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arXiv:0711.3477 · The European Physical Journal Special Topics

Gaussian entanglement of symmetric two-mode Gaussian states

Paulina Marian, Tudor A. Marian

Published Jun 17, 2008• 9 citations
arXivHTMLPDF
DOI
10.1140/epjst/e2008-00731-x

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We evaluate a Gaussian entanglement measure for a symmetric two-mode Gaussian state of the quantum electromagnetic field in terms of its Bures distance to the set of all separable Gaussian states. The required minimization procedure was considerably simplified by using the remarkable properties of the Uhlmann fidelity as well as the standard form II of the covariance matrix of a symmetric state. Our result for the Gaussian degree of entanglement measured by the Bures distance depends only on the smallest symplectic eigenvalue of the covariance matrix of the partially transposed density operator. It is thus consistent to the exact expression of the entanglement of formation for symmetric two-mode Gaussian states. This non-trivial agreement is specific to the Bures metric.
The entanglement of multi-mode Gaussian state in continuous variable system is investigated. In recent studies, the determination in entanglement of a general two-mode Gaussian state depend on purity measurement, and some studies have already extended it to the case of quasi-symmetry multi-mode Gaussian state. In this paper, entanglement is quantified by logarithmic negativity. The study focuses on the entanglement of quasi-symmetric eight-mode Gaussian state. Our aim is to find the analytic description of the correlation between the quasi-symmetric Gaussian state and two-mode Gaussian state. The results show that: the correlation is not unique. A two-mode Gaussian state may correspond to many different multi-mode Gaussian states, and these multi-mode Gaussian states have the same entanglement information.