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arXiv:1105.6321 · Physical Review A

Entanglement of formation for a family of (2⊗d) -dimensional systems

F. Lastra, C. E. López, Luis Roa, J. C. Retamal

Published Feb 15, 2012• 18 citations
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Abstract

Entanglement of formation for arbitrary mixed states in a ($2\ensuremath{\bigotimes}2$)-dimensional Hilbert space is well known. For higher-dimensional mixed states analytical expressions for quantifying entanglement exist for a particular family of states. In this work, we calculate analytically the entanglement of formation for a family of bipartite ($2\ensuremath{\bigotimes}d$)-dimensional mixed states which can be obtained from tripartite $2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}d$ pure states following the entanglement relations obtained by Koashi and Winter [Phys. Rev. A 69, 022309 (2004)].

DOI
10.1103/physreva.85.022320

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Entanglement of formation for arbitrary mixed states in a (2\ensuremath⨂22\ensuremath{\bigotimes}22\ensuremath⨂2)-dimensional Hilbert space is well known. For higher-dimensional mixed states analytical expressions for quantifying entanglement exist for a particular family of states. In this work, we calculate analytically the entanglement of formation for a family of bipartite (2\ensuremath⨂d2\ensuremath{\bigotimes}d2\ensuremath⨂d)-dimensional mixed states which can be obtained from tripartite 2\ensuremath⨂2\ensuremath⨂d2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}d2\ensuremath⨂2\ensuremath⨂d pure states following the entanglement relations obtained by Koashi and Winter [Phys. Rev. A 69, 022309 (2004)].
By investigating the convex property of the function RRR, which appeared when computing the entanglement of formation for isotropic states [Phys. Rev. Lett. 85, 2625 (2000)] and in a tight lower bound of entanglement of formation for arbitrary bipartite mixed states [Phys. Rev. Lett. 95, 210501 (2005)], we show analytically that the results in these papers are valid not only for dimensions two and three but for any dimensions.
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Entanglement of formation quantifies the entanglement of a state in terms of the entropy of entanglement of the least entangled pure state needed to prepare it. An analytical expression for this measure exists only for special cases, and finding a closed formula for an arbitrary state remains an open problem. In this work we focus on two-mode Gaussian states, and we derive narrow upper and lower bounds for the measure that get tight for several special cases. Further, we show that the problem of calculating the actual value of the entanglement of formation for arbitrary two-mode Gaussian states reduces to a trivial single parameter optimization process, and we provide an efficient algorithm for the numerical calculation of the measure.