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arXiv:1306.3248 · Physical Review A

Detecting initial system-environment correlations: Performance of various distance measures for quantum states

Steffen Wißmann, Bruno Leggio, Heinz‐Peter Breuer

Published Aug 12, 2013• 20 citations
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Abstract

We study the time evolution of four distance measures in the presence of initial system-environment correlations. It is well known that the trace distance between two quantum states of an open system may increase due to initial correlations, which leads to a breakdown of the contractivity of the reduced dynamics. Here we compare and analyze, for two different models, the time evolution of the trace distance, the Bures metric, the Hellinger distance, and the Jensen-Shannon divergence regarding an increase above their initial values, witnessing initial correlations. This work generalizes, deepens, and corrects a part of the study performed by Dajka et al. [Phys. Rev. A 84, 032120 (2011)] and thereby reveals generic features of the considered distance measures with respect to the capability of detecting initial system-environment correlations.

DOI
10.1103/physreva.88.022108

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We study the time evolution of four distance measures in the presence of initial system-environment correlations. It is well known that the trace distance between two quantum states of an open system may increase due to initial correlations, which leads to a breakdown of the contractivity of the reduced dynamics. Here we compare and analyze, for two different models, the time evolution of the trace distance, the Bures metric, the Hellinger distance, and the Jensen-Shannon divergence regarding an increase above their initial values, witnessing initial correlations. This work generalizes, deepens, and corrects a part of the study performed by Dajka et al. [Phys. Rev. A 84, 032120 (2011)] and thereby reveals generic features of the considered distance measures with respect to the capability of detecting initial system-environment correlations.
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.
We study the evolution of a general open quantum system when the system and its environment are initially correlated. We show that the trace distance between two states of the open system can increase above its initial value, and derive tight upper bounds for the growth of the distinguishability of open-system states. This represents a generalization of the contraction property of quantum dynamical maps. The obtained inequalities can be interpreted in terms of the exchange of information between the system and the environment, and lead to a witness for system-environment correlations which can be determined through measurements on the open system alone.
A quantum system inevitably interacts with its surroundings. In general, one does not have detailed information on an environment. Identifying the environmental features can help us to control the environment and its effects on the dynamics of an open system. Here, we consider a tripartite system and introduce a witness for the initial correlations among environments by means of the concept of the trace distance. Due to the existence of the initial environmental correlations, a tight upper bound is obtained for the growth of the trace distance of open quantum system states. Therefore, the initial correlations among the environments subject to particular conditions can be detected by measurements on the open system.
Intriguing features of the distance between two arbitrary states of an open quantum system are identified that are induced by initial system-environment correlations. As an example, we analyze a qubit dephasingly coupled to a bosonic environment. Within tailored parameter regimes, initial correlations are shown to substantially increase the distance between two qubit states evolving to long-time-limit states according to exact non-Markovian dynamics. It exemplifies the breakdown of the distance contractivity of the reduced dynamics.
Consider an open quantum system which interacts with its environment. Assuming that the experimenter has access only to the system, an interesting question is whether it is possible to detect initial correlations between the system and the environment by performing measurements only on the system. Various methods have been proposed to detect correlations by local measurements on the system. After reviewing these methods, we will show that initial correlations between the system and the environment are always detectable. In particular, we will show that one can always find a unitary evolution, for the whole system-environment, such that the trace distance method, proposed to witness correlations locally, succeeds. We also find the condition for existence of the optimal unitary evolution, for which the entire correlation is locally detectable. Next, we address the case where the system and the environment interact through a time-independent Hamiltonian. For this case we will see that if the initial correlation can be detected locally at some time ttt, then it can be detected for almost all the other times too. On the other hand, we see that one can find cases for which initial correlations between the system and the environment always remain undetectable even though the unitary evolution, generated by the Hamiltonian, is not factorized.
Employing the trace distance as a measure for the distinguishability of quantum states, we study the influence of initial correlations on the dynamics of open systems. We concentrate on the Jaynes-Cummings model for which the knowledge of the exact joint dynamics of system and reservoir allows the treatment of initial states with arbitrary correlations. As a measure for the correlations in the initial state we consider the trace distance between the system-environment state and the product of its marginal states. In particular, we examine the correlations contained in the thermal equilibrium state for the total system, analyze their dependence on the temperature and on the coupling strength, and demonstrate their connection to the entanglement properties of the eigenstates of the Hamiltonian. A detailed study of the time dependence of the distinguishability of the open system states evolving from the thermal equilibrium state and its corresponding uncorrelated product state shows that the open system dynamically uncovers typical features of the initial correlations.