Most distant elements from tensor-product channels are factorized unitaries

Determine whether, for all tensor-product quantum channels E ⊗ F on finite-dimensional systems, the channel that maximizes the base norm (diamond norm) distance from E ⊗ F is always a product of unitary channels U ⊗ V; equivalently, ascertain whether the most distinguishable element from a factorized channel is necessarily factorized and unitary.

Background

Using the established link between boundariness and optimal distinguishability, multiplicativity of boundariness would imply that the most distinguishable element from x ⊗ y is x0 ⊗ y0, where x0 and y0 are the respective most distinguishable elements from x and y. For channels, the paper proves that the most distinguishable element from an interior single channel is a unitary channel.

In the summary, the authors state that, if boundariness were multiplicative for channels, factorized unitaries would be the most distant elements for factorized channels. However, they explicitly leave open whether this actually holds in general, making it a concrete unresolved question about the structure of optimal discrimination for composite channels.

References

For channels this would mean that factorized unitaries are the most distant ones for all factorized channels. However, whether this is the case is left open.

— Exploring boundaries of quantum convex structures: special role of unitary processes  (1504.00477 - Puchała et al., 2015) in Section VII (Summary)