant in Applications of this formula to circuit and waveguide characterizing the scale of the leakage. By definition QED quantum computation schemes can, respectively, Eq. (4.29), L−1 has the same block upper-triangular be found in Refs. [61,62]. structure as L from Eq. (3.3). This fact conspires with NS case.—In this case, we have to use the formula for PVP ¼ 0 to allow us to ignore L and write PΨ from Eq. (3.23), restated below: −1 Q∞jρðtÞ⟫ ¼ −L Vjρ∞⟫: ð4:31Þ P P ⊗ ϱ ⟫⟪ Ψ ¼ DFS j ax Paxj; ð4:25Þ Therefore, the relevant gap is the nonzero eigenvalue of P with DFSð·Þ¼PDFS · PDFS being the superoperator pro- L with the smallest absolute value. However, we now jection on the DFS part, P being the operator projection ax show how the spectrum of L is actually contained in on the auxiliary part, and P ¼ P ⊗ P . Direct multi- DFS ax the spectrum of L þ L. Recalling the block upper- plication yields triangular structure of L from Eq. (3.3), one can establish L W P VP ⟪ V ϱ ⟫ ⊗ ϱ ⟫⟪ that its eigenvalues must consist of eigenvalues of , ¼ Ψ Ψ ¼ Paxj j ax j ax Paxj; ð4:26Þ L, and L. However, evolution of the two coherence blocks is decoupled, L ¼ L þ L (see Appendix C), where the evolution within the auxiliary part is trivial and and eigenvalues of L come in pairs. Therefore, one can evolution within the DFS part is generated by the effective Δ DFS Hamiltonian W: then define the effective dissipative gap edg to be the nonzero eigenvalue of L þ L with the smallest abso- ⟪ V ϱ ⟫ − ϱ ≡ − lute value. Paxj j ax ¼ i½Traxf axVg; · i½W;· : ð4:27Þ As a brief aside, we mention that the piece L is also not −1 ϱ relevant in a term P∞VL VP∞ [135–137] that acts on To better reveal the effect of ax, it is worthwhile to express AsðHÞ and is second order in the perturbation. Since V as a sum of tensor productsP of various DFS and −1 ι ⊗ ι PVP ¼ 0, one can reduce this term to P∞VL VPΨ. auxiliary Hamiltonians: V ¼ ιV Vax. The effective Hamiltonian then becomes However, we cannot replace the remaining P∞ with PΨ V X since two actions of can take the state from to . ϱ ι ι DFS case.—Recall that now all of is stationary W ¼ Traxf axVaxgV : ð4:28Þ ι (provided that H∞ ¼ 0). We show that for certain DFS Δ cases, edg is the excitation gap of a related Hamiltonian. In words, W is a linear combination of Hamiltonian Such DFS cases are those where L [Eq. (A3)] can be perturbations Vι on the DFS, with each perturbation written without a Hamiltonian part,
041031-14 GEOMETRY AND RESPONSE OF LINDBLADIANS PHYS. REV. X 6, 041031 (2016) X 1 l l† l† l l† l L ρ κ 2 ρ − ρ − ρ 10 Δedg ð Þ¼2 lð F F F F F F Þ; ð4:32Þ l 8 Δdg and where DFS states are annihilated by the jump oper- 6 l l l ators, F jψ ki¼0. This implies that F ¼ PF P ¼ 0 P 4 d−1 ψ ψ Δ (with P ¼ k¼0 j kih kj). We now determine edg for such systems. Since there is no evolution in , 2 L ¼ 0. Borrowing from Appendix C and using the above assumptions, 0.5 1.0 1.5 2.0
X FIG. 3. Plot of the effective dissipative gap Δedg, the nonzero 1 l† l L ρ − κ ρ eigenvalue of L with smallest real part, and the dissipative gap ð Þ¼ 2 lP ðF F Þ: ð4:33Þ l Δdg versus α for the Lindbladian with jump operator 2 2 F ¼ a − α . One can see that Δedg ≥ Δdg. From this, we can extract the decoherence [138] or parent [31] Hamiltonian: thereby covering all L. Namely, just like Hamiltonian 1 X ≡ κ l† l perturbations V, jump operator perturbations induce unitary Hedg 2 lF F : ð4:34Þ l evolution within AsðHÞ and the leakage scale associated Δ ≥ Δ with them is still edg dg. The (zero-energy) ground states of Hedg are exactly the Returning to Eq. (4.4), the action of the perturbation to ψ DFS states j ki [31,138] and the excitation gap of Hedg first order in g is characterized by Δ is edg. 1 δL ρ ≡ Y ρ κ ρ † − † † ρ 1. Example: Driven two-photon absorption ð Þ ð Þ¼ F f þ H:c: 2 ff F þ F f; g ; As an example of the above DFS simplification, consider ð4:37Þ the bosonic Lindbladian [61,71,139,140] with one jump 2 − α2 κ 1 α ∈ R operator F ¼ a and rate ¼ , where , κ † ≡ † α with being the rate corresponding to the jump operator F ½a; a ¼I and n a a. For sufficiently large , this l Lindbladian possesses a DFS spanned by the bosonic (we ignore the index for clarity). We hope to invoke the clean-leak property [Eq. (2.11)] once again, but the first coherent states jαi and j− αi. All states orthogonal to α term on the right-hand side of the above equation acts j i constitute the decaying subspace . The decoherence ρ Hamiltonian is readily calculated to be simultaneously and nontrivially on both sides of . There is thus a possibility that one can reach when acting with Y ρ∞ F 0 1 2 2 †2 4 on a steady state . However, the condition ¼ from H ¼ ½nðn − 1Þ − α ða þ a Þþα : ð4:35Þ † edg 2 Proposition 1 implies that PðFρ∞f Þ is zero for all f,so one can still substitute PΨ for P∞: Δ The excitation gap of Hedg ( edg) is plotted in Fig. 3 versus α, along with Δ and the eigenvalue of L with smallest dg P∞YP∞jρ∞⟫ ¼ PΨYPΨjρ∞⟫: ð4:38Þ real part. One can see that for α > 1.5, the dissipative gap of L is smaller and does not coincide with the energy scale P YP 0 governing leakage. Furthermore, the fact that ¼ allows us to ignore in determining the leakage energy scale associated with D. Jump operator perturbations these jump operator perturbations. We finish with calculat- ing the corresponding effective Hamiltonian for the most Having covered Hamiltonian perturbations, let us return general cases. to jump operator perturbations of the Lindbladian Eq. (3.2). NS case.—Having eliminated the influence of the Recall from Eq. (4.3b) that decaying subspace , we can now repeat the calculation done for Hamiltonian perturbations using the NS projection → F F þ gðtÞf; ð4:36Þ Eq. (4.25), yielding with f ∈ OpðHÞ, not necessarily Hermitian. It was first P YP ⟪ Y ϱ ⟫ ⊗ ϱ ⟫⟪ shown in Ref. [60] that such perturbations actually induce Ψ Ψ ¼ Paxj j ax j ax Paxj: ð4:39Þ unitary evolution on NS blocks of those Lindbladians that do not possess a nontrivial decaying subspace (P ¼ I). After some algebra, the DFS part reduces to Hamiltonian ≠ ⟪ Y ϱ ⟫ − Here, we extend this interesting result to cases where P I, form [60]: Paxj j ax ¼ i½Y;· , where
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i † † a time-dependent Lindbladian LðsÞ, where the end time T Y ≡ κTr fϱ ðF f − f FÞg: ð4:40Þ 2 ax ax is infinite in the adiabatic limit. For all s, we define a continuous and differentiable family of instantaneous — Multiblock case. We now sketch the calculation of asymptotic subspaces with corresponding asymptotic both Hamiltonian and jump operator perturbations, projections δL V Y ¼ þ , for the most general case of housing X multiple NS blocks. Once again, we can get rid of the PðsÞ ΨðsÞ⟫⟪ μ P P ∞ ¼ j μ J ðsÞj; ð5:1Þ decaying subspace and substitute Ψ for ∞. In addition, μ since PΨ does not have any presence except within the (gray) NS blocks of [(see Fig. 1), PΨ will not project ðsÞ ðsÞ steady-state basis elements Ψμ [such that LðsÞjΨμ ⟫¼0], onto any coherences between the NS blocks. The contrib- μ μ and conserved quantities J ðsÞ [such that ⟪J ðsÞjLðsÞ¼0]. uting part of P∞δLP∞ thus consists of the Hamiltonian and The dimension of the instantaneous subspaces (i.e., the jump operator perturbations projected onto each NS block. P ðsÞ Combining the effective Hamiltonians arising from V and Y rank of ∞ ) is assumed to stay constant during this L s [respectively, Eqs. (4.27) and (4.40)], the effective evolution evolution. In other words, the zero eigenvalue of ð Þ is ϰ isolated from all other eigenvalues at all points s by the within the DFS part of each NS block (indexed by )is Δ generated by the Hamiltonian dissipative gap dg (analogous to the excitation gap in Hamiltonian systems). We further assume that s ∈ ½0; 1 M ðϰÞ ðϰÞ ðϰÞ i † † parametrizes a path in a space of control parameters , X ≡ Trax ϱax V þ κðFf − fFÞ : ð4:41Þ 2 whose coordinate basis is fxαg. In other words, we can parametrize In fact, the unprojected Hamiltonian, 1 1 X ∂ ∂ x˙ ∂ i t ¼ s ¼ α α; ð5:2Þ ≡ κ † − † T T X V þ 2 ðF f f FÞ; ð4:42Þ α has previously been introduced (Ref. [70], Theorem 5) as the where ∂s is the derivative along the path, ∂α ≡∂=∂xα are operator resulting from joint adiabatic variation of the derivatives in various directions in parameter space, and Hamiltonian and jump operators of L. It is thus not surprising x˙ ≡ dxα α ds are (unitless) parameter velocities. that the effect of perturbations to the Hamiltonian and jump Following Ref. [72], starting with an initially steady state ρ H operators on ∞ is X projected onto Asð Þ. jρð0Þ⟫ ∈ AsðHÞ, adiabatic perturbation theory is an expan- sion of the equation of motion V. ADIABATIC RESPONSE 1 ∂ jρðsÞ⟫ ¼ LðsÞjρðsÞ⟫ ð5:3Þ We now apply the four-corners decomposition to adia- T s batic perturbation theory. Here, the leading-order term governs adiabatic evolution within AsðHÞ while all other in a series in 1=T. Each term in the expansion is further terms are nonadiabatic corrections. We show that for a divided using the decomposition I ¼ P∞ þ Q∞ into terms cyclic adiabatic deformation of steady AsðHÞ, the holon- inside and outside the instantaneous AsðHÞ. This allows omy is unitary. We also determine that the energy scale one to derive both the adiabatic limit (when T → ∞) and all governing nonadiabatic corrections is once again governed corrections. The Oð1=TÞ expansion for the final state from Δ by the effective dissipative gap edg. Theorem 6 of Ref. [72] reads 1 0 −1 ˙ ðsÞ 0 A. Decomposing the adiabatic formula jρðsÞ⟫¼Uðs; Þjρð0Þ⟫þ L ðsÞP∞ Uðs; Þjρð0Þ⟫ Z T First, let us briefly recall the setup of the standard 1 s ðs;rÞ ˙ −1 ˙ ðrÞ ðr;0Þ adiabatic limit for Lindbladians (see Sec. II D for a þ drU fP∞L P∞g U jρð0Þ⟫; ð5:4Þ reference list). Readers who are unfamiliar are encouraged T 0 to read about the closely related Hamiltonian-based adia- where all quantities in curly brackets are functions of r, batic limit in Appendix E1. Unlike adiabatic evolution of ˙ −1 P∞ ≡∂ P∞, Q∞ ≡ I − P∞, and L is the instantaneous “non-Hermitian Hamiltonian” systems, Lindbladian adia- s inverse [Eq. (4.29)]. The superoperator batic evolution always obeys the rules of quantum mechan- Z ics (i.e., is completely-positive and trace-preserving). s 0 ˙ ðrÞ ðrÞ Throughout this entire section, we assume that AsðHÞ is Uðs;s Þ ¼ P exp P∞ P∞ dr ð5:5Þ 0 steady (H∞ ¼ 0) but note that this analysis can be extended s H “ to non-steady Asð Þ by carefully including a dynamical ðs0Þ P∞ H phase” contribution from H∞. Recall that a system parallel transports states in Opð Þ to states in ðsÞ evolves in a rescaled time s ≡ t=T ∈ ½0; 1 according to P∞ OpðHÞ and is a path-ordered product of exponentials
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˙ of the adiabatic connection P∞P∞, the generator of and corresponding adiabatic connection adiabatic evolution (see Appendix E). μ Like the Kubo formula, all terms can be interpreted when Aα;μν ≡ ⟪J j∂αΨν⟫: ð5:10Þ read from right to left. The first term in Eq. (5.4) represents μ adiabatic evolution of AsðHÞ, the (second) leakage term Note that Aα is a real matrix since fJ ; Ψνg are Hermitian. quantifies leakage of jρð0Þ⟫ out of AsðHÞ, and the (last) The connection transforms as a gauge potential under μ −1 ν tunneling term represents interference coming back into jΨμ⟫ → jΨν⟫Rνμ and ⟪J j → Rμν ⟪J j for any AsðHÞ from outside. This term is a continuous sum of R ∈ GL½dim AsðHÞ; R : adiabatically evolved steady states that are perturbed by ˙ −1 ˙ −1 −1 P∞L P∞ at all points r ∈ ½0;s during evolution. Aα → R AαR þ R ∂αR: ð5:11Þ Because of its dependence on the spectrum of L, this term needs to be minimized to determine the optimal adiabatic Upon evolution in the closed path, the density matrix path through AsðHÞ [141]. Notice also the similarity transforms as between the leakage term and the leakage term Eq. (4.31) of the Kubo formula. Motivated by this, we proceed to apply dX2−1 dX2−1 ð0Þ ð0Þ the four-corners decomposition to all three terms. jρð0Þ⟫ ¼ cμjΨμ ⟫ → BμνcνjΨμ ⟫; ð5:12Þ μ¼0 μ;ν¼0
B. Evolution within AsðHÞ equivalent to the operator representation. We study both Let us now assume a closed path [LðsÞ¼Lð0Þ]. representations below, showing that the holonomy is However, due to the geometry of the parameter space M, unitary for all AsðHÞ. the state may be changed (e.g., acquire a Berry phase). In First, let us remove the decaying subspace from both the adiabatic limit [according to Eq. (5.4)], an initial steady representations of the connection by applying the clean- state evolves in closed path C as jρð0Þ⟫ → Ujρð0Þ⟫, leak property Eq. (2.11). Simplifying Aα turns out to be acquiring a holonomy similar to calculating the effective Hamiltonian perturba- W H I tion within Asð Þ in Sec. IV. By Eq. (2.11), ð1;0Þ ˙ U ≡ U ¼ P exp P∞P∞ds : ð5:6Þ μ μ Aα μν ≡ ⟪J ∂αΨν⟫ ⟪J ∂αΨν⟫: 5:13 C ; j ¼ j ð Þ
The above expression acts on the steady-state basis For the operator representation, one first applies Eq. (2.11) ðs¼0Þ to the parallel transport condition Eq. (5.8): elements Ψμ used to express X 0 P ∂ ρ⟫ P ∂ ρ⟫ ð0Þ ¼ ∞j s ¼ Ψj s : ð5:14Þ jρð0Þ⟫ ¼ cμjΨμ ⟫; ð5:7Þ μ Then, one uses this condition to obtain an equation of motion for ρ: so we deem it the operator representation of the holonomy ˙ U and connection P∞P∞. ˙ j∂ ρ⟫ ¼ðI − PΨÞj∂ ρ⟫ ¼ PΨPΨjρ⟫: ð5:15Þ Instead of looking at how the basis elements evolve, let s s us instead express the effect of the holonomy on the The last equality above can be checked by expressing both coordinates cμ of the state above. This can be done by sides in terms of the steady-state basis elements Ψμ and generalizing the Hamiltonian analysis of Appendix E to conserved quantities Jμ. For a closed path, the solution to Lindbladians [68,72], which produces a parallel transport this equation of motion is then the same holonomy, but now condition with the minimal projection PΨ instead of the asymptotic projection P∞: P∞∂ jρ⟫ ¼ 0 ð5:8Þ s I ˙ characterizing the Lindbladian adiabatic limit. After U ¼ P exp PΨPΨds : ð5:16Þ C expressing ∂s in terms of the various ∂α’s [Eq. (5.2)], this condition provides an equation of motion for the coordinate The holonomy U thus does not depend on the piece P∞P vector cμ. Solving this equation yields (what we call) the associated with the decaying subspace. coordinate representation of the holonomy, DFS case.—Since PΨ ¼ P, the operator representa- X I tion allows us to readily extract the DFS case. The ψ B ¼ P exp − Aαdxα ; ð5:9Þ (unitary)P holonomy for a set of states j ki (with α C d−1 ψ ψ P ¼ k¼0 j kih kj) is determined by the adiabatic
041031-17 ALBERT, BRADLYN, FRAAS, and JIANG PHYS. REV. X 6, 041031 (2016) connections of the states themselves, namely, PP˙ and its However, such deformations do not change the dimen- corresponding superoperator form sion d2 of AsðHÞ and thus do not close the dissipative gap. To account for such deformations in the one NS ˙ ˙ ˙ PPðρÞ¼PPρP þ PρPP: ð5:17Þ block case, the path can be partitioned into segments of constant rankfPg and the connection calculation below This result is known [67] and is a cornerstone of can be applied to each segment. reservoir-engineered holonomic quantum computation Simplifying Eq. (5.13) by invoking the reference basis Ψ (see example below). We study this case in the coordinate structure of fJ; g from Eq. (5.19) yields representation in Appendix E2. A A~DFS Aax − ~ DFS ⊗ ϱ ⟫⟪ Aax Unique state case.—Now, the only conserved quantity is α ¼ α þ α ¼ i½Aα ; · j ax Paxjþ α ; the identity J ¼ I, so it is easy to show that ð5:21Þ A ⟪ ∂ ϱ⟫ ∂ ϱ 0 α ¼ Ij α ¼ Trf α g¼ : ð5:18Þ where the DFS effective Hamiltonian is [69] The unique steady state can never acquire a Berry phase. ~ DFS ≡ ¯ ⊗ ϱðsÞ While the Hamiltonian formalism does yield a Berry Aα TraxfðPDFS ax ÞGαg; ð5:22Þ phase for a unique ground state, that (overall) phase disappears when the state is written as a density matrix. and the second term is the nax-dependent constant Since the Lindbladian formalism deals with density ax ðsÞ matrices, one never encounters an overall phase. Aα;μν ¼ −∂α ln nax δμν: ð5:23Þ NS case.—For this case, the NS factors into a DFS and an auxiliary part for each s ∈ ½0; 1 . The DFS part is The first term clearly leaves the auxiliary part invariant mapped into a reference DFS spanned by a (parameter- and generates unitary evolution within the DFS part of Ψ¯ DFS⟫ d2−1 independent Hermitian matrix) basis fj μ gμ¼0 . [Note the NS. We can thus see that DFS holonomies can be ¯ DFS DFS ðsÞ that, in general, jΨμ ⟫ ≠ jΨμ ðs ¼ 0Þ⟫ since s para- influenced by ϱax . We will see that the second term’s ¯ DFS metrizes a particular path in M while fjΨμ ⟫g is fixed.] only role is to preserve the trace for open paths. ΨðsÞ We let SðsÞ [with SðρÞ ≡ SρS†] be the unitary operator that Sticking with the convention that 0 is traceful and the ðsÞ simultaneously maps the instantaneous basis elements traceless Ψμ≠0 carry the DFS Bloch vector, we notice that ðsÞ A jΨμ ⟫ into the reference DFS basis and diagonalizes α transforms as a gauge potential under orthogonal Bloch 2 ðsÞ vector rotations R ∈ SOðd − 1Þ: ϱax . Similarly, this SðsÞ will factor the instantaneous μ conserved quantities ⟪J ðsÞj into a DFS part and the Ψ ⟫ → Ψ ⟫R μ≠0⟫ → ν≠0⟫R ðsÞ j μ≠0 j ν≠0 νμ and jJ jJ νμ: identity Pax on the auxiliary space. Therefore, we define the family of instantaneous minimal projections as ð5:24Þ
ðsÞ ðsÞ ðsÞ ‡ In addition, one can internally rotate ϱ without mixing Ψμ P ¼ SðsÞðP¯ ⊗ jϱ ⟫⟪P jÞS ðsÞ; ð5:19Þ ax Ψ DFS ax ax Ψ S with ν≠μ. Under such a transformation ax,
P 2 P¯ d −1 Ψ¯ DFS⟫⟪Ψ¯ DFS ⟫ ¯ ¯ † where DFSð·Þ¼ μ¼0 j μ μ j · ¼ PDFS · PDFS ϱ ¯ DFS Rax axRax x jΨμ⟫ → S jΨμ⟫ ¼ S Ψμ ⊗ ⟫ ð5:25Þ is the superoperator projection onto the α-independent ax n DFS reference basis. The generators of motion ax for some R ∈ U d , and the connection transforms as † ax ð axÞ Gα ≡ iS ∂αS and Gα ≡ −i½Gα; · ð5:20Þ an Abelian gauge potential: can mix up the DFS with the auxiliary part, generating μ ‡ Aα μν → Aα μν þ ⟪JjSax∂αS jΨν⟫: ð5:26Þ novel dissipation-assisted adiabatic dynamics. ; ; ax ϱðsÞ ðsÞ We note that ax (and, therefore,pffiffiffiffiffiffiffiffiffiffiffi Pax ) can change rank Plugging Eq. (5.21) into the Lindblad holonomy ðsÞ ðsÞ ϱ2 PðsÞ Aax (dax ) and purity (nax ¼ Tr ax), provided that Ψ Eq. (5.9), we can see that α is proportional to the identity remains differentiable. For example, one can imagine matrix (of the space of coefficients cμ) and thus can be ðsÞ factored out. Therefore, ϱax to be a thermal state associated with some H Hamiltonian on ax whose rank jumps from one to X I d as the temperature is turned up from zero. This B ∂ x BDFS ax ¼ exp α ln naxd α ; ð5:27Þ ðsÞ ðsÞ α C implies that P and thus P⊞ can change rank also.
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BDFS ϱ α α where is the unitary ax-enhanced holonomy asso- holonomy. For example, if 0 is fixed and 1 is varied ~ DFS iϕ ciated with A . The first term in the above product for an in a closed loop far away from α0, then jα1i → e jα1i, 1 0 ϕ open path s ∈ ½0; 1 is simply nð Þ=nð Þ, providing the where is twice the area (in phase space) enclosed by the ax ax path. This scheme can be generalized to obtain universal proper rescaling of the coefficients cμ to preserve the trace quantum computation on superpositions of coherent states of jρð0Þ⟫ [142]. For a closed path, this term vanishes (since B BDFS of multiple modes [71]. nax is real and positive) and ¼ . Thus, the holonomy after a closed-loop traversal of one NS block is unitary. As H Multiblock case.—The generalization to multiple NS C. Leakage out of ð Þ blocks is straightforward: the reference basis now consists We now return to the adiabatic response formula (5.4) to of multiple blocks. Recall that Jμ do not have presence in apply the four-corners decomposition to the Oð1=TÞ non- the off-diagonal parts neighboring the NS blocks [Fig. 1(b)] adiabatic corrections. By Eq. (4.29), L−1 has the same and that the only NS block that ∂αΨμ has presence in is that block upper-triangular structure as L from Eq. (3.3). The of Ψμ. Therefore, each NS block is imparted with its own derivative of the asymptotic projection has partition unitary holonomy. 2 ˙ ˙ ˙ 3 ðPΨÞ PP∞P PP∞P ˙ 6 ˙ ˙ 7 1. Adiabatic curvature P∞ ¼ 4 PPΨP 0 PP∞P 5: ð5:31Þ
The adiabatic connection Aα [Eq. (5.10)] can be used to 000 define an adiabatic curvature defined on the parameter ˙ space induced by the steady states. For simply connected One can interpret P∞ as a perturbation, analogous to V ˙ parameter spaces M [145], the adiabatic curvature can be from Sec. IV, and observe from the above partition that P∞ ˙ shown to generate the corresponding holonomy. More does not connect block diagonal spaces: PP∞P ¼ 0.In precisely, the Ambrose-Singer theorem (Ref. [148], ˙ ðrÞ addition, whenever P∞ acts on a parallel transported state Theorem 10.4) implies that the holonomy for an infini- PðrÞ H xð0Þ living in Opð Þ, only the first column in the above tesimal closed path C with base point α is the adiabatic P˙ P ð0Þ partition ( ∞ ) is relevant. These observations result in curvature at xα . One can alternatively use a generalization −1 −1 ˙ L → L and the replacement of two factors of P∞ with of Stokes’s theorem to non-Abelian connections [149] to P˙ P˙ express the holonomy in terms of a “surface-ordered” Ψ in Eq. (5.4). (We cannot replace the remaining ∞ P P˙ P ∂ μ integral of the corresponding adiabatic curvature, since ∞ contains contributions from sJ.) ∂ generalizing the Abelian case, Eq. (E19). Letting ½αAβ ¼ ∂ − ∂ 1 αAβ βAα, the curvature is ðs;0Þ −1 ˙ ðsÞ ðs;0Þ jρðsÞ⟫ ¼ U jρð0Þ⟫ þ L ðsÞPΨ U jρð0Þ⟫ Z T F ≡∂ A A A 1 s αβ;μν ½α β ;μν þ½ α; β μν: ð5:28Þ ðs;rÞ ˙ −1 ˙ ðrÞ ðr;0Þ þ drU fP∞L PΨg U jρð0Þ⟫: T 0 NS case.—Using the NS adiabatic connection Eq. (5.21) ax ð5:32Þ and remembering that ∂αAβ is symmetric in α, β, the adiabatic curvature for one NS block, Using the results of Sec. IV C, the energy scale governing ~DFS ~DFS ~DFS the leading-order nonadiabatic corrections is once again the F αβ μν ∂ αAβ μν Aα ; Aβ ; 5:29 ; ¼ ½ ; þ½ μν ð Þ Δ — adiabatic dissipative gap edg the nonzero eigenvalue of L L is just the curvature associated with the connection A~ DFS. þ with the smallest real part. A similar result is shown for the leakage term in the Supplemental Material of Ref. [69]. In addition, the tunneling term, which is similar 2. Example: Driven two-photon absorption −1 to the second-order perturbative correction P∞VL VP∞ A concrete example of Lindbladian-assisted holonomic we discuss in Sec. IV C, does not contain contributions H manipulation of Asð Þ is a generalized version of the from L. driven two-photon absorption example from Sec. IV C. One can generalize the jump operator to VI. LINDBLADIAN QUANTUM GEOMETRIC TENSOR F ¼ða − α0Þða − α1Þ; ð5:30Þ Here, we introduce the Lindbladian QGT Q where α0, α1 are complex. For the well-separated case and explicitly calculate it for the unique state and NS (jα0 − α1j ≫ 1), the DFS is spanned by coherent states jα0i block cases. The antisymmetric part of the QGT is equal and jα1i. After adiabatically traversing a closed loop in the to the curvature F generated by the connection A (see parameter space of the two α’s, the DFS acquires a Sec. VB1). The symmetric part of the QGT produces a
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Q Q generalized metric tensor for Lindbladian steady-state αβ consists of parts symmetric ( ðαβÞ) and antisymmetric Q α β subspaces. We review the Hamiltonian QGT and cover ( ½αβ )in , . From the third form, it is evident that its in detail the DFS case in Appendix F. Most of the relevant antisymmetric part is exactly the adiabatic curvature F αβ from quantities for the Hamiltonian, degenerate Hamiltonian Eq. (5.28) (cf. Ref. [70], Proposition 13). The rest of this or DFS, and NS cases are summarized in Table I.We section is devoted to calculating the symmetric part and its introduce other geometric quantities in Appendix G, corresponding metric on M, which is defined as the trace TR including an alternative geometric tensor Qalt whose (i.e., trace in superoperator space) of the QGT’s symmetric curvature is different from the adiabatic curvature from part, Sec. VB1, but whose metric appears in the Lindbladian adiabatic path length. dX2−1 M ≡ P ∂ P ∂ P Q In Sec. V, we show, using the operator representation αβ TRf Ψ ðα Ψ βÞ Ψg¼ ðαβÞ;μμ: ð6:3Þ of the adiabatic connection and the conditions Eqs. (2.9) μ¼0 and (2.11), that the minimal projection PΨ ¼ P∞P (and not P∞) generates adiabatic evolution within AsðHÞ. Before proving this is a metric for some of the relevant Following this, we define cases, let us first reveal how such a structure corresponds to an infinitesimal distance between adiabatically connected Qαβ ≡ PΨ∂αPΨ∂βPΨPΨ ð6:1Þ Lindbladian steady states by adapting results from non- Hermitian Hamiltonian systems [150–152]. The zero P μ P Ψ ⟫⟪ eigenspace of L is diagonalized by right and left to be the associated QGT. While Ψ ¼ μj μ Jj is not μ μ eigenmatrices Ψμ⟫ and ⟪J , respectively. In accordance always Hermitian due to J ≠ Ψμ (e.g., in the NS case), we j j Ψ show that the QGT nevertheless remains a meaningful with this duality between and J, we introduce an ρˆ ⟫ geometric quantity. Looking at the matrix elements of associated operator j ∞ [151,152], Qαβ and explicitly plugging in the instantaneous PΨ dX2−1 dX2−1 [Eq. (5.19)] yields the following three forms: μ jρ∞⟫ ¼ cμjΨμ⟫ ↔ jρˆ∞⟫ ≡ cμjJ⟫; ð6:4Þ μ μ¼0 μ¼0 Qαβ;μν ≡ ⟪Jj∂αPΨ∂βPΨjΨν⟫ ð6:2aÞ
μ for every steady-state subspace operator jρ∞⟫. This allows ¼ ⟪∂αJ jðI − PΨÞj∂βΨν⟫ ð6:2bÞ us to define a modified inner product ⟪AbjB⟫ for matrices A μ μ and B living in the steady-state subspace. Since Ψμ and J ¼ ∂αAβ;μν þðAαAβÞμν − ⟪Jj∂α∂βΨν⟫; ð6:2cÞ μ are biorthogonal (⟪JjΨν⟫ ¼ δμν), this inner product is with Aα the Lindblad adiabatic connection Eq. (5.10).Since surprisingly equivalent to the Hilbert-Schmidt inner prod- μ uct ⟪A B⟫. However, the infinitesimal distance is not the Aα;μν are real and fJ ; Ψνg are Hermitian, the matrix elements j are all real. From its second form, one easily deduces that the same: −1 QGT transforms as Qαβ → R QαβR for any basis trans- ⟪∂ ρˆ ∂ ρ ⟫ ≠ ⟪∂ ρ ∂ ρ ⟫ formationR ∈ GL½dim AsðHÞ; R [see Eq. (5.11)]. The QGT s ∞j s ∞ s ∞j s ∞ : ð6:5Þ
TABLE I. Summary of quantities defined in Secs. V and VI.
Hamiltonians: Operator notation Hamiltonians: Superoperator notation Lindbladians: One NS block ϱ ψ DFS DFS † DFS ax State basis j ki¼DFS states Ψμ ¼ðΨμ Þ ∈ spanfjψ kihψ ljg jΨμ⟫ ¼jΨμ ⟫ ⊗ j ⟫ P P P nax P d−1 ψ ψ P d2−1 ΨDFS⟫⟪ΨDFS P d2−1 Ψ ⟫⟪ μ DFS ¼ k¼0 j kih kj DFS ¼ μ¼0 j μ μ j ∞ ¼ μ¼0 j μ J j ¼ PΨ þ P∞P DFS ψ ∂ ψ ADFS ⟪ΨDFS ∂ ΨDFS⟫ A ⟪ μ ∂ Ψ ⟫ Connection Aα;kl ¼ ih kj α li α;μν ¼ μ j α ν α;μν ¼ J j α ν ~DFS ax ¼ Aα;μν þ Aα;μν Curvature DFS ∂ DFS − DFS DFS F DFS ∂ ADFS ADFS ADFS F αβ ∂ αAβ Aα; Aβ Fαβ ¼ ½αAβ i½Aα ;Aβ αβ ¼ ½α β þ½ α ; β ¼ ½ þ½ ∂ A~DFS A~DFS A~DFS ¼ ½α β þ½ α ; β DFS ψ ∂ ∂ ψ F DFS ⟪ΨDFS ∂ P ∂ P ΨDFS⟫ F ⟪ μ ∂ P ∂ P Ψ ⟫ Fαβ;kl ¼h kj ½αPDFS β PDFSj li αβ;μν ¼ μ j ½α DFS β DFSj ν αβ;μν ¼ Jj ½α Ψ β Ψj ν DFS ψ ∂ ∂ ψ QDFS ⟪ΨDFS ∂ P ∂ P ΨDFS⟫ Q ⟪ μ ∂ P ∂ P Ψ ⟫ QGT Qαβ;kl ¼h kj αPDFS βPDFSj li αβ;μν ¼ μ j α DFS β DFSj ν αβ;μν ¼ Jj α Ψ β Ψj ν − ∂ DFS − DFS DFS ∂ ADFS ADFSADFS ∂ A A A ¼ i αAβ;kl ðAα Aβ Þkl ¼ α β;μν þð α β Þμν ¼ α β;μν þð α βÞμν DFS DFS μ −hψ kj∂α∂βψ li −⟪Ψμ j∂α∂βΨν ⟫ −hJj∂α∂βΨνi DFS ∂ ∂ MDFS P ∂ P ∂ P M P ∂ P ∂ P Metric Mαβ ¼TrfPDFS ðαPDFS βÞPDFSg αβ ¼ TRf DFS ðα DFS βÞ DFSg αβ ¼ TRf Ψ ðα Ψ βÞ Ψg
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1 2 The symmetric part Q αβ shows up in precisely this ð Þ ð Þ ð Þ Mαβ ¼ Mαβ þ Mαβ ; modified infinitesimal distance. Using Eq. (6.4), the paral- Mð1Þ 2 ⟪ ¯ ⊗ ϱ − ¯ ⊗ ⟫ lel transport condition Eq. (5.8), and parametrizing ∂s in αβ ¼ d PDFS axjGðαðI PDFS PaxÞGβÞ ; ∂ ’ terms of the α s [Eq. (5.2)] yields Mð2Þ 2 ⟪ P¯ ⋆ ⊗ O ⟫ αβ ¼ d Gðαj DFS axjGβÞ ; ð6:10Þ
2 1 X dX−1 P¯ ⋆ ⟪∂ ρˆ ∂ ρ ⟫ Q x˙ x˙ with projection DFS consisting of only traceless DFS s ∞j s ∞ ¼ ðαβÞ;μν α βcμcν; ð6:6Þ ¯ 1 ¯ 2 generators (we set ΨDFS ¼ pffiffi P ), α;β μ;ν¼0 0 d DFS
dX2−1 as evidenced by the second form Eq. (6.2b) of the Lindblad P¯ ⋆ ≡ Ψ¯ DFS⟫⟪Ψ¯ DFS P¯ − Ψ¯ DFS⟫⟪Ψ¯ DFS DFS j μ μ j¼ DFS j 0 0 j; QGT. Tracing the symmetric part over the steady-state μ¼1 subspace gives the metric Mαβ. ð6:11Þ Unique state case.—Here, things simplify significantly, yet the obtained metric turns out to be novel nonetheless. and auxiliary superoperator defined (for all auxiliary P ϱ⟫⟪ O ≡ − ⟪ϱ ⟫ ϱ The asymptotic projection is Ψ ¼j Pj and a straight- operators A)as axðAÞ ðA axjA Þ ax. forward calculation using Eq. (6.2b) yields The quantity Mαβ is clearly real and symmetric in α, β, so to show that it is a (semi)metric, we need to prove M ⟪∂ ∂ ϱ⟫ positivity [wαMαβwβ ≥ 0, with sum over α, β implied, αβ ¼ ðαPj βÞ : ð6:7Þ for all vectors w in the tangent space TMðxÞ at a x ∈ M ϱ P point [148]]. Since ax is positive definite, one ϱ dϱ−1 λ ψ ψ Using the eigendecomposition ¼ k¼0 kj kih kj, can show that
1 dϱ−1 ð Þ X wαMαβ wβ ¼ 4d⟪OjO⟫ ≥ 0; ð6:12Þ Mαβ 2 λ ∂ αψ Q ∂β ψ ; 6:8 ¼ kh ð kj j Þ ki ð Þ ffiffiffiffiffiffi k¼0 − ¯ ⊗ w ¯ ⊗ pϱ with O ¼ðI PDFS PaxÞðGα αÞðPDFS axÞ.For Mð2Þ P¯ ⋆ − ∂ ψ ∂ ψ the second term αβ ,wecanseethat DFS is positive where Q ¼ I P and h ðα kjQj βÞ ki is the Fubini-Study O semidefinite since it is a projection. We show that ax is metric corresponding to the eigenstate jψ i. In words, Mαβ k positive semidefinite by utilizing yet another inner prod- is the sum of the eigenstate Fubini-Study metrics weighed uct associated with open systems [52].First,notethat by their respective eigenvalues or populations. If ϱ is pure, then it is clear that Mαβ reduces to the Fubini-Study metric. ⟪ O ⟫ ϱ † − ϱ 2 Aj axjA ¼ Trf axA Ag jTrf axAgj : ð6:13Þ Finally, if ϱ is full rank, then P ¼ I and Mαβ ¼ 0. This means that the metric is nonzero only for those ϱ that are ϱ ⟪A B⟫ ≡ ϱ A†B Since ax is full rank, j ϱax Trf ax g is a valid not full rank. inner product [52] and ⟪AjO jA⟫ ≥ 0 is merely a state- — ax NS case. Recall from Eq. (5.19) that adiabatic evolu- ment of the Cauchy-Schwarz inequality associated with tion on the NS is parametrized by the instantaneous this inner product. For Hermitian A,Eq.(6.13) reduces to minimal projections ⟪ ϱ ⟫ the variance of Aj ax . ð1Þ Roughly speaking, the first term Mαβ describes how PðsÞ S P¯ ⊗ ϱðsÞ⟫⟪ ðsÞ S‡ Ψ ¼ ðsÞð DFS j ax Pax jÞ ðsÞ; ð6:9Þ much the DFS and auxiliary parts mix and the second term ð2Þ Mαβ describes how much they leave the block while P 2 P¯ d −1 Ψ¯ DFS⟫⟪Ψ¯ DFS ⟫ ¯ ¯ ð2Þ where DFSð·Þ¼ μ¼0 j μ μ j · ¼ PDFS · PDFS moving in parameter space. For the DFS case, Mαβ ¼ 0 is the superoperator projection onto the xα-independent O 0 (due to ax ¼ for that case) and the metric reduces to the DFS reference basis. We remind the reader (see Sec. VB) standard DFS metric covered in Appendix F. For the unique that the only assumption of such a parametrization is that Mð2Þ P¯ ⋆ ðsÞ state case, αβ is also zero (due to DFS not containing the state jρ∞ ⟫ is unitarily equivalent (via unitary S)toa any traceful DFS elements and thus reducing to zero when tensor product of a DFS state and auxiliary part for all ¯ 1 Mð2Þ points s ∈ ½0; 1 in the path. PDFS ¼ ). The mixing term αβ is thus nonzero only in We can simplify Mαβ and show that it is indeed a metric. the NS block case. In the reference basis decomposition of PΨ from Eq. (6.9), † † the operators Gα ≡ iS ∂αS [with SðsÞjρ⟫ ≡ jSρS ⟫] generate motion in parameter space. After significant VII. OUTLOOK simplification, one can express Mαβ in terms of these This work examines the properties of asymptotic (e.g., generators: steady-state) subspaces of Lindbladians, comparing them
041031-21 ALBERT, BRADLYN, FRAAS, and JIANG PHYS. REV. X 6, 041031 (2016) to analogous subspaces of Hamiltonian systems. We [167]) to verify whether a given projection is equal to P that characterize such subspaces as “not very different” from does not rely on diagonalization. their Hamiltonian cousins in terms of their geometrical and Lastly, the properties of Lindbladian eigenmatrices can response properties. A quantitative description of our be extended to eigenmatrices of more general quantum results is found in Sec. II. channels [79,83–85]. Statements similar to Proposition 2 While we focus on response to Hamiltonian perturba- exist for fixed points of quantum channels [79,88], and tions within first order and evolution within the adiabatic their extension to rotating points will be a subject of future limit, it would be interesting to apply our results further to work. These results may also be useful in determining Lindbladian perturbations [127], second-order perturbative properties of asymptotic algebras of observables [168,169] effects [137,153], and corrections to adiabatic evolution. and properties of quantum jump trajectories when the While several elements of this study consider asymptotic Lindbladian is “unraveled” [170,171]. subspaces consisting of only one block of steady states, it is not unreasonable to imagine that the aforementioned second-order and/or nonadiabatic effects could produce ACKNOWLEDGMENTS transfer of information between two or more blocks. Similar to the first-order case, we anticipate that jump V. V. A. acknowledges N. Read, L. I. Glazman, S. M. operator perturbations may provide alternative ways to Girvin, F. Ticozzi, A. Rivas, A. del Campo, L. Viola, K.-J. generate second-order effects [137,153], which are cur- Noh, and C. Shen for stimulating discussions and thanks rently only producible with Hamiltonian perturbations. R. T. Brierley, A. Alexandradinata, Z. K. Minev, J. I. Recently developed diagrammatic series aimed at deter- Väyrynen, J. Höller, and A. Dua for useful comments mining perturbed steady states [154] (see also Ref. [155]) on the manuscript. V. V. A. was supported by the NSF may benefit from the four-corners decomposition (when- GRFP under Grant No. DGE-1122492 through part of this ever the unperturbed steady state is not full rank). work. L. J. is supported by the ARO, AFOSR MURI, ARL It has recently been postulated [156] that Lindbladian CDQI program, the Alfred P. Sloan Foundation, and the metastable states also possess the same structure as the David and Lucile Packard Foundation. steady states. This may mean that our results regarding conserved quantities (which are dual to the steady states) also apply to the pseudoconserved quantities (dual to the APPENDIX A: LINDBLADIANS AND metastable states). DOUBLE-KET NOTATION We obtain a Lindblad generalization of the quantum geometric tensor for Hamiltonian systems [74]. The 1. Introduction to Lindbladians Lindblad QGT encodes both the adiabatic curvature of Lindbladians operate on the space of (linear) operators the steady-state subspace and also a novel metric which on H,orOpðHÞ ≡ H ⊗ H⋆ [172,173] (also known as generalizes the Fubini-Study metric for Hamiltonians. This Liouville space [129], von Neumann space, or Hilbert- metric will be examined in future work, particularly to see Schmidt space [174]). This space is also a Hilbert space whether it reveals information about bounds on conver- when endowed with the Hilbert-Schmidt inner product and gence rates [157–160]. It remains to be seen whether the Frobenius norm (for N ≡ dim H < ∞). An operator A in scaling behavior of the metric is correlated with phase quantum mechanics is thus both in the space of operators stability [117–120] and phase transitions [114–116] for acting on ordinary states and in the space of vectors acted Lindbladian phases with nonequilibrium steady states. It on by superoperators. We denote the two respective cases would also be of interest to see whether the adiabatic as Ajψi and OjA⟫ (for jψi ∈ H and for a superoperator O). curvature is related to the Uhlmann phase [161] and various While (strictly speaking) jA⟫ is an N2-by-1 vector and A is mixed state Chern numbers [162–164]. an N-by-N matrix, they are isomorphic, and so we define We show that the dissipative gap of Lindbladians is not OjA⟫, OðAÞ, and jOðAÞ⟫ by their context. always relevant in linear response and in corrections to For A, B ∈ OpðHÞ, the Hilbert-Schmidt inner product adiabatic evolution. In fact, another scale, the effective and Frobenius norm are, respectively, dissipative gap, is the relevant energy scale for those pffiffiffiffiffiffiffiffiffiffiffiffiffiffi processes. It would be of interest to determine how the ⟪AjB⟫ ≡ TrfA†Bg and ∥A∥ ≡ ⟪AjA⟫: ðA1Þ effective gap scales with system size in physically relevant dissipative systems [31,135,136,165]. At this point, the only way to find the projection P onto The inner product allows one to define an adjoint operation ‡ † the range of the steady states of a Lindbladian L is to that complements the adjoint operation on matrices in H diagonalize L [79]. It could be of interest to determine Opð Þ: whether diagonalization of L is necessary for determining ⟪ O ⟫ ⟪ O ⟫ ⟪O‡ ⟫ P. Interestingly, there exists an algorithm [166] (see also Aj ðBÞ ¼ Aj jB ¼ ðAÞjB : ðA2Þ
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Writing O as an N2-by-N2 matrix (see Ref. [50], 2. Double-bra-ket basis for steady states ‡ Appendix A, for the explicit form), O is just the conjugate We now bring in intuition from Hamiltonian-based transpose of that matrix. For example, if Oð·Þ¼A · B, then quantum mechanics by writing the eigenmatrices as ‡ † † one can use Eq. (A1) to verify that O ð·Þ¼A · B . Similar vectors using double-ket notation. First, we introduce to the Hamiltonian description of quantum mechanics, O is some bases for OpðHÞ, with which we can build bases ‡ −1 Hermitian if O ¼ O. For example, all projections P⊞ from H ϕ N for Asð Þ. Given any orthonormal basis fj kigk¼0 for the Eq. (2.2) are Hermitian. system Hilbert space H, one can construct the correspond- A Lindbladian acts on a density matrix ρ as ing orthonormal (under the trace) outer product basis for OpðHÞ, 1X L ρ − ρ κ 2 lρ l† − l† lρ − ρ l† l ð Þ¼ i½H; þ2 lð F F F F F F Þ; l Φ ⟫ N−1 Φ ≡ ϕ ϕ fj kl gk;l¼0; where kl j kih lj: ðA7Þ ðA3Þ The analogy with quantum mechanics is that the matrices with Hamiltonian H, jump operators Fl ∈ OpðHÞ, and real Φ ↔ Φ ⟫ Φ† ↔ ⟪Φ kl j kl and kl klj are vectors in the vector nonzero rates κl. References [170,175,176] describe the space OpðHÞ and superoperators O are linear operators on conditions on a system and reservoir for which Lindbladian those vectors. Furthermore, one can save an index and use evolution captures the dynamics of the system. The form of † properly normalized Hermitian matrices Γκ ¼ Γκ to form the Lindbladian Eq. (A3) is not unique due to the following N2−1 an orthonormal basis fjΓκ⟫gκ 0 : “gauge” transformation (for complex gl), ¼ † X ⟪ΓκjΓλ⟫ ≡ TrfΓκ Γλg¼TrfΓκΓλg¼δκλ: ðA8Þ → − i κ ⋆ l − l† H H 2 lðglF glF Þ; l For example, an orthonormal Hermitian matrix basis for l l F → F þ glI; ðA4Þ OpðHÞ with H two dimensional consists of the identity matrix and the three Pauli matrices, all normalized pffiffiffi that allows parts of the Hamiltonian to be included in the by 1= 2. jump operators (and vice versa) while keeping L invariant. It is easy to see that the coefficients in the expansion of Note that there exists a unique “gauge” in which Fl are any Hermitian operator in such a matrix basis are real. For traceless (Ref. [2], Theorem 2.2). It is easy to determine example, the coefficients cκ in the expansion of a density how an observable A ∈ OpðHÞ evolves (in the Heisenberg matrix, picture) using the definition of the adjoint Eq. (A2) and cyclic permutations under the trace: NX2−1 jρ⟫ ¼ cκjΓκ⟫ with cκ ¼ ⟪Γκjρ⟫; ðA9Þ 1X κ¼0 L‡ −H κ 2 l† l − l† l ðAÞ¼ ðAÞþ2 lð F AF fF F ;AgÞ: ðA5Þ l are clearly real and represent the components of a gener- alized Bloch (coherence) vector [177,178]. Furthermore, The superoperator Hð·Þ ≡ −i½H; · corresponding to the defining Hamiltonian is therefore anti-Hermitian because we have “ ” † absorbed the i in its definition. Oκλ ≡ ⟪ΓκjOjΓλ⟫ ≡ TrfΓκ OðΓλÞg ðA10Þ Time evolution of states is determined by the equation ∂ ρ ⟫ L ρ ⟫ ≥ 0 j t ðtÞ ¼ j ðtÞ , so for t , for any superoperator O, one can write
ρ ⟫ tL ρ ⟫ 2 j ðtÞ ¼ e j in ; ðA6Þ NX−1 O ¼ OκλjΓκ⟫⟪Γλj: ðA11Þ ρ κ;λ¼0 with in being the initial state. The norm of a wave function corresponds to the trace of ρ (⟪Ijρ⟫); it is preserved under O “ ” both Hamiltonian and Lindbladian evolution. It is easy to There are many physical for which the matrix O check that the exponential of any superoperator of the elements κλ are real. For example, it is easy to show H H ≡ − above form preserves both trace [⟪IjLjρ⟫ ¼ 0 with I the that matrix elements κλ, where ð·Þ i½H; · , are real identity of OpðHÞ] and Hermiticity [LðA†Þ¼½LðAÞ †,as using cyclic permutations under the trace and Hermiticity Γ’ can be verified from Eq. (A3)]. However, the norm or purity of the s: of ρ (⟪ρjρ⟫ ¼ Trfρ2g) is not always preserved under H⋆ Γ Γ − Γ Γ H Lindbladian evolution. κλ ¼ iTrf λ½H; κ g ¼ iTrf κ½H; λ g ¼ κλ: ðA12Þ
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This calculation easily extends to all Hermiticity-preserv- Proposition 2.—The left eigenmatrices of L correspond- ing O, i.e., superoperators such that OðA†Þ¼½OðAÞ † for ing to pure imaginary eigenvalues iΔ are all operators A. P Given a Lindbladian, one can provide necessary and Δμ Δμ ⟪J j¼⟪J j P − L ; ðB2Þ sufficient conditions under which it generates Hamiltonian L − iΔP time evolution. This early key result in open quantum Δμ systems [5] can be used to determine whether a perturbation where ⟪J j are left eigenmatrices of L. generates unitary evolution (Proposition 3). We now Proof.—For a left eigenmatrix ⟪JΔμj with eigenvalue iΔ, proceed to state and prove it as well as the other two propositions in the main text. L‡jJΔμ⟫ ¼ −iΔjJΔμ⟫:
Now partition this eigenvalue equation using the projec- APPENDIX B: PROOFS OF PROPOSITIONS tions fP; P; Pg. Taking the ‡ of the partitioned L 1, 2, AND 3 from Eq. (3.3) results in 2 32 3 — H ‡ Δμ Proposition 1. Let fP; Qg be projections on and L 00jJ ⟫ P P P P 6 76 7 f ; ; ; g be their corresponding projections on ‡ Δμ 6 ‡ ‡ 76 Δμ 7 L jJ ⟫ ¼ 6 PL P L 0 76 J ⟫ 7: OpðHÞ. Then, 4 54 j 5 P L‡P P L‡P L‡ Δμ l jJ ⟫ ∀l∶F ¼ 0; X i l† l H ¼ − κlF F: The eigenvalue equation is then equivalent to the following 2 Δμ l three conditions on the components of J :
— P H Δμ ‡ Δμ Proof. By definition Eq. (2.1), Opð Þ is the small- −iΔJ ¼ LðJ Þ; ðB3aÞ est subspace of OpðHÞ containing all asymptotic states. Therefore, all states evolving under L converge to states in Δμ ‡ Δμ ‡ Δμ −iΔJ ¼ PL PðJ ÞþLðJ Þ; ðB3bÞ POpðHÞ as t → ∞ (Ref. [78], Theorem 2-1). This implies invariance, i.e., states ρ ¼ PðρÞ remain there under Δμ ‡ Δμ ‡ Δμ −iΔJ ¼ PL PðJ ÞþL ðJ Þ application of L: ‡ Δμ þ PL PðJ Þ: ðB3cÞ LðρÞ¼LPðρÞ¼PLPðρÞ: ðB1Þ We now examine them in order. P l† Δμ Applying , we get (i) Condition Eq. (B3a) implies that ½F ;J ¼0 for X all l. [This part is essentially the Lindblad version of l l† PLPðρÞ¼ κlFρF ¼ 0; a similar statement for quantum channels (Ref. [79], l Lemma 5.2). Another way to prove this is to apply “well-known” algebra decomposition theorems (see, since the projections are mutually orthogonal. Taking the e.g., Ref. [9], Theorem 5)]. To show this, we use the trace, dissipation function J associated with L [1].For X ∈ P H l† l some A Opð Þ, ⟪IjPLPjρ⟫ ¼ κlTrfρF Fg¼0: l ‡ ‡ ‡ J ðAÞ ≡ L ðA†AÞ − L ðA†ÞA − A†L ðAÞ X If ρ is a full-rank density matrix (rank ρ Tr P ), then l † l f g¼ f g ¼ κl½F;A ½F;A : each summand above is non-negative (since κl > 0 and l l† l F F are positive semidefinite). Thus, the only way for Δμ† −Δμ l† l the above to hold for all ρ is for F F ¼ 0 for all l, which Using Eq. (B3a) and remembering that J ¼ J , l J Δμ implies that F ¼ 0. Applying P to Eq. (B1) and the two expressions for ðJ Þ imply that l simplifying using F ¼ 0 gives X ‡ Δμ† Δμ l Δμ † l Δμ L ðJ J Þ¼ κl½F ;J ½F ;J : ðB4Þ 1 X P LP ρ ρ − κ l† l 0 l ð Þ¼P iH 2 lF F ¼ ; l We now take the trace using the full-rank steady- implying the condition on H. ■ state density matrix:
041031-24 GEOMETRY AND RESPONSE OF LINDBLADIANS PHYS. REV. X 6, 041031 (2016) X ρ ⟫ P ρ ⟫ ≡ Ψ ⟫ ‡ Δμ ‡ Δμ j ss ¼ j ss cμj 0μ : ½L þ iΔP ðJ Þ¼−PL ðJ Þ: μ ‡ Now, we can show that the operator L þ iΔP Such an asymptotic state is simply jρ∞⟫ from is invertible when restricted to POpðHÞ using a Eq. (3.14b) with cΔμ ¼ δΔ0cμ and cμ ≠ 0. It is full proof by contradiction similar to the one used to rank because it is a linear superposition of projec- Δμ tions on eigenstates of H∞, and such projections prove Eq. (B6). Inversion gives a formula for J provide a basis for all diagonal matrices of which is used along with Eq. (B6) to obtain the POpðHÞ. Taking the trace of the left-hand side statement. ■ of Eq. (B4) yields Proposition 3.—The matrix Lκλ ¼ ⟪ΓκjLjΓλ⟫ is real. Moreover, ⟪ρ L‡ Δμ† Δμ ⟫ ⟪L ρ Δμ† Δμ⟫ 0 ssj ðJ J Þ ¼ ð ssÞjJ J ¼ ; Lλκ ¼ −Lκλ⇔L ¼ −i½H; · with Hamiltonian H: ðB7Þ implying that the trace of the right-hand side is zero: X κ ρ l Δμ † l Δμ 0 lTrf ss½F;J ½F;J g ¼ : Proof.—To prove reality, use the definition of the l adjoint of L, Hermiticity of Γκ, and cyclicity under the trace: Each summand above is non-negative (since κl > 0,
the commutator products are positive semidefinite, ⋆ ‡ ρ Lκλ ¼ ⟪ΓλjL jΓκ⟫ ¼ ⟪LðΓλÞjΓκ⟫ ¼ ⟪ΓκjLjΓλ⟫ ¼ Lκλ: and ss is positive definite). Thus, the only way for l Δμ † l Δμ the above to hold is for ½F;J ½F;J ¼0, l Δμ ⇐ L which implies that F and J commute for all Assume generates unitary evolution. Then there Δμ† −Δμ exists a Hamiltonian H such that LjΓκ⟫ ¼ −ij½H; Γκ ⟫ l; Δ; μ. If we once again remember that J ¼ J and L is antisymmetric: and that the eigenvalues come in pairs Δ, then
l† Δμ l Δμ ½F ;J ¼½F;J ¼0: ðB5Þ Lλκ ¼ −iTrfΓλ½H; Γκ g ¼ iTrfΓκ½H; Γλ g ¼ −Lκλ:
⇒ (An alternative way to prove this part is to observe (ii) Now consider condition Eq. (B3b). The first term on that all eigenvalues of L lie on the imaginary axis and the right-hand side can be obtained from Eq. (C5) use Theorem 18-3 in Ref. [78].) Assume Lκλ is anti- and is as follows: X symmetric, so L‡ ¼ −L. Then the dynamical semigroup ‡ Δμ l† Δμ l Δμ l† l L PL PðJ Þ¼ κlðF J F − J F FÞ fet ; t ≥ 0g is isometric (norm-preserving): let t ≥ 0 and lX jA⟫ ∈ OpðHÞ and observe that l† Δμ l l† l Δμ þ κlðF J F − F FJ Þ: l ⟪etLðAÞjetLðAÞ⟫ ¼ ⟪Aje−tLetLjA⟫ ¼ ⟪AjA⟫: This term is identically zero due to Eq. (B5), Δμ L‡ − L reducing condition Eq. (B3b) to ðJ Þ¼ i Since it is clearly invertible, et ∶OpðHÞ → OpðHÞ is a Δμ ΔJ . We now show that this implies surjective map. All surjective isometric one-parameter dynamical semigroups can be expressed as etLðρÞ¼ P Δμ⟫ 0 † jJ ¼ ðB6Þ UtρUt , with Ut belonging to a one-parameter unitary group fU ; t ∈ Rg acting on H (Ref. [5], Theorem 6). Δμ t for all Δ and μ. By contradiction, assume J ð≠ 0Þ By Stone’s theorem on one-parameter unitary groups, −iHt is a left eigenmatrix of L. Then there must exist a there then exists a Hamiltonian H such that Ut ¼ e 0 0 corresponding right eigenmatrix ΨΔμ ¼ PðΨΔμÞ and LðρÞ¼−i½H; ρ . ■ since the sets of Ψ and J are biorthogonal (see, e.g., Ref. [78], Theorem 18). However, all right eigen- matrices are contained in POpðHÞ by definition APPENDIX C: FOUR-CORNERS Δμ DECOMPOSITION OF L Eq. (2.1), so we have a contradiction and J ¼ 0. (iii) Finally, consider condition Eq. (B3c). Applying Based on conditions Eq. (2.1) and after simplifications Eq. (B6) removes the last term on the right-hand due to Proposition 1, the nonzero elements of Eq. (3.3) side of that condition and simplifies it to acting on a Hermitian matrix ρ ¼ ρ þ ρ þ ρ are
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1 X L ρ − ρ κ 2 l ρ l† − l† l ρ − ρ l† l ð Þ¼ i½H; þ2 lð F F F F F FÞ; ðC1Þ l 1 X L ρ − ρ − ρ κ 2 l ρ l† − l† l ρ − ρ l† l ð Þ¼ iðH HÞþ2 l½ F F F F ðF F Þ ; ðC2Þ l
† LðρÞ¼½LðρÞ ; ðC3Þ 1 X L ρ − ρ κ 2 l ρ l† − l† l ρ − ρ l† l ð Þ¼ i½H; þ2 l½ F F ðF F Þ ðF F Þ ; ðC4Þ l X l l† l† l PLPðρÞ¼ κlðFρF − ρF FÞþH:c:; ðC5Þ l X l l† l† l PLPðρÞ¼ κlðFρF − F FρÞþH:c:; ðC6Þ l X l l† PLPðρÞ¼ κlFρF : ðC7Þ l
l† l l† l l† l X Note that ðF F Þ ¼ F F þ F F and that evolution 1 hνi¼ n ω m þ ; μ ; ðD3Þ of coherences decouples: PLP ¼ PLP ¼ 0. Since F c 2 l m F ≠ 0, L [see Eq. (C4)]isnot of Lindblad form and instead ensures decay of states in (see Ref. [80], m sums over the occupied Landau levels, and nFðϵ; μÞ is the Proposition 3, and Ref. [171], Lemma 4). Fermi distribution function.
APPENDIX D: CONDUCTIVITY FOR A THERMAL LINDBLADIAN APPENDIX E: HAMILTONIAN ADIABATIC THEORY Here, we compute the conductivity for the thermal Landau Lindbladian for noninteracting particles, making Here, we review the adiabatic (Berry) connection for a contact with Example 9 of Ref. [70]. The thermal Hamiltonian system and the DFS case. Lindbladian consists the Hamiltonianffiffiffiffiffi Eq. (4.8)ffiffiffiffiffialong with p2γ ¯ p2¯γ † two jump operators, Fi ¼ bi and Fi ¼ bi , and the 1. Hamiltonian case effective temperature is given by First, let us review two important consequences of the (Hamiltonian) quantum mechanical adiabatic theorem. In 1 γ β ≡ this work, the adiabatic theorem holds by assumption. We ω log γ¯ : ðD1Þ c do not make the adiabatic approximation to HðsÞ [and, later, LðsÞ [179]] since it is not sufficient for the adiabatic For this portion, we consider noninteracting particles. This theorem to hold; see Ref. [180] and references therein. is necessary in order to interpret the Lindblad operator as Adiabatic evolution can be thought of as either (1) being representing coupling to a thermal bath. We work in the generated by an effective operator [181] or (2) generating grand-canonical ensemble: our initial steady state is transport of vectors in parameter space, leading to abelian βðH−μNÞ ρ∞ ∝ e , where N is the number operator and μ is [182–185] or non-abelian [186] holonomies. We loosely the chemical potential lying in a gap between the Landau follow the excellent expositions in Chap. 2.1.2 of ¯ levels. Since Fi is quite similar to Fi, its contribution is Ref. [187] and Sec. 9 of Ref. [188]. We conclude with a calculated in analogous fashion and the temperature- summary of four different ways [Eqs. (E17)–(E20)]of dependent conductivity tensor is writing holonomies for the nondegenerate case. ðtÞ Let jψ 0 i be the instantaneous unique (up to a phase) ωT hνiωcðiωδςτ − ωc½1 − iðγ − γ¯Þ 2 ϵςτÞ ωc zero-energy ground state of a Hamiltonian HðtÞ.We σςτ ω; β ; ð Þ¼ 2π ω2 − ω2 ðD2Þ ð T cÞ assume that the ground state is separated from all other eigenstates of HðtÞ by a nonzero excitation gap for all times where we define ωT ¼ ω þ iðγ − γ¯Þ, the thermally aver- of interest. Let us also rescale time (s ¼ t=T) such that the aged filling factor is given by exact state jψðsÞi evolves according to
041031-26 GEOMETRY AND RESPONSE OF LINDBLADIANS PHYS. REV. X 6, 041031 (2016) ˙ 1 ∂ jψðsÞi ¼ P0P0jψðsÞi ðE9Þ ∂ jψðsÞi ¼ −iHðsÞjψðsÞi: ðE1Þ s T s and effective adiabatic evolution operator The adiabatic theorem states that jψðsÞi [with Z 0 s ðs¼ Þ ðsÞ ˙ 0 jψð0Þi ¼ jψ 0 i] remains an instantaneous eigenstate of U ¼ P exp P0P0ds : ðE10Þ HðsÞ (up to a phase θ) in the limit as T → ∞, with 0 ðsÞ ðsÞ ðsÞ corrections of order Oð1=TÞ. Let P0 ¼jψ 0 ihψ 0 j be We now assume that s parametrizes a path in the the projection onto the instantaneous ground state. In the parameter space M of some external time-dependent param- adiabatic limit, eters of HðsÞ. For simplicity, we assume that M is simply ˙ connected [145]. By writing P0 and P0 in terms of jψ 0i and ψ iθðsÞ ψ ðsÞ j ðsÞi ¼ e j 0 i; ðE2Þ explicitly differentiating, the adiabatic Schrödinger ð0Þ and the initial projection P0 evolves into equation (E9) becomes
0 † ðsÞ ð Þ ∂sjψi¼ðI − P0Þ∂sjψi: ðE11Þ P0 ¼ UadðsÞP0 UadðsÞ. ðE3Þ This implies a parallel transport condition, The adiabatic evolution operator Uad is determined by the Kato equation 0 ¼ P0∂sjψi¼hψj∂sψijψi; ðE12Þ ∂ U −iKU ; s ad ¼ ad ðE4Þ which describes how to move the state vector from one point ˙ in M to another. The particular condition resulting from with the so-called Kato Hamiltonian [181] (P0 ≡∂sP0) adiabatic evolution eliminates any first-order deviation from ˙ K ¼ i½P0;P0 : ðE5Þ the unit overlap between nearby adiabatically evolving states [189]: The adiabatic operator Uad can be shown to satisfy Eq. (E3) (see Ref. [187], Proposition 2.1.1) using hψðs þ δsÞjψðsÞi ¼ 1 þ Oðδs2Þ: ðE13Þ ˙ ˙ P0P0P0 ¼ Q0P0Q0 ¼ 0: ðE6Þ Therefore, we show two interpretations stemming from the adiabatic theorem. The first is that adiabatic evolution of ˙ The P0P0P0 ¼ 0 is a key consequence of the idempotence ψ ψ 0 ψ ðs¼0Þ ˙ j ðsÞi [with j ð Þi¼j 0 i] is generated (in the ordinary of projections while Q0P0Q0 ¼ 0 is obtained by applica- ˙ quantum mechanical sense) by the P0P0 piece of the Kato tion of the no-leak property Eq. (2.9); both are used Hamiltonian K. The second is that adiabatic evolution throughout the text. The adiabatic evolution operator is ψ − realizes parallel transport of j ðsÞi along a curve in then a product of exponentials of iK ordered along the parameter space. As we show now, either framework can path s0 ∈ ½0;s (with path ordering denoted by P): Z be used to determine the adiabatically evolved state and the s resulting Berry phase. P ˙ 0 UadðsÞ¼ exp ½P0;P0 ds : ðE7Þ xα 0 We now define a coordinate basis f g for the parameter space M. In other words, Because of the intertwining property Eq. (E3), UadðsÞ 1 1 X ð0Þ ðsÞ ∂ ∂ x˙ ∂ simultaneously transfers states in P H to P H and states t ¼ s ¼ α α; ðE14Þ 0 0 T T ð0Þ ðsÞ α in Q0 H to Q0 H (with Q0 ≡ I − P0) without mixing the ˙ ∂ ∂ ≡∂ ∂x two subspaces during the evolution. The term P0P0 in where s is the derivative along the path, α = α are Eq. (E5) is responsible for generating the adiabatic evolu- derivatives in various directions in parameter space, and ˙ x˙ ≡ dxα tion of P0H, while the term P0P0 generates adiabatic α ds are (unitless) parameter velocities. Combining evolution of Q0H. To see this, observe that the adiabatically Eqs. (E2) and (E14) with the parallel transport condition ψ ψ ðs¼0Þ ∈ ðsÞH Eq. (E12) yields evolving state j ðsÞi ¼ UadðsÞj 0 i P0 obeys the X Schrödinger equation ˙ 0 ¼ P0∂sjψi¼i xαð∂αθ − Aα;00Þjψi; ðE15Þ ˙ α ∂sjψðsÞi¼½P0;P0 jψðsÞi: ðE8Þ where the adiabatic (Berry) connection Aα 00 ¼ ihψ 0j∂αψ 0i Applying Eq. (E6), the second term in the commutator ; is a vector (gauge) potential in parameter space. The can be removed without changing the evolution. Since reason we can think of Aα 00 as a gauge potential is we are interested only in adiabatic evolution of the zero- ; H because it transforms as one under gauge transformations eigenvalue subspace P0 (and not its complement), iϑ jψ 0i → e jψ 0i, where ϑ ∈ R: we can simplify Uad by removing the second term in the Kato Hamiltonian. This results in the adiabatic → − ∂ ϑ Schrödinger equation Aα;00 Aα;00 α : ðE16Þ
041031-27 ALBERT, BRADLYN, FRAAS, and JIANG PHYS. REV. X 6, 041031 (2016) Y ðsÞ These structures arise because the adiabatic theorem has U ¼ P P0 ; ðE20Þ furnished for us a vector bundle over the parameter-space s∈C manifold M [188,189]. More formally, given the trivial Q bundle M × H (where at each point in M we have a copy of where P denotes a continuous product ordered from the full Hilbert space H), the projection P0 defines a right to left along the path C [see Eq. (47) of Ref. [72] and (possibly nontrivial) sub-bundle of M × H (in this case, a Proposition 1 of Ref. [60]]. This form of the holonomy line bundle, since P0 is rank one). The trivial bundle has should be reminiscent of the Pancharatnam phase [183,187] a covariant derivative ∇α ≡∂α with an associated connec- and, more generally, of a dynamical quantum Zeno effect tion that can be taken to vanish. The Berry connection Aα;00 (Refs. [63,64,66]; see also Refs. [97,98]). is then simply the connection associated with the covariant derivative P0∇α induced on the sub-bundle defined by P0. 2. DFS case The Berry connection describes what happens to the We briefly provide, in addition to Eq. (5.17), another initial state vector as it is parallel transported. It may happen proof of unitarity of the holonomy for the DFS case. Here, that the vector does not return to itself after transport around we do not need the reference basis of Sec. VB, so we let DFS ¯ DFS μ DFS a closed path in parameter space (due to, e.g., curvature or jΨμ ⟫ ≡ SðsÞjΨμ ⟫ and the same for ⟪J j¼⟪Ψμ j. M nonsimple connectedness of ). Given an initial condition Now Aα from Eq. (5.13) reduces to the coordinate form of θ 0 0 ð Þ¼ , the parallel transport condition Eq. (E15) the DFS connection: uniquely determines how θ will change during adiabatic ADFS ⟪ΨDFS ∂ ΨDFS⟫ ΨDFS∂ ΨDFS traversal of a path C parametrized by s ∈ ½0; 1 , i.e., from a α;μν ¼ μ j α ν ¼ Trf μ α ν g: ðE21Þ ðs¼0Þ ð1Þ ð1Þ ð0Þ point xα ∈ M to xα . For a closed path (xα ¼ xα ) and Although this can be equivalently expressed using the assuming Aα 00 is defined uniquely for the whole path DFS ψ ∂ ψ ; Wilczek-Zee adiabatic connection [186] Akl ¼ih kj α li, [146], the state transforms as jψð0Þi → Bjψð0Þi, with we briefly examine the superoperator counterpart. Sticking resulting gauge-invariant holonomy (here, Berry phase) with the convention that I X 1 1 Xd−1 ≡ x ΨDFS ≡ ffiffiffi ffiffiffi ψ ψ B exp i Aα;00d α : ðE17Þ 0 p PDFS ¼ p j kih kj α C d d k¼0 Alternatively, we can use Eq. (E14) and the Schrödinger is the only traceful element and using Eq. (E6), ψ 0 → ψ 0 ΨDFS∂ ΨDFSΨDFS 0 ADFS ADFS 0 equation (E9): j ð Þi Uj ð Þi, with holonomy 0 α 0 0 ¼ ,weseethat α;μ0 ¼ α;0μ ¼ for I DFS X all μ. Thus, Aα consists of a direct sum of zero with a 2 U ≡ P exp ∂αP0P0dxα : ðE18Þ ðd − 1Þ-dimensional antisymmetric matrix acting on the α C DFS Bloch vector components fjΨμ≠0 ⟫g. Since the latter is Since the geometric and Kato Hamiltonian formulations of antisymmetric, the holonomy is unitary. adiabatic evolution are equivalent, Eqs. (E17) and (E18) Formally, letting OpðHÞ⋆ be the space of traceless P offer two ways to get to the same answer. They reveal two d-dimensional Hermitian matrices, DFS defines a sub- ⋆ DFS representations of the Berry connection and holonomy: the bundle of the trivial bundle M ×OpðHÞ and Aα is the P ∂ coordinate representation fiAα;00;Bg, which determines connection associated with the covariant derivative DFS α evolution of θ from Eq. (E2), and the operator representa- induced on that sub-bundle. tion f∂αP0P0;Ug, which determines evolution of jψ 0i [see Proposition 1.2 of Ref. [190] and Eq. (5) of Ref. [111]]. APPENDIX F: HAMILTONIAN QGT Despite the latter being a path-ordered product of matrices, Here, we review the Hamiltonian quantum geometric it simplifies to the Berry phase in the case of closed paths. tensor. Some relevant quantities for the Hamiltonian, For completeness, we also state an alternative form for degenerate Hamiltonian or DFS, and NS cases are sum- each holonomy representation [Eqs. (E17) and (E18)]. If marized in Table I. there are two or more parameters, then the coordinate representation can be expressed in terms of the (here, 1. Hamiltonian case Abelian) Berry curvature Fαβ;00 ≡∂αAβ;00 − ∂βAα;00 using Stokes’s theorem: First, let us review the nondegenerate Hamiltonian case before generalizing to the degenerate Hamiltonians in oper- X ZZ i ator or superoperator form. We recommend Ref. [191] for a B ¼ exp Fαβ 00dxαdxβ ; ðE19Þ 2 ; more detailed exposition. α;β S Continuing from Sec. E1, we begin with an where S is a surface whose boundary is the contour C. The instantaneous zero-energy state jψ 0i and projection operator representation can also be written as a product of P0 ¼jψ 0ihψ 0j, which are functions of a vector of the path-dependent projections P0: control parameters fxαg. The distance between the
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ðsÞ ðsþδsÞ 0 ∂ ψ ψ ∂ ψ ψ ψ ∂ ψ projections P0 and P0 along a path parametrized by ¼ βh kj li¼h β kj liþh kj β li; ðsÞ s ∈ ½0; 1 (with parameter vectors xα at each s)is ∂αhψ kj∂βψ li¼h∂αψ kj∂βψ liþhψ kj∂α∂βψ li: ðF5Þ governed by the QGT: The Berry curvature is the part of the QGTantisymmetric in α, ψ ∂ ∂ ψ β DFS DFS Qαβ;00 ¼h 0j αP0 βP0j 0iðF1aÞ (here, also the imaginary part of the QGT): Fαβ ¼ iQ½αβ . From Eq. (F4c), we recover the form of the DFS Berry curvature. ¼h∂αψ 0jðI − P0Þj∂βψ 0i: ðF1bÞ The symmetric part of the QGT appears in the infini- tesimal distance between nearby parallel transported rays The latter form can be obtained from the former by explicit (i.e., states of arbitrary phase) ψðsÞ and ψðs þ δsÞ in the differentiation of P0 and ∂αP0∂βP0 ¼ð∂αP0Þð∂βP0Þ by degenerate subspace: convention. The I − P0 term makes Qαβ;00 invariant upon iϑ the gauge transformations jψ 0i → e jψ 0i. The tensor can be h∂ ψj∂ ψi¼h∂ ψjðI − P Þj∂ ψi; ðF6Þ split into symmetric and antisymmetric parts, s s s DFS s where we use the parallel transport condition 2 − Qαβ;00 ¼ Mαβ;00 iFαβ;00; ðF2Þ ∂ ψ 0 ∂ PDFSj s i¼ . Expanding s into parameterP derivatives d−1 using Eq. (E14) and writing out jψi¼ 0 ckjψ ki yields which coincide with its real and imaginary parts. The anti- k¼ symmetric part is none other than the adiabatic or Berry 1 X Xd−1 DFS ⋆ curvature from Eq. (E19). The symmetric part is the quantum h∂ ψj∂ ψi¼ Q x˙ αx˙ βc c : ðF7Þ s s 2 ðαβÞ;kl k l Fubini-Study metric tensor [74], α;β k;l¼0
∂ ∂ ∂ ∂ Mαβ;00 ¼ TrfP0 ðαP0 βÞP0g¼Trf αP0 βP0g; ðF3Þ The corresponding Fubini-Study metric on the parameter M DFS space is QðαβÞ traced over the degenerate subspace: where AðαBβÞ ¼ AαBβ þ AβBα, and the latter form can be obtained using P0∂αP0P0 ¼ 0. This quantity is Xd−1 α β DFS ≡ DFS ⟪ ∂ ∂ ⟫ manifestly symmetric in , and real; it is also non- Mαβ QðαβÞ;kk ¼ PDFSj ðαPDFS βÞPDFS : ðF8Þ negative when evaluated in parameter space (see k¼0 Ref. [192], Appendix D). All of this reasoning easily extends to the superoperator DFS formalism (jψ i → jΨμ ⟫). The superoperator QGT cor- 2. DFS case k responding to QDFS can be written as For degenerate Hamiltonian systems [192,193] and in the DFS case, the QGT QDFS is a tensor in both parameter QDFS ⟪ΨDFS ∂ P ∂ P ΨDFS⟫ αβ;μν ¼ μ j α DFS β DFSj ν (α, β) and state (k, l) indices and can be written as ∂ ADFS ADFSADFS − ⟪ΨDFS ∂ ∂ ΨDFS⟫ ¼ α β;μν þð α β Þμν μ j α β ν ; DFS ψ ∂ ∂ ψ Qαβ;kl ¼h kj αPDFS βPDFSj liðF4aÞ ðF9Þ
DFS ∂ ψ − ∂ ψ where Aα is the adiabatic connection Eq. (E21).The ¼h α kjðI PDFSÞj β li; ðF4bÞ DFS QGT is a real matrix (since Aα is real) and consists of P −1 QDFS QDFS α d ψ ψ parts symmetric ( ðαβÞ) and antisymmetric ( ½αβ )in , where PDFS ¼ k¼0 j kih kj is the projection onto the degenerate zero eigenspace of HðsÞ. Since projections are β. Observing the second line of Eq. (F9), it should be F DFS QDFS invariant under changes of basis of their constituents, it is easy to see that the Berry curvature αβ ¼ ½αβ .The DFS † DFS easy to see that Qαβ → R Qαβ R under DFS changes of symmetric part of the superoperator QGT appears in the basis jψ ki → jψ liRlk for R ∈ UðdÞ. Notice that the QGT in infinitesimal Hilbert-Schmidt distance (Ref. [174], Eq. (F4b) consists of overlaps between states outside of the Sec. 14.3) between nearby parallel transported DFS zero eigenspace. For our applications, we write the QGT in a states ρðsÞ and ρðs þ δsÞ: third way such that it consists of overlaps within the zero eigenspace only: ⟪∂ ρ ∂ ρ⟫ ⟪∂ ρ I − P ∂ ρ⟫ s j s ¼ s jð DFSÞj s ; ðF10Þ
DFS − ∂ DFS − DFS DFS − ψ ∂ ∂ ψ P ∂ ρ⟫ 0 Qαβ;kl ¼ i αAβ;kl ðAα Aβ Þkl h kj α β li; ðF4cÞ where we use the parallel transport condition DFSj s ¼ . Similar manipulations as withP the operator QGT, DFS ρ⟫ d2−1 ΨDFS⟫ where Aα is the DFS Berry connection, and we use including the expansion j ¼ μ¼0 cμj ν , yield
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1 X dX2−1 Consider the Frobenius norm Eq. (A1) of Uð1Þ.By DFS ⟪∂ ρ∞j∂ ρ∞⟫ ¼ Q x˙ αx˙ βcμcν: ðF11Þ s s 2 ðαβÞ;μν expanding the definition of the path-ordered exponential, α β μ ν 0 ; ; ¼ one can show that ∥Uð1Þ∥ ≤ expðSÞ with path length Z The corresponding superoperator metric 1 ˙ L0 ≡ ∥P0P0∥ds: ðG3Þ MDFS ≡ P ∂ P ∂ P 0 αβ TRf DFS ðα DFS βÞ DFSg; ðF12Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∥ ∥ † ∂ where TR is the trace in superoperator space, is the Remembering that A ¼ TrfA Ag and writing s in symmetric part of the superoperator QGT traced over terms of parameter derivatives, we see that the Fubini- the degenerate subspace. Since OpðHÞ¼H ⊗ H⋆,itis Study metric appears in the path length: MDFS not surprising that αβ is proportional to the operator 1 X DFS ∥ ˙ ∥2 x˙ x˙ metric Mαβ : P0P0 ¼ 2 Mαβ;00 α β: ðG4Þ α;β dX2−1 MDFS QDFS 2 DFS Therefore, the shortest path between states in Hilbert space αβ ¼ ðαβÞ;μμ ¼ dMαβ : ðF13Þ μ¼0 projects to a geodesic in parameter space satisfying the Euler-Lagrange equations associated with the metric Mαβ;00 APPENDIX G: OTHER GEOMETRIC TENSORS and minimizing the path length [see, e.g., Ref. [148], Eq. (7.58)] (with sum implied) In Sec. VI, we show that the antisymmetric part of the Z rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi QGT 1 1 ˙ ˙ L0 ¼ Gαβ;00xαxβds: ðG5Þ 0 2 Q ¼ PΨ∂PΨ∂PΨPΨ; In Hamiltonian systems, the adiabatic path length appears corresponds to the curvature F associated with the in bounds on corrections to adiabatic evolution (Ref. [196], adiabatic connection A from Sec. V. We thus postulate Theorem 3; see also Ref. [192]). This path length is also that this QGT and its corresponding symmetric part applicable when one wants to simulate adiabatic evolution should be relevant in determining distances between in a much shorter time (counterdiabatic or superadiabatic adiabatically connected Lindbladian steady states. dynamics [197–199] or shortcuts to adiabaticity [200,201]) However, the story does not end there as there are ˙ two more tensorial quantities that can be defined using by explicitly engineering the Kato Hamiltonian i½P0;P0 the steady-state subspace. The first is an extension of from Eq. (E5). Qalt the Fubini-Study metric to non- or pseudo-Hermitian The tensor αβ arises in the computation of the corre- Hamiltonians [151,152,194,195] (different from sponding Lindbladian adiabatic path length, Ref. [150]) that can also be generalized to Lindblad Z 1 ˙ systems; we do not further comment on it here. The L ≡ ∥PΨPΨ∥ds; ðG6Þ second is the alternative geometric tensor, 0 ˙ alt ‡ ‡ where the superoperator norm of PΨPΨ is the analogue Q ≡ PΨ∂PΨ∂PΨPΨ. ðG1Þ of thep operatorffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Frobenius norm from Eq. (A1): ‡ We show that Qalt appears in a bound on the adiabatic ∥O∥ ≡ TRfO Og, where O is a superoperator. This path length for Lindbladian systems, which has tradi- path length provides an upper bound on the norm of the tionally been used to determine the shortest possible Lindblad adiabatic evolution superoperator Eq. (5.6): M distance between states in a parameter space . Here, Z we introduce the adiabatic path length, generalize it to 1 ð1;0Þ ˙ Qalt U ¼ P exp PΨPΨds : ðG7Þ Lindbladians, and comment on . 0 The adiabatic path length for Hamiltonian systems quantifies the distance between two adiabatically con- ðs¼0Þ ð1Þ Using properties of norms and assuming one NS block, it is nected states jψ i and jψ i. The adiabatic evolution 0 0 straightforward to show that operator (derived in Sec. E1) for an arbitrary path s ∈ ½0; 1 ψ ð0Þ Z rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and for initial zero-energy state j 0 i is 1 1 ∥Uð1;0Þ∥ ≤ Malt x˙ x˙ Z expðLÞ with L ¼ dax αβ α βds 1 0 2 ð1Þ ˙ U ¼ P exp P0P0ds : ðG2Þ 0 ðG8Þ
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(with sum over α, β implied). The metric governing this [6] T. Banks, L. Susskind, and M. E. Peskin, Difficulties for path length turns out to be the Evolution of Pure States into Mixed States, Nucl. Phys. B244 X , 125 (1984). [7] P. Zanardi and M. Rasetti, Noiseless Quantum Codes, Malt ⟪ϱ ϱ ⟫ Qalt : αβ ¼ axj ax ðαβÞ;μμ ðG9Þ Phys. Rev. Lett. 79, 3306 (1997). α β ; [8] D. A. Lidar, I. L. Chuang, and K. B. Whaley, Decoherence-Free Subspaces for Quantum Computation, For a unique steady state ϱ, this alternative metric Phys. Rev. Lett. 81, 2594 (1998). reduces to the Hilbert-Schmidt metric [9] E. Knill, R. Laflamme, and L. Viola, Theory of Quantum Error Correction for General Noise, Phys. Rev. Lett. 84, Malt ⟪∂ ϱ ∂ ϱ⟫ 2525 (2000). αβ ¼ ðα j βÞ : ðG10Þ [10] J. Kempe, D. Bacon, D. A. Lidar, and K. B. Whaley, Theory of Decoherence-Free Fault-Tolerant Universal Note the subtle difference between this metric and the QGT Quantum Computation, Phys. Rev. A 63, 042307 metric [Eq. (6.7)] (2001). [11] S. Diehl, E. Rico, M. A. Baranov, and P. Zoller, Topology M ⟪∂ ∂ ϱ⟫ by Dissipation in Atomic Quantum Wires 7 αβ ¼ ðαPj βÞ : ðG11Þ , Nat. Phys. , 971 (2011). This difference is precisely due to the absence of ϱ in the [12] C.-E. Bardyn, M. A. Baranov, E. Rico, A. Imamoglu, P. ϱ Zoller, and S. Diehl, Majorana Modes in Driven- left eigenmatrices J. For the QGT metric, is never in Dissipative Atomic Superfluids with a Zero Chern Number, the same trace twice, while for the alternative metric, the Phys. Rev. Lett. 109, 130402 (2012). ‡ presence of PΨ yields such terms. We note that for a pure [13] J. Dengis, R. König, and F. Pastawski, An Optimal steady state ϱ ¼ P (with P being rank one), both metric Dissipative Encoder for the Toric Code, New J. Phys. tensors reduce to the Fubini-Study metric: 16 013023 (2014). [14] P. G. Kwiat, A. J. Berglund, J. B. Altepeter, and A. G. White, Experimental Verification of Decoherence-Free M Malt ⟪∂ ∂ ⟫ αβ ¼ αβ ¼ ðαPj βÞP : ðG12Þ Subspaces, Science 290, 498 (2000). [15] M. Bourennane, M. Eibl, S. Gaertner, C. Kurtsiefer, A. ϱ 1 Cabello, and H. Weinfurter, Decoherence-Free Quantum Another notable example is the DFS case ( ax ¼ ). In μ Ψ — Information Processing with Four-Photon Entangled that case, J ¼ μ the QGT and alternative tensor States, Phys. Rev. Lett. 92, 107901 (2004). become equal (Qalt ¼ Q). Therefore, it is the presence [16] L. Viola, E. M. Fortunato, M. A. Pravia, E. Knill, R. ϱ M Malt of ax that allows for two different metrics αβ and αβ. Laflamme, and D. G. Cory, Experimental Realization of However, for the NS case, the “alternative” curvature Noiseless Subsystems for Quantum Information Process- ing 293 Qalt F , Science , 2059 (2001). ½αβ ;μν does not reduce to the adiabatic curvature αβ;μν [17] N. Boulant, L. Viola, E. M. Fortunato, and D. G. Cory, associated with the connection Aα (unlike the QGT Experimental Implementation of a Concatenated Quantum curvature). How this subtle difference between Qαβ Error-Correcting Code., Phys. Rev. Lett. 94, 130501 alt (2005). and Qαβ for the NS and unique steady-state cases is [18] D. Kielpinski, V. Meyer, M. A. Rowe, C. A. Sackett, W. M. relevant in determining distances between adiabatic steady Itano, C. Monroe, and D. J. Wineland, A Decoherence- states of Lindbladians should be a subject of future Free Quantum Memory Using Trapped Ions, Science 291, investigation. 1013 (2001). [19] C. F. Roos, M. Chwalla, K. Kim, M. Riebe, and R. Blatt, Designer Atoms’ for Quantum Metrology, Nature (London) 443, 316 (2006). [20] J. T. Barreiro, P. Schindler, O. Gühne, T. Monz, M. [1] G. Lindblad, On the Generators of Quantum Dynamical Chwalla, C. F. Roos, M. Hennrich, and R. Blatt, Exper- Semigroups, Commun. Math. Phys. 48, 119 (1976). imental Multiparticle Entanglement Dynamics Induced by [2] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Decoherence, Nat. Phys. 6, 943 (2010). Completely Positive Dynamical Semigroups of N-Level [21] J. T. Barreiro, M. Müller, P. Schindler, D. Nigg, T. Monz, Systems, J. Math. Phys. (N.Y.) 17, 821 (1976). M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller, and R. [3] A. A. Belavin, B. Ya. Zel’dovich, A. M. Perelomov, and Blatt, An Open-System Quantum Simulator with Trapped V. S. Popov, Relaxation of Quantum Systems with Ions, Nature (London) 470, 486 (2011). Equidistant Spectra, Sov. Phys. JETP 29, 145 (1969). [22] P. Schindler, M. Müller, D. Nigg, J. T. Barreiro, E. A. [4] R. H. Lehmberg, Radiation from an N-Atom System. I. Martinez, M. Hennrich, T. Monz, S. Diehl, P. Zoller, and General Formalism, Phys. Rev. A 2, 883 (1970). R. Blatt, Quantum Simulation of Dynamical Maps with [5] A. Kossakowski, On Quantum Statistical Mechanics of Trapped Ions, Nat. Phys. 9, 361 (2013). Non-Hamiltonian Systems, Rep. Math. Phys. 3, 247 [23] T. Pruttivarasin, M. Ramm, S. G. Porsev, I. I. Tupitsyn, (1972). M. S. Safronova, M. A. Hohensee, and H. Häffner,
041031-31 ALBERT, BRADLYN, FRAAS, and JIANG PHYS. REV. X 6, 041031 (2016)
Michelson-Morley Analogue for Electrons Using Trapped [39] N. Y. Yao, A. V. Gorshkov, C. R. Laumann, A. M. Läuchli, Ions to Test Lorentz Symmetry, Nature (London) 517, 592 J. Ye, and M. D. Lukin, Realizing Fractional Chern (2015). Insulators in Dipolar Spin Systems, Phys. Rev. Lett. [24] Z. Leghtas, S. Touzard, I. M. Pop, A. Kou, B. Vlastakis, A. 110, 185302 (2013). Petrenko, K. M. Sliwa, A. Narla, S. Shankar, M. J. [40] E. Kapit, M. Hafezi, and S. H. Simon, Induced Self- Hatridge, M. Reagor, L. Frunzio, R. J. Schoelkopf, M. Stabilization in Fractional Quantum Hall States of Light, Mirrahimi, and M. H. Devoret, Confining the State of Light Phys. Rev. X 4, 031039 (2014). to a Quantum Manifold by Engineered Two-Photon Loss, [41] J. C. Budich, P. Zoller, and S. Diehl, Dissipative Prepa- Science 347, 853 (2015). ration of Chern Insulators, Phys. Rev. A 91, 042117 [25] B. Kraus, H. Büchler, S. Diehl, A. Kantian, A. Micheli, (2015). and P. Zoller, Preparation of Entangled States by [42] S. Caspar, F. Hebenstreit, D. Mesterházy, and U. J. Wiese, Quantum Markov Processes, Phys. Rev. A 78, 042307 Dynamics of Dissipative Bose-Einstein Condensation, (2008). Phys. Rev. A 93, 021602 (2016). [26] F. Verstraete, M. M. Wolf, and J. I. Cirac, Quantum [43] M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala, J. T. Computation and Quantum-State Engineering Driven by Barreiro, S. Nascimbène, N. R. Cooper, I. Bloch, and N. Dissipation, Nat. Phys. 5, 633 (2009). Goldman, Measuring the Chern Number of Hofstadter [27] M. J. Kastoryano, F. Reiter, and A. S. Sorensen, Dissipa- Bands with Ultracold Bosonic Atoms, Nat. Phys. 11, 162 tive Preparation of Entanglement in Optical Cavities, (2014). Phys. Rev. Lett. 106, 090502 (2011). [44] T. Li, L. Duca, M. Reitter, F. Grusdt, E. Demler, M. Endres, [28] H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, M. Schleier-Smith, I. Bloch, and U. Schneider, Exper- J. M. Petersen, J. I. Cirac, and E. S. Polzik, Entanglement imental Reconstruction of Wilson Lines in Bloch Bands, Generated by Dissipation and Steady State Entanglement Science 352, 1094 (2016). of Two Macroscopic Objects, Phys. Rev. Lett. 107, 080503 [45] M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider, J. (2011). Catani, C. Sias, P. Zoller, M. Inguscio, M. Dalmonte, and [29] M. Müller, S. Diehl, G. Pupillo, and P. Zoller, in Advances L. Fallani, Observation of Chiral Edge States with Neutral in Atomic, Molecular, and Optical Physics, edited by P. Fermions in Synthetic Hall Ribbons, Science 349, 1510 Berman, E. Arimondo, and C. Lin (Academic Press, (2015). New York, 2012), Vol. 61, pp. 1–80. [46] J. Kaczmarczyk, H. Weimer, and M. Lemeshko, Dissipa- [30] G. Morigi, J. Eschner, C. Cormick, Y. Lin, D. Leibfried, tive Preparation of Antiferromagnetic Order in the Fermi- and D. J. Wineland, Dissipative Quantum Control of a Hubbard Model, New J. Phys. 18 093042 (2016). Spin Chain, Phys. Rev. Lett. 115, 200502 (2015). [47] I. Carusotto, D. Gerace, H. E. Tureci, S. De Liberato, C. [31] F. Iemini, D. Rossini, R. Fazio, S. Diehl, and L. Ciuti, and A. Imamoğlu, Fermionized Photons in an Array Mazza, Dissipative Topological Superconductors in of Driven Dissipative Nonlinear Cavities, Phys. Rev. Lett. Number-Conserving Systems, Phys. Rev. B 93, 115113 103, 033601 (2009). (2016). [48] A. Tomadin, V. Giovannetti, R. Fazio, D. Gerace, I. [32] P. D. Johnson, F. Ticozzi, and L. Viola, General Fixed Carusotto, H. E. Türeci, and A. Imamoglu, Signatures of Points of Quasi-Local Frustration-Free Quantum Semi- the Superfluid-Insulator Phase Transition in Laser-Driven groups: From Invariance to Stabilization, Quantum Inf. Dissipative Nonlinear Cavity Arrays, Phys. Rev. A 81, Comput. 16, 0657 (2016). 061801(R) (2010). [33] T. Prosen, Matrix Product Solutions of Boundary Driven [49] I. Carusotto and C. Ciuti, Quantum Fluids of Light, Rev. Quantum Chains, J. Phys. A 48, 373001 (2015). Mod. Phys. 85, 299 (2013). [34] M. Žnidarič, Dissipative Remote-State Preparation in [50] V. V. Albert and L. Jiang, Symmetries and Conserved an Interacting Medium, Phys. Rev. Lett. 116, 030403 Quantities in Lindblad Master Equations, Phys. Rev. A 89, (2016). 022118 (2014). [35] F. Schwarz, M. Goldstein, A. Dorda, E. Arrigoni, A. [51] B. Buča and T. Prosen, A Note on Symmetry Reductions of Weichselbaum, and J. von Delft, Lindblad-Driven Discre- the Lindblad Equation: Transport in Constrained Open tized Leads for Non-Equilibrium Steady-State Transport in Spin Chains, New J. Phys. 14, 073007 (2012). Quantum Impurity Models: Recovering the Continuum [52] R. Alicki, On the Detailed Balance Condition for Limit, arXiv:1604.02050. Non-Hamiltonian Systems, Rep. Math. Phys. 10, 249 [36] S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. Büchler, (1976). and P. Zoller, Quantum States and Phases in Driven Open [53] W. De Roeck and C. Maes, Steady State Fluctuations of the Quantum Systems with Cold Atoms, Nat. Phys. 4, 878 Dissipated Heat for a Quantum Stochastic Model, Rev. (2008). Math. Phys. 18, 619 (2006). [37] H. Weimer, M. Müller, I. Lesanovsky, P. Zoller, and H. P. [54] V. Jakšić, C.-A. Pillet, and M. Westrich, Entropic Fluc- Büchler, A Rydberg Quantum Simulator, Nat. Phys. 6, 382 tuations of Quantum Dynamical Semigroups, J. Stat. Phys. (2010). 154, 153 (2014). [38] N. R. Cooper and J. Dalibard, Reaching Fractional Quan- [55] P. Strasberg, G. Schaller, T. Brandes, and M. Esposito, tum Hall States with Optical Flux Lattices, Phys. Rev. Lett. Thermodynamics of Quantum-Jump-Conditioned Feed- 110, 185301 (2013). back Control, Phys. Rev. E 88, 062107 (2013).
041031-32 GEOMETRY AND RESPONSE OF LINDBLADIANS PHYS. REV. X 6, 041031 (2016)
[56] M. A. Nielsen and I. L. Chuang, Quantum Computation [74] J. P. Provost and G. Vallee, Riemannian Structure on and Quantum Information (Cambridge University Press, Manifolds of Quantum States, Commun. Math. Phys. Cambridge, England, 2011). 76, 289 (1980). [57] H. Lu, C. Liu, D.-S. Wang, L.-K. Chen, Z.-D. Li, [75] M. V. Berry, in Geometric Phases in Physics, edited by F. X.-C. Yao, L. Li, N.-L. Liu, C.-Z. Peng, B. C. Sanders, Wilczek and A. Shapere (World Scientific, Singapore, Y.-A. Chen, and J.-W. Pan, Universal Digital Photonic 1989), Chap. 1. Single-Qubit Quantum Channel Simulator, arXiv: [76] R. Kubo, Statistical-Mechanical Theory of Irreversible 1505.02879. Processes. I. General Theory and Simple Applications to [58] B. M. Terhal, Quantum Error Correction for Quantum Magnetic and Conduction Problems, J. Phys. Soc. Jpn. 12, Memories, Rev. Mod. Phys. 87, 307 (2015). 570 (1957). [59] P. Zanardi and L. Campos Venuti, Coherent Quantum [77] L. P. Kadanoff and P. C. Martin, Hydrodynamic Equations Dynamics in Steady-State Manifolds of Strongly and Correlation Functions, Ann. Phys. (N.Y.) 24, 419 Dissipative Systems, Phys. Rev. Lett. 113, 240406 (1963). (2014). [78] B. Baumgartner and H. Narnhofer, Analysis of Quantum [60] P. Zanardi and L. Campos Venuti, Geometry, Robustness, Semigroups with GKS-Lindblad Generators II. General, J. and Emerging Unitarity in Dissipation-Projected Dynam- Phys. A 41, 32 (2008). ics, Phys. Rev. A 91, 052324 (2015). [79] R. Blume-Kohout, H. K. Ng, D. Poulin, and L. Viola, [61] M. Mirrahimi, Z. Leghtas, V. V. Albert, S. Touzard, Information-Preserving Structures: A General Framework R. J. Schoelkopf, L. Jiang, and M. H. Devoret, Dynami- for Quantum Zero-Error Information, Phys. Rev. A 82, cally Protected Cat-Qubits: A New Paradigm for Univer- 062306 (2010). sal Quantum Computation, New J. Phys. 16, 045014 [80] F. Ticozzi and L. Viola, Quantum Markovian Subsystems: (2014). Invariance, Attractivity, and Control, IEEE Trans. Autom. [62] V. Paulisch, H. J. Kimble, and A. González-Tudela, Uni- Control 53, 2048 (2008). versal Quantum Computation in Waveguide QED Using [81] J. Deschamps, F. Fagnola, E. Sasso, and V. Umanita’, Decoherence Free Subspaces, New J. Phys. 18, 043041 Structure of Uniformly Continuous Quantum Markov (2016). Semigroups, Rev. Math. Phys. 28, 1650003 (2016). [63] P. Facchi and S. Pascazio, Quantum Zeno Subspaces, Phys. [82] G. Lindblad, A General No-Cloning Theorem, Lett. Math. Rev. Lett. 89, 080401 (2002). Phys. 47, 189 (1999). [64] F. Schäfer, I. Herrera, S. Cherukattil, C. Lovecchio, F. S. [83] R. Blume-Kohout, H. K. Ng, D. Poulin, and L. Viola, Cataliotti, F. Caruso, and A. Smerzi, Experimental Reali- Characterizing the Structure of Preserved Information in zation of Quantum Zeno Dynamics, Nat. Commun. 5, 3194 Quantum Processes, Phys. Rev. Lett. 100, 030501 (2008). (2014). [84] B. Baumgartner and H. Narnhofer, The Structures of State [65] A. Signoles, A. Facon, D. Grosso, I. Dotsenko, S. Haroche, Space Concerning Quantum Dynamical Semigroups, Rev. J.-M. Raimond, M. Brune, and S. Gleyzes, Confined Math. Phys. 24, 1250001 (2012). Quantum Zeno Dynamics of a Watched Atomic Arrow, [85] R. Carbone and Y. Pautrat, Irreducible Decompositions Nat. Phys. 10, 715 (2014). and Stationary States of Quantum Channels, Rep. Math. [66] C. Arenz, D. Burgarth, P. Facchi, V. Giovannetti, H. Phys. 77, 293 (2016). Nakazato, S. Pascazio, and K. Yuasa, Universal Control [86] M. M. Wolf, Quantum Channels & Operations Guided Induced by Noise, Phys. Rev. A 93, 062308 (2016). Tour (2010), http://www‑m5.ma.tum.de/foswiki/pub/M5/ [67] A. Carollo, M. F. Santos, and V. Vedral, Coherent Quan- Allgemeines/MichaelWolf/QChannelLecture.pdf. tum Evolution via Reservoir Driven Holonomies, Phys. [87] A. Shabani and D. A. Lidar, Theory of Initialization-Free Rev. Lett. 96, 020403 (2006). Decoherence-Free Subspaces and Subsystems, Phys. Rev. [68] M. S. Sarandy and D. A. Lidar, Abelian and Non-Abelian A 72, 042303 (2005). Geometric Phases in Adiabatic Open Quantum Systems, [88] G. I. Cirillo and F. Ticozzi, Decompositions of Hilbert Phys. Rev. A 73, 062101 (2006). Spaces, Stability Analysis and Convergence Probabilities [69] O. Oreshkov and J. Calsamiglia, Adiabatic Markovian for Discrete-Time Quantum Dynamical Semigroups, J. Dynamics, Phys. Rev. Lett. 105, 050503 (2010). Phys. A 48, 085302 (2015). [70] J. E. Avron, M. Fraas, and G. M. Graf, Adiabatic [89] J. Novotný, G. Alber, and I. Jex, Asymptotic Properties of Response for Lindblad Dynamics, J. Stat. Phys. 148, Quantum Markov Chains, J. Phys. A 45, 485301 (2012). 800 (2012). [90] M. Jakob and S. Stenholm, Variational Functions in [71] V. V. Albert, C. Shu, S. Krastanov, C. Shen, R.-B. Liu, Degenerate Open Quantum Systems, Phys. Rev. A 69, Z.-B. Yang, R. J. Schoelkopf, M. Mirrahimi, M. H. 042105 (2004). Devoret, and L. Jiang, Holonomic Quantum Control with [91] L. M. Sieberer, M. Buchhold, and S. Diehl, Keldysh Field Continuous Variable Systems, Phys. Rev. Lett. 116, Theory for Driven Open Quantum Systems, Rep. Prog. 140502 (2016). Phys. 79 096001 (2016). [72] J. E. Avron, M. Fraas, G. M. Graf, and P. Grech, Adiabatic [92] S. Lloyd and L. Viola, Engineering Quantum Dynamics, Theorems for Generators of Contracting Evolutions, Phys. Rev. A 65, 010101 (2001). Commun. Math. Phys. 314, 163 (2012). [93] E. Andersson and D. K. L. Oi, Binary Search Trees for [73] P. Zanardi and M. Rasetti, Holonomic Quantum Compu- Generalized Measurements, Phys. Rev. A 77, 052104 tation, Phys. Lett. A 264, 94 (1999). (2008).
041031-33 ALBERT, BRADLYN, FRAAS, and JIANG PHYS. REV. X 6, 041031 (2016)
[94] D.-S. Wang and B. C. Sanders, Quantum Circuit Design [115] P. Zanardi, P. Giorda, and M. Cozzini, Information- for Accurate Simulation of Qudit Channels, New J. Phys. Theoretic Differential Geometry of Quantum Phase Tran- 17, 043004 (2015). sitions, Phys. Rev. Lett. 99, 100603 (2007). [95] C. Shen et al., Quantum Channel Construction with [116] M. Kolodrubetz, V. Gritsev, and A. Polkovnikov, Classi- Circuit Quantum Electrodynamics (to be published). fying and Measuring Geometry of a Quantum Ground [96] R. Azouit, A. Sarlette, and P. Rouchon, Adiabatic Elimi- State Manifold, Phys. Rev. B 88, 064304 (2013). nation for Open Quantum Systems with Effective Lindblad [117] R. Roy, Band Geometry of Fractional Topological Insula- Master Equations, arXiv:1603.04630. tors, Phys. Rev. B 90, 165139 (2014). [97] J. Anandan and Y. Aharonov, Geometric Quantum Phase [118] E. Dobardžić, M. V. Milovanović, and N. Regnault, Geo- and Angles, Phys. Rev. D 38, 1863 (1988). metrical Description of Fractional Chern Insulators Based [98] A. Beige, D. Braun, B. Tregenna, and P. L. Knight, on Static Structure Factor Calculations, Phys. Rev. B 88, Quantum Computing Using Dissipation to Remain in a 115117 (2013). Decoherence-Free Subspace, Phys. Rev. Lett. 85, 1762 [119] T. S. Jackson, G. Möller, and R. Roy, Geometric Stability (2000). of Topological Lattice Phases, Nat. Commun. 6, 8629 [99] W. K. Abou Salem, On the Quasi-Static Evolution of (2015). Nonequilibrium Steady States, Ann. Inst. Henri Poincaré [120] D. Bauer, T. S. Jackson, and R. Roy, Quantum Geometry 8, 569 (2007). and Stability of the Fractional Quantum Hall Effect [100] A. Joye, General Adiabatic Evolution with a Gap Con- in the Hofstadter Model, Phys. Rev. B 93, 235133 dition, Commun. Math. Phys. 275, 139 (2007). (2016). [101] J. Schmid, Adiabatic Theorems With and Without Spectral [121] L. Banchi, P.Giorda, and P.Zanardi, Quantum Information- Gap Condition for Non-Semisimple Spectral Values, Geometry of Dissipative Quantum Phase Transitions, Phys. arXiv:1401.0089. Rev. E 89, 022102 (2014). [102] L. Campos Venuti, T. Albash, D. A. Lidar, and P. Zanardi, [122] U. Marzolino and T. Prosen, Quantum Metrology with Adiabaticity in Open Quantum Systems, Phys. Rev. A 93, Nonequilibrium Steady States of Quantum Spin Chains, 032118 (2016). Phys. Rev. A 90, 062130 (2014). [103] E. B. Davies and H. Spohn, Open Quantum Systems with [123] J. H. Wei and Y. Yan, Linear Response Theory for Time-Dependent Hamiltonians and Their Linear Re- Quantum Open Systems, arXiv:1108.5955. sponse, J. Stat. Phys. 19, 511 (1978). [124] H. Z. Shen, W. Wang, and X. X. Yi, Hall Conductance and [104] P. Thunstrom, J. Aberg, and E. Sjoqvist, Adiabatic Topological Invariant for Open Systems, Sci. Rep. 4, 6455 Approximation for Weakly Open Systems, Phys. Rev. A (2014). 72, 022328 (2005). [125] M. Ban, Linear Response Theory for Open Quantum [105] J. P. Pekola, V. Brosco, M. Möttönen, P. Solinas, and A. Systems within the Framework of the ABL Formalism, Shnirman, Decoherence in Adiabatic Quantum Evolution: Quantum Stud. Math. Found. 2, 51 (2015). Application to Cooper Pair Pumping, Phys. Rev. Lett. 105, [126] R. Chetrite and K. Mallick, Quantum Fluctuation Rela- 030401 (2010). tions for the Lindblad Master Equation, J. Stat. Phys. 148, [106] J. Anandan and Y. Aharonov, Geometry of Quantum 480 (2012). Evolution, Phys. Rev. Lett. 65, 1697 (1990). [127] L. Campos Venuti and P. Zanardi, Dynamical Response [107] T. Neupert, C. Chamon, and C. Mudry, Measuring the Theory for Driven-Dissipative Quantum Systems, Phys. Quantum Geometry of Bloch Bands with Current Noise, Rev. A 93, 032101 (2016). Phys. Rev. B 87, 245103 (2013). [128] J. E. Gough, T. S. Ratiu, and O. G. Smolyanov, Noether’s [108] J. E. Avron and R. Seiler, Quantization of the Hall Theorem for Dissipative Quantum Dynamical Semi- Conductance for General, Multiparticle Schrödinger Groups, J. Math. Phys. (N.Y.) 56, 022108 (2015). Hamiltonians, Phys. Rev. Lett. 54, 259 (1985). [129] S. Mukamel, Principles of Nonlinear Optical Spectroscopy [109] D. Xiao, M.-C. Chang, and Q. Niu, Berry Phase Effects (Oxford University Press, New York, 1995). on Electronic Properties, Rev. Mod. Phys. 82, 1959 (2010). [130] B. M. Villegas-Martínez, F. Soto-Eguibar, and H. M. [110] N. Read and E. H. Rezayi, Hall Viscosity, Orbital Spin, Moya-Cessa, Application of Perturbation Theory to and Geometry: Paired Superfluids and Quantum Hall a Master Equation, Adv. Math. Phys. 2016, 9265039 Systems, Phys. Rev. B 84, 085316 (2011). (2016). [111] J. E. Avron, M. Fraas, G. M. Graf, and O. Kenneth, [131] I. A. Dmitriev, A. D. Mirlin, and D. G. Polyakov, Cyclotron- Quantum Response of Dephasing Open Systems, New J. Resonance Harmonics in the ac Response of a 2D Electron Phys. 13, 053042 (2011). Gas with Smooth Disorder, Phys. Rev. Lett. 91, 226802 [112] V. Gritsev and A. Polkovnikov, Dynamical Quantum Hall (2003). Effect in the Parameter Space, Proc. Natl. Acad. Sci. [132] N. Schine, A. Ryou, A. Gromov, A. Sommer, and J. U.S.A. 109, 6457 (2012). Simon, Synthetic Landau Levels for Photons, Nature 534, [113] B. Bradlyn, M. Goldstein, and N. Read, Kubo Formulas for 671 (2016). Viscosity: Hall Viscosity, Ward Identities, and the Relation [133] There can be, in general, a set of exactly degenerate with Conductivity, Phys. Rev. B 86, 245309 (2012). ground states. However, for fractional quantum Hall [114] L. Campos Venuti and P. Zanardi, Quantum Critical systems, these states cannot be connected by local Scaling of the Geometric Tensors, Phys. Rev. Lett. 99, operators, and, hence, they do not play a role in our 095701 (2007). linear response calculation.
041031-34 GEOMETRY AND RESPONSE OF LINDBLADIANS PHYS. REV. X 6, 041031 (2016)
[134] W. Kohn, Cyclotron Resonance and de Haas–van Alphen [153] F. Reiter and A. S. Sorensen, Effective Operator Formal- Oscillations of an Interacting Electron Gas, Phys. Rev. ism for Open Quantum Systems, Phys. Rev. A 85, 032111 123, 1242 (1961). (2012). [135] Z. Cai and T. Barthel, Algebraic versus Exponential [154] A. C. Y. Li, F. Petruccione, and J. Koch, Resummation Decoherence in Dissipative Many-Particle Systems, Phys. for Nonequilibrium Perturbation Theory and Applica- Rev. Lett. 111, 150403 (2013). tion to Open Quantum Lattices, Phys. Rev. X 6, 021037 [136] M. Žnidarič, Relaxation Times of Dissipative Many- (2016). Body Quantum Systems, Phys. Rev. E 92, 042143 [155] A. C. Y. Li, F. Petruccione, and J. Koch, Perturbative (2015). Approach to Markovian Open Quantum Systems, Sci. Rep. [137] P. Zanardi, J. Marshall, and L. Campos Venuti, Dissipative 4, 4887 (2014). Universal Lindbladian Simulation, Phys. Rev. A 93, [156] K. Macieszczak, M. Guta, I. Lesanovsky, and J. P. 022312 (2016). Garrahan, Towards a Theory of Metastability in Open [138] R. I. Karasik, K.-P. Marzlin, B. C. Sanders, and K. B. Quantum Dynamics, Phys. Rev. Lett. 116, 240404 (2016). Whaley, Criteria for Dynamically Stable Decoherence- [157] A. del Campo, I. L. Egusquiza, M. B. Plenio, and S. F. Free Subspaces and Incoherently Generated Coherences, Huelga, Quantum Speed Limits in Open System Dynamics, Phys. Rev. A 77, 052301 (2008). Phys. Rev. Lett. 110, 050403 (2013). [139] E. Hach III and C. Gerry, Generation of Mixtures of [158] P. Rouchon and A. Sarlette, in Proceedings of the Schrödinger-Cat States from a Competitive Two-Photon 52nd IEEE Conference on Decision and Control (IEEE, Process, Phys. Rev. A 49, 490 (1994). New York, 2013), pp. 6568–6573. [140] R. Azouit, A. Sarlette, and P. Rouchon, Convergence and [159] J. Jing, L.-A. Wu, and A. del Campo, Fundamental Speed Adiabatic Elimination for a Driven Dissipative Quantum Limits to the Generation of Quantumness, arXiv:1510.01106. Harmonic Oscillator,in54th IEEE Conference on Deci- [160] D. P. Pires, M. Cianciaruso, L. C. Céleri, G. Adesso, and sion and Control (CDC), Osaka, 2015 (IEEE, New York, D. O. Soares-Pinto, Generalized Geometric Quantum 2015). Speed Limits, Phys. Rev. X 6, 021031 (2016). [141] J. E. Avron, M. Fraas, G. M. Graf, and P. Grech, Optimal [161] O. Viyuela, A. Rivas, and M. A. Martin-Delgado, Time Schedule for Adiabatic Evolution, Phys. Rev. A 82, Uhlmann Phase as a Topological Measure for One- 040304 (2010). Dimensional Fermion Systems, Phys. Rev. Lett. 112, [142] For open paths, B is related to noncyclic geometric phases 130401 (2014). in other dissipative systems [see, e.g., Ref. [143], Eq. (47)] [162] A. Rivas, O. Viyuela, and M. A. Martin-Delgado, Density- and non-Hermitian systems [144]. Matrix Chern Insulators: Finite-Temperature Generaliza- [143] N. A. Sinitsyn, The Stochastic Pump Effect and Geometric tion of Topological Insulators, Phys. Rev. B 88, 155141 Phases in Dissipative and Stochastic Systems, J. Phys. A (2013). 42, 193001 (2009). [163] J. C. Budich and B. Trauzettel, From the Adiabatic [144] D. Viennot, A. Leclerc, G. Jolicard, and J. P. Killingbeck, Theorem of Quantum Mechanics to Topological States Consistency between Adiabatic and Non-Adiabatic Geo- of Matter, Phys. Status Solidi RRL 7, 109 (2013). metric Phases for Non-Self-Adjoint Hamiltonians, J. Phys. [164] Z. Huang and D. P. Arovas, Topological Indices for Open A 45, 335301 (2012). and Thermal Systems via Uhlmann’s Phase, Phys. Rev. [145] If M is not simply connected (i.e., has holes), then the Lett. 113, 076407 (2014). Berry phase may contain “topological” contributions. Such [165] H. Wilming, M. J. Kastoryano, A. H. Werner, and J. Eisert, effects are responsible for anyonic statistics (see, e.g., Emergence of Spontaneous Symmetry Breaking in Ref. [146], Sec. I.B) and can produce Berry phases even Dissipative Lattice Systems, arXiv:1602.01108. for a one-dimensional parameter space [147]. [166] F. Ticozzi, R. Lucchese, P. Cappellaro, and L. Viola, [146] N. Read, Non-Abelian Adiabatic Statistics and Hall Hamiltonian Control of Quantum Dynamical Semigroups: Viscosity in Quantum Hall States and px þ ipy Paired Stabilization and Convergence Speed, IEEE Trans. Superfluids, Phys. Rev. B 79, 045308 (2009). Autom. Control 57, 1931 (2012). [147] J. Zak, Berry’s Phase for Energy Bands in Solids, Phys. [167] S. Ying, Y. Feng, N. Yu, and M. Ying, Reachability Rev. Lett. 62, 2747 (1989). Probabilities of Quantum Markov Chains, Lect. Notes [148] M. Nakahara, Geometry, Topology, and Physics, 2nd ed. Comput. Sci. 8052, 334 (2013). (IOP Publishing, Bristol, England, 2003). [168] A. Dhahri, F. Fagnola, and R. Rebolledo, The [149] I. Y. Aref’eva, Non-Abelian Stokes Formula, Theor. Math. Decoherence-Free Subalgebra of a Quantum Markov Phys. 43, 353 (1980). Semigroup with Unbounded Generator, Inf. Dim. Anal. [150] A. I. Nesterov, Non-Hermitian Quantum Systems and Quant. Prob. Rel. Topics 13, 413 (2010). Time-Optimal Quantum Evolution, SIGMA 5,69 [169] S. Alipour, D. Chruscinski, P. Facchi, G. Marmo, S. (2009). Pascazio, and A. T. Rezakhani, Dynamically Contracted [151] D. C. Brody and E.-M. Graefe, Information Geometry of Algebra of Observables for Dissipative Quantum Systems, Complex Hamiltonians and Exceptional Points, Entropy arXiv:1511.08132. 15, 3361 (2013). [170] H. M. Wiseman and G. J. Milburn, Quantum Measurement [152] D. C. Brody, Biorthogonal Quantum Mechanics, J. Phys. and Control (Cambridge University Press, Cambridge, A 47, 035305 (2014). England, 2009).
041031-35 ALBERT, BRADLYN, FRAAS, and JIANG PHYS. REV. X 6, 041031 (2016)
[171] T. Benoist, C. Pellegrini, and F. Ticozzi, Exponential [187] D. Chruscinski and A. Jamiolkowski, Geometric Phases in Stability of Subspaces for Quantum Stochastic Master Classical and Quantum Physics (Birkhauser, Boston, Equations, arXiv:1512.00732. 2004). [172] J. E. Tyson, Operator-Schmidt Decompositions and the [188] J. E. Avron, in Mesoscopic Quantum Physics, Proceedings Fourier Transform, with Applications to the Operator- of Les Houches Summer School, Session LXI, edited by Schmidt Numbers of Unitaries, J. Phys. A 36, 10101 E. Akkermans, G. Montambaux, J. Pichard, and J. (2003). Zinn-Justin, (Elsevier, Amsterdam, 1995). [173] M. Zwolak and G. Vidal, Mixed-State Dynamics in One- [189] B. Simon, Holonomy, the Quantum Adiabatic Theorem, Dimensional Quantum Lattice Systems: A Time-Dependent and Berry’s Phase, Phys. Rev. Lett. 51, 2167 (1983). Superoperator Renormalization Algorithm, Phys. Rev. [190] J. E. Avron, L. Sadun, J. Segert, and B. Simon, Chern Lett. 93, 207205 (2004). Numbers, Quaternions, and Berry’s Phases in Fermi [174] I. Bengtsson and K. Zyczkowski, Geometry of Quantum Systems, Commun. Math. Phys. 124, 595 (1989). States: An Introduction to Quantum Entanglement [191] M. Kolodrubetz, P. Mehta, and A. Polkovnikov, Geometry (Cambridge University Press, Cambridge, England, 2006). and Non-Adiabatic Response in Quantum and Classical [175] Á. Rivas, A. D. K. Plato, S. F. Huelga, and M. B. Plenio, Systems, arXiv:1602.01062. Markovian Master Equations: A Critical Study, New J. [192] A. T. Rezakhani, D. F. Abasto, D. A. Lidar, and P. Zanardi, Phys. 12, 113032 (2010). Intrinsic Geometry of Quantum Adiabatic Evolution and [176] C. Gardiner and P. Zoller, Quantum Noise (Springer, Quantum Phase Transitions, Phys. Rev. A 82, 012321 Berlin, 2000). (2010). [177] S. G. Schirmer and X. Wang, Stabilizing Open Quantum [193] Y.-Q. Ma, S. Chen, H. Fan, and W.-M. Liu, Abelian and Systems by Markovian Reservoir Engineering, Phys. Rev. Non-Abelian Quantum Geometric Tensor, Phys. Rev. B A 81, 062306 (2010). 81, 245129 (2010). [178] K. Alicki and R. Lendi, Quantum Dynamical Semigroups [194] A. Mostafazadeh, Quantum Brachistochrone Problem and and Applications (Springer-Verlag, Berlin, 2007). the Geometry of the State Space in Pseudo-Hermitian [179] M. S. Sarandy and D. A. Lidar, Adiabatic Approximation Quantum Mechanics, Phys. Rev. Lett. 99, 130502 (2007). in Open Quantum Systems, Phys. Rev. A 71,012331 [195] A. Mostafazadeh, Hamiltonians Generating Optimal- (2005). Speed Evolutions, Phys. Rev. A 79, 014101 (2009). [180] J. Ortigoso, Quantum Adiabatic Theorem in Light of the [196] S. Jansen, M.-B. Ruskai, and R. Seiler, Bounds for the Marzlin-Sanders Inconsistency, Phys. Rev. A 86, 032121 Adiabatic Approximation with Applications to Quantum (2012). Computation, J. Math. Phys. (N.Y.) 48, 102111 (2007). [181] T. Kato, On the Adiabatic Theorem of Quantum Mechan- [197] M. Demirplak and S. A. Rice, Adiabatic Population ics, J. Phys. Soc. Jpn. 5, 435 (1950). Transfer with Control Fields, J. Phys. Chem. A 107, [182] S. I. Vinitskii, V. L. Derbov, V. M. Dubovik, B. L. 9937 (2003). Markovski, and Y. P. Stepanovskii, Topological Phases [198] R. Lim and M. V. Berry, Superadiabatic Tracking of in Quantum Mechanics and Polarization Optics, Sov. Quantum Evolution, J. Phys. A 24, 3255 (1991). Phys. Usp. 33, 403 (1990). [199] G. Vacanti, R. Fazio, S. Montangero, G. M. Palma, M. [183] S. Pancharatnam, Generalized Theory of Interference, Paternostro, and V. Vedral, Transitionless Quantum and Its Applications, Proc. Indian Acad. Sci. A 44, 247 Driving in Open Quantum Systems, New J. Phys. 16, (1956). 053017 (2014). [184] M. V. Berry, Quantal Phase Factors Accompanying [200] M. V. Berry, Transitionless Quantum Driving, J. Phys. A Adiabatic Changes, Proc. R. Soc. A 392, 45 (1984). 42, 365303 (2009). [185] Y.Aharonov and J. Anandan, Phase Change during a Cyclic [201] E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, Quantum Evolution, Phys. Rev. Lett. 58, 1593 (1987). A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, [186] F. Wilczek and A. Zee, Appearance of Gauge Structure in and J. G. Muga, Advances In Atomic, Molecular, and Simple Dynamical Systems, Phys. Rev. Lett. 52, 2111 Optical Physics (Elsevier, New York, 2013), Vol. 62, (1984). Chap. 2, pp. 117–169.
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