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arXiv:quant-ph/0602031 · Physical Review A

Connecting the generalized robustness and the geometric measure of entanglement

Daniel Cavalcanti

Published Apr 20, 2006• 44 citations
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Abstract

The main goal of this paper is to provide a connection between the generalized robustness of entanglement $({R}_{g})$ and the geometric measure of entanglement $({E}_{GME})$. First, we show that the generalized robustness is always higher than or equal to the geometric measure. Then we find a tighter lower bound to ${R}_{g}(\ensuremath{\rho})$ based only on the purity of $\ensuremath{\rho}$ and its maximal overlap to a separable state. As we will see it is also possible to express this lower bound in terms of ${E}_{GME}$.

DOI
10.1103/physreva.73.044302

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The main goal of this paper is to provide a connection between the generalized robustness of entanglement (Rg)({R}_{g})(Rg​) and the geometric measure of entanglement (EGME)({E}_{GME})(EGME​). First, we show that the generalized robustness is always higher than or equal to the geometric measure. Then we find a tighter lower bound to Rg(\ensuremathρ){R}_{g}(\ensuremath{\rho})Rg​(\ensuremathρ) based only on the purity of \ensuremathρ\ensuremath{\rho}\ensuremathρ and its maximal overlap to a separable state. As we will see it is also possible to express this lower bound in terms of EGME{E}_{GME}EGME​.
In the quest to completely describe entanglement in the general case of a finite number of parties sharing a physical system of finite-dimensional Hilbert space an entanglement magnitude is introduced for its pure and mixed states: robustness. It corresponds to the minimal amount of mixing with locally prepared states which washes out all entanglement. It quantifies in a sense the endurance of entanglement against noise and jamming. Its properties are studied comprehensively. Analytical expressions for the robustness are given for pure states of two-party systems, and analytical bounds for mixed states of two-party systems. Specific results are obtained mainly for the qubit-qubit system (qubit denotes quantum bit). As by-products local pseudomixtures are generalized, a lower bound for the relative volume of separable states is deduced, and arguments for considering convexity a necessary condition of any entanglement measure are put forward.
As two of the most important entanglement measures--the entanglement of formation and the entanglement of distillation--have so for been limited to bipartite settings, the study of other entanglement measures for multipartite systems appears necessary. Here, connections between two other entanglement measures--the relative entropy of entanglement and the geometric measure of entanglement--are investigated. It is found that for arbitrary pure states the latter gives rise to a lower bound on the former. For certain pure states, some bipartite and some multipartite, this lower bound is saturated, and thus their relative entropy of entanglement can be found analytically in terms ot their known geometric measure of entanglement. For certain mixed states, upper bounds on the relative entropy of entanglement are also established. Numerical evidence strongly suggests that these upper bounds are tight, i.e., they are actually the relative entropy of entanglement.
As two of the most important entanglement measures--the entanglement of formation and the entanglement of distillation--have so far been limited to bipartite settings, the study of other entanglement measures for multipartite systems appears necessary. Here, connections between two other entanglement measures--the relative entropy of entanglement and the geometric measure of entanglement--are investigated. It is found that for arbitrary pure states the latter gives rise to a lower bound on the former. For certain pure states, some bipartite and some multipartite, this lower bound is saturated, and thus their relative entropy of entanglement can be found analytically in terms of their known geometric measure of entanglement. For certain mixed states, upper bounds on the relative entropy of entanglement are also established. Numerical evidence strongly suggests that these upper bounds are tight, i.e., they are actually the relative entropy of entanglement.
As two of the most important entanglement measures---the entanglement of formation and the entanglement of distillation---have so far been limited to bipartite settings, the study of other entanglement measures for multipartite systems appears necessary. Here, connections between two other entanglement measures---the relative entropy of entanglement and the geometric measure of entanglement---are investigated. It is found that for arbitrary pure states the latter gives rise to a lower bound on the former. For certain pure states, some bipartite and some multipartite, this lower bound is saturated, and thus their relative entropy of entanglement can be found analytically in terms of their known geometric measure of entanglement. For certain mixed states, upper bounds on the relative entropy of entanglement are also established. Numerical evidence strongly suggests that these upper bounds are tight, i.e., they are actually the relative entropy of entanglement.
One of the biggest problems in the theory of quantum information is the quantification of amount of entanglement in an arbitrary multipartite mixed state. Different axiomatic and operational measures were proposed so far. In this work we will establish a connection between geometric measure of entanglement proposed in [Phys. Rev. A 68, 042307 (2003)] and measures based on fidelity. One result will be, that geometric and revised geometric measure of entanglement proposed in [J. Phys. A: Math. Theor. 40, 3507 (2007)] are equal. Further a useful expression for fidelity is derived.