Constructing Entanglement Witnesses for States in Infinite-Dimensional Bipartite Quantum Systems

In this paper, two approaches of constructing entanglement witnesses for finite- or infinite-dimensional bipartite quantum systems are presented. Let H (A) and H (B) be complex Hilbert spaces and {E (k) } and {F (k) } be sequences of self-adjoint Hilbert-Schmidt operators on H (A) and H (B) , respectively, such that Tr(E-k(+) F-tau) = delta(k tau). Then W = I - Sigma(k) E-k circle times F-k is an entanglement witness on H-A circle times H-B if W not greater than or equal to 0. If rho is an entangled state and tau (0) is the nearest separable state to rho under the Hilbert-Schmidt norm, then W=c (0) I+tau (0)-rho with c (0)=Tr[tau (0)(rho-tau (0))] is an entanglement witness.