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 HA and HB be complex Hilbert spaces and {Ek} and {Fk} be sequences of self-adjoint Hilbert-Schmidt operators on HA and HB, respectively, such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathrm{Tr}(E^{\dag}_{k}E_{l})=\mathrm{Tr}(F^{\dag}_{k}F_{l})=\delta_{kl}$\end{document}. Then W=I−∑kEk⊗Fk is an entanglement witness on HA⊗HB if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$W\not\geq 0$\end{document}. If ρ is an entangled state and τ0 is the nearest separable state to ρ under the Hilbert-Schmidt norm, then W=c0I+τ0−ρ with c0=Tr[τ0(ρ−τ0)] is an entanglement witness.