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Physical Review A

Non-Markovian environments and entanglement preservation

Jackson Tan, Thi Ha Kyaw, Ye Yeo

Published Jun 21, 2010• 29 citations
Publisher

Abstract

Using the Shabani-Lidar post-Markovian master equation, we derive non-Markovian generalizations of important quantum decohering operations on single qubits. When environmental memory effects are being taken into account, both single-qubit coherence and two-qubit entanglement may be preserved over a longer period of time, in contrast to the corresponding situations where these are totally neglected. We argue that recognizing the fact that every environment is inherently non-Markovian could be the key to the resolution of the issue of entanglement sudden death.

DOI
10.1103/physreva.81.062119

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Using the Shabani-Lidar post-Markovian master equation, we derive non-Markovian generalizations of important quantum decohering operations on single qubits. When environmental memory effects are being taken into account, both single-qubit coherence and two-qubit entanglement may be preserved over a longer period of time, in contrast to the corresponding situations where these are totally neglected. We argue that recognizing the fact that every environment is inherently non-Markovian could be the key to the resolution of the issue of entanglement sudden death.
A post-Markovian master equation has been recently proposed as a tool to describe the evolution of a system coupled to a memory-keeping environment [A. Shabani and D. A. Lidar, Phys. Rev. A 71, 020101(R) (2005)]. For a single qubit affected by appropriately chosen environmental conditions, the corresponding dynamics is always legitimate and physical. Here we extend such a situation to the case of two qubits, only one of which experiences the environmental effects. We show how, despite the innocence of such an extension, the introduction of the second qubit should be done cum grano salis to avoid consequences such as the breaking of the positivity of the associated dynamical map. This hints at the necessity of using care when adopting phenomenologically derived models for evolutions occurring outside the Markovian framework.
The non-Markovian dynamics of quantum entanglement is studied by the Shabani—Lidar master equation when one of entangled quantum systems is coupled to a local reservoir with memory effects. The completely positive reduced dynamical map can be constructed in the Kraus representation. Quantum entanglement decays more slowly in the non-Markovian environment. The decoherence time for quantum entanglement can be markedly increased with the change of the memory kernel. It is found out that the entanglement sudden death between quantum systems and entanglement sudden birth between the system and reservoir occur at different instants.
In this paper we demonstrate that two commonly used phenomenological post-Markovian quantum master equations can be derived without using any perturbative approximation. A system coupled to an environment characterized by self-classical configurational fluctuations, the latter obeying a Markovian dynamics, defines the underlying physical model. Both Shabani-Lidar equation [A. Shabani and D. A. Lidar, Phys. Rev. A 71, 020101(R) (2005)] and its associated approximated integrodifferential kernel master equation are obtained by tracing out two different bipartite Markovian Lindblad dynamics where the environment fluctuations are taken into account by an ancilla system. Furthermore, conditions under which the non-Markovian system dynamics can be unraveled in terms of an ensemble of measurement trajectories are found. In addition, a non-Markovian quantum jump approach is formulated. Contrary to recent analysis [L. Mazzola, E. M. Laine, H. P. Breuer, S. Maniscalco, and J. Piilo, Phys. Rev. A 81, 062120 (2010)], we also demonstrate that these master equations, even with exponential memory functions, may lead to non-Markovian effects such as an environment-to-system backflow of information if the Hamiltonian system does not commutate with the dissipative dynamics.
We study the entanglement preservation of two qubits locally interacting with their reservoirs. We show that the existence of a bound state of the qubit and its reservoir and the non-Markovian effect are two essential ingredients and their interplay plays a crucial role in preserving the entanglement in the steady state. When the non-Markovian effect is neglected, the entanglement sudden death (ESD) is reproduced. On the other hand, when the non-Markovian is significantly strong but the bound state is absent, the phenomenon of the ESD and its revival is recovered. Our formulation presents a unified picture about the entanglement preservation and provides a clear clue on how to preserve the entanglement in quantum information processing.
We investigate the sudden birth and sudden death of entanglement of two qubits interacting with un-correlated structured reservoirs. The system is initially prepared in two-qubit extended Werner-like state. We work out the dependence of the entanglement dynamics on both non-Markovian environments and the purity of initial state, and show that non-Markovian environments and the purity can control the time of the two-qubit entanglement sudden death and the reservoirs' entanglement sudden birth. Furthermore, under the conditions of different purity and initial entanglement, the revival of qubits' entanglement can manifest before, simultaneously or even after the disentanglement of their corresponding reservoirs.
We theoretically study the non-Markovian dynamics of qubit systems coupled to nonequilibrium environments with nonstationary and non-Markovian statistical properties. The reduced density matrix of the single qubit system satisfies a closed third-order differential equation with all the higher-order environmental correlations taken into account and the reduced density matrix of the two qubit system can be expressed as the Kraus representation in terms of the tensor products of the single qubit Kraus operators. We derive the relation between the entanglement and nonlocality of the two qubit system which are both closely associated with the decoherence function. We identify the threshold values of the decoherence function to ensure the existences of the concurrence and nonlocal quantum correlations for a given evolution time when the two qubit system is initially prepared in the composite Bell states and the extended Werner states, respectively. It is shown that the environmental nonstationary feature can suppress the decoherence and disentanglement dynamics and can reduce the coherence and entanglement revivals in non-Markovian dynamics regime. In addition, it is shown that the environmental non-Markovian feature can enhance the coherence revivals in the single decoherence dynamics and the entanglement revivals in the two qubit disentanglement dynamics, respectively. Furthermore, the environmental nonstationary and non-Markovian features can enhance the nonlocality of the two qubit system.