extremal quantum channel

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The set of all quantum channels on $\mathcal{M}_{d}$ is convex and compact meaning it may be decomposed as

$T=\sum_{i}p_{i}T_{i}$

where the $p$‘s are probabilities and the $T_{i}$’s are $extremal$ unital channels, that is channels that may not be further decomposed.

Channels with a single Kraus operator are pure channels and the extremal points in the convex set of channels are precisely the pure channels. Here $T$ represents the set of $all$ channels on the particular space, not necessarily copies of the same one, i.e. the $T_{i}$ may not represent the same channel.

$T$, with Kraus operators $\{A\}_i$, is extremal if and only if the set

$\left \{A_{k}^{\dagger}A_{l} \right \}_{k,l\ldots N}$

is linearly independent.

In the case where $T$ is unital, it is extremal if and only if the set

$\left \{A_{k}^{\dagger}A_{l} \oplus A_{l}A_{k}^{\dagger} \right \}_{k,l \ldots N}$

is linearly independent.

Ian Durham: One major conundrum is to determine whether extremality is preserved over tensor products, i.e. given an extremal quantum channel, if you take $n$ copies of it (which amounts to tensoring it $n$ times with itself), as a whole are these $n$ copies still extremal? It would be nice to see if category theory can shed some light on this problem since it is at the root of a particularly gnarly problem in quantum information theory..

Christian B. Mendl and Michael M. Wolf, *Unital Quantum Channels - Convex Structure and Revivals of Birkhoff’s Theorem* (pdf).

L. J. Landau and R. F. Streater, *On Birkhoff ’s theorem for doubly stochastic completely positive maps of matrix algebras*, Lin. Alg. Appl., 193:107–127, 1993.

Revised on March 28, 2010 02:23:10
by Eric Forgy
(61.92.132.172)