Ruskai’s conjecture on convex decomposition into generalized extreme dimension-altering channels

Establish whether every completely positive trace-preserving map from the set of density operators on an n-dimensional Hilbert space to the set of density operators on an m-dimensional Hilbert space admits a convex decomposition into m generalized extreme channels in S_{n,m}^{≤n}, each having Kraus rank at most n, with nonnegative weights summing to one.

Background

The paper studies convex decompositions of dimension-altering (DA) quantum channels, focusing on decompositions into extreme or generalized extreme channels. A central theme is Ruskai’s conjecture, originally proposed for general qudit channels and extended here to the DA setting, stating that any channel from qunits to qumits can be written as a convex sum of m generalized extreme channels of rank at most n.

The authors provide circuit constructions and numerical simulations for low-dimensional cases, offering support for the conjecture but not a general proof. Special cases (e.g., n=m=2 and m=2) are known to be proved, whereas the general case remains conjectural.

References

The extreme-channel decomposition is also motivated by Ruskai's conjecture, which states that a quantum channel from qu$n$its to qu$m$its can be decomposed as a convex sum of $m$ “generalized” extreme channels, each of which is at most of rank $n$.

— Convex decomposition of dimension-altering quantum channels  (1510.01040 - Wang, 2015) in Section 1 (Introduction)