Non-Markovianity-Assisted Steady State Entanglement
Open accessAbstract
We analyze the steady state entanglement generated in a coherently coupled dimer system subject to dephasing noise as a function of the degree of Markovianity of the evolution. By keeping fixed the effective noise strength while varying the memory time of the environment, we demonstrate that non-Markovianity is an essential, quantifiable resource that may support the formation of steady state entanglement whereas purely Markovian dynamics governed by Lindblad master equations lead to separable steady states. This result illustrates possible mechanisms leading to long-lived entanglement in purely decohering, possibly local, environments. We present a feasible experimental demonstration of this noise assisted phenomenon using a system of trapped ions.
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TL;DR
The paper shows that, for a coherently coupled dimer subject to dephasing noise of fixed effective strength, non-Markovianity can be the key resource that supports steady-state entanglement, whereas purely Markovian Lindblad dynamics lead to separable steady states. It also outlines a trapped-ion implementation of the proposed noise-assisted phenomenon.
Key findings
- Keeping the effective dephasing rate fixed while varying the underlying dynamics, the authors find that non-Markovianity is an essential, quantifiable resource for steady-state entanglement; purely Markovian dynamics governed by Lindblad master equations give separable steady states.
- For sufficiently small \(f\) in the weakly non-Markovian regime, the dimer steady state approaches the singlet state, with entanglement “close to one e-bit”; for large \(f\), the steady state is maximally mixed and separable.
- The non-Markovianity measure \(D_{\mathrm{NM}}\) decreases monotonically with \(f\), becoming zero for sufficiently large \(f\), where the dynamical map is divisible and the evolution is fully Markovian.
- At finite temperature with average photon number \(n_{\mathrm{th}}=0.1\), corresponding to about 100 mK for GHz frequencies, the steady-state entanglement reaches 0.85 for \(f=0.01\). Asymmetry \(g_1\neq g_2\) does not reduce the singlet overlap below 90% up to ratios \(g_2/g_1=2\).
Methods
- Theoretical model: a two-site dimer with exchange coupling \(J\), each site coupled to a localized damped mode via \(g_j \sigma_j^z(a_j+a_j^\dagger)\), with the local modes damped by a Markovian bath at rate \(\kappa\); tracing out the local mode yields generally non-Markovian dimer dynamics.
- Parametrization \(g=\sqrt{f}g_0\) and \(\kappa=f\kappa_0\) keeps the noise strength \(g^2/\kappa=g_0^2/\kappa_0\) fixed while varying \(f\) from Markovian to non-Markovian regimes; the analysis uses the non-Markovianity measure \(D_{\mathrm{NM}}\) from [18] and logarithmic negativity for entanglement, restricted to the one-excitation subspace and truncating each oscillator after Fock layer 2.
- Experimental proposal: trapped ions, with electronic degrees of freedom as the dimer, motional modes as the environment, a Sørensen-Mølmer gate for the exchange coupling, a dispersive \(\sigma_z(a+a^\dagger)\) coupling via a far off-resonant standing wave, and laser cooling of an auxiliary third ion to damp motional modes.
Limitations
- Stated: the authors work in the regime \(2g<\kappa\) (weakly non-Markovian), where an effective dephasing rate is still reasonable; for \(g>\kappa\), coherent oscillations occur and the definition of an effective dephasing rate becomes meaningless.
- Stated: finite-temperature results are not shown in the figures; the paper notes that numerical methods able to incorporate temperature dependence and exactly simulate structured spectral densities would be required to fully address realistic biological systems at physiological temperatures.
- Evident: the oscillator is truncated after Fock layer 2, justified by the probability of a single excitation in either mode never exceeding 10% in the considered parameter regime, and the dynamics is confined to the one-excitation sector; the experimental realization is proposed, not performed.
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- Findingown data§ Non-Markovianity assisted Steady State Entanglement · p. 3
For the selected parameter regime, the non-Markovianity measure D_NM decreases monotonically with f.
“The value of D NM decreases monotonically with f for the selected parameter regime.”
- Also stated in: § Non-Markovianity assisted Steady State Entanglement · p. 2 · boosted
“The monotonic change in Markovianity of the noise with the parameter f is shown in Fig. 2, where an explicit measure of non-Markovianity is considered.”
Quote in the paperSubstantial claim - Also stated in: § Non-Markovianity assisted Steady State Entanglement · p. 2 · boosted
- Findingown data§ Non-Markovianity assisted Steady State Entanglement · p. 3
For an average photon number n_th = 0.1 (corresponding to about 100 mK for GHz frequencies), the steady state entanglement reaches the value 0.85 for f = 0.01.
“For an average photon number of n th = 0 . 1, which corresponds to a temperature of about 100 mK for GHz frequencies, we obtain that the steady state entanglement reaches the value 0.85 for f = 0 . 01 (Finite T results are not shown in the Figures).”
Quote in the paperNot a substantial claim - Findingreasoning§ Non-Markovianity assisted Steady State Entanglement · p. 3
In the limit of sufficiently small f, and in the regime where the local mode is quasi-resonant with the dimer eigenstates (Ω ~ J), the dimer steady state approaches the singlet state.
“In the limit of sufficiently small f , and in the regime where the local mode is quasi-resonant with the dimer eigenstates, so that Ω ∼ J , one finds that the above expression tends to unity. Hence, in this picture, the system is in weak interaction with a composite environment (local mode+Markovian reservoir) and relaxes towards its ground state, which in this case is the singlet state.”
Quote in the paperSubstantial claim - Findingcitation§ Non-Markovianity assisted Steady State Entanglement · p. 1
Purely Markovian dephasing at the same effective rate leads the dimer to a maximally mixed separable steady state with no long-term entanglement.
“whereas purely Markovian dynamics governed by Lindblad master equations lead to separable steady states.”
- Also stated in: § Non-Markovianity assisted Steady State Entanglement · p. 3 · plain
“When weakly coupled to a purely Markovian environment and experiencing dephasing at the same rate as before, the steady state of the dimer system tends towards the maximally mixed state and no entanglement survives in the long term.”
- Also stated in: § Non-Markovianity assisted Steady State Entanglement · p. 3 · plain
“When the evolution is strictly Markovian, the steady state is separable while in the presence of non-Markovianity, the dimer system contains quantum correlations that steadily become close to one e-bit for f sufficiently small.”
Quote in the paperSubstantial claim0 of 2 sources checked - Also stated in: § Non-Markovianity assisted Steady State Entanglement · p. 3 · plain
- Findingcitation§ Non-Markovianity assisted Steady State Entanglement · p. 3
For sufficiently small f, the dimer steady state contains quantum correlations that steadily become close to one e-bit.
“When the evolution is strictly Markovian, the steady state is separable while in the presence of non-Markovianity, the dimer system contains quantum correlations that steadily become close to one e-bit for f sufficiently small.”
Quote in the paperSubstantial claim
- Findingcitation§ Non-Markovianity assisted Steady State Entanglement · p. 3
When f decreases but remains in the weakly non-Markovian regime, the steady state approaches the singlet state, which is the lowest energy eigenstate of the coherent evolution.
“When f decreases but remaining in the weakly non-Markovian regime, the steady state approaches the singlet state, which in this case is the lowest energy eigenstate of the coherent evolution.”
Quote in the paperSubstantial claim - Premisecitation§ Non-Markovianity assisted Steady State Entanglement · p. 1
The effective dephasing rate γ_eff is proportional to g^2/κ in the limit where the coherent coupling g is smaller than the decay κ.
“In the limit where the coherent coupling g is smaller than the decay κ we find that the effective dephasing rate γ eff , which quantifies the strength of the system-environment coupling, is proportional to the ratio g 2 /κ (see [14] for the case of cavity QED).”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [14] A. Imamoglu, Phys. Rev. A 50 , 3650 (1994) and references therein. source
Not checked in the free run.
- Interpretationreasoning§ Non-Markovianity assisted Steady State Entanglement · p. 4
Combining the three elements (direct coupling, dispersive coupling, and laser cooling) yields the dynamical Hamiltonian discussed in Eq. (2) which underlies the non-Markovianity driven steady state entanglement in a dimer system.
“Combining these three elements yields the dynamical Hamiltonian discussed in Eq. (2) which underlies the non-Markovianity driven steady state entanglement in a dimer system.”
Quote in the paperSubstantial claim - Interpretationreasoning§ Non-Markovianity assisted Steady State Entanglement · p. 4
The scheme for the excitation-preserving coupling couples to all the modes of the ion crystal, which may lead to correlations between the environments acting on separate parts of the dimer.
“This scheme couples to all the modes of the ion crystal which may lead to correlations between the environments acting on separate parts of the dimer.”
Quote in the paperSubstantial claim - Interpretationreasoning§ Non-Markovianity assisted Steady State Entanglement · p. 1
A feasible experimental demonstration of this noise-assisted phenomenon is possible using a system of trapped ions.
“We present a feasible experimental demonstration of this noise assisted phenomenon using a system of trapped ions.”
- Also stated in: § Non-Markovianity assisted Steady State Entanglement · p. 3 · hedged
“This realization employs proven elements of ion trap technology and offers, in principle, the possibility for control of all the relevant parameters in the Hamiltonian Eq. (2).”
Quote in the paperNot a substantial claim - Also stated in: § Non-Markovianity assisted Steady State Entanglement · p. 3 · hedged
- Interpretationreasoning§ Non-Markovianity assisted Steady State Entanglement · p. 2
Keeping the noise strength g^2/κ fixed while varying f from values much larger than 1 to much smaller than 1 leads to Markovian and non-Markovian dynamics, respectively, with the same effective local dephasing rate γ_eff.
“In this way, the noise strength g 2 /κ = g 2 0 /κ 0 is kept fixed while varying f from values much larger than 1 to much smaller than 1 leads to, respectively, Markovian and non-Markovian dynamics with, crucially, the same effective local dephasing rate γ eff .”
Quote in the paperSubstantial claim - Interpretationown data§ Non-Markovianity assisted Steady State Entanglement · p. 1
In a coherently coupled dimer system subject to dephasing noise, non-Markovianity is an essential, quantifiable resource that can support the formation of steady state entanglement, even when the environments act locally.
“By keeping fixed the noise strength while varying the underlying dynamics, we demonstrate that non-Markovianity is an essential, quantifiable resource that may support the formation of steady state entanglement whereas purely Markovian dynamics governed by Lindblad master equations lead to separable steady states.”
- Also stated in: § Non-Markovianity assisted Steady State Entanglement · p. 1 · boosted
“We show that the presence of steady state entanglement can be unambiguously linked to an increasing degree of non-Markovianity, even when the environments are acting locally.”
- Also stated in: § Non-Markovianity assisted Steady State Entanglement · p. 4 · boosted
“We have demonstrated that, when subject to system-environment interactions of the same strength, the non-Markovian character of the noise can be the crucial property that leads to steady state entanglement where purely Markovian noise would result in the complete destruction of entanglement.”
Quote in the paperSubstantial claim - Also stated in: § Non-Markovianity assisted Steady State Entanglement · p. 1 · boosted
- Premisereasoning§ Non-Markovianity assisted Steady State Entanglement · p. 1
For 2g < κ (weakly non-Markovian regime), the definition of an effective dephasing rate γ_eff is still reasonable for times exceeding κ^-1 when the decay is well approximated by an exponential.
“For 2 g < κ , a regime which we denote as weakly non-Markovian, the definition of an effective dephasing rate γ eff is still reasonable for times exceeding κ -1 when the decay is well approximated by an exponential.”
Quote in the paperSubstantial claim
- May combine two claims. 5
- Classified as probably not a substantial claim. 4
May combine two claims.
Findingown data§ Non-Markovianity assisted Steady State Entanglement · p. 3For sufficiently large f, the dynamical map is divisible, D_NM is zero, and the dimer's evolution is fully Markovian.
“For f sufficiently large, the dynamical map is divisible, D NM is zero and the dimer's evolution is fully Markovian.”
- Also stated in: § Non-Markovianity assisted Steady State Entanglement · p. 2 · boosted
“showing that the domain of large f leads to the dimer's dynamics to be fully Markovian.”
Quote in the paperSubstantial claim- Also stated in: § Non-Markovianity assisted Steady State Entanglement · p. 2 · boosted
May combine two claims.
Findingnone given§ Non-Markovianity assisted Steady State Entanglement · p. 3At finite temperature, the steady state entanglement is decreased but remains finite.
“At a finite temperature, the steady state entanglement is decreased but remains finite.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim.
Premisecitation§ Non-Markovianity assisted Steady State Entanglement · p. 2The 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 .”
Quote in the paperSubstantial claim0 of 2 sources checked- cites [18] A. Rivas, S. F. Huelga and M. B. Plenio, Phys. Rev. Lett. 105 , 050403 (2010). Other measures of non-Markovianity have been put forward in the literature (see H.-P. Breuer, E.-M. Laine, J. Piilo, Phys. Rev. Lett. 103 , 210401 (2009) and M. M. Wolf, J. Eisert, T. S. Cubitt and J. I. Cirac, Phys. Rev. Lett. 101 , 150402 (2008)). We have chosen our own due to its simplicity, given that it does not require further optimization. source
Not checked in the free run.
- cites [19] A. Jamiosuppress lkowski, Rep. Math. Phys. 3 , 275 (1972); M.-D. Choi, Lin. Alg. and Appl. 10 , 285 (1975). source
Not checked in the free run.
May combine two claims.
Premisecitation§ Non-Markovianity assisted Steady State Entanglement · p. 4The Sørensen-Mølmer gate scheme does not require the center-of-mass mode to be in its ground state, increasing the robustness of the scheme.
“The added advantage of this scheme is the fact that it does not require the center-of-mass mode to be in its ground state thus increasing the robustness of the scheme.”
Quote in the paperSubstantial claim0 of 2 sources checkedMay combine two claims.
Premisenone given§ Non-Markovianity assisted Steady State Entanglement · p. 1In the limit g > κ, significant coherent oscillations can occur and the definition of an effective dephasing rate becomes meaningless.
“In the limit where g > κ significant coherent oscillations can occur and the definition of an effective dephasing rate becomes meaningless.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Premisenone given§ Non-Markovianity assisted Steady State Entanglement · p. 4Damping of motional degrees of freedom can be achieved by applying laser cooling to an auxiliary third ion that couples to all the motional modes of the system.
“Damping of motional degrees of freedom can be achieved, of course, by applying laser cooling to an auxiliary third ion that couples to all the motional modes of the system.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Study design and conductown data§ Non-Markovianity assisted Steady State Entanglement · p. 2Each oscillator was truncated after Fock layer 2 because the probability to have a single excitation in either mode never exceeds 10% in the parameter regime considered.
“We truncated each oscillator after Fock layer 2 as the probability to have a single excitation in either mode never exceeds 10% in the parameter regime that we consider here.”
Quote in the paperNot a substantial claimMay combine two claims.
Premisereasoning§ Non-Markovianity assisted Steady State Entanglement · p. 1When 2g << κ, the dephasing is exponential and the dynamics of the system can be reproduced by a Lindblad master equation with the effective decay rate γ_eff.
“When 2 g ≪ κ the dephasing is exponential and the dynamics of the system can be reproduced by a Lindblad master equation with the effective decay rate γ eff .”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation§ Non-Markovianity assisted Steady State Entanglement · p. 3The direct coupling between the two qubits can be implemented using a Sørensen-Mølmer gate.
“For the implementation of the direct coupling between the two qubits we make use of a Sørensen-Mølmer gate [21, 22].”
Quote in the paperNot a substantial claim0 of 2 sources checked
The authors state this as an assumption.
The local modes are damped by a conventional Markovian bath.
“We will assume that the local modes are damped by a conventional Markovian bath so that the global time evolution of the dimer and the vibrational modes is described by a Markovian master equation with a Liouvillian part accounting for the damping of the localized modes at a certain rate κ .”
The authors state this as an assumption.
The dimer dynamics is confined to the one excitation sector.
“Motivated by the situations frequently encountered in biomolecular complexes where external illumination is either weak and/or doubly excited states are strongly suppressed [15], we will focus here on situations where the dimer dynamics is confined to the one excitation sector so that the entire dynamics for the two sites takes place in the subspace spanned by the states | 01 〉 , | 10 〉 .”
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?
Non-Markovian effects become important when the relaxation time of the system is comparable to the relaxation time of the environment.
“Non-Markovian (or memory) effects become important when the relaxation time of the system is comparable to the relaxation time of the environment [25,26].”
From Quantum Speed Limit for Non-Markovian Dynamics (2013) · cites this paper as [26]
Multisite networks may exhibit more complex behaviour due to interplay between constructive and destructive interference effects and environmental noise.
“While linear networks can already exhibit interesting noise assisted transport phenomena [16, 107-111], multisite networks may exhibit more complex behaviour which arises due to the interplay between a wealth of constructive and destructive interference effects in a quantum dynamical system on the one hand and environmental noise on the other hand [16, 62, 67-70, 77, 112].”
From Vibrations, quanta and biology (2013) · cites this paper as [70]
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References 30 · Cited by 203
- Authors
- Huelga, S. F., Rivas, Á., & Plenio, M. B.
- Affiliations
- Universität Ulm (DE)
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2012-04-20
- Volume
- 108
- Issue
- 16
- Pages
- 160402-160402
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W1981105843
- Semantic Scholar
- 9ff16b13520c7ba4ebe2f5ae586f9acc13f23673
- Cited by
- 203
- References
- 30
- Citations by year
- 2026: 4 · 2025: 15 · 2024: 12 · 2023: 7 · 2022: 14 · 2021: 12 · 2020: 6 · 2019: 18 · 2018: 15 · 2017: 14 · 2016: 19 · 2015: 22 · 2014: 25 · 2013: 15 · 2012: 5
- FWCI
- 21.10
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- arXiv (Cornell University) (publishedVersion)
- PubMed (publishedVersion)
- 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 › Atomic and Molecular Physics, and Optics › Quantum optics and atomic interactions
- Keywords
- non-Markovianity, steady-state entanglement, dimer systems, dephasing noise, memory time, Lindblad master equation, trapped ions, decoherence, quantum resource theory
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