nLab
extremal quantum channel

Extremal quantum channels

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

$$T=\sum _i{p}_i{T}_i$$T=\sum_{i}p_{i}T_{i}

where the $p$’s are probabilities and the $T_i$’s are $\mathrm{extremal}$ 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 $\mathrm{all}$ channels on the particular space, not necessarily copies of the same one, i.e. the $T_i$ may not represent the same channel.

General extremality

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

$${\left\{A_k^{†}{A}_l\right\}}_{k,l\dots N}$$\left \{A_{k}^{\dagger}A_{l} \right \}_{k,l\ldots N}

is linearly independent.

Unital extremality

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

$${\{A_k^{†}{A}_l\oplus A_l{A}_k^{†}\}}_{k,l\dots N}$$\left \{A_{k}^{\dagger}A_{l} \oplus A_{l}A_{k}^{\dagger} \right \}_{k,l \ldots N}

is linearly independent.

Discussion

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..

References

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.