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Concepts of quantum non-Markovianity: a hierarchy

Li Li, Michael J. W. Hall, Howard M. Wiseman

arXiv:1712.08879v3quant-ph

TL;DR

The report addresses the lack of a consistent, rigorous meaning for quantum Markovianity by comparing classical and quantum processes and defining many related concepts in a unified framework. It derives hierarchy relations among these concepts and concludes that quantum Markovianity is context-dependent, while noting that some conditions are idealizations rather than exact properties of physical systems.

  • Problem

    Quantum Markovianity is used inconsistently and unrigorously, while quantum processes have inherent differences from classical stochastic processes that complicate characterizing non-Markovianity.

  • Method

    The report rigorously defines many existing and newly proposed Markov-related concepts for open quantum systems within a unified, general framework.

  • Results

    The concepts form a nontrivial hierarchy, showing that quantum Markovianity has multiple possible physical interpretations rather than one universally appropriate definition.

  • Takeaways & Limitations

    Quantum non-Markovianity should be assessed relative to the specific Markov-related condition relevant to the context.

  • Takeaways & Limitations

    Past–future independence and the quantum regression formula are not strictly satisfied by physical continuous-time systems because they idealize zero bath correlation time or related conditions.

Abstract

from arXiv · show

Markovian approximation is a widely-employed idea in descriptions of the dynamics of open quantum systems (OQSs). Although it is usually claimed to be a concept inspired by classical Markovianity, the term quantum Markovianity is used inconsistently and often unrigorously in the literature. In this report we compare the descriptions of classical stochastic processes and quantum stochastic processes (as arising in OQSs), and show that there are inherent differences that lead to the non-trivial problem of characterizing quantum non-Markovianity. Rather than proposing a single definition of quantum Markovianity, we study a host of Markov-related concepts in the quantum regime. Some of these concepts have long been used in quantum theory, such as quantum white noise, factorization approximation, divisibility, Lindblad master equation, etc.. Others are first proposed in this report, including those we call past-future independence, no (quantum) information backflow, and composability. All of these concepts are defined under a unified framework, which allows us to rigorously build hierarchy relations among them. With various examples, we argue that the current most often used definitions of quantum Markovianity in the literature do not fully capture the memoryless property of OQSs. In fact, quantum non-Markovianity is highly context-dependent. The results in this report, summarized as a hierarchy figure, bring clarity to the nature of quantum non-Markovianity.

1. Introduction

Classical Markovianity has a clear mathematical formulation, but extending it to open quantum systems is nontrivial because quantum observables and system–environment correlations introduce distinct issues. The report therefore organizes multiple quantum Markov-related concepts into a hierarchy rather than selecting one universal definition.

  • Classical foundations: Classical Markovianity formalizes the condition that future states depend statistically only on the current state.The condition is particularly natural for discrete-time stochastic processes.
  • From classical to quantum: Quantum generalization is difficult because quantum observables are operators, and even restricted sets may not admit a classical probability description.Open-system coupling also produces correlations between system and environment, leading to non-unitary reduced dynamics.
  • Quantum Markov-related concepts: The report reviews existing methods and criteria, including factorization, quantum white noise, master equations, dynamical decoupling, and quantum unravellings.It formalizes existing concepts, introduces new ones, and establishes connections among them.
  • Competing approaches: State-based approaches use system evolution and distinguishability, whereas other approaches explicitly examine system–environment interactions or interventions.State-based approaches can ignore environmental correlations that are relevant to broader notions of non-Markovianity.
  • Central conclusion: Quantum Markovianity is presented as context-dependent, so rigorous analysis must identify which mathematical condition is relevant before testing an evolution.The report argues that no single concept should universally bear the name quantum Markovianity.
  • Hierarchy and scope: The report defines its concepts in a unified, general framework and derives their nontrivial hierarchical relations, summarized in Figure 1.The figure distinguishes concepts requiring interaction knowledge from those formulated solely through the system dynamical map.

2. Open quantum systems

The report models an open quantum system together with its environment as an isolated composite system evolving unitarily, then obtains system dynamics by tracing out the environment. This produces completely positive, trace-preserving dynamical maps and supports both Schrödinger- and Heisenberg-picture descriptions.

  • Composite-system model: The system–environment Hilbert space is Hs ⊗ He, and the total Hamiltonian contains system, environment, and interaction terms.The interaction-term split is not unique because system- or environment-only Hermitian terms can be redistributed.
  • Total evolution: The total evolution is generated by a possibly time-dependent Hamiltonian and may include singular or kicked limits used to describe quantum white noise.Definitions are formulated where appropriate for both discrete and continuous time.
  • Initial conditions and correlations: The initial system–environment state is assumed factorizable, although interaction typically generates correlations and entanglement at later times.These correlations make the reduced system state mixed and represent decoherence when attention is restricted to the system.
  • Reduced dynamics: The system state is obtained by taking the partial trace over the environment, yielding reduced dynamics from the composite unitary evolution.This reduction is useful because the system evolution is often the quantity of interest, while exact total-state analysis can be intractable.
  • Dynamical maps: The resulting dynamical map is linear, trace-preserving, positive, and completely positive, and therefore maps system states to valid quantum states.Complete positivity ensures validity even when the system is considered jointly with an ancilla.
  • Representations and framework: Equivalent descriptions can be given in Schrödinger, interaction, and Heisenberg pictures, with the latter evolving operators while leaving the state fixed.The report uses this general formalism as the basis for defining Markov-related concepts and deriving their hierarchy.

3. Formalizations of quantum Markovianity

The report formalizes several quantum Markovianity-related concepts, including factorization approximation, quantum white noise, past–future independence, and quantum regression. It emphasizes that these concepts differ in scope and that idealized conditions may only approximate physical dynamics.

  • 3.1. Factorization approximation: Factorization approximation treats the joint system–environment state as approximately factorized when system–environment coupling is weak.This is typically motivated by environments that are much larger than the system.
  • 3.1. Factorization approximation: Under factorization approximation, the environment cannot store information about the system, so the system state alone carries information about its past.This motivates interpreting factorization approximation as a memoryless condition.
  • 3.1. Factorization approximation: The formalized factorization approximation replaces approximate factorization with an equality and requires the environment state to be independent of the initial system state, making it a strong condition.The report includes this condition in its hierarchy because of its implications for other Markovianity concepts.
  • 3.2. Quantum white noise: Quantum white-noise dynamics can be represented in the Heisenberg picture using Hudson–Parthasarathy evolution driven by bath noise operators.The noise statistics are fixed by the initial bath state and are independent of the system, with the description becoming exact for a delta-function bath correlation.
  • 3.3. Past–future independence: Past–future independence models the environment as initially uncorrelated components that interact with the system sequentially, separating already-interacted past parts from incoming future parts.The concept is broader than cases in which a differential equation for the system state exists.
  • 3.3. Past–future independence: Past–future independence requires all three defining conditions together; failure of any one can permit strongly non-Markovian evolution.In continuous time, exact past–future independence requires zero bath correlation time, so it is an idealization that can hold approximately when correlations are sufficiently short.
  • 3.4.1. Quantum regression formula: The quantum regression formula is weaker than factorization approximation because it captures a similar memoryless aspect only for two-time correlation functions.Its generalized dynamical map replaces the joint state at the earlier time by a factorized state.
  • 3.4.1. Quantum regression formula: Quantum regression and other Markovian assumptions are not strictly exact for continuous-time physical systems, but may accurately describe idealized models such as white-noise environments.The report therefore treats the practical validity range of such models as the relevant issue.

3.5. System interventions

System interventions expose memory effects by testing whether past operations alter future evolution, while environment interventions test whether removing system–environment correlations changes that evolution. These criteria distinguish several context-dependent forms of quantum Markovianity, including dynamical-decoupling failure, composability, no information backflow, and no quantum information backflow.

  • System interventions: Dynamical decoupling fails completely when decoherence is generated by the dynamical maps, and quantum white noise defeats decoupling regardless of pulse number.Decoupling also cannot remove noise above the control frequency.
  • System interventions: Under quantum white noise or past-future independence, failure of dynamical decoupling is a logical consequence, whereas successful decoupling is evidence of non-Markovianity.The paper states only a sufficient condition toward decoupling failure, not implications from failure to other Markovianity concepts.
  • Environment interventions: Environment interventions remove selected system–environment correlations and test whether information encoded in those correlations can flow back into the system.If the system evolution is unchanged, the corresponding backflow is absent under that intervention.
  • Composability: Composability uses an environment reset and means future system evolution matches the evolution obtained after applying the factorization approximation.It is sufficient for no information backflow but depends on the specific reset channel.
  • Information backflow: No information backflow generalizes composability by allowing any replacement channel that removes all system–environment correlations.No quantum information backflow is weaker: classical information may still return, so it is not proposed as a full definition of quantum Markovianity.
  • Quantum unravellings: Environment measurements can recover information lost through tracing out the bath while preserving the average system evolution, enabling repeated conditional quantum trajectories under stronger requirements.A single-time pure conditioned state is universal for open quantum systems; multi-time preservation is the Markovianity-relevant requirement.

3.8. Concepts deriving from the dynamical map

The paper examines quantum Markovianity through mathematical properties of dynamical maps, including divisibility, semigroups, Lindblad equations, distinguishability, and Monte Carlo wave-function concepts. These properties are related but do not uniformly capture the full memoryless behavior of open quantum systems.

  • Overview: Map-based definitions are mathematically simpler but correspond to classical concepts that are strictly weaker than classical Markovianity.The approach focuses on dynamical-map structure without specifying system–environment interaction details.
  • Divisibility: Divisibility requires a CPTP map connecting every earlier system state to its corresponding later state, without requiring memory of the state before the earlier time.It generalizes composability and is distinct from infinite divisibility.
  • Dynamical semigroups: Dynamical semigroups impose a composition law that makes the system evolution time-homogeneous and can apply to discrete or continuous time.For continuous time, the semigroup has a generator when the relevant limit exists.
  • Master equations: A time-independent generator yields a homogeneous master equation, while the corresponding memory-kernel formulation can include non-Markovian memory effects.Under suitable conditions, the memory-kernel equation reduces to a time-local differential equation generated by L.
  • GKS-Lindblad equations: A negative canonical decoherence rate, γ_k(t) < 0, represents quantum non-Markovianity when the time-dependent GKS-Lindblad concept is adopted.The time-dependent GKS-Lindblad equation is regarded as the differential form of divisibility.
  • System state distinguishability: Trace-distance monotonicity provides an easily evaluated distinguishability criterion, but it is strictly weaker than the general no-information-backflow definition.The associated interpretation is that information does not flow back from the environment to the system about its earlier history.
  • Monte Carlo wave-function simulations: The Monte Carlo wave-function concept links divisibility and pure-state unravelling in the hierarchy but implies neither one, making it distinct from pure-state unravelling.This distinction matters when interpreting pure-state trajectories in non-Markovian settings.

4. Building the hierarchy

The hierarchy establishes one-way relations among quantum Markovianity concepts, showing that several familiar conditions are sufficient but not necessary for broader or related properties. These results connect factorization, regression, past–future independence, interventions, information backflow, unravellings, divisibility, and master equations.

  • Factorization, regression, and past–future independence: FA is sufficient but not necessary for QRF and GQRF, while PFI is also sufficient but not necessary for GQRF.The AFL model satisfies QRF but fails FA and GQRF; PFI likewise implies QRF without implying FA.
  • Environment interventions and information backflow: FA implies composability, and QRF implies composability, which in turn implies NIB.These implications connect environment-factorization and correlation-function criteria to information-flow criteria.
  • Quantum white noise and master equations: QWN is sufficient but not necessary for PFI and for a time-dependent GKS–Lindblad equation.PFI is presented as broader than QWN, while the hierarchy separately relates GQRF to failure of dynamical decoupling.
  • System interventions and divisibility: GQRF is sufficient but not necessary for QRF and sufficient for failure of dynamical decoupling, while NIB is sufficient but not necessary for divisibility.The AFL model provides a counterexample to the converse between GQRF and QRF; dynamical decoupling can therefore diagnose failure of GQRF.
  • Unravelling connections: PFI is sufficient but not necessary for MPU, and MPU is sufficient but not necessary for a pure unravelling; a pure unravelling is sufficient for NQIB.The converse from MPU to PFI fails in a two-environment feedback model, and PFI can support infinitely many pure unravellings for nontrivial dynamics.
  • Divisibility, master equations, and simulation: A dynamical semigroup is necessary but not sufficient for a strict GKS–Lindblad master equation, whereas PU is sufficient but not necessary for MCWF simulation.For continuous-time evolutions, the semigroup implication holds when the dynamical map is norm continuous; MCWF can also apply to explicitly non-Markovian systems.

5. An analogous classical hierarchy

The classical hierarchy is simpler than the quantum one because classical Markovianity has a direct probabilistic definition, while several quantum concepts merge or lack classical analogues. Its relations distinguish multi-time regression conditions from weaker transition-matrix properties and connect classical concepts to corresponding quantum notions.

  • 5. An analogous classical hierarchy: The classical hierarchy is simpler because some quantum concepts are inapplicable classically or merge with one another.
  • 5. An analogous classical hierarchy: Classical Markovianity is rigorously defined by the condition that future states depend statistically only on the current state.
  • 5.1.1. Classical Markovianity: GCRF is equivalent to classical Markovianity, with recursive factorization extending the two-time condition to arbitrary numbers of times.
  • 5.1.3. Classical regression formula: If CRF_n holds, then CRF_n−1 holds, but the reverse implication fails in general, forming a strict hierarchy of classical regression formulas.
  • 5.1.4. Chapman-Kolmogorov equation: CKE and classical divisibility are weaker than classical Markovianity and than CRF_n for n ≥2.
  • 5.1.5. Classical divisibility: Classical distinguishability decrease is equivalent to classical divisibility, unlike the corresponding quantum concepts, because classical processes lack the distinction between completely positive and merely positive maps.

6. Concluding remarks

The report develops a hierarchy of rigorously defined Markov-related concepts rather than selecting one definition of quantum Markovianity. It emphasizes that quantum Markovianity is context-dependent, while acknowledging that the hierarchy is incomplete and contains open questions.

  • The report defines many Markov-related concepts for open quantum systems within a very general framework.
  • These concepts divide into interaction-dependent criteria and criteria based solely on the system dynamical map.
  • The report treats quantum Markovianity as highly context-dependent rather than identifying any single concept with it.
  • The resulting hierarchy clarifies why different perspectives produce different measures of quantum non-Markovianity.
  • The hierarchy is not claimed to be complete, and several relations remain conjectural or open.

From GQRF to PFI

The report leaves open whether GQRF implies PFI for time-discrete evolutions.

  • The authors conjecture that GQRF might imply PFI for time-discrete evolutions.
  • This implication is considered where PFI can be strictly fulfilled.
  • The necessary direction of Theorem 3 remains open.

From Composability to QRF

The report conjectures that composability does not imply QRF, but no concrete counterexample has been found.

  • QRF is described as prima facie more general than composability.
  • The authors conjecture that the reverse direction of Theorem 7 does not hold.
  • A concrete counterexample is still missing, leaving the composability-to-QRF direction open.

From NQIB to PU

The report leaves open whether NQIB can imply PU because the concepts are defined over different measurement structures.

  • It is unclear whether NQIB can imply PU.
  • NQIB is defined for three times, whereas PU uses bath measurements at multiple times.
  • NQIB has not been generalized to multiple times in the manner used for composability and NIB.
  • The implication remains open because no counterexample has been found.

From NIB to composability

The report conjectures that no information backflow is necessary but not sufficient for composability, although no counterexample has been found.

  • No information backflow is characterized as a stronger version of composability under the stated definitions.NIB requires only an environment state satisfying Eq. (51), whereas composability requires unitary equivalence to the initial environment state.
  • The authors therefore conjecture that NIB is only a necessary condition for composability.The conjecture remains unsupported by a counterexample.

GQRF and MPU

The relation between GQRF and MPU remains unresolved within the hierarchy: MPU does not imply GQRF, while the converse is unknown.

  • MPU does not imply GQRF, as shown by the counterexample used in Theorem 10.
  • Whether GQRF implies MPU is left as an open question.

Appendix B. FA, PFI and QRF

Appendix B supplies proof details for hierarchy results connecting factorization approximation and past-future independence to quantum regression formulas.

  • Appendix B. FA, PFI and QRF: The appendix provides the promised details for the proofs of Theorems 1 and 3.
  • FA and QRF: Theorem 1 derives QRF from FA.
  • PFI and QRF: Theorem 3 derives QRF from PFI.
  • The displayed derivations are identified as reproducing Eq. (35).

Appendix C. The AFL model and its relations to FA, QRF and GQRF

The AFL model is a qubit coupled to a single environment field mode and provides an example that satisfies QRF while failing FA and GQRF.

  • Model setup: The AFL model couples a qubit to a single environment field mode through a time-independent Hamiltonian.The environment is formulated as a quantum superposition in the report’s version of the model.
  • Model setup: The initial combined state is assumed to be a product of the system state and a pure field state.
  • Reduced dynamics: Despite the single-mode environment, tracing over the field yields a dephasing-like master equation for the system.
  • Relations to FA: The AFL model fails FA because an initially pure system becomes mixed while the total state remains pure and therefore entangled.
  • Relations to QRF: The model strictly satisfies QRF, including the correlation functions involving two system operators.The stated sufficient-and-necessary condition is strictly satisfied for the relevant eigenvalue choices.
  • Relations to GQRF: The AFL model does not satisfy GQRF, with failure exhibited by a three-operator correlation function and an explicit parameter choice.

Appendix D. Proof of Lemma 1

The proof shows that if the entangling alternative fails, the supported output states must have a restricted factorisable structure, forcing the second property of Lemma 1. Mixedness and an isometric relation between the states then follow.

  • Proof of Lemma 1: If property 2 fails, the unitary maps every supported product input to an unentangled state.The argument restricts the second subsystem to the support of ρb and considers all pure states there.
  • Proof of Lemma 1: The resulting subspace is isomorphic to the support of ρb and consists solely of factorisable states.It is formed by applying the unitary to the fixed pure state on subsystem a and every vector in the support of ρb.
  • Proof of Lemma 1: Closure under superposition forces all factorisable states in this subspace to share one fixed factor, up to scalar multipliers.A superposition of two factorisable states remains factorisable only when one of their factors is proportional to the corresponding fixed factor.
  • Proof of Lemma 1: The subspace therefore contains normalized states of only one form, represented by unitary images of product states.This is the structural restriction used to derive the alternatives in the lemma.
  • Proof of Lemma 1: The first alternative would make ρ′a pure and contradict the lemma’s assumption, so the second alternative must hold.Preservation of inner products then maps orthonormal states in the supported subspace to an orthonormal set in Ha.
  • Proof of Lemma 1: Because ρa is mixed, ρb is also mixed and is related to ρ′a by an isometry.This establishes the lemma’s first property after excluding the first possibility.

Appendix E. Hierarchy of classical regression formulas

The appendix constructs classical processes showing that the regression formulas form a strict hierarchy: lower-order formulas can hold while higher-order formulas fail. A generalized block construction extends this separation to arbitrary orders and can be made stationary.

  • Hierarchy of classical regression formulas: The classical regression formula CRFn implies CRFn−1, but the converse does not hold.Choosing tn−1 = tn gives the forward implication, while the counterexamples establish strictness.
  • Hierarchy of classical regression formulas: Feller’s example satisfies CRF2 but not CRF3, motivating a different construction that generalizes to all orders.The discussion restricts attention to discrete classical stochastic processes and discrete times.
  • Hierarchy of classical regression formulas: A three-variable ±1 construction has correlated variables whose lower-order marginals become statistically independent after summing over variables.This provides the basic block used to build an infinite process.
  • Hierarchy of classical regression formulas: For the block process, CRF2 is valid but CRF3 fails, even when statistically independent blocks are inserted so consecutive time differences become arbitrarily small.Thus failure of the higher-order formula is not removed by shrinking the spacing between consecutive variables.
  • Hierarchy of classical regression formulas: The generalized M-variable construction satisfies CRFn for every n < N but fails for n = N.It uses independent blocks of length M with joint distributions containing N-variable correlations.
  • Hierarchy of classical regression formulas: Averaging the generalized process over M consecutive time shifts extends the example to a stationary process.The construction therefore separates arbitrary regression orders within a stationary setting as well.
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