Existence of channels with superlinear Hadamard-approximation scaling

Construct a family of quantum channels L_p, parameterized by a noise level p in [0,1], such that either the Hadamard parameter Had(L_p) or the alternative Hadamard parameter ε(L_p) scales as O(p^a) with a>1, enabling nontrivial single-letter upper bounds on classical capacity via approximate Hadamardness.

Background

The paper introduces two Hadamard approximation parameters—Had(·) based on diamond-distance proximity to a Hadamard channel, and ε(·) based on proximity of the complementary channel to an entanglement-breaking degrading map. For useful continuity bounds, these parameters must scale faster than linearly (a>1) in p.

The authors were unable to find channels achieving this scaling, and establishing such examples would broaden the applicability of approximate-Hadamard techniques to produce strong upper bounds.