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 $\backslash varrho\_\{A,\; B\}$ is the [[density matrix]] of a system $(A,B)$ composed of two subsystems $A$ and $B$, then the CMI Entanglement $E\_\{CMI\}$ of system $(A,B)$ is defined by $Eq.(1)\backslash qquad\; E\_\{CMI\}(\backslash varrho\_\{A,\; B\})\; =\; \backslash frac\{1\}\{2\}\backslash min\_\{\backslash varrho\_\{A,B,\backslash Lambda\}\backslash in\; K\}S(A:B\; |\; \backslash Lambda)$, where $K$ is the set of all density matrices $\backslash varrho\_\{A,\; B,\; \backslash Lambda\}$ for a tripartite system $(A,B,\backslash Lambda)$ such that $\backslash varrho\_\{A,\; B\}=tr\_\backslash Lambda\; (\backslash varrho\_\{A,\; B,\; \backslash Lambda\})$. Thus, CMI Entanglement is defined as an extremum of a functional $S(A:B\; |\; \backslash Lambda)$ of $\backslash varrho\_\{A,\; B,\; \backslash Lambda\}$. 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 $\backslash varrho\_\{A,\; B\}$ is a pure state, $E\_\{CMI\}(\backslash varrho\_\{A,\; B\})=S(\backslash varrho\_\{A\})=S(\backslash varrho\_\{B\})$, in agreement with the definition of [[Entanglement of Formation]] for pure states. ==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 | Mutual Information]], a measure of correlations, as $Eq.(2)\backslash qquad\; H(A:B)\; =H(A)\; +\; H(B)\; -\; H(A,\; B)\backslash ,$. For three random variables $A,B,C$, it defines the CMI as . It can be shown that $H(A\; :\; B\; |\; \backslash Lambda)\backslash geq\; 0$. Now suppose $\backslash varrho\_\{A,\; B,\; \backslash Lambda\}$ is the density matrix for a tripartite system $(A,B,\backslash Lambda)$. We will represent the partial trace of $\backslash varrho\_\{A,\; B,\; \backslash Lambda\}$ with respect to one or two of its subsystems by $\backslash varrho\_\{A,\; B,\; \backslash Lambda\}$ with the symbol for the traced system erased. For example, $\backslash varrho\_\{A,\; B\}=\; trace\_\backslash Lambda(\backslash varrho\_\{A,\; B,\; \backslash Lambda\})$. One can define a quantum analogue of Eq.(2) by $Eq.(4)\backslash qquad\; S(A:B)\; =\; S(\backslash varrho\_\{A\})\; +\; S(\backslash varrho\_\{B\})\; -S(\backslash varrho\_\{A,\; B\})\; \backslash ,$, and a quantum analogue of Eq.(3) by $Eq.(5)\backslash qquad\; S(A:B|\backslash Lambda)\; =\; S(\backslash varrho\_\{A,\; \backslash Lambda\})\; +S(\backslash varrho\_\{B,\; \backslash Lambda\})\; -S(\backslash varrho\_\backslash Lambda)\; -S(\backslash varrho\_\{A,\; B,\; \backslash Lambda\})\; \backslash ,$. It can be shown that $S(A\; :\; B\; |\; \backslash Lambda)\backslash geq\; 0$. This inequality is often called the [[strong sub-additivity]] property of quantum entropy. Consider three random variables $A,B,\; \backslash Lambda$ with probability distribution $P\_\{A,B,\backslash Lambda\}(a,b,\; \backslash lambda)$, which we will abbreviate as $P(a,b,\; \backslash lambda)$. For those special $P(a,\; b,\; \backslash lambda)$ of the form $Eq.(6)\backslash qquad\; P(a,\; b,\; \backslash lambda)\; =\; P(a|\; \backslash lambda)\; P(b\; |\; \backslash lambda)\; P(\backslash lambda)\backslash ,$, 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. [[Image:Bnet_fan2.png|frame|Fig.1: Bayesian Network representation of Eq.(6)]] One can define a classical CMI Entanglement by $Eq.(7)\backslash qquad\; E\_\{CMI\}(\; P\_\{A,B\})\; =\; \backslash min\_\{P\_\{A,\; B,\; \backslash Lambda\}\backslash in\; K\}\; H(A:\; B\; |\backslash Lambda)$, where $K$ is the set of all probability distributions $P\_\{A,B,\backslash Lambda\}$ in three random variables $A,B,\backslash Lambda$, such that $\backslash sum\_\backslash lambda\; P\_\{A,B,\backslash Lambda\}(a,\; b,\backslash lambda)=P\_\{A,B\}(a,b)\backslash ,$for all $a,b$. Because, given a probability distribution $P\_\{A,B\}$, one can always extend it to a probability distribution $P\_\{A,B,\; \backslash Lambda\}$ that satisfies Eq.(6), it follows that the classical CMI Entanglement, $E\_\{CMI\}(\; P\_\{A,B\})$, is zero for all $P\_\{A,B\}$. The fact that $E\_\{CMI\}(\; P\_\{A,B\})$ always vanishes is an important motivation for the definition of $E\_\{CMI\}(\; \backslash varrho\_\{A,B\})$. 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 $|\backslash lambda>$ 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 $\backslash varrho\_\{A,B\}\; =\; \backslash sum\_\backslash lambda\; \backslash varrho\_A^\backslash lambda\; \backslash varrho\_B^\backslash lambda\; w\_\backslash lambda\; \backslash ,$, which is a [[Separable_and_entangled_states |separable state]]. Therefore, $E\_\{CMI\}(\backslash varrho\_\{A,\; B\})$ given by Eq.(1) vanishes for all separable states. When $\backslash varrho\_\{A,\; B\}$ is a pure state, one gets $E\_\{CMI\}(\backslash varrho\_\{A,\; B\})=S(\backslash varrho\_\{A\})=S(\backslash varrho\_\{B\})$. This agrees with the definition of [[Entanglement of Formation]] for pure states, as given in '''Ben96'''. Next suppose $|\backslash psi\_\{A,B\}^\backslash lambda>$ 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 $\backslash varrho\_\{A,\; B,\; \backslash Lambda\}$ that are elements of $K$ and have the special form . 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 $\backslash varrho\_\{A,\; B,\; \backslash Lambda\}$ was created. $K$ represents total ignorance. ==History== Classical CMI, given by Eq.(3), first entered Information Theory lore, shortly after Shannon's seminal 1948 paper and at least as early as 1954 in '''McG54'''. The quantum CMI, given by Eq.(5), was first defined by Cerf and Adami in '''Cer96'''. However, it appears that Cerf and Adami did not realize the relation of CMI to entanglement or the possibility of obtaining a measure of quantum entanglement based on CMI; this can be inferred, for example, from a later paper, '''Cer97''', where they try to use $S(A|B)$ instead of CMI to understand entanglement. The first paper to explicitly point out a connection between CMI and quantum entanglement appears to be '''Tuc99'''. The final definition Eq.(1) of CMI entanglement was first given by Tucci in a series of 6 papers. (See, for example, Eq.(8) of '''Tuc02''' and Eq.(42) of '''Tuc01a'''). In '''Tuc00b''', he pointed out the classical probability motivation of Eq.(1), and its connection to the definitions of Entanglement of Formation for pure and mixed states. In '''Tuc01a''', he presented an algorithm and computer program, based on the [[Rate distortion theory|Arimoto-Blahut method]] of information theory, for calculating CMI entanglement numerically. In '''Tuc01b''', he calculated CMI entanglement analytically, for a mixed state of two [[qubits]]. In '''Hay03''', Hayden, Jozsa, Petz and Winter explored the connection between quantum CMI and [[Separable_and_entangled_states|separability]]. 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. As first shown in '''Ben96''', Entanglement of Formation does not increase under [[LOCC]] (Local Operations and Classical Communication). In '''Chr03''', Christandl and Winter showed that CMI Entanglement also does not increase under LOCC, by adapting '''Ben96''' arguments about Entanglement of Formation. In '''Chr03''', they also proved many other interesting inequalities concerning CMI Entanglement, and explored its connection to other measures of entanglement. The name '''Squashed Entanglement''' first appeared in '''Chr03'''. In '''Chr05''', Christandl and Winter calculated analytically the CMI Entanglement of some interesting states. In '''Ali03''', Alicki and Fannes proved the continuity of CMI Entanglement. ==References== *'''Ali03''' R. Alicki, M. Fannes, ``Continuity of quantum mutual information", [http://arxiv.org/abs/quant-ph/0312081 quant-ph/0312081] *'''Ben96''' C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, W.K. Wootters, ``Mixed State Entanglement and Quantum Error Correction", [http://arxiv.org/abs/quant-ph/quant-ph/9604024 quant-ph/9604024] *'''Cer96''' N. J. Cerf, C. Adami, ``Quantum Mechanics of Measurement", [http://arxiv.org/abs/quant-ph/9605002 quant-ph/9605002] *'''Cer97''' N.J. Cerf, C. Adami, R.M. Gingrich, ``Quantum conditional operator and a criterion for separability", [http://arxiv.org/abs/quant-ph/9710001 quant-ph/9710001] *'''Chr03''' M. Christandl, A. Winter, ``Squashed Entanglement - An Additive Entanglement Measure", [http://arxiv.org/abs/quant-ph/0308088 quant-ph/0308088] *'''Chr05''' M. Christandl, A. Winter, ``Uncertainty, Monogamy, and Locking of Quantum Correlations", [http://arxiv.org/abs/quant-ph/0501090 quant-ph/0501090] *'''Chr06''' M. Christandl, Ph.D. Thesis, [http://arxiv.org/abs/quant-ph/0604183 quant-ph/0604183] *'''Hay03''' P. Hayden, R. Jozsa, D. Petz, A. Winter, ``Structure of states which satisfy strong subadditivity of quantum entropy with equality" [http://arxiv.org/abs/quant-ph/0304007 quant-ph/0304007] *'''McG54''' W.J. McGill, ``Multivariate Information Transmission", IRE Trans. Info. Theory '''4'''(1954) 93-111. *'''Tuc99''' R.R. Tucci, ``Quantum Entanglement and Conditional Information Transmission", [http://arxiv.org/abs/quant-ph/9909041 quant-ph/9909041] *'''Tuc00a''' R.R. Tucci,``Separability of Density Matrices and Conditional Information Transmission", [http://arxiv.org/abs/quant-ph/0005119 quant-ph/0005119] *'''Tuc00b''' R.R. Tucci, ``Entanglement of Formation and Conditional Information Transmission", [http://arxiv.org/abs/quant-ph/0010041 quant-ph/0010041] *'''Tuc01a''' R.R. Tucci, ``Relaxation Method For Calculating Quantum Entanglement", [http://arxiv.org/abs/quant-ph/0101123 quant-ph/0101123] *'''Tuc01b''' R.R. Tucci, ``Entanglement of Bell Mixtures of Two Qubits", [http://arxiv.org/abs/quant-ph/0103040 quant-ph/0103040] *'''Tuc02''' R.R. Tucci, ``Entanglement of Distillation and Conditional Mutual Information", [http://arxiv.org/abs/quant-ph/0202144 quant-ph/0202144] [[Category:quantum Information Theory]]