Some Characterizations for Partial Classical Correlation States in Multipartite Quantum Systems

One crucial problem is to characterize and quantify the correlations in bipartite or multipartite states in the theory of quantum information. In this paper, we consider m-partite composite quantum systems H=H1⊗H2⊗⋯⊗Hm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=H_{1}\otimes H_{2}\otimes \cdots \otimes H_{m}$$\end{document} with dimH<+∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dim H<+\infty $$\end{document}. We first introduce a concept of k-classical correlation states (1≤k≤m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le k\le m$$\end{document})(for short single classical correlation states when k=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=1$$\end{document}, and fully classical correlation states when k=m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=m$$\end{document}); Next we give some mathematical representation for partial classical correlation states and a necessary condition for fully-classical correlation states; Finally, we present a correlation measure for k-classical correlation states based on the angle between quantum states, and prove that this correlation measure is well-defined, which possesses the basic physical properties of correlation measure.