Strictness and equality in bounds for Schur channel approximations

Establish whether, for every Schur channel Φ on L(Hd), the lower bound (1/2)·D(Φ, Ad) ≤ D(Φ, Conv(Ud)) is strict and the upper bound D(Φ, Conv(Ud)) ≤ D(Φ, Ad) is an equality, where Ad denotes mixtures of diagonal unitary channels and Conv(Ud) denotes mixtures of all unitary channels.

Background

The authors derive two-sided bounds relating the trace distance between a Schur channel Φ and the convex hull of unitary channels Conv(Ud) to the distance between Φ and Ad, the convex hull of diagonal unitary channels. Specifically, they prove 1/2·D(Φ, Ad) ≤ D(Φ, Conv(Ud)) ≤ D(Φ, Ad).

They conjecture a sharpening of these bounds—namely, strictness of the lower bound and equality of the upper bound—but explicitly note that this remains unsettled.

References

It seems quite likely that in Eq. (8) the first inequality should be strict and the second one should be an equality. However, this is still an unsettled issue.

— Bounds on the distance between a unital quantum channel and the convex hull of unitary channels, with applications to the asymptotic quantum Birkhoff conjecture  (1201.1172 - Yu et al., 2012) in Following Theorem 3, Section V