Multiplicativity of boundariness for quantum channels

Establish whether boundariness is multiplicative for all finite-dimensional quantum channels; specifically, prove or refute the identity b(E ⊗ F) = b(E) b(F) for all completely positive trace-preserving maps E and F acting on finite-dimensional Hilbert spaces.

Background

The paper introduces boundariness b(·) as a measure of how an element of a compact convex set can be decomposed into boundary elements and connects it operationally to optimal distinguishability via the base norm. For quantum devices, the authors show that the most distinguishable channel from any interior channel is always unitary and derive an explicit formula for channels: b(F) = [max_U λ1(J_F^{-1} J_U)]^{-1}.

In Section V, the authors prove sub-multiplicativity of boundariness for channels, b(E ⊗ F) ≤ b(E) b(F), and multiplicativity for states and observables. They present numerical evidence suggesting multiplicativity may also hold for channels, but they do not have a proof. They further show multiplicativity in a special case (a qubit channel tensored with an erasure channel mapping to the maximally mixed state). This leaves the general multiplicativity question for channels unresolved.