Foundations and measures of quantum non-Markovianity
Open accessAbstract
The basic features of the dynamics of open quantum systems, such as the dissipation of energy, the decay of coherences, the relaxation to an equilibrium or non-equilibrium stationary state, and the transport of excitations in complex structures are of central importance in many applications of quantum mechanics. The theoretical description, analysis and control of non-Markovian quantum processes play an important role in this context. While in a Markovian process an open system irretrievably loses information to its surroundings, non-Markovian processes feature a flow of information from the environment back to the open system, which implies the presence of memory effects and represents the key property of non-Markovian quantum behavior. Here, we review recent ideas developing a general mathematical definition for non-Markoviantiy in the quantum regime and a measure for the degree of memory effects in the dynamics of open systems which are based on the exchange of information between system and environment. We further study the dynamical effects induced by the presence of system-environment correlations in the total initial state and design suitable methods to detect such correlations through local measurements on the open system.
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TL;DR
This review defines quantum non-Markovianity through a backflow of information, quantified by temporal increases in the trace distance between open-system states, and presents a corresponding measure \(N(\Phi)\). It also derives bounds and a local witness for initial system-environment correlations.
Key findings
- A quantum process is non-Markovian iff there exists a pair of initial states whose trace distance increases at some time \(t>0\); all divisible dynamical maps are Markovian, and non-Markovian processes require non-divisible maps and at least one temporarily negative rate \(\gamma_i(t)\) in the time-local generator. The converse is not true: non-divisible processes can still be Markovian if negative-rate channels are overcompensated by positive ones.
- The measure \(N(\Phi)\) is the maximum over initial state pairs of the total increase of trace distance over all intervals where \(\sigma>0\). For the qubit decay model with Lorentzian spectral density, there is a threshold \(\gamma_0=\lambda/2\): below it \(N(\Phi)=0\), while for \(\gamma_0>\lambda/2\) the map is non-divisible and non-Markovian, with non-Markovianity increasing monotonically with coupling; optimal initial pairs are antipodal points on the Bloch equator (\(a=0\), \(|b|=1\)).
- Initial correlations: an increase of the open-system trace distance over its initial value implies that the initial environmental states are different or at least one total initial state is correlated. If a local operation generates a reference state with the same environmental marginals, any such increase witnesses correlations in \(\rho_1(0)\); the increase is bounded by the correlations \(D(\rho,\rho_S\otimes\rho_E)\).
- Experiments with photons in optical fibers or birefringent materials—system as polarization, environment as frequency/mode—determined the measure, realized the maximization over initial states, and observed the Markovian-to-non-Markovian transition via tomography; local detection of initial correlations has also been experimentally reported.
Methods
- Review article, not primary data: recapitulates open quantum systems, completely positive dynamical maps, semigroups, Lindblad generators, time-local master equations, and divisibility.
- Uses trace distance as state distinguishability and information-flow quantifier; defines non-Markovianity and its measure, then analyzes a qubit decaying into a bosonic reservoir with Lorentzian spectral density and exponential correlation function as the main example.
- Derives general inequalities for initial system-environment correlations and a local-operation scheme to witness them, citing experimental implementations.
Limitations
- The paper states that necessary and sufficient conditions for complete positivity of the time-local generator with time-dependent rates remain an important unsolved problem, and that non-divisibility does not imply non-Markovianity in the trace-distance sense.
- The paper states alternatives and modifications: relative entropy is often infinite and can cause singularities; trace distance can be extremely difficult for infinite-dimensional systems, with no analytical formula even for Gaussian states; Hilbert-Schmidt distance cannot be used because trace-preserving operations are not contractive for it.
- Evident from the text: as a review, it presents no new experimental data, and its experimental claims rely on cited photon experiments and model examples.
Model-written summary. Check it against the paper before relying on it.
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Claims the paper makes
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- Findingnone given§ D. Time-local master equations · p. 12
The structure (40) does not guarantee complete positivity of the dynamics.
“The structure (40) takes into account the Hermiticity and trace preservation of the dynamics, but does not guarantee its complete positivity.”
Quote in the paperSubstantial claim - Findingreasoning§ E. Divisibility of dynamical maps · p. 15
Any increase of |G(t)| leads to a negative decay rate in the corresponding generator.
“This shows that any increase of | G ( t ) | leads to a negative decay rate in the corresponding generator (47)”
Quote in the paperSubstantial claim - Findingreasoning§ B. Definition of non-Markovian quantum dynamics · p. 21
For all divisible dynamical maps the trace distance decreases monotonically.
“This shows that for all divisible dynamical maps the trace distance decreases monotonically and that, therefore, these processes are Markovian.”
Quote in the paperSubstantial claim - Findingreasoning§ E. Divisibility of dynamical maps · p. 15
The dynamical map of the model is divisible if and only if |G(t)| is a monotonically decreasing function of time.
“Thus we see that the dynamical map of the model is divisible if and only if | G ( t ) | is a monotonically decreasing function of time.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [7] source
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- Findingreasoning§ C. Construction of a measure for the degree of non-Markovianity · p. 23
There is a threshold γ0 = λ/2 for the system-reservoir coupling below which N(Φ) = 0.
“There is thus a threshold γ 0 = λ/ 2 for the system-reservoir coupling below which N (Φ) = 0.”
Quote in the paperSubstantial claim
- Findingreasoning§ B. Definition of non-Markovian quantum dynamics · p. 19
For an open system with initial states ρ1S(0) and ρ2S(0), the trace distance between the time-evolved states can never be larger than the trace distance between the initial states.
“Since the dynamical maps Φ( t, 0) are completely positive and trace preserving, the trace distance between the time evolved states can never be larger than the trace distance between the initial states,”
Quote in the paperSubstantial claim - Findingreasoning§ D. Time-local master equations · p. 12
If the rates γ_i(t) are positive for all times, the resulting dynamics is completely positive.
“the resulting dynamics is indeed completely positive, since the generator is then in Lindblad form for each fixed t ≥ 0.”
Quote in the paperSubstantial claim - Findingreasoning§ C. Construction of a measure for the degree of non-Markovianity · p. 22
For the example dynamical map, the trace distance increases at time t if and only if the function |G(t)| increases at that time.
“We conclude from this equation that the trace distance increases at time t if and only if the function | G ( t ) | increases at this point of time.”
Quote in the paperSubstantial claim - Findingreasoning§ B. Quantum dynamical maps · p. 7
The model of a two-state system decaying into a bosonic reservoir allows an analytical expression for the dynamical map.
“The model therefore allows to derive an analytical expression for the dynamical map (10).”
Quote in the paperNot a substantial claim- The authors state this as an assumption: The total system S+E is closed and follows a unitary dynamics.
- The authors state this as an assumption: System S and environment E are statistically independent at the initial time, i.e., the initial total system state is a tensor product state.
- Findingreasoning§ A. General relations describing the effect of initial correlations · p. 26
In the special case of an initial pair given by a correlated state ρ1(0) and the uncorrelated product of its marginals ρ2(0)=ρ1S(0)⊗ρ1E(0), the increase of the trace distance between ρ1S(t) and ρ2S(t) is bounded from above by the amount of correlations in the initial state ρ1(0).
“Thus, the increase of the trace distance between ρ 1 S ( t ) and ρ 2 S ( t ) (which is zero initially) is bounded from above by the amount of correlations in the initial state ρ 1 (0).”
Quote in the paperSubstantial claim - Findingreasoning§ A. General relations describing the effect of initial correlations · p. 24
One can always introduce a linear map on the total system's state space that takes any initial state of the total system to the corresponding reduced open-system state at time t.
“This map takes any initial state ρ (0) of the total system to the corresponding reduced open system state ρ S ( t ) at time t .”
Quote in the paperSubstantial claim - Findingreasoning§ A. General relations describing the effect of initial correlations · p. 24
Since unitary transformations and partial traces are completely positive and trace-preserving maps, the composite maps Λ(t,0) are again completely positive and trace-preserving.
“Since unitary transformations and partial traces are CPT maps, the composite maps Λ( t, 0) are again CPT maps.”
Quote in the paperSubstantial claim - Findingreasoning§ B. Witnessing system-environment correlations by local operations · p. 26
Any dynamical increase of the trace distance between the open-system states over the initial value implies the presence of correlations in the initial state ρ1(0).
“This inequality shows that any dynamical increase of the trace distance between the open system states over the initial value implies the presence of correlations in the initial state ρ 1 (0).”
Quote in the paperSubstantial claim - Findingreasoning§ A. General relations describing the effect of initial correlations · p. 25
The increase of the trace distance between the open-system states is bounded from above by I(ρ1(0),ρ2(0)), which is the distinguishability of the total initial states minus the distinguishability of the initial open-system states.
“The increase of the trace distance between the open system states is thus bounded from above by the quantity I ( ρ 1 (0) , ρ 2 (0)) on the right-hand side of this equation. This quantity represents the distinguishability of the total initial states minus the distinguishability of the initial open system states.”
Quote in the paperSubstantial claim - Findingreasoning§ A. General relations describing the effect of initial correlations · p. 24
The distinguishability of the open-system states can never be larger than the distinguishability of the total initial states.
“which means that the distinguishability of the open system states can never be larger than the distinguishability of the total initial states.”
Quote in the paperSubstantial claim - Premisecitation§ C. Completely positive semigroups · p. 10
The standard Markov condition alone does not guarantee that the Markovian master equation provides a reasonable description of the dynamics.
“Moreover, the standard Markov condition (34) alone does not guarantee, in general, that the Markovian master equation provides a reasonable description of the dynamics, a situation which can occur, for example, for finite and/or structured reservoirs [16, 17].”
Quote in the paperSubstantial claim0 of 2 sources checked - Premisecitation§ B. Definition of non-Markovian quantum dynamics · p. 21
The converse of the statement that non-Markovian processes are non-divisible is not true: there exist non-divisible dynamical processes which are Markovian in the sense of the paper's definition.
“However, it is important to note that the converse of this statement is not true, i.e., there exist non-divisible dynamical processes which are Markovian in the sense of the above definition.”
Quote in the paperSubstantial claim0 of 3 sources checked - Interpretationreasoning§ B. Definition of non-Markovian quantum dynamics · p. 20
All divisible families of dynamical maps are Markovian, including the class of quantum dynamical semigroups.
“In particular, it implies that all divisible families of dynamical maps are Markovian, including of course the class of quantum dynamical semigroups.”
Quote in the paperSubstantial claim - Premisecitation§ C. Construction of a measure for the degree of non-Markovianity · p. 23
For γ0 > λ/2, the non-Markovianity increases monotonically with increasing system-environment coupling.
“One finds, as expected, that for γ 0 > λ/ 2 the non-Markoviantiy increases monotonically with increasing system-environment coupling [7].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [7] source
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- Interpretationreasoning§ E. Divisibility of dynamical maps · p. 15
The non-divisibility of the dynamical map is equivalent to the occurrence of a temporarily negative rate in the time-local master equation.
“and illustrates the equivalence of the non-divisibility of the dynamical map and the occurrence of a temporarily negative rate in the time-local master equation demonstrated above.”
Quote in the paperSubstantial claim - Premisecitation§ E. Divisibility of dynamical maps · p. 14
Divisibility of a family of dynamical maps is equivalent to positive rates in the time-local master equation.
“Thus we see that divisibility is equivalent to positive rates in the time-local master equation [7].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [7] source
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- Interpretationreasoning§ B. Definition of non-Markovian quantum dynamics · p. 20
The fact that the trace distance cannot exceed its initial value does not imply that the trace distance between the time-evolved open-system states is a monotonically decreasing function of time.
“Of course, this general feature does not imply that D ( ρ 1 S ( t ) , ρ 2 S ( t )) is a monotonically decreasing function of time.”
Quote in the paperSubstantial claim - Interpretationreasoning§ B. Definition of non-Markovian quantum dynamics · p. 20
The contraction property guarantees that the maximal amount of information the system can recover from the environment is the amount of information that earlier flowed out of the system.
“On the other hand, the contraction property of Eq. (67) guarantees that the maximal amount of information the system can recover from the environment is the amount of information earlier flowed out the system.”
Quote in the paperSubstantial claim - Premisecitation§ I. INTRODUCTION · p. 2
Markovian processes tend to continuously reduce the trace distance and hence the distinguishability between any pair of physical states.
“Markovian processes tend to continuously reduce the trace distance and, hence, the distinguishability between any pair of physical states”
Quote in the paperSubstantial claim0 of 3 sources checked- cites [8] Helstrom C W, 1976, Quantum Detection and Estimation Theory
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- cites [9] Holevo A S, 1972, Trans. Moscow Math. Soc.
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- cites [10] Hayashi M, 2006, Quantum Information
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- Interpretationreasoning§ I. INTRODUCTION · p. 2
Memory effects emerge through a recycling of information such that earlier states of the open system influence its future states.
“Memory effects thus emerge through a recycling of information such that earlier states of the open system influence its future states.”
Quote in the paperSubstantial claim - Interpretationreasoning§ B. Definition of non-Markovian quantum dynamics · p. 20
The invariance of the trace distance under unitary transformations indicates that information is preserved under the dynamics of closed systems.
“The invariance under unitary transformations (66) indicates that information is preserved under the dynamics of closed systems.”
Quote in the paperSubstantial claim - Interpretationreasoning§ I. INTRODUCTION · p. 2
The characteristic feature of a non-Markovian quantum process is the increase of the distinguishability, i.e., a reversed flow of information from the environment back to the open system.
“In view of this interpretation the characteristic feature of a nonMarkovian quantum process is the increase of the distinguishability, i.e., a reversed flow of information from the environment back to the open system.”
Quote in the paperSubstantial claim - Interpretationreasoning§ A. Trace distance and distinguishability of quantum states · p. 18
The trace distance between two quantum states can be interpreted as a measure of their distinguishability, representing the bias in favor of correct state identification achievable by an optimal strategy.
“The trace distance D ( ρ 1 , ρ 2 ) can therefore be interpreted as a measure for the distinguishability of the quantum states ρ 1 and ρ 2 [8-10].”
Quote in the paperSubstantial claim0 of 3 sources checked- cites [8] Helstrom C W, 1976, Quantum Detection and Estimation Theory
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- cites [9] Holevo A S, 1972, Trans. Moscow Math. Soc.
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- cites [10] Hayashi M, 2006, Quantum Information
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- Premisecitation§ A. Trace distance and distinguishability of quantum states · p. 18
All trace-preserving completely positive maps are contractions of the trace distance.
“More generally, all trace preserving and completely positive maps, i.e., all trace preserving quantum operations Λ are contractions of the trace distance,”
Quote in the paperSubstantial claim0 of 2 sources checked- cites [10] Hayashi M, 2006, Quantum Information
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- cites [11] Nielsen M A, 2000, Quantum Computation and Quantum Information
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- Interpretationreasoning§ B. Definition of non-Markovian quantum dynamics · p. 21
For non-divisible but Markovian processes, the effect of decay channels with a negative rate is overcompensated by channels with a positive decay rate, resulting in a total information flow directed from the open system to the environment.
“For such processes the effect of the contribution of the decay channels with a negative rate is overcompensated by the channels with a positive decay rate, resulting in a total information flow directed from the open system to the environment.”
Quote in the paperSubstantial claim - Interpretationreasoning§ D. Time-local master equations · p. 10
Even the presence of strong memory effects does not exclude the description of the dynamics in terms of a quantum master equation which is local in time.
“However, even the presence of strong memory effects does not exclude the description of the dynamics in terms of a quantum master equation which is local in time, as may be seen from the following simple argument.”
Quote in the paperSubstantial claim - Premisecitation§ I. INTRODUCTION · p. 2
The gain or loss of information can be quantified through the dynamics of the trace distance between a pair of quantum states of the open system.
“The gain or loss of information can be quantified through the dynamics of the trace distance between a pair of quantum states of the open system.”
Quote in the paperSubstantial claim0 of 2 sources checked - Premisecitation§ I. INTRODUCTION · p. 2
The trace distance between quantum states can be interpreted as a measure for the distinguishability of the states.
“It is known that this distance measure for quantum states can be interpreted as a measure for the distinguishability of the states”
Quote in the paperSubstantial claim0 of 3 sources checked- cites [8] Helstrom C W, 1976, Quantum Detection and Estimation Theory
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- cites [9] Holevo A S, 1972, Trans. Moscow Math. Soc.
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- cites [10] Hayashi M, 2006, Quantum Information
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- Interpretationreasoning§ B. Definition of non-Markovian quantum dynamics · p. 19
No completely positive and trace-preserving quantum operation can increase the distinguishability between quantum states.
“An important conclusion from this interpretation is that according to the contraction property of Eq. (67) no completely positive and trace preserving quantum operation can increase the distinguishability between quantum states.”
Quote in the paperSubstantial claim - Interpretationreasoning§ I. INTRODUCTION · p. 2
The reduction of trace distance in Markovian processes means there is a flow of information from the open system to its environment.
“Markovian processes tend to continuously reduce the trace distance and, hence, the distinguishability between any pair of physical states, which means that there is a flow of information from the open system to its environment.”
Quote in the paperSubstantial claim - Premisecitation§ D. Time-local master equations · p. 11
For an analytic time-dependence, the inverse of Φ(t,0) and the generator K(t) do exist apart from isolated singularities of the time axis.
“However, for an analytic time-dependence the inverse of Φ( t, 0) and the generator K ( t ) do exist apart from isolated singularities of the time axis [22].”
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- Interpretationreasoning§ B. Definition of non-Markovian quantum dynamics · p. 21
Information lost from the open system is not necessarily contained completely in the reduced states of the environment; it can also be carried by system-environment correlations.
“This does not imply that the information lost is now contained completely in the reduced states of the environment, but instead, this information can also be carried by the system-environment correlations.”
Quote in the paperSubstantial claim - Interpretationcitation§ Abstract · p. 1
Non-Markovian processes feature a flow of information from the environment back to the open system.
“non-Markovian processes feature a flow of information from the environment back to the open system”
Quote in the paperSubstantial claim0 of 2 sources checked - Premisecitation§ C. Construction of a measure for the degree of non-Markovianity · p. 23
A careful preparation of the initial environmental state allows control of the information flow between the system and its environment and observation of the transition from the Markovian to the non-Markovian regime through quantum state tomography on the open system.
“It was shown further that a careful preparation of the initial environmental state allows to control the information flow between the system and its environment and to observe the transition from the Markovian to the non-Markovian regime through quantum state tomography carried out on the open system.”
Quote in the paperSubstantial claim0 of 2 sources checked - Interpretationreasoning§ V. CONCLUSIONS · p. 28
Given that the initial correlations were created in the past from a product state through a system-environment interaction, the increase of the trace distance of the open-system states over its initial value signifies that the open system regains information which was lost previously.
“Given that the initial correlations were created in the past from a product state through a system-environment interaction, the increase of the trace distance of the open system states over its initial value signifies that the open system regains information which was lost previously.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The initial correlations were created in the past from a product state through a system-environment interaction.
- Premisecitation§ C. Construction of a measure for the degree of non-Markovianity · p. 23
Two experiments determining the non-Markovianity measure have been carried out employing photons moving in optical fibers or birefringent materials.
“Quite recently, two experiments determining this quantity have been carried out employing photons moving in optical fibers or birefringent materials [44, 45].”
Quote in the paperNot a substantial claim0 of 2 sources checked - Interpretationcitation§ C. Construction of a measure for the degree of non-Markovianity · p. 23
The non-Markovianity measure represents a novel experimentally accessible quantity.
“The non-Markovianity measure (76) represents a novel experimentally accessible quantity.”
Quote in the paperSubstantial claim - Interpretationreasoning§ A. General relations describing the effect of initial correlations · p. 26
Equation (87) represents a generalization of the contraction property of dynamical maps on the reduced state space.
“Thus, Eq. (87) represents a generalization of the contraction property of dynamical maps on the reduced state space.”
Quote in the paperNot a substantial claim - Interpretationreasoning§ A. General relations describing the effect of initial correlations · p. 25
The quantity I(ρ1(0),ρ2(0)) can be interpreted as the information on the total system states which is outside the open system, i.e., inaccessible through local measurements on the open system.
“It can be interpreted as the information on the total system states which is outside the open system, i.e., which is inaccessible through local measurements on the open system.”
Quote in the paperNot a substantial claim - Premisecitation§ V. CONCLUSIONS · p. 28
A disadvantage of the relative entropy is that it is often infinite, leading to singularities of the measure.
“However, a disadvantage of the relative entropy is given by the fact that it is often infinite, leading to singularities of the measure [5].”
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- Premisecitation§ V. CONCLUSIONS · p. 28
The Bures distance or the fidelity leads to useful lower and upper bounds for the trace distance.
“It seems that in those cases it is much easier to work with the Bures distance or the fidelity which also leads to useful lower and upper bounds for the trace distance [55].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [55] source
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- Premisecitation§ B. Witnessing system-environment correlations by local operations · p. 27
By taking E to be a pure dephasing operation, the scheme enables the local detection of nonclassical correlations, i.e., of total initial states with nonzero quantum discord.
“Moreover, by taking E to be a pure dephasing operation the scheme enables the local detection of nonclassical correlations, i.e., of total initial states with nonzero quantum discord [51].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [51] Laine E-M, 2010, Europhys. Lett.
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- Interpretationreasoning§ V. CONCLUSIONS · p. 27
The concept of non-Markovianity based on trace distance allows a natural extension to the case of initial system-environment correlations.
“As we have seen this concept allows a natural extension to the case of initial system-environment correlations.”
Quote in the paperSubstantial claim - Premisecitation§ C. Construction of a measure for the degree of non-Markovianity · p. 23
A complete determination of the non-Markovianity measure is experimentally possible, including the realization of the maximization over initial states in the definition.
“It has been demonstrated that a complete determination of the measure is experimentally possible, including the realization of the maximization over initial states in the definition (76).”
Quote in the paperSubstantial claim0 of 2 sources checked - Interpretationreasoning§ A. General relations describing the effect of initial correlations · p. 25
The increase of the distinguishability of the open-system states, and hence the flow of information to the open system, is bounded by the information which is outside the open system at the initial time.
“Thus we see that the increase of the distinguishability of the open system states and, hence, the flow of information to the open system is bounded by the information which is outside the open system at the initial time.”
Quote in the paperSubstantial claim - Premisecitation§ A. General relations describing the effect of initial correlations · p. 24
In the general case of correlated initial states, a quantum dynamical map can only be defined for a restricted set of states.
“While in the general case of correlated initial states a quantum dynamical map can only be defined for a restricted set of states [47], we can of course always introduce a linear map”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [47] source
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- Interpretationcitation§ B. Witnessing system-environment correlations by local operations · p. 26
Equation (87) leads to an experimentally realizable scheme for the local detection of correlations in an unknown total system's initial state ρ1(0).
“Equation (87) leads to an experimentally realizable scheme for the local detection of correlations in an unknown total system's initial state ρ 1 (0) as follows [46].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [46] source
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- May combine two claims. 24
- Classified as probably not a substantial claim. 22
May combine two claims.
Findingcitation§ C. Construction of a measure for the degree of non-Markovianity · p. 23The maximum in the non-Markovianity measure is attained for a = 0 and |b| = 1, meaning that the optimal pairs of initial states correspond to antipodal points on the equator of the Bloch sphere representing the qubit.
“Moreover, it is easy to show with the help of Eq. (78) that the maximum in Eq. (76) is attained for a = 0 and | b | = 1 [38]. This means that the optimal pairs of initial states correspond to antipodal points on the equator of the Bloch sphere representing the qubit.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [38] source
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Findingreasoning§ C. Construction of a measure for the degree of non-Markovianity · p. 23In the strong-coupling regime γ0 > λ/2, |G(t)| oscillates, showing non-monotonic behavior; the dynamical map is no longer divisible and the process is non-Markovian.
“However, in the strong coupling regime, γ 0 > λ/ 2, the function | G ( t ) | starts to oscillate, showing a non-monotonic behavior. Consequently, the dynamical map is then no longer divisible and the process is non-Markovian.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Findingnone given§ A. Trace distance and distinguishability of quantum states · p. 17The trace distance between two states is zero if and only if the states are identical, and reaches its upper bound of one if and only if the states are orthogonal.
“Of course, we have D ( ρ 1 , ρ 2 ) = 0 if and only if ρ 1 = ρ 2 , while the upper bound is reached, i.e., D ( ρ 1 , ρ 2 ) = 1 if and only if ρ 1 and ρ 2 are orthogonal, which means that the supports of ρ 1 and ρ 2 are orthogonal.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Findingnone given§ A. Trace distance and distinguishability of quantum states · p. 17The trace distance satisfies the triangular inequality.
“and satisfies the triangular inequality:”
Quote in the paperNot a substantial claimMay combine two claims.
Findingreasoning§ A. Trace distance and distinguishability of quantum states · p. 19State identification with certainty can be achieved if and only if the trace distance is one, i.e., if and only if the states are orthogonal.
“We further see that a state identification with certainty, P max = 1, can be achieved if and only if D ( ρ 1 , ρ 2 ) = 1, i.e., if and only if ρ 1 and ρ 2 are orthogonal, in which case Bob's optimal strategy is to measure the projection Π onto the support of ρ 1 (or of ρ 2 ).”
Quote in the paperNot a substantial claimMay combine two claims.
Findingreasoning§ C. Construction of a measure for the degree of non-Markovianity · p. 22For the example, the process is non-Markovian, N(Φ) > 0, if and only if the dynamical map is non-divisible, which is equivalent to a temporarily negative rate γ(t) in the time-local master equation.
“It follows that the process is non-Markovian, N (Φ) > 0, if and only if the dynamical map is non-divisible, which in turn is equivalent to a temporarily negative rate γ ( t ) in the time-local master equation with generator (47).”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Findingnone given§ A. Trace distance and distinguishability of quantum states · p. 17The trace distance is symmetric in its input arguments.
“Moreover, D ( ρ 1 , ρ 2 ) is obviously symmetric in the input arguments”
Quote in the paperNot a substantial claimMay combine two claims.
Findingreasoning§ C. Construction of a measure for the degree of non-Markovianity · p. 23For an exponential correlation function in the small-coupling regime γ0 < λ/2, G(t) decreases monotonically; the dynamical map is divisible, the rate γ(t) is always positive, and the process is Markovian.
“For small couplings, γ 0 < λ/ 2, this function decreases monotonically. The dynamical map is then divisible, the rate γ ( t ) is always positive, and the process is Markovian.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Findingnone given§ A. Trace distance and distinguishability of quantum states · p. 18The trace distance is invariant under unitary transformations.
“The trace distance is invariant under unitary transformations U ,”
Quote in the paperNot a substantial claimMay combine two claims.
Findingreasoning§ D. Time-local master equations · p. 11The generator K(t) of the time-local master equation must preserve Hermiticity and trace, and hence must have the general form (40).
“From these requirements it follows that the generator must be of the following most general form,”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Findingnone given§ A. Trace distance and distinguishability of quantum states · p. 18The trace distance is sub-additive with respect to tensor products of states.
“The trace distance is sub-additive with respect to tensor products of states which means that”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Findingnone given§ A. Trace distance and distinguishability of quantum states · p. 18The trace distance can be represented as the maximum of a certain functional over all projection operators.
“The trace distance can be represented as the maximum of a certain functional:”
Quote in the paperNot a substantial claimMay combine two claims.
Findingreasoning§ A. General relations describing the effect of initial correlations · p. 26A dynamical increase of the trace distance of the open-system states over the initial value implies that the corresponding initial environmental states are different or that at least one of the total initial states is correlated.
“implies that the corresponding initial environmental states ρ 1 , 2 E (0) are different or that at least one of the total initial states ρ 1 , 2 (0) is correlated.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ A. General relations describing the effect of initial correlations · p. 26In the case of two uncorrelated states with the same environmental state, the right-hand side of inequality (87) vanishes, leading to the contraction property for completely positive and trace-preserving dynamical maps on the reduced state space.
“Wefurther remark that in the case of two uncorrelated states with the same environmental state, i.e., ρ 1 , 2 (0) = ρ 1 , 2 S (0) ⊗ ρ E (0), the right-hand side of the inequality (87) vanishes, and we are led again to the contraction property (72) for CPT dynamical maps on the reduced state space.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation§ B. Quantum dynamical maps · p. 6For a Hilbert space with finite dimension N_S = dim H_S, complete positivity is equivalent to N_S-positivity.
“for a Hilbert space with finite dimension N S = dim H S complete positivity is equivalent to N S -positivity [13].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [13] source
Not checked in the free run.
Classified as probably not a substantial claim.
Interpretationreasoning§ B. Definition of non-Markovian quantum dynamics · p. 21If information flows from the environment back to the system, the distinguishability of the open-system states increases because information is regained by the open system, carried by environmental degrees of freedom or by system-environment correlations.
“And vice versa, if we say that information flows from the environment back to the system, this means that the distinguishability of the open system states increases because information is regained by the open system, information which was carried by the environmental degrees of freedom or by the correlations between the degrees of freedom of system and environment (see also Sec. IV).”
Quote in the paperSubstantial claimClassified as probably not a substantial claim.
Premisenone given§ B. Quantum dynamical maps · p. 6Quantum dynamical maps are completely positive.
“An important further property of dynamical maps is that they are not only positive but also completely positive.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The total system S+E is closed and follows a unitary dynamics.
- The authors state this as an assumption: System S and environment E are statistically independent at the initial time, i.e., the initial total system state is a tensor product state.
Probably not a substantial claim: may not pass S4 (based).
Interpretationreasoning§ Abstract · p. 1The backflow of information from the environment to the open system implies the presence of memory effects.
“which implies the presence of memory effects”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation§ B. Quantum dynamical maps · p. 6A linear map Φ is completely positive if and only if it admits a Kraus representation.
“We recall that a linear map Φ is completely positive if and only if it admits a Kraus representation [12]”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [12] source
Not checked in the free run.
May combine two claims.
Premisenone given§ D. Time-local master equations · p. 11The inverse map Φ^{-1}(t,0) is generally not completely positive, and even not positive.
“In general, the inverse map is not only not completely positive, but even not positive.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim.
Interpretationreasoning§ Abstract · p. 1The backflow of information from the environment to the open system represents the key property of non-Markovian quantum behavior.
“represents the key property of non-Markovian quantum behavior”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S2 (checkable).
Interpretationreasoning§ B. Definition of non-Markovian quantum dynamics · p. 21The precise meaning of information loss from the open system is that the available relative information on the considered pair of states, which enables one to distinguish them, is lost when only measurements on the open system's degrees of freedom can be performed.
“The more precise meaning of this statement is that the available relative information on the considered pair of states, that is, the information which enables one to distinguish these states is lost from the open system, i.e., when only measurements on the open system's degrees of freedom can be performed.”
Quote in the paperNot a substantial claimClassified as probably not a substantial claim. May combine two claims.
Interpretationreasoning§ B. Definition of non-Markovian quantum dynamics · p. 20A decrease of distinguishability signifies information flowing from the system to the environment, while an increase signifies information flowing from the environment back to the system.
“When a quantum process reduces the distinguishability of states, information is flowing from the system to the environment. Correspondingly, an increase of the distinguishability signifies that information flows from the environment back to the system.”
Quote in the paperSubstantial claim- The authors state this as an assumption: For Markovian processes, any two quantum states become less and less distinguishable under the dynamics, leading to a perpetual loss of information into the environment.
May combine two claims.
Premisecitation§ C. Completely positive semigroups · p. 10Large couplings or interactions with low-temperature reservoirs can lead to strong correlations resulting in long memory times and in a failure of the Markov approximation.
“However, large couplings or interactions with lowtemperature reservoirs can lead to strong correlations resulting in long memory times and in a failure of the Markov approximation.”
Quote in the paperNot a substantial claim0 of 2 sources checkedClassified as probably not a substantial claim. May combine two claims.
Premisecitation§ A. Trace distance and distinguishability of quantum states · p. 17The trace distance is well defined and finite for all pairs of quantum states and provides a metric on the space of physical states.
“The trace distance is well defined and finite for all pairs of quantum states and provides a metric on the space S ( H ) of physical states.”
Quote in the paperSubstantial claim0 of 2 sources checked- cites [10] Hayashi M, 2006, Quantum Information
Not checked in the free run.
- cites [11] Nielsen M A, 2000, Quantum Computation and Quantum Information
Not checked in the free run.
May combine two claims.
Interpretationreasoning§ B. Definition of non-Markovian quantum dynamics · p. 21Non-Markovian quantum processes must necessarily be described by non-divisible dynamical maps and by time-local master equations whose generator involves at least one temporarily negative rate γi(t).
“Hence, we see that non-Markovian quantum processes must necessarily be described by non-divisible dynamical maps and by time-local master equations whose generator (40) involves at least one temporarily negative rate γ i ( t ).”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation§ C. Completely positive semigroups · p. 9The Gorini-Kossakowski-Sudarshan-Lindblad theorem states that L is the generator of a semigroup of completely positive quantum dynamical maps if and only if it has the Lindblad form.
“The famous Gorini-Kossakowski-Sudarshan-Lindblad theorem [1, 2] states that L is the generator of a semigroup of completely positive quantum dynamical maps if and only if it has the following form,”
Quote in the paperNot a substantial claim0 of 2 sources checkedProbably not a substantial claim: may not pass S4 (based).
Interpretationreasoning§ Abstract · p. 1In a Markovian process, an open system irretrievably loses information to its surroundings.
“While in a Markovian process an open system irretrievably loses information to its surroundings”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationreasoning§ C. Construction of a measure for the degree of non-Markovianity · p. 22For a non-Markovian process described by a family of dynamical maps, information must flow from the environment to the system for some interval of time, and thus σ > 0 for this time interval.
“For a non-Markovian process described by a family of dynamical maps Φ( t, 0) information must flow from the environment to the system for some interval of time and thus we must have σ > 0 for this time interval.”
Quote in the paperSubstantial claim- The authors state this as an assumption: For Markovian processes, any two quantum states become less and less distinguishable under the dynamics, leading to a perpetual loss of information into the environment.
May combine two claims.
Interpretationreasoning§ C. Construction of a measure for the degree of non-Markovianity · p. 23Experimental quantification and control of memory effects in open quantum systems is feasible.
“This means that an experimental quantification and control of memory effects in open quantum systems is indeed feasible, which could be useful in the development of quantum memory and communication devices.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim. May combine two claims.
Interpretationreasoning§ V. CONCLUSIONS · p. 27The key feature of quantum non-Markovianity is the temporal increase of distinguishability, which can be interpreted as a backflow of information from the environment to the open system.
“According to our definition the key feature of quantum non-Markovianity is the temporal increase of the distinguishability which can be interpreted as a backflow of information from the environment to the open system.”
Quote in the paperSubstantial claim- The authors state this as an assumption: For Markovian processes, any two quantum states become less and less distinguishable under the dynamics, leading to a perpetual loss of information into the environment.
Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Premisecitation§ B. Witnessing system-environment correlations by local operations · p. 27The fact that the scheme enables local detection of nonclassical correlations has been shown in Ref. [52], where a statistical approach to initial correlations on the basis of random matrix theory has also been developed.
“This fact has been shown very recently in Ref. [52] where also a statistical approach to initial correlations on the basis of random matrix theory has been developed.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [52] source
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation§ V. CONCLUSIONS · p. 28The relative entropy represents a possibility for an alternative distance measure that is natural both from an information-theoretic perspective and from the point of view of nonequilibrium thermodynamics, since the negative rate of change of the entropy relative to an invariant thermal equilibrium state can be interpreted as entropy production.
“In particular, the relative entropy represents a possibility which is natural both from an information theoretic perspective and from the point of view of nonequilibrium thermodynamics, since the negative rate of change of the entropy relative to an invariant thermal equilibrium state can be interpreted as entropy production [53, 54].”
Quote in the paperNot a substantial claim0 of 2 sources checkedMay combine two claims.
Premisecitation§ C. Construction of a measure for the degree of non-Markovianity · p. 23In these experiments, the open system is provided by the polarization degrees of freedom of the photons, while the environment is given by their frequency (mode) degrees of freedom.
“In these experiments the open system is provided by the polarization degrees of freedom of the photons, while the environment is given by their frequency (mode) degrees of freedom.”
Quote in the paperNot a substantial claim0 of 2 sources checkedProbably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation§ B. Witnessing system-environment correlations by local operations · p. 27Experimental realizations of the scheme for local detection of initial correlations have been reported recently.
“In fact, experimental realizations of the scheme have been reported recently [48, 49].”
Quote in the paperNot a substantial claim0 of 2 sources checkedMay combine two claims.
Interpretationreasoning§ B. Witnessing system-environment correlations by local operations · p. 26If the left-hand side of the inequality is greater than zero, then either ρ1(0) or ρ2(0) must be correlated; if ρ1(0) was uncorrelated, then ρ2(0) must also be uncorrelated because it is obtained from ρ1(0) by a local operation. Thus any increase of the trace distance of the open-system states over the initial value witnesses correlations in ρ1(0).
“In fact, if one finds that the left-hand side of the inequality is greater than zero, then either ρ 1 (0) or ρ 2 (0) must be correlated. If ρ 1 (0) was uncorrelated, then also ρ 2 (0) must be uncorrelated since it is obtained from ρ 1 (0) through application of a local operation. Thus, any increase of the trace distance of the open system states over the initial value witnesses correlations in ρ 1”
Quote in the paperSubstantial claimMay combine two claims.
Premisecitation§ V. CONCLUSIONS · p. 28The Hilbert-Schmidt distance cannot be used for the quantification of non-Markovianity because trace-preserving quantum operations are, in general, not contractive for this metric.
“The Hilbert-Schmidt distance which is technically much easier to deal with cannot be used for the quantification of non-Markovianity because trace preserving quantum operations are, in general, not contractive for this metric [56].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [56] Modi K Brodutch A Cable H Paterek T Vedral V, 2011
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing).
Interpretationnone given§ B. Witnessing system-environment correlations by local operations · p. 27There is no principal restriction on the operation E used to generate the second state ρ2(0), which makes the scheme very flexible in practice.
“Moreover, there is no principal restriction on the operation E used to generate the second state ρ 2 (0), which makes the scheme very flexible in practice.”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationreasoning§ V. CONCLUSIONS · p. 27While a quantum dynamical map acting on the open system's state space does in general not exist in the case of initial correlations, the trace distance between pairs of states of the open system leads to a dynamical witness for the presence of initial correlations in the total system state.
“While a quantum dynamical map acting on the open system's state space does in general not exist in this case, the trace distance between pairs of states of the open system leads to a dynamical witness for the presence of initial correlations in the total system state.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ B. Witnessing system-environment correlations by local operations · p. 27The strategy for local detection of initial correlations requires only local control and measurements of the open quantum system; it neither demands knowledge about the structure of the environment or of the system-environment interaction, nor a full knowledge of the initial system-environment state ρ1(0).
“We note that this strategy for the local detection of initial correlations requires only local control and measurements of the open quantum system. It neither demands knowledge about the structure of the environment or of the system-environment interaction, nor a full knowledge of the initial system-environment state ρ 1 (0).”
Quote in the paperSubstantial claimMay combine two claims.
Premisecitation§ V. CONCLUSIONS · p. 28Both the quantum measure for non-Markovianity and the witness for initial system-environment correlations have been demonstrated to be experimentally measurable quantities.
“Both the quantum measure for non-Markovianity and the witness for initial systemenvironment correlations studied here have been demonstrated very recently to be experimentally measurable quantities [44, 45, 48, 49].”
Quote in the paperSubstantial claim0 of 3 sources checked
The authors state this as an assumption.
The total system S+E is closed and follows a unitary dynamics.
“We will assume here that the total system S + E is closed and follows a unitary dynamics described by some unitary time evolution operator”
The authors state this as an assumption.
System S and environment E are statistically independent at the initial time, i.e., the initial total system state is a tensor product state.
“system S and environment E are statistically independent at the initial time, i.e., that the initial total system state represents a tensor product state”
The authors state this as an assumption.
The construction of a completely positive quantum dynamical map acting on the open system's state space is based on the assumption of an uncorrelated total initial state.
“the construction of a completely positive quantum dynamical map Φ( t, 0) acting on the open system's state space is based on the assumption of an uncorrelated total initial state.”
The authors state this as an assumption.
For Markovian processes, any two quantum states become less and less distinguishable under the dynamics, leading to a perpetual loss of information into the environment.
“Our definition for quantum non-Markovianity is based on the idea that for Markovian processes any two quantum states become less and less distinguishable under the dynamics, leading to a perpetual loss of information into the environment.”
The authors state this as an assumption.
The initial correlations were created in the past from a product state through a system-environment interaction.
“Given that the initial correlations were created in the past from a product state through a system-environment interaction”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The von Neumann trace inequality holds for any complex n by n matrices with descending singular values.
“Inequality (7) holds for any complex n  n matrices A 1 and A 2 with descending singular values, 1;1 ! . . .! 1;n and 2;1 ! . . .! 2;n .”
From Quantum Speed Limit for Non-Markovian Dynamics (2013) · cites this paper as [24]
If the two operators are simple positive functions of density operators acting on the same Hilbert space, the von Neumann trace inequality remains true for arbitrary dimensions.
“If A 1 and A 2 are simple (positive) functions of density operators acting on the same Hilbert space, Eq. ( 7) remains true for arbitrary dimensions [24].”
From Quantum Speed Limit for Non-Markovian Dynamics (2013) · cites this paper as [24]
The citation lists haven’t been fetched yet
References 62 · Cited by 225
Research works printed but not linked: 3
- Holevo A S, 1972, Trans. Moscow Math. Soc., 26, 133
- Tang J-S, 2012, Europhys. Lett., 97
- Laine E-M, 2010, Europhys. Lett., 92
- Authors
- Breuer, H.-P.
- Affiliations
- University of Freiburg (DE)
- Corresponding author
- Breuer, Heinz-Peter
- ORCID
- Breuer, Heinz-Peter
- Venue
- Journal of Physics B: Atomic, Molecular and Optical Physics
- Publisher
- IOP Publishing
- Published
- 2012-07-30
- Volume
- 45
- Issue
- 15
- Pages
- 154001
- ISSN
- 0953-4075 (ISSN-L)
- 1361-6455
- OpenAlex
- W2169844369
- Semantic Scholar
- 06ff5153e0454a1f85801357824ffe9a5877cecc
- Cited by
- 225
- References
- 62
- Citations by year
- 2026: 10 · 2025: 15 · 2024: 16 · 2023: 11 · 2022: 24 · 2021: 22 · 2020: 11 · 2019: 16 · 2018: 14 · 2017: 19 · 2016: 15 · 2015: 17 · 2014: 21 · 2013: 13 · 2012: 1
- FWCI
- 8.56
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Journal of Physics B Atomic Molecular and Optical Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- FreiDok plus (Universitätsbibliothek Freiburg) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Deutscher Akademischer Austauschdienst
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Advanced Thermodynamics and Statistical Mechanics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Keywords
- quantum non-Markovianity, open quantum systems, memory effects, system-environment correlations, information flow, non-Markovianity measures, energy dissipation, non-equilibrium stationary states, local measurements
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