Topology of extreme quantum channels in higher-dimensional domains

Characterize the geometry and topology of the set of extreme points of the convex set of completely positive trace-preserving maps from a finite-dimensional complex Hilbert space X to a finite-dimensional complex Hilbert space Y, and of its closure, when the domain dimension satisfies d_X \geq 2.

Background

The paper studies the hom-sets of completely positive trace-preserving maps through their extreme points and the closure of those extreme points. When the domain is one-dimensional, the extreme points are pure states and are homeomorphic to a complex projective space, so their topology is known. The authors explicitly identify the corresponding problem for domains of dimension at least two as unresolved.

References

The case $d_X=1$ is already understood, but for $d_X \geq 2$ it is still an open problem.

— On quantum channels: extreme points, topology, and categorical properties  (2608.20689 - Silva, 21 Aug 2026) in Chapter 2, Section "CPTP maps with 1-dimensional domain"

For CPTP maps, the extreme points don't need to be closed, so we also have the closure of the set of extreme points to study. Ruskai showed that this closure is the set of all CPTP maps with Choi-rank at most the dimension of the domain . There she also posed the study of this set as an open problem of Quantum Information Theory.

— On quantum channels: extreme points, topology, and categorical properties  (2608.20689 - Silva, 21 Aug 2026) in Chapter 1, Section "Research objectives"; Chapter 1, Section "Organization of the text"