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arXiv:quant-ph/0602137 · Physical Review A

R function related to entanglement of formation

Shao-Ming Fei, Xianqing Li‐Jost

Published Feb 10, 2006• 17 citations
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Abstract

By investigating the convex property of the function $R$, 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.

DOI
10.1103/physreva.73.024302

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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.
We give an explicit expression for the entanglement of formation for isotropic density matrices in arbitrary dimensions in terms of the convex hull of a simple function. For two qutrit isotropic states we determine the convex hull and we have strong evidence for its exact form for arbitrary dimension. Unlike for two qubits, the entanglement of formation for two qutrits or more is found to be a nonanalytic function of the maximally entangled fraction in the regime where the density matrix is entangled.
We study the entanglement R\'enyi $\ensuremath{\alpha}$ entropy $(\mathrm{ER}\ensuremath{\alpha}\mathrm{E})$ as the measure of entanglement. Instead of a single quantity in standard entanglement quantification for a quantum state by using the von Neumann entropy for the well-accepted entanglement of formation (EoF), the $\mathrm{ER}\ensuremath{\alpha}\mathrm{E}$ gives a continuous spectrum parametrized by variable $\ensuremath{\alpha}$ as the entanglement measure, and it reduces to the standard EoF in the special case $\ensuremath{\alpha}\ensuremath{\rightarrow}1$. The $\mathrm{ER}\ensuremath{\alpha}\mathrm{E}$ provides more information in entanglement quantification and can be used, for example, to determine the convertibility of entangled states by local operations and classical communication. A series of results is obtained: (i) we show that the $\mathrm{ER}\ensuremath{\alpha}\mathrm{E}$ of two states, which can be mixed or pure, may be incomparable, in contrast to the fact that there always exists an order for the EoF of two states; (ii) similar to the case of EoF, we study in a fully analytical way the $\mathrm{ER}\ensuremath{\alpha}\mathrm{E}$ for arbitrary two-qubit states, the Werner states, and isotropic states in the general ddd dimension; and (iii) we provide a proof of the previous conjecture for the analytical functional form of the EoF of isotropic states in the arbitrary ddd dimension.
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)].