Do factorizable maps satisfy the Asymptotic Quantum Birkhoff Conjecture (AQBC)?

Determine whether every factorizable unital quantum channel (completely positive, trace-preserving, and unital) satisfies the Asymptotic Quantum Birkhoff Conjecture; specifically, ascertain whether for any factorizable map Φ on L(Hd), the asymptotic distance lim_{n→∞} D(Φ⊗n, Conv(U(Hd⊗n))) equals 0.

Background

The Asymptotic Quantum Birkhoff Conjecture (AQBC) posits that for any unital channel Φ the n-fold tensor power can be approximated arbitrarily well by a mixture of unitary channels on the larger space, i.e., lim_{n→∞} D(Φ⊗n, Conv(U(Hd⊗n))) = 0. Haagerup and Musat disproved AQBC by constructing counterexamples, all of which are non-factorizable maps.

The authors note that it remains unknown whether the AQBC might still hold for the subclass of factorizable maps. Notably, if all factorizable maps did satisfy AQBC, then the Connes embedding problem would have a positive answer, underscoring the significance of resolving this question.

References

The interesting thing here is that all these counterexamples are non- factorizable maps, and it remained unknown whether any facterizable map would fulfill AQBC. This problem was signified in the arXiv version of Ref. [11] by establishing the following surprising connection: If all factorizable maps satisfy AQBC, then the Connes embedding problem has a positive answer.

— 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 Section I. Introduction