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arXiv:1905.06198 · Journal of Physics A Mathematical and Theoretical

Nearly Markovian maps and entanglement-based bound on corresponding non-Markovianity

Sreetama Das, Sudipto Singha Roy, Samyadeb Bhattacharya, Ujjwal Sen

Published Aug 13, 2021• 5 citations
arXivHTMLPDF

Abstract

Abstract We identify a set of dynamical maps of open quantum system, and refer to them as ‘ ϵ -Markovian’ maps. It is constituted of maps which, in a higher dimensional system–environment Hilbert space, possibly violate Born approximation but only a ‘little’. We characterize the ‘ ϵ -nonmarkovianity’ of a general dynamical map by the minimum distance of that map from the set of ϵ -Markovian maps. We analytically derive an inequality which gives a bound on the ϵ -nonmarkovianity of the dynamical map, in terms of an entanglement-like resource generated between the system and its ‘immediate’ environment. In the special case of a vanishing ϵ , this inequality gives a relation between the ϵ -nonmarkovianity of the reduced dynamical map on the system and the entanglement generated between the system and its immediate environment. We numerically investigate the behavior of the similar distant based measures of non-Markovianity for classes of amplitude damping and phase damping channels.

DOI
10.1088/1751-8121/ac1d8b

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Abstract We identify a set of dynamical maps of open quantum system, and refer to them as ‘ ϵ -Markovian’ maps. It is constituted of maps which, in a higher dimensional system–environment Hilbert space, possibly violate Born approximation but only a ‘little’. We characterize the ‘ ϵ -nonmarkovianity’ of a general dynamical map by the minimum distance of that map from the set of ϵ -Markovian maps. We analytically derive an inequality which gives a bound on the ϵ -nonmarkovianity of the dynamical map, in terms of an entanglement-like resource generated between the system and its ‘immediate’ environment. In the special case of a vanishing ϵ , this inequality gives a relation between the ϵ -nonmarkovianity of the reduced dynamical map on the system and the entanglement generated between the system and its immediate environment. We numerically investigate the behavior of the similar distant based measures of non-Markovianity for classes of amplitude damping and phase damping channels.
An alternative non-Markovianity measure for open quantum processes is proposed, which takes advantage of the nonincreasing property of relative entropy of coherence under the incoherent completely positive and trace-preserving maps in the extended Hilbert space constituted by the open system and its ancillary. By applying the proposed measure to some typical noisy channels, we find that for phase damping and amplitude damping channels it is equivalent to the three previous measures of non-Markovianity, i.e., the measures based on the quantum trace distance, dynamical divisibility, and quantum mutual information. For the random unitary channel, however, these measures do not coincide exactly, and the proposed measure in the witness of Markovianity is more general than the measures based on quantum trace distance and dynamical divisibility but overlaps partly with the measure based on quantum mutual information.
We introduce a non-Markovianity measure for continuous-variable open quantum systems based on the idea put forward in H.-P. Breuer et al. [Phys. Rev. Lett. 103, 210401 (2009);], that is, by quantifying the flow of information from the environment back to the open system. Instead of the trace distance we use here the fidelity to assess distinguishability of quantum states. We employ our measure to evaluate non-Markovianity of two paradigmatic Gaussian channels: the purely damping channel and the quantum Brownian motion channel with Ohmic environment. We consider different classes of Gaussian states and look for pairs of states maximizing the backflow of information. For coherent states we find simple analytical solutions, whereas for squeezed states we provide both exact numerical and approximate analytical solutions in the weak coupling limit.
Although a number of measures for quantum non-Markovianity have been proposed recently, it is still an open question whether these measures directly characterize the memory effect of the environment, i.e., the dependence of a quantum state on its past in a time evolution. In this paper, we present a criterion and propose a measure for non-Markovianity with clear physical interpretations of the memory effect. The non-Markovianity is defined by the inequality $T({t}_{2},{t}_{0})\ensuremath{\ne}T({t}_{2},{t}_{1})T({t}_{1},{t}_{0})$ in terms of memoryless dynamical map TTT introduced in this paper. This definition is conceptually distinct from that based on divisibility used by Rivas et al. [Phys. Rev. Lett. 105, 050403 (2010)], whose violation is manifested by noncomplete positivity of the dynamical map. We demonstrate via a typical quantum process that without Markovian approximation, nonzero memory effects (non-Markovianity) always exist even if the non-Markovianity is zero by the other non-Markovianity measures.
Abstract Based on the nonincreasing property of quantum coherence via skew information under incoherent completely positive and trace-preserving maps, we propose a non-Markovianity measure for open quantum processes. As applications, by applying the proposed measure to some typical noisy channels, we find that it is equivalent to the three previous measures of non-Markovianity for phase damping and amplitude damping channels, i.e. the measures based on the quantum trace distance, dynamical divisibility, and quantum mutual information. For the random unitary channel, it is equivalent to the non-Markovianity measure based on l 1 norm of coherence for a class of output states and it is incompletely equivalent to the measure based on dynamical divisibility. We also use the modified Tsallis relative entropy of coherence to detect the non-Markovianity of dynamics of quantum open systems, the results show that the modified Tsallis relative entropy of coherence are more comfortable than the original Tsallis relative entropy of coherence for small .
We test the conjecture that non-Markovian dynamics may help to protect quantum entanglement. To this end, we address an open system whose environment behaves either as a quantal system or as an effective reservoir, depending on the regime chosen to rule the system–environment dynamics. This allows us to study entanglement dynamics within Markovian and non-Markovian regimes. We focus on the recently introduced definition of quantum non-Markovianity that is based on the trace distance between pairs of initial states. The corresponding quantum measure for assessing the non-Markovianity of a quantum map Φ is submitted to test. We report cases for which signals non-Markovianity, even though Φ belongs to a purely dissipative regime. We argue that is not generally applicable to assess the non-Markovianity of a quantum process, because information is not always encoded in a way that depends on the distinguishability between states. When entanglement is used as the carrier of information, non-Markovianity should be diagnosed in terms of an entanglement measure. We thus compare the dynamics of Wooters concurrence and of trace distance between states, showing that they may behave qualitatively alike but leading to conflicting diagnosis about the backflow of information from the environment to the system. In contrast to other studies, in our case entanglement is transferred only to potential carriers of information, thereby excluding paradoxical situations that involve entangled reservoirs.
Abstract We propose an alternative non-Markovianity measure of open quantum systems, which is based on max-information. The max-information is a one-shot generalization of the mutual information derived from the max-relative entropy. This operational interpretation provides a new site for viewing the non-Markovian process. As applications, some typical noisy channels have been investigated: for phase damping and amplitude damping channels, non-Markovianity measure based on max-information is equivalent to the three previous non-Markovianity measures, i.e. the measures based on the distinguishability, entanglement of environment and the mutual information. For the random unitary channel, the proposed measurement in the witness of Markovianity is equivalent to the measures based on quantum trace distance and thus overlaps partly with the measurement based on the mutual information. It shows that max-information may be a better candidate for correlation-based measurement of non-Markovian than the mutual information.