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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Squashed entanglement ： ウィキペディア英語版 Squashed entanglement Squashed entanglement, also called CMI entanglement (CMI can be pronounced "see me"), is an information theoretic measure of quantum entanglement for a bipartite quantum system. If $\backslash varrho\_$ is the density matrix of a system $(A,B)$ composed of two subsystems $A$ and $B$, then the CMI entanglement $E\_$ of system $(A,B)$ is defined by where $K$ is the set of all density matrices $\backslash varrho\_$ for a tripartite system $(A,B,\backslash Lambda)$ such that $\backslash varrho\_=tr\_\backslash Lambda\; (\backslash varrho\_)$. Thus, CMI entanglement is defined as an extremum of a functional $S(A:B\; |\; \backslash Lambda)$ of $\backslash varrho\_$. We define $S(A:B\; |\; \backslash Lambda)$, the quantum Conditional Mutual Information (CMI), below. A more general version of Eq.(1) replaces the ``min" (minimum) in Eq.(1) by an ``inf" (infimum). When $\backslash varrho\_$ is a pure state, $E\_(\backslash varrho\_)=S(\backslash varrho\_)=S(\backslash varrho\_)$, in agreement with the definition of entanglement of formation for pure states. Here $S(\backslash varrho)$ is the Von Neumann entropy of density matrix $\backslash varrho$. ==Motivation for definition of CMI entanglement== CMI entanglement has its roots in classical (non-quantum) information theory, as we explain next. Given any two random variables $A,B$, classical information theory defines the mutual information, a measure of correlations, as For three random variables $A,B,\backslash Lambda$, it defines the CMI as It can be shown that $H(A\; :\; B\; |\; \backslash Lambda)\backslash geq\; 0$. Now suppose $\backslash varrho\_$ is the density matrix for a tripartite system $(A,B,\backslash Lambda)$. We will represent the partial trace of $\backslash varrho\_$ with respect to one or two of its subsystems by $\backslash varrho\_$ with the symbol for the traced system erased. For example, $\backslash varrho\_=\; trace\_\backslash Lambda(\backslash varrho\_)$. One can define a quantum analogue of Eq.(2) by and a quantum analogue of Eq.(3) by It can be shown that $S(A\; :\; B\; |\; \backslash Lambda)\backslash geq\; 0$. This inequality is often called the strong-subadditivity property of quantum entropy. Consider three random variables $A,B,\; \backslash Lambda$ with probability distribution $P\_(a,b,\; \backslash lambda)$, which we will abbreviate as $P(a,b,\; \backslash lambda)$. For those special $P(a,\; b,\; \backslash lambda)$ of the form it can be shown that $H(A:\; B\; |\backslash Lambda)=0$. Probability distributions of the form Eq.(6) are in fact described by the Bayesian network shown in Fig.1. One can define a classical CMI entanglement by where $K$ is the set of all probability distributions $P\_$ in three random variables $A,B,\backslash Lambda$, such that $\backslash sum\_\backslash lambda\; P\_(a,\; b,\backslash lambda)=P\_(a,b)\backslash ,$for all $a,b$. Because, given a probability distribution $P\_$, one can always extend it to a probability distribution $P\_$ that satisfies Eq.(6), it follows that the classical CMI entanglement, $E\_(\; P\_)$, is zero for all $P\_$. The fact that $E\_(\; P\_)$ always vanishes is an important motivation for the definition of $E\_(\; \backslash varrho\_)$. We want a measure of quantum entanglement that vanishes in the classical regime. Suppose $w\_\backslash lambda$ for $\backslash lambda=1,2,...,dim(\backslash Lambda)$ is a set of non-negative numbers that add up to one, and $|\backslash lambda\backslash rangle$ for $\backslash lambda=1,2,...,dim(\backslash Lambda)$ is an orthonormal basis for the Hilbert space associated with a quantum system $\backslash Lambda$. Suppose $\backslash varrho\_A^\backslash lambda$ and $\backslash varrho\_B^\backslash lambda$, for $\backslash lambda=1,2,...,dim(\backslash Lambda)$ are density matrices for the systems $A$ and $B$, respectively. It can be shown that the following density matrix satisfies $S(A:\; B\; |\backslash Lambda)=0$. Eq.(8) is the quantum counterpart of Eq.(6). Tracing the density matrix of Eq.(8) over $\backslash Lambda$, we get $\backslash varrho\_\; =\; \backslash sum\_\backslash lambda\; \backslash varrho\_A^\backslash lambda\; \backslash varrho\_B^\backslash lambda\; w\_\backslash lambda\; \backslash ,$, which is a separable state. Therefore, $E\_(\backslash varrho\_)$ given by Eq.(1) vanishes for all separable states. When $\backslash varrho\_$ is a pure state, one gets $E\_(\backslash varrho\_)=S(\backslash varrho\_)=S(\backslash varrho\_)$. This agrees with the definition of entanglement of formation for pure states, as given in Ben96. Next suppose $|\backslash psi\_^\backslash lambda\backslash rangle$ for $\backslash lambda=1,2,...,dim(\backslash Lambda)$ are some states in the Hilbert space associated with a quantum system $(A,B)$. Let $K$ be the set of density matrices defined previously for Eq.(1). Define $K\_o$ to be the set of all density matrices $\backslash varrho\_$ that are elements of $K$ and have the special form $\backslash varrho\_\; =\; \backslash sum\_\backslash lambda|\backslash psi\_^\backslash lambda\backslash rangle\; \backslash langle\; \backslash psi\_^\backslash lambda|\; w\_\backslash lambda\; |\backslash lambda\backslash rangle\backslash langle\backslash lambda|\backslash ,$. It can be shown that if we replace in Eq.(1) the set $K$ by its proper subset $K\_o$, then Eq.(1) reduces to the definition of entanglement of formation for mixed states, as given in Ben96. $K$ and $K\_o$ represent different degrees of knowledge as to how $\backslash varrho\_$ was created. $K$ represents total ignorance. Since CMI entanglement reduces to entanglement of formation if one minimizes over $K\_o$ instead of $K$, one expects that CMI entanglement inherits many desirable properties from entanglement of formation. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Squashed entanglement」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.