Ruskai–Audenaert conjecture on t-term convex decompositions into generalized extreme channels

Establish whether every completely positive trace-preserving map from C^{s×s} to C^{t×t} can be written as a convex combination of at most t generalized extreme channels, i.e., channels with Kraus rank at most s, thereby proving or refuting the conjecture of Ruskai and Audenaert.

Background

After proving that any channel admits a convex decomposition into extreme channels and giving a general Carathéodory-type bound, the authors highlight a more refined conjecture due to Ruskai and Audenaert. This conjecture proposes that only t generalized extreme channels suffice to represent any s-to-t channel.

Confirming this conjecture would significantly sharpen known bounds on convex decompositions and directly impact the efficiency of circuit-based implementations of quantum channels.

References

It is conjectured by Ruskai and Audenaert that k\leqslant t if we allow convex combinations of channels \mathcal{E}j\in\mathcal{E}{s,t,\leqslant s} (note that \mathcal{E}_{s,t,\leqslant s} is equal to the closure of the set of all s to t extreme channels). However, as far as we know, this remains unproven.

— Smooth manifold structure for extreme channels  (1610.02513 - Iten et al., 2016) in Remark after Theorem “Convex decomposition,” Section 4 (Decomposition of channels into Convex Combinations of Extreme Channels)