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Physical Review A

Trace distance in stochastic dephasing with initial correlation

Masashi Ban, Sachiko Kitajima, Fumiaki Shibata

Published Oct 19, 2011• 7 citations
Publisher

Abstract

The time evolution of the trace distance between quantum states of a qubit which is placed under the influence of stochastic dephasing is investigated within the framework of the stochastic Liouville equation. When stochastic dephasing is subject to the homogeneous Gauss-Markov process, the trace distance is exactly calculated in the presence of the initial correlation between the qubit and the stochastic process, where the stochastic process is inevitably a nonstationary process. It is found that even the initial correlation with the classical environment can make the trace distance greater than the initial value if stochastic dephasing causes the slow modulation of the qubit.

DOI
10.1103/physreva.84.042115

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The time evolution of the trace distance between quantum states of a qubit which is placed under the influence of stochastic dephasing is investigated within the framework of the stochastic Liouville equation. When stochastic dephasing is subject to the homogeneous Gauss-Markov process, the trace distance is exactly calculated in the presence of the initial correlation between the qubit and the stochastic process, where the stochastic process is inevitably a nonstationary process. It is found that even the initial correlation with the classical environment can make the trace distance greater than the initial value if stochastic dephasing causes the slow modulation of the qubit.
The reduced time evolution of a single qubit under the influence of a classical environment is studied by means of the stochastic Liouville equation. The qubit initially correlates with the environment, which is created by a preparation process of an initial state. The phase fluctuation caused by the environment is characterized by means of the Gauss-Markov process and the two-state-jump Markov process. It is shown that depending on values of the parameters, the initial correlation enhances or reduces the decoherence of the qubit. In the case of the slow modulation, the effect of the initial preparation of the qubit becomes significant and it cannot be ignored in the reduced time evolution of the qubit.
Time evolution of a quantum system which is influenced by a stochastically fluctuating environment is studied by means of the stochastic Liouville equation. The two different types of the stochastic Liouville equation and their relation are discussed. The stochastic Liouville equation is shown to be derived from the quantum master equation of the Lindblad under certain conditions. Relaxation processes of single and bipartite quantum systems which are initially correlated with a stochastic environment are investigated. It is shown the possibility that the stochastic fluctuation can create coherence and entanglement of a quantum system with the assistance of the initial correlation. The results are examined in the pure dephasing processes of qubits, which are caused by the nonstationary Gauss-Markov process and two-state jump Markov process.
Time evolutions of quantum correlation, classical correlation and total correlation of two qubits are investigated when the two qubits are placed under the influence of classical phase fluctuations with correlation. Stochastic variables that describe the phase fluctuations are correlated and subject to the stationary Gauss–Markov process. The model includes the local and global dephasing models. It is shown that the quantum correlation measured by the quantum discord is increased by the correlation between the stochastic variables in the initial time region while the classical correlation and the total correlation are not. Furthermore the entanglement, the optimal fidelity of the quantum teleportation and the violation of the Bell inequality are investigated.
Decoherence of qubits under the influence of random phase fluctuations is studied by means of the stochastic Liouville equation, where the phase fluctuations are characterized by means of the Gauss-Markov process and the two-state-jump Markov process. We investigate how an observation of a stochastic variable during the time evolution influences the decoherence of the qubit system. We examine the decays of distinguishability and entanglement of qubit states. It is shown that the decoherence of qubits can be suppressed by the observation of the stochastic variable. It is also found that the suppression of the decoherence does not depend on the observed value.
The time evolution of the trace distance between two states of an open quantum system may increase due to initial system-environment correlations, thus exhibiting a breakdown of distance contractivity of the reduced dynamics. We analyze how the time evolution of the distance depends on the chosen distance measure. Here we elucidate the behavior of the trace distance, the Hilbert-Schmidt distance, the Bures distance, the Hellinger distance, and the quantum Jensen-Shannon divergence for two system-environment setups, namely a qubit bilinearly coupled to an infinite and a finite-size environment with the latter composed of harmonic oscillators.