When different entanglement witnesses detect the same entangled states
- 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, Vol. 82, Issue 5; Other Information: DOI: 10.1103/PhysRevA.82.052301; (c) 2010 The American Physical Society; ISSN 1050-2947
- Country of Publication:
- United States
- Language:
- English
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