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arXiv:1007.4498 · Physical Review A

Kraus representation of quantum evolution and fidelity as manifestations of Markovian and non-Markovian forms

A. K. Rajagopal, A. R. Usha Devi, R. W. Rendell

Published Oct 12, 2010• 117 citations
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Abstract

It is shown that the fidelity of the dynamically evolved system with its earlier time-density matrix provides a signature of non-Markovian dynamics. Also, the fidelity associated with the initial state and the dynamically evolved state is shown to be larger in the non-Markovian evolution compared to that in the corresponding Markovian case. Starting from the Kraus representation of quantum evolution, the Markovian and non-Markovian features are discerned in its short-time structure. These two features are in concordance with each other and they are illustrated with the help of four models of interaction of the system with its environment.

DOI
10.1103/physreva.82.042107

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It is shown that the fidelity of the dynamically evolved system with its earlier time-density matrix provides a signature of non-Markovian dynamics. Also, the fidelity associated with the initial state and the dynamically evolved state is shown to be larger in the non-Markovian evolution compared to that in the corresponding Markovian case. Starting from the Kraus representation of quantum evolution, the Markovian and non-Markovian features are discerned in its short-time structure. These two features are in concordance with each other and they are illustrated with the help of four models of interaction of the system with its environment.
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.
The question, whether an open system dynamics is Markovian or non-Markovian can be answered by studying the direction of the information flow in the dynamics. In Markovian dynamics, information must always flow from the system to the environment. If the environment is interacting with only one of the subsystems of a bipartite system, the dynamics of the entanglement in the bipartite system can be used to identify the direction of information flow. Here we study the dynamics of a two-level system interacting with an environment, which is also a heat bath, and consists of a large number of two-level quantum systems. Our model can be seen as a close approximation to the `spin bath' model at low temperatures. We analyze the Markovian nature of the dynamics, as we change the coupling between the system and the environment. We find the Kraus operators of the dynamics for certain classes of couplings. We show that any form of time-independent or time-polynomial coupling gives rise to non-Markovianity. Also, we witness non-Markovianity for certain parameter values of time-exponential coupling. Moreover, we study the transition from non-Markovian to Markovian dynamics as we change the value of coupling strength.
Starting from unitary dynamics we study the evolution in time of a non-relativistic quantum system that exchanges energy with a thermal reservoir of harmonic oscillators. System and reservoir are assumed to be initially decorrelated. Reservoir correlation functions are factorized by means of a Kraus compliant version of Wick's theorem. As a result, the non-Markovian Kraus map for the system density operator can be completely expressed in terms of system potentials and reservoir pair correlation functions. An infinite hierarchy for the evolution operators of the Kraus map is derived. The system density operator is obtained as a time-ordered exponential containing a non-Markovian counterpart of the standard Markovian generator for dissipative dynamics. We specify a condition on this non-Markovian generator for return to thermal equilibrium. We set up a non-Markovian perturbation theory that preserves both trace and positivity of the system density operator.
The question, whether an open system dynamics is Markovian or non-Markovian can be answered by studying the direction of the information flow in the dynamics. In Markovian dynamics, information must always flow from the system to the environment. If the environment is interacting with only one of the subsystems of a bipartite system, the dynamics of the entanglement in the bipartite system can be used to identify the direction of information flow. Here we study the dynamics of a two-level system interacting with an environment, which is also a heat bath, and consists of a large number of two-level quantum systems. Our model can be seen as a close approximation to the 'spin bath' model at low temperatures. We analyze the Markovian nature of the dynamics, as we change the coupling between the system and the environment. We find the Kraus operators of the dynamics for certain classes of couplings. We show that any form of time-independent or time-polynomial coupling gives rise to non-Markovianity. Also, we witness non-Markovianity for certain parameter values of time-exponential coupling. Moreover, we study the transition from non-Markovian to Markovian dynamics as we change the value of coupling strength.
We theoretically study the non-Markovian disentanglement dynamics of a two-qubit system coupled to nonequilibrium environments with nonstationary and non-Markovian random telegraph noise statistical properties. 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 an arbitrary evolution time when the two-qubit system is initially prepared in the composite Bell states and the Werner states, respectively. It is shown that the environmental nonequilibrium feature can suppress the disentanglement dynamics and reduce the entanglement revivals in non-Markovian dynamics regime. In addition, the environmental nonequilibrium feature can enhance the nonlocality of the two-qubit system. Moreover, the entanglement sudden death and rebirth phenomena and the transition between quantum and classical nonlocalities closely depend on the parameters of the initial states and the environmental parameters in nonequilibrium environments.
It is known that the dynamical evolution of a system, from an initial tensor product state of system and environment, to any two later times, t1, t2 (t2 > t1), are both completely positive (CP) but in the intermediate times between t1 and t2 it need not be CP. This reveals the key to the Markov (if CP) and non Markov (if it is not CP) avataras of the intermediate dynamics. This is brought out here in terms of the quantum stochastic map A and the associated dynamical map B—without resorting to master equation approaches. We investigate these features with four examples which have entirely different physical origins: 1) A two qubit Werner state map with time dependent noise parameter; 2) Phenomenological model of a recent optical experiment (Nature Physics, 7, 931 (2011)) on the open system evolution of photon polarization; 3) Hamiltonian dynamics of a qubit coupled to a bath of N qubits; 4) Two qubit unitary dynamics of Jordan et al. (Phys. Rev. A 70, 052110 (2004) with initial product states of qubits. In all these models, it is shown that the positivity/negativity of the eigenvalues of intermediate time dynamical B map determines the Markov/non-Markov nature of the dynamics.