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Cited by 1,037Quantum Information and CryptographyQuantum Mechanics and ApplicationsAdvanced Thermodynamics and Statistical Mechanics

Entanglement and Non-Markovianity of Quantum Evolutions

Open access
Rivas, Ángel, Huelga, Susana F., Plenio, Martin B.·2010·Physical Review Letters
DOI 10.1103/physrevlett.105.050403
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Abstract

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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Written from the paper’s full text.

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.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1103/physrevlett.105.050403

Cites

21 of 21 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 9 cited sources checked · 0 supported · 0 flagged

  • Findingown data
    § Entanglement and non-Markovianity of quantum evolutions · p. 2

    For α large enough, the value of I(E) itself increases with increasing temperature.

    “the same happens with the value of I ( E ) itself for α large enough.”
    Quote in the paperNot a substantial claim
    • 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.
  • Findingown data
    § Entanglement and non-Markovianity of quantum evolutions · p. 2

    In the limit α → 0, the evolution approaches a Markovian evolution independently of the value of T.

    “It is clear from Fig. 2 that in the limit α → 0 we approach a Markovian evolution independently of the value of T .”
    Quote in the paperSubstantial claim
    • 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.
  • Findingown data
    § Entanglement and non-Markovianity of quantum evolutions · p. 2

    For a super-Ohmic spectral density, the specific value of I(E) for given T and α depends strongly on the bath spectral function.

    “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.”
    Quote in the paperNot a substantial claim
    • 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.
  • Findingown data
    § Entanglement and non-Markovianity of quantum evolutions · p. 2

    The onset of deviations from Markovianity for stronger coupling strengths is temperature-dependent; the value of α for which I(E) becomes nonzero increases with increasing temperature.

    “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”
    Quote in the paperSubstantial claim
    • 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.
  • Findingreasoning
    § Entanglement and non-Markovianity of quantum evolutions · p. 4

    For a qubit undergoing pure dephasing dρ/dt = γ(t)(σ_z ρ σ_z - ρ), I is proportional to the area of γ(t) below zero.

    “Therefore, for this kind of evolution, I is proportional to the area of γ ( t ) which is below zero.”
    Quote in the paperSubstantial claim
  • Classified as probably not a substantial claim. 6
  • The quote only partly states this claim. 1
  • May combine two claims. 6

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Findingnone given
    § Entanglement and non-Markovianity of quantum evolutions · p. 2

    The shape of the curves does not depend significantly on the squeezing parameter r.

    “although the shape of the curves does not depend significatively on r .”
    Quote in the paperNot a substantial claim
    • 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 quote only partly states this claim.

    Interpretationreasoning
    § Entanglement and non-Markovianity of quantum evolutions · p. 4

    The proposed measure provides a necessary and sufficient condition to quantify strictly non-Markovianity when the structural form of the dynamical map is known.

    “the other provides a measure which quantify strictly the non-Markovianity, provided that the structural form of the dynamical map is known.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § Entanglement and non-Markovianity of quantum evolutions · p. 4

    g(t) ≥ 0, with g(t) = 0 if and only if E(t+ε,t) is CP.

    “noticing that g ( t ) ≥ 0, with g ( t ) = 0 if and only if E ( t + ǫ,t ) is CP.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § Entanglement and non-Markovianity of quantum evolutions · p. 4

    E(t+ε,t) is CP if and only if f_NCP(t+ε,t) = 1; otherwise f_NCP(t+ε,t) > 1.

    “Therefore, E ( t + ǫ,t ) is CP if and only if f NCP ( t + ǫ, t ) = 1, otherwise f NCP ( t + ǫ, t ) > 1.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Interpretationreasoning
    § Entanglement and non-Markovianity of quantum evolutions · p. 2

    The proposed measure I(E) can be particularly useful in the study of infinite dimensional systems where the computation of the exact dynamical map is often difficult.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationcitation
    § Entanglement and non-Markovianity of quantum evolutions · p. 2

    For non-Markovian evolutions, strict monotonicity of entanglement with an ancilla does not hold; environmental correlations can lead to bipartite entanglement increasing and decreasing as a function of time.

    “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”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [11] source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation
    § Entanglement and non-Markovianity of quantum evolutions · p. 1

    The structure of semigroups was analyzed in detail previously, concluding that a quantum system will undergo Markovian dynamics provided that its evolution satisfies a Master equation of the standard Lindblad form.

    “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”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [8] source

      Not checked in the free run.

  • Classified as probably not a substantial claim. May combine two claims.

    Interpretationreasoning
    § Entanglement and non-Markovianity of quantum evolutions · p. 4

    The integral I = ∫ g(t) dt can be taken as a measure of non-Markovianity, and is finite if g(t) decreases fast enough.

    “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.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim. May combine two claims.

    Study design and conductreasoning
    § Entanglement and non-Markovianity of quantum evolutions · p. 2

    In the simulation, the number of bath oscillators was chosen so that the final time is shorter than the recurrence time of the bath but large enough to capture the general properties of the dynamics.

    “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.”
    Quote in the paperSubstantial claim
    • 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.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § Entanglement and non-Markovianity of quantum evolutions · p. 1

    In interacting many-body systems, the subsystem's coupling strength may be comparable to the coupling to the bath.

    “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].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [4] source

      Not checked in the free run.

10 of 10

  • 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”
4 of 4

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 Lindblad equation describes Markovian dynamics for Γ_k ≥ 0, and when the loss rate Γ_k becomes negative it can also describe non-Markovian dynamics.

    “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

  • The results are applicable to a wide variety of scenarios including bounding decoherence rates, quantum speed limits in dissipative state preparation, quantum computation and simulation assisted by dissipation.

    “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]

  • The measure D_NM provides a necessary and sufficient condition for a given evolution to depart from strict Markovianity by evaluating whether the associated dynamical map is completely positive.

    “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]

  • The two measures of quantum non-Markovianity considered rely on the violation of the quantum analog of the classical properties of P-divisibility and monotonic decrease in time of the Kolmogorov distance.

    “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]

  • The two criteria of CP-divisibility and trace distance are at the basis of two recently introduced measures of non-Markovianity for open quantum systems [12,13].

    “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]

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

    Minor

    The 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.

  • Entanglement derivative need not be strictly negative for Markovian evolution

    Minor

    The identity evolution is Markovian but leaves entanglement constant, giving a zero derivative rather than an always-negative one.

2 of 2

Cites

Flagged first, then by citations

1 of the 22 references couldn’t be matched to a known paper.

Research works printed but not linked: 2

  1. A. Kossakowski, 1972, Bull. Acad. Polon. Sci., Sér. Sci. Math., 20, 1021
  2. M. Reed, 1975, Methods of Modern Mathematical Physics II

Cited by

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Related

Authors
Rivas, Á., Huelga, S. F., & Plenio, M. B.
Affiliations
  • University of Hertfordshire (GB)
  • Universität Ulm (DE)
ORCID
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
DOI
10.1103/physrevlett.105.050403
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
Licence
link.aps.org/licenses/aps-default-license (vor)
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 › Computer Science › Artificial Intelligence › 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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