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International Journal of Theoretical Physics

Distance Between Qubit States with Initial System-Environment Correlation

Masashi Ban, Sachiko Kitajima, Fumiaki Shibata

Published Mar 8, 2012• 9 citations
Publisher
DOI
10.1007/s10773-012-1121-y

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Correlations in a multi-qubit state

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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 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.