Constructing Entanglement Witnesses for Infinite-Dimensional Systems

We generalize the results in Yu and Liu (Phys. Rev. Lett. 95, 150504, 2005) and Hou and Guo (Int. J. Theor. Phys. 50, 1245–1254, 2011) to infinite-dimensional systems and answer a problem raised in the second paper. Consider a bipartite system H ⊗ K with dimH = dimK = ∞. We show that (1) for any orthonormal sequences{Ek}k=1∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{E_{k}\}_{k = 1}^{\infty }$\end{document} and{Fk}k=1∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{F_{k}\}_{k = 1}^{\infty }$\end{document} consist of observables respectively inC2(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {C}_{2}(H)$\end{document} andC2(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {C}_{2}(K)$\end{document}, if∑kEk⊗Fk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\sum }_{k} E_{k} \otimes F_{k}$\end{document} converges under the weak operator topology and ifW=I-∑kEk⊗Fk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$W=I-{\sum }_{k} E_{k}\otimes F_{k}$\end{document} is not positive, then W is a decomposable entanglement witness; (2) every state ρ of system H ⊗ K has a Schmidt decompositionρ=∑kδkEk⊗Fk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\rho = {\sum }_{k} \delta _{k} E_{k} \otimes F_{k}$\end{document} with {Ek} and {Fk} orthonormal sequences of observables.