Quantum mutual information

The '''quantum mutual information''' of a [[bipartite]] state $\backslash rho\_\{AB\}$ is defined as :$I\backslash left(A:B\backslash right):=S\backslash left(A\backslash right)+S\backslash left(B\backslash right)-S\backslash left(AB\backslash right)$ where $S(A)$ and $S(B)$ are the [[Von Neumann entropy|Von Neumann entropies]] of $\backslash rho\_A$ and $\backslash rho\_B$, obtained by [[tracing out]] system "B" and "A" respectively from $\backslash rho\_\{AB\}$. $S(AB)$ is the Von Neumann entropy of the total state. The quantity is formally equivalent to the [[classical mutual information]] with the [[Shannon entropy]] changed to its quantum counterpart. ==Properties== The quantum mutual information is always non-negative $I\backslash big(A:B)\; \backslash geq\; 0$. It is monotonic under the action of quantum channels (CPTP maps), that is $I\backslash big(A:\backslash Lambda\; B\backslash big)\; \backslash leq\; I(A:B)$. As the [[partial trace]] is a CPTP map, this means that $I\backslash big(A:\; B\backslash big)\; \backslash leq\; I(A:BC)$ for any tripartite state $\backslash rho\_\{ABC\}$. ==Uses== When maximized over input states, it gives the [[entanglement assisted classical capacity]] of a memoryless quantum channel. That is, :$C\_E\backslash big(\backslash Lambda\backslash big)\; =\; \backslash max\_\{\backslash rho\_\{RQ\}\}\; I(R:\backslash Lambda\; Q)$ where $I\backslash big(R:\backslash Lambda\; Q\backslash big)$ is the quantum mutual information of the state $\backslash big(\backslash mathbb\{I\}\_R\; \backslash otimes\; \backslash Lambda\_Q\backslash big)\backslash rho\_\{RQ\}$. When the maximization of the state $\backslash rho\_\{RQ\}$ is taken over all separable states, then the maximized quantum mutual information is equivalent to the Holevo capacity, and thus :$C\backslash big(\backslash Lambda\backslash big)\; =\; \backslash lim\_\{n\backslash rightarrow\; \backslash infty\}\; \backslash max\_\{\backslash rho\_\{RQ\}\; \backslash in\; \backslash mathcal\{D\}\}\; \backslash frac\{1\}\{n\}I(R:\backslash Lambda^\{\backslash otimes\; n\}\; Q)$ for $\backslash mathcal\{D\}$ the set of all separable states between systems $R$ and $Q$. Whether or not the classical capacity of a memoryless channel can be expressed in the unregularized form $C(\backslash Lambda)\; =\; \backslash max\_\{\backslash rho\_\{RQ\}\; \backslash in\; \backslash mathcal\{D\}\}\; I(R:\backslash Lambda\; Q)$ is the [[additivity]] problem. {{stub}} [[Category:Quantum Information Theory]] [[Category:Quantum Communication]] [[Category:Handbook of Quantum Information]]