When different entanglement witnesses detect the same entangled states

Journal Article · Mon Nov 15 04:00:00 EST 2010 · Physical Review. A

DOI:https://doi.org/10.1103/PHYSREVA.82.052301· OSTI ID:21528580

Jinchuan, Hou; Yu, Guo ^[1]

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1. Department of Mathematics, Shanxi University, Taiyuan 030006, P. R. China and Department of Mathematics, Taiyuan University of Technology, Taiyuan 030024 (China)

The question of under what conditions different witnesses (e.g., W{sub 1},W{sub 2}) may detect some common entangled states [i.e., there exists some state {rho} so that Tr(W{sub 1{rho}})<0 and Tr(W{sub 2{rho}})<0] is answered for both finite-dimensional and infinite-dimensional bipartite systems. Finitely many different witnesses W{sub 1},W{sub 2},...,W{sub n} can detect some common entangled states if and only if {Sigma}{sub i=1}{sup n}d{sub i}W{sub i} is still a witness for any nonnegative numbers d{sub 1},d{sub 2},...,d{sub n} with {Sigma}{sub i=1}{sup n}d{sub i}=1; they cannot detect any common entangled state if and only if {Sigma}{sub i=1}{sup n}c{sub i}W{sub i} is a positive operator for some nonnegative numbers c{sub 1},c{sub 2},...,c{sub n} with {Sigma}{sub i=1}{sup n}c{sub i}=1. For two witnesses W{sub 1} and W{sub 2} more can be said. First, W{sub 1} and W{sub 2} can detect the same set of entangled states if and only if W{sub 1}=aW{sub 2} for some number a>0. Second, W{sub 2} can detect more entangled states than W{sub 1} can if and only if W{sub 1}=aW{sub 2}+D for some number a>0 and a positive operator D. As an application, some characterizations of the optimal witnesses are given and some structural properties of the decomposable optimal witnesses are presented.

• OSTI ID: 21528580
• Journal Information: Physical Review. A, Journal Name: Physical Review. A Journal Issue: 5 Vol. 82; ISSN 1050-2947; ISSN PLRAAN
• Country of Publication: United States
• Language: English

Related Subjects

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GENERAL PHYSICS
DETECTION
MATHEMATICAL OPERATORS
QUANTUM ENTANGLEMENT
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