Coherent information

The '''coherent information''' is a quantity used in [[quantum communication]] theory. It is a property of a [[quantum state]] and a [[quantum channel]] $\backslash Lambda$; intuitively, it attempts to describe how much of the [[quantum information]] in the state will remain after the state goes through the channel. In this sense, it is analogous to the [[classical mutual information]] of [[classical information theory]]. As the classical mutual information gives the capacity of a [[classical channel]], the coherent information gives the [[capacity]] of a [[quantum channel]]. The coherent information is written $I(A\backslash rangle\; B)$. ==Definition== The coherent information is defined as $I(A\backslash rangle\; B)\; \backslash equiv\; S(B)\; -\; S(AB)$ where $S(B)$ is the [[Von Neumann entropy]] of the output state $\backslash rho\_B$ and $S(AB)$ is the entropy of the channel [[environment]] (equivalently, the joint entropy of the output state, and the [[purification]] $\backslash rho\_A$) of the input state. One may think of inputting part of a pure state through the channel. The part which remains at the input is $\backslash rho\_A$, and the channel produces $\backslash rho\_B$ at the output. ==Properties== ==Uses== The coherent information, when maximised over input states $\backslash rho\_A$, gives the [[quantum capacity]] of the [[quantum channel]]. This is known as the [[LSD Theorem]] after Lloyd, Shor and Devetak. The quantity is not additive, and one must go to large block size (i.e. many quantum uses of the channel). The coherent information, gives also the [[one way distillable entanglement]] when maximised over [[local operations]] performed by [[Alice]], and if maximised over [[LOCC]] operations, it gives the [[two-way distillable entanglement]]. ==History== The coherent information was introduced by [[Benjamin Schumacher]] and [[Michael A. Nielsen]] in a 1996 paper ''Quantum data processing and error correction'', which appeared in Physical Review A. ==See Also== * [[quantum conditional entropy]] ==References== * Nielsen, Michael A. and Isaac L. Chuang (2000). ''Quantum Computation and Quantum Information'', Cambridge University Press, ISBN 0-521-63503-9 * Nielsen, Michael A. and Benjamin Schumacher (1996). Quantum data processing and error correction. ''Physical Review A.'', '''54''' (4), 2629-2635. {{stub}} [[Category:Quantum Information Theory]] [[Category:Quantum Communication]] [[Category:Handbook of Quantum Information]]