Determine the distance to the quantum Schrödinger bridge

Determine the distance between a given nonunital quantum channel and the unital channel produced from it by the quantum Schrödinger bridge construction, particularly in the relevant Choi relative-entropy geometry.

Background

The paper relates the problem of finding a unital channel closest to a given channel to operator scaling and the quantum Schrödinger bridge problem. For classical stochastic matrices, iterative Sinkhorn scaling can find a closest doubly stochastic matrix in an appropriate relative-entropy formulation.

For quantum channels, operator scaling and quantum Schrödinger bridge methods construct candidate unital channels, but the paper notes that operator scaling need not converge to the Choi-relative-entropy minimizer. It also states that the distance between the original nonunital channel and its quantum Schrödinger bridge remains unresolved.

References

The quantum Schrödinger bridge approach likewise constructs a family of unital channels from a given nonunital channel cite{georgiou_positive_2015}, but the distance between the original channel and its ``bridge'' remains unexplored.

— Dynamical resource theory of time-reversal symmetry breaking  (2609.36408 - Miyazaki et al., 29 Sep 2026) in Section 3.3, final paragraph of “Relative entropies of T-violation: general properties”