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arXiv:1005.0364 · Physical Review A

Distance growth of quantum states due to initial system-environment correlations

Jerzy Dajka, J. Łuczka

Published Jul 30, 2010• 85 citations
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Abstract

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.

DOI
10.1103/physreva.82.012341

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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.
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 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.
We investigate the preparation of a target initial state for a two-level (qubit) system from a system-environment equilibrium or correlated state by an external field. The system-environment equilibrium or correlated state results from the inevitable interaction of the system with its environment. An efficient method in an extended auxiliary Liouville space is introduced to describe the dynamics of the non-Markovian open quantum system in the presence of a strong field and an initial system-environment correlation. By using the time evolutions of the population difference, the state trajectory in the Bloch sphere representation, and the trace distance between two reduced system states of the open quantum system, the effect of initial system-environment correlations on the preparation of a system state is studied. We introduce an upper bound and a lower bound for the trace distance within our perturbation formalism to describe the diverse behaviors of the dynamics of the trace distance between various correlated states after the system state preparation. These bounds, that are much more computable than similar bounds in the literature, give a sufficient condition and a necessary condition for the increase of the trace distance and are related to the witnesses of non-Markovianity and initial system-bath correlation.
We investigate the conditions under which the trace distance between two different states of a given open system increases in time due to the interaction with an environment, therefore signaling non-Markovianity. We find that the finite-time difference in trace distance is bounded by two sharply defined quantities that are strictly linked to the occurrence of system-environment correlations created throughout their interaction and affecting the subsequent evolution of the system. This allows us to shed light on the origin of non-Markovian behaviors in quantum dynamics. We best illustrate our findings by tackling two physically relevant examples: a non-Markovian dephasing mechanism that has been the focus of a recent experimental endeavor and the open-system dynamics experienced by a spin connected to a finite-size quantum spin chain.
The exact reduced dynamics for two independent qudits dephasingly coupled to a common zero-temperature, super-Ohmic, bosonic environment is studied. It is shown that the qudit-qudit subsystem initially prepared in a separable state evolves into an entangled state. Details of the entanglement origination depend on both the coupling strength and the spectral properties of the environment as well as on the dimension of the qudit; the environment-induced entanglement can survive in the long-time limit. The entanglement dynamics of initially maximally entangled qudit-qudit states is analyzed for two examples: the qubit-qubit and qutrit-qutrit systems. The theory is applied to negative partial transposition entangled states of a two-qutrit system, which are shown to be distillable (for the assumed initial state) for arbitrary environment parameters.
We determine the total state dynamics of a dephasing open quantum system using the standard environment of harmonic oscillators. Of particular interest are random unitary approaches to the same reduced dynamics and system–environment correlations in the full model. Concentrating on a model with an at times negative dephasing rate, the issue of 'non-Markovianity' will also be addressed. Crucially, given the quantum environment, the appearance of non-Markovian dynamics turns out to be accompanied by a loss of system–environment correlations. Depending on the initial purity of the qubit state, these system–environment correlations may be purely classical over the whole relevant time scale, or there may be intervals of genuine system–environment entanglement. In the latter case, we see no obvious relation between the build-up or decay of these quantum correlations and 'non-Markovianity'.