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Entanglement and Non-Markovianity of Quantum Evolutions
Open accessAbstract
We address the problem of quantifying the non-markovian character of quantum time evolutions of general systems in contact with an environment. We introduce two different measures of non-markovianity that exploit the specific traits of quantum correlations and are suitable for opposite experimental contexts. When complete tomographic knowledge about the evolution is available, our measure provides a necessary and sufficient condition to quantify strictly the non-markovianity. In the opposite case, when no information whatsoever is available, we propose a sufficient condition for non-markovianity. Remarkably, no optimization procedure underlies our derivation, which greatly enhances the practical relevance of the proposed criteria.
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TL;DR
The paper introduces two measures of non-Markovianity for quantum evolutions that avoid optimization: an entanglement-based sufficient condition usable when the dynamics is unknown, and a necessary-and-sufficient CP-divisibility measure when the dynamical map can be reconstructed. It illustrates the approach with a damped harmonic oscillator coupled to Ohmic and super-Ohmic baths and with a qubit pure-dephasing model.
Key findings
- A sufficient condition for non-Markovianity is \(I(\mathcal{E})>0\), obtained by preparing a maximally entangled system-ancilla state and checking for non-monotonic decay of entanglement; Markovian evolutions give \(I(\mathcal{E})=0\).
- When the dynamical map is known, non-Markovianity is equivalent to the existence of times \(t,\epsilon\) such that \(\mathcal{E}(t+\epsilon,t)\) is not completely positive; the proposed measure uses \(f_{\mathrm{NCP}}\), \(g(t)\ge 0\), and \(I=\int g(t)\,dt\), with normalized \(D_{\mathrm{NM}}=I/(I+1)\) so that \(D_{\mathrm{NM}}=0\) for Markovian and \(D_{\mathrm{NM}}\to 1\) for \(I\to\infty\).
- For a damped harmonic oscillator with Ohmic bath \(J(\omega)=\alpha\omega e^{-\omega/\omega_c}\), the simulations show that \(\alpha\to 0\) approaches Markovian evolution independently of temperature \(T\), while the onset of non-Markovianity is \(T\)-dependent: the coupling \(\alpha\) at which \(I(\mathcal{E})\) becomes nonzero increases with increasing temperature, as does \(I(\mathcal{E})\) for sufficiently large \(\alpha\).
- A super-Ohmic bath \(J(\omega)=\alpha\omega^3 e^{-\omega/\omega_c}\) shows analogous behavior, but the specific value of \(I(\mathcal{E})\) for given \(T\) and \(\alpha\) depends strongly on the bath spectral function; for qubit pure dephasing \(d\rho/dt=\gamma(t)(\sigma_z\rho\sigma_z-\rho)\), \(I\) is proportional to the area of \(\gamma(t)\) below zero, and if negative \(\gamma(t)\) does not tend to zero fast enough then \(I\to\infty\), as with \(\gamma(t)\sim\tan(t)\).
Methods
- Theoretical proposal with two criteria: an entanglement-based sufficient condition for unknown dynamics, and a necessary-and-sufficient condition based on complete positivity of infinitesimal dynamical maps when the evolution can be exactly reconstructed, e.g. by process tomography or a microscopic model.
- For the unknown-dynamics measure, a maximally entangled state \(|\Phi\rangle=\frac{1}{\sqrt{d}}\sum_{n=0}^{d-1}|n\rangle|n\rangle\) is prepared between the system and an isolated ancilla, and the entanglement is monitored over \([t_0,t_{\max}]\).
- Illustrative simulations use a single damped harmonic oscillator coupled to a bath under the rotating wave approximation, with Ohmic and super-Ohmic spectral densities, several temperatures, and varying coupling strengths; entanglement is quantified by logarithmic negativity for Gaussian states.
Limitations
- The paper states that the entanglement-based criterion is only sufficient: there can be non-Markovian quantum evolutions that remain undetected because \(I(\mathcal{E})>0\) is not necessary.
- The necessary-and-sufficient measure requires exact reconstruction of the dynamical map; if \(\mathcal{E}(t,0)\) is not invertible for larger times, \(\mathcal{E}(t+\epsilon,t)\) cannot be defined unequivocally, and the integral measure is finite only if \(g(t)\) decreases fast enough, which “will not be always the case.”
- The simulations use a discrete and finite number of bath oscillators, which the authors say affects the results; they choose parameters so that \(t_{\max}\) is shorter than the bath recurrence time. Evident from the text: the work is theoretical/simulated and does not report experimental validation.
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A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“the same happens with the value of I ( E ) itself for α large enough.”
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passes- The authors state this as an assumption: The rotating wave approximation (RWA) is valid in the model.
- The authors state this as an assumption: The bath is initially in a thermal state.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“It is clear from Fig. 2 that in the limit α → 0 we approach a Markovian evolution independently of the value of T .”
Load-bearingStrength:boosted· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: The rotating wave approximation (RWA) is valid in the model.
- The authors state this as an assumption: The bath is initially in a thermal state.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“An analogous situation is encountered in the presence of a super-Ohmic spectral density J ( ω ) = αω 3 e -ω/ω c , as shown in figure 3; however, the specific value of I ( E ) for given T and α depends strongly on the bath spectral function.”
Strength:boosted· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passes- The authors state this as an assumption: The rotating wave approximation (RWA) is valid in the model.
- The authors state this as an assumption: The bath is initially in a thermal state.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“whereas the on-set of deviations from Markovianity for stronger coupling strengths are T -dependent, so that the value of α for which the measure I ( E ) becomes nonzero increases with increasing temperature”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: The rotating wave approximation (RWA) is valid in the model.
- The authors state this as an assumption: The bath is initially in a thermal state.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Therefore, for this kind of evolution, I is proportional to the area of γ ( t ) which is below zero.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“then it is possible to prove that if and only if L t can be written in the standard form (1), with H ( t ), γ k ( t ) ≥ 0 and V k ( t ) potentially time dependent, the family of propagators E ( t 2 ,t 1 ) = T exp ( ∫ t 2 t 1 L τ dτ ) satisfying the composition law”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“in such a way that if the evolution of the system is Markovian the derivative of E [ ρ SA ( t )] is always negative and I ( E ) = 0.”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesNone of the sources cited for this claim have been checked yet.
In the paper
“The actual dynamics of any real open quantum system is expected to deviate to some extent from the idealized Markovian evolution that arises from the conditions of weak (or singular) coupling to a memoryless reservoir [1, 2].”
Load-bearingStrength:hedged· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“There can be however non-Markovian quantum evolutions that remain undetected by the proposed measure”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“However if and only if there exist times t and ǫ such that E ( t + ǫ,t ) is not CP, the dynamics will be non-Markovian.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“although the shape of the curves does not depend significatively on r .”
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✗ Based: failsProbably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
- The authors state this as an assumption: The rotating wave approximation (RWA) is valid in the model.
- The authors state this as an assumption: The bath is initially in a thermal state.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“the other provides a measure which quantify strictly the non-Markovianity, provided that the structural form of the dynamical map is known.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe quote only partly states this claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“noticing that g ( t ) ≥ 0, with g ( t ) = 0 if and only if E ( t + ǫ,t ) is CP.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Therefore, E ( t + ǫ,t ) is CP if and only if f NCP ( t + ǫ, t ) = 1, otherwise f NCP ( t + ǫ, t ) > 1.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Note that since the knowledge of the exact form of the dynamics is not necessary to measure I ( E ) , this method can be particularly useful in the study of infinite dimensional systems, where the computation of the exact dynamical map is often difficult.”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
In the paper
“However, if the evolution is non-Markovian, the requirement of strict monotonicity does no longer hold [11], as environmental correlations can lead to bipartite entanglement to be increased and decreased as a function of time”
What the sources say
- Not checked yetcites [11] source
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
In the paper
“The structure of this kind of semigroups was analyzed in detail some time ago [8], concluding that a quantum system will undergo a Markovian dynamics provided that its evolution satisfies a Master equation of the standard”
What the sources say
- Not checked yetcites [8] source
Strength:boosted· not yet calibratedCausal:conditional· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“can be taken as a measure of non-Markovianity, and as long as g ( t ) decreases fast enough (this will not be always the case) will be finite.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim. It may state two separate claims as one.
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In the paper
“In the simulation, of course, the discrete and finite number of oscillators in the bath will affect the results, but in order to avoid this effect we have taken a number of oscillators distributed in a range of frequencies (see details on the plots) such that our final time t max is shorter than the recurrence time of the bath but large enough to capture the general properties of the dynamics.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim. It may state two separate claims as one.
- The authors state this as an assumption: The rotating wave approximation (RWA) is valid in the model.
- The authors state this as an assumption: The bath is initially in a thermal state.
In the paper
“This is particularly the case when considering interacting many-body systems, where the subsystem's coupling strength may be comparable to the coupling to the bath [4].”
What the sources say
- Not checked yetcites [4] source
Strength:hedged· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
The paper relies on this without support.
An accurate microscopic model of the system-bath interaction may be unfeasible, especially for many-body systems.
“The exact details of what makes a given quantum evolution non-Markovian may be complicated, and in many cases, especially when thinking about many-body systems, an accurate microscopic model of the system-bath interaction may actually be unfeasible.”
The authors state this as an assumption.
The rotating wave approximation (RWA) is valid in the model.
“we have assumed the validity of the rotating wave approximation (RWA).”
The authors state this as an assumption.
The bath is initially in a thermal state.
“The bath is assumed to be initially in a thermal state ρ B = exp( -H B /T ) / tr [exp ( -H B /T )]”
The paper relies on this without support.
The ancillary system remains isolated from decoherence sources.
“an ancillary system which has to remain isolated from the decoherence sources”
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 cited source says what the claim attributes to it.
In the citing paper
“The above equation describes Markovian dynamics for Γ k ≥ 0 . Whenever the loss rate Γ k becomes negative (Fleming and Hu, 2012; Rivas et al. , 2010), Eq. (52) can also describe non-Markovian dynamics (Bastidas et al. , 2018; Kyaw et al. , 2020a; Tan et al. , 2010).”
From Noisy intermediate-scale quantum algorithms (2022) · cites this paper as Rivas et al., 2010
In the citing paper
“Our results are applicable to a wide variety of scenarios including bounding decoherence rates [40], and quantum speed limits in dissipative state preparation [41], quantum computation and simulation assisted by dissipation [42].”
From Quantum Speed Limits in Open System Dynamics (2013) · cites this paper as [40]
In the citing paper
“This measure provides a necessary and sufficient condition for a given evolution to depart from strict Markovianity by means of evaluating whether or not the associated dynamical map E ( t + ǫ,t ) is completely positive (CP) for any ǫ .”
From Non-Markovianity-Assisted Steady State Entanglement (2012) · cites this paper as [19]
In the citing paper
“The two measures that we will consider here [12, 13] actually respectively rely on the violation of the quantum analog of the classical properties of P-divisibility and monotonic decrease in time of the Kolmogorov distance, that we have considered in Sect. III and hold true for the Markovian case, while also other approaches have been considered [11].”
From Markovianity and non-Markovianity in quantum and classical systems (2011) · cites this paper as [13]
In the citing paper
“These two criteria are at the basis of two recently introduced measures of non-Markovianity for open quantum systems [12, 13], which we here compare considering a quantum counterpart of classical semiMarkov processes.”
From Markovianity and non-Markovianity in quantum and classical systems (2011) · cites this paper as [13]
In the citing paper
“In particular, as shown by Eq. (64) and Eq. (67) and discussed thoroughly in Sect. IV B 3, we put into evidence that the measure based on CP-divisibility [13] gives the same infinite value to quite different time evolutions, at variance with the measure based on the dynamics of the trace distance [12], which assigns to them different weights.”
From Markovianity and non-Markovianity in quantum and classical systems (2011) · cites this paper as [13]
In the citing paper
“Recent work has demonstrated that the retention of quantum dynamics in biological systems is intricately connected with environmental fluctuations taking place at biologically relevant length and time scales [51–55].”
From The origins of quantum biology (2018) · cites this paper as [53]
In the citing paper
“Note that the shift is independent of the temperature, and although its effect is typically small (e.g.[3,18]), we will not neglect it in our study.”
From Markovian master equations: a critical study (2010) · cites this paper as [18]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Unknown-dynamics criterion is called both necessary and sufficient
MinorThe paper says this criterion can miss some non-Markovian evolutions, and later explicitly calls a positive result sufficient for non-Markovianity. Calling it a necessary condition for detecting deviations contradicts those statements.
“This will allow us to give a necessary condition to measure deviations from Markovianity even when the actual form of the dynamics is completely unknown, although some non-Markovian evolutions may be undetected.”
“There can be however non-Markovian quantum evolutions that remain undetected by the proposed measure, given that the requirement I ( E ) > 0 is a sufficient condition for deviation from Markovianity.”
Entanglement derivative need not be strictly negative for Markovian evolution
MinorThe identity evolution is Markovian but leaves entanglement constant, giving a zero derivative rather than an always-negative one.
“if the evolution of the system is Markovian the derivative of E [ ρ SA ( t )] is always negative and I ( E ) = 0.”
“1 is trivially Markovian”
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Metadata
- Authors
- Rivas, Á., Huelga, S. F., & Plenio, M. B.
- Affiliations
- University of Hertfordshire (GB)
- Universität Ulm (DE)
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2010-07-28
- Volume
- 105
- Issue
- 5
- Pages
- 050403-050403
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W2087654101
- Semantic Scholar
- 650d8d49ad99f9052b54cd06292f0aa97673ea14
- Cited by
- 1,037
- References
- 22
- Citations by year
- 2026: 67 · 2025: 83 · 2024: 63 · 2023: 59 · 2022: 67 · 2021: 73 · 2020: 75 · 2019: 72 · 2018: 62 · 2017: 68 · 2016: 67 · 2015: 64 · 2014: 69 · 2013: 50 · 2012: 58
- FWCI
- 49.75
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- University of Hertfordshire Research Archive (University of Hertfordshire) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- 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
- Keywords
- non-Markovianity, quantum entanglement, open quantum systems, quantum correlations, non-Markovianity measures, quantum evolution, system-environment interaction, necessary and sufficient conditions, quantum state tomography
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