Quantum soft-covering lemma with applications to rate-distortion coding, resolvability and identification via quantum channels

TL;DR

This paper introduces a quantum soft-covering lemma and demonstrates its applications to rate-distortion coding, channel resolvability, and identification capacities, advancing quantum information theory.

Contribution

It presents a one-shot quantum covering lemma based on smooth min-entropies and applies it to multiple quantum channel problems, including resolvability and identification capacity separation.

Findings

Established a quantum soft-covering lemma using decoupling techniques.

Derived bounds for quantum channel resolvability in one-shot and asymptotic regimes.

Proved the separation of simultaneous and unrestricted quantum channel identification capacities.

Abstract

We propose a quantum soft-covering problem for a given general quantum channel and one of its output states, which consists in finding the minimum rank of an input state needed to approximate the given channel output. We then prove a one-shot quantum covering lemma in terms of smooth min-entropies by leveraging decoupling techniques from quantum Shannon theory. This covering result is shown to be equivalent to a coding theorem for rate distortion under a posterior (reverse) channel distortion criterion by two of the present authors. Both one-shot results directly yield corollaries about the i.i.d. asymptotics, in terms of the coherent information of the channel. The power of our quantum covering lemma is demonstrated by two additional applications: first, we formulate a quantum channel resolvability problem, and provide one-shot as well as asymptotic upper and lower bounds. Secondly, we provide new upper bounds on the unrestricted and simultaneous identification capacities of quantum channels, in particular separating for the first time the simultaneous identification capacity from the unrestricted one, proving a long-standing conjecture of the last author.