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Acta Sinica Quantum Optica

Relative Entropy of Entanglement for a Two-parameter Class ofStates in 2n Quantum System

Di Yao-Min

Published Jan 1, 2008• 0 citations
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Abstract

Any arbitrary state in bipartite quantum system can be transformed into a two-parameter class of states by means of local quantum operation and classical communication.In this paper,the quantum relative entropy of entanglement for this kind of states is calculated,the nearest separable states are found and the analytic expression of the entanglement is obtained.The comparation with two other measures of entanglement,the negativity and the entanglement of formation,is also made.The sequence of three measures from high to low is the negativity,the entanglement of formation and the relative entropy of entanglement.

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Any arbitrary state in bipartite quantum system can be transformed into a two-parameter class of states by means of local quantum operation and classical communication.In this paper,the quantum relative entropy of entanglement for this kind of states is calculated,the nearest separable states are found and the analytic expression of the entanglement is obtained.The comparation with two other measures of entanglement,the negativity and the entanglement of formation,is also made.The sequence of three measures from high to low is the negativity,the entanglement of formation and the relative entropy of entanglement.
Entanglement for a three-parameter class of states in bipartite quantum system is investigated.The concurrence for these states in 2n quantum system is calculated and their lower bound of the entanglement of formation is discussed.The negativity for these states in mn(2≤m≤n) quantum system is calculated and the analytic expressions are obtained.Investigation reveals that for θ=π/4,entanglement of the states reaches its maximum,and the states come down to the two-parameter class of states researched by some authors.For these three-parameter states in 2n quantum system,all PPT states are separable,and all NPT states are entangled.But for higher dimensional three-parameter states,the similar conclusion does not exist.
We exhibit a two-parameter class of states ρ (α,γ) in a 2 ⊗ n quantum system for n ⩾ 3, which can be obtained from an arbitrary state by means of local quantum operations and classical communication, and which are invariant under all bilateral operations on a 2 ⊗ n quantum system. We calculate the negativity of ρ (α,γ) , and a lower bound and a tight upper bound on its entanglement of formation. It follows from this calculation that the entanglement of formation of ρ (α,γ) cannot exceed its negativity.
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.