Tight bound on the number of generalized extreme channels needed for convex decompositions

Determine a tight upper bound, as a function of input dimension s and output dimension t, on the minimal number of generalized extreme channels (i.e., completely positive trace-preserving maps in the closure of the set of extreme channels, equivalently channels with Kraus rank at most s) required to express an arbitrary completely positive trace-preserving map from C^{s×s} to C^{t×t} as a convex combination of such generalized extreme channels.

Background

The paper studies the smooth manifold structure of the set of extreme quantum channels and shows, among other results, that every channel can be decomposed as a convex combination of extreme channels. Such decompositions are relevant for practical implementations of channels in quantum information.

Beyond the existence of decompositions, a central unresolved question concerns how many generalized extreme channels are needed to represent any channel. Generalized extreme channels are defined as those lying in the closure of the set of extreme channels (equivalently, channels of Kraus rank at most s). Establishing a tight bound would clarify the complexity of channel synthesis and optimization.