Equality of quantum channel resolvability rate and quantum capacity

Determine whether, for every quantum channel N, the asymptotic resolvability rate R(N)—defined as the smallest input-rank rate that suffices to approximate any output state of N to vanishing trace distance—equals the weak converse quantum transmission capacity Q(N), i.e., show R(N) = Q(N).

Background

The paper introduces a quantum channel resolvability problem, defining the rate as the minimal input-state rank needed to approximate a given channel output, and establishes upper and lower bounds in one-shot and asymptotic regimes. In particular, the authors relate resolvability to transmission capacities: they prove an upper bound via the strong converse quantum capacity and provide a lower bound via smooth min-entropy.

Motivated by these bounds, they pose a conjecture relating the asymptotic resolvability rate to the standard (weak converse) quantum capacity—also referred to as the quantum transmission capacity—thus asking whether resolvability is exactly governed by the same quantity that characterizes reliable quantum communication.