Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0308031 · Physical Review A

Measures of entanglement in multipartite bound entangled states

Tzu-Chieh Wei, Joseph B. Altepeter, Paul M. Goldbart, William J. Munro

Published Aug 27, 2004• 39 citations
arXivHTMLPDF

Abstract

Bound entangled states are states that are entangled but from which no entanglement can be distilled if all parties are allowed only local operations and classical communication. However, in creating these states one needs nonzero entanglement resources to start with. Here, the entanglement of two distinct multipartite bound entangled states is determined analytically in terms of a geometric measure of entanglement and a related quantity. The results are compared with those for the negativity and the relative entropy of entanglement.

DOI
10.1103/physreva.70.022322

Related papers

Connections between relative entropy of entanglement and geometric measure of entanglement

Tzu-Chieh Wei, Marie Ericsson, Paul M. Goldbart +1 more

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.

Quantum Information and Computation•Jul 1, 2004•58 citations

Connections between relative entropy of entanglement and geometric measure of entanglement

Tzu-Chieh Wei, Marie Ericsson, Paul M. Goldbart +1 more

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.

arXiv:quant-ph/0405002•Apr 30, 2004•0 citations

Connections between relative entropy of entanglement and geometric measure of entanglement

Tzu-Chieh Wei, Marie Ericsson, Paul M. Goldbart +1 more

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.

Quantum Information and Computation•Jul 1, 2004•44 citations

Entanglement bound for multipartite pure states based on local measurements

Li-zhen Jiang, Xiaoyu Chen, Tian-Yu Ye

An entanglement bound based on local measurements is introduced for multipartite pure states. It is the upper bound of the geometric measure and the relative entropy of entanglement. It is the lower bound of the minimal-measurement entropy. For pure bipartite states, the bound is equal to the entanglement entropy. The bound is applied to pure tripartite qubit states and the exact tripartite relative entropy of entanglement is obtained for a wide class of states.

arXiv:1104.3280•Physical Review A•Oct 5, 2011•2 citations

Relative Entropy of Entanglement and Restricted Measurements

Marco Piani

We introduce variants of relative entropy of entanglement based on the optimal distinguishability from unentangled states by means of restricted measurements. In this way we are able to prove that the standard regularized entropy of entanglement is strictly positive for all multipartite entangled states. This implies that the asymptotic creation of a multipartite entangled state by means of local operations and classical communication always requires the consumption of a nonlocal resource at a strictly positive rate.

arXiv:0904.2705•Physical Review Letters•Oct 14, 2009•84 citations

Application of the geometric measure of entanglement to three-qubit mixed states

Tzu-Chieh Wei, Paul M. Goldbart

The geometric measure of entanglement, originated by Shimony and by Barnum and Linden, is determined for a family of tripartite mixed states consisting of aribitrary mixtures of GHZ, W, and inverted-W states. For this family of states, other measures of entanglement, such as entanglement of formation and relative entropy of entanglement, have not been determined. The results for the geometric measure of entanglement are compared with those for negativity, which are also determined. The results for the geometric measure of entanglement and the negativity provide examples of the determination of entanglement for nontrivial mixed multipartite states.

arXiv:quant-ph/0303158•Mar 26, 2003•0 citations
Bound entangled states are states that are entangled but from which no entanglement can be distilled if all parties are allowed only local operations and classical communication. However, in creating these states one needs nonzero entanglement resources to start with. Here, the entanglement of two distinct multipartite bound entangled states is determined analytically in terms of a geometric measure of entanglement and a related quantity. The results are compared with those for the negativity and the relative entropy of entanglement.
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.
We introduce variants of relative entropy of entanglement based on the optimal distinguishability from unentangled states by means of restricted measurements. In this way we are able to prove that the standard regularized entropy of entanglement is strictly positive for all multipartite entangled states. This implies that the asymptotic creation of a multipartite entangled state by means of local operations and classical communication always requires the consumption of a nonlocal resource at a strictly positive rate.
The geometric measure of entanglement, originated by Shimony and by Barnum and Linden, is determined for a family of tripartite mixed states consisting of aribitrary mixtures of GHZ, W, and inverted-W states. For this family of states, other measures of entanglement, such as entanglement of formation and relative entropy of entanglement, have not been determined. The results for the geometric measure of entanglement are compared with those for negativity, which are also determined. The results for the geometric measure of entanglement and the negativity provide examples of the determination of entanglement for nontrivial mixed multipartite states.