Practical finite-dimensional counterexample construction

Construct a simple closed-form or practically computable finite-dimensional quantum-channel counterexample exhibiting nonadditivity of minimum output entropy using permutation-generated channels.

Background

The paper replaces Haar-random unitaries with random permutations and uses the deterministic O’Donnell–Wu procedure to establish an asymptotic derandomization. However, the resulting dimensions are enormous, and the construction does not yield a simple formula or an instance that can currently be computed or implemented in practice. This problem concerns turning the existential and asymptotic construction into a practically usable counterexample.

References

Thus the random construction can be derandomized in an asymptotic algorithmic sense, although finding a simple closed-form or practically computable counterexample remains open.

— Superadditivity of classical communication over quantum channels via random and deterministic permutations  (2608.25961 - Lovitz et al., 26 Aug 2026) in Abstract

Therefore, it remains a major challenge whether a practical search is successful.

— Superadditivity of classical communication over quantum channels via random and deterministic permutations  (2608.25961 - Lovitz et al., 26 Aug 2026) in Section 1, subsection “Sketch proof of the main result”