Error Exponent and Strong Converse for Quantum Soft Covering

TL;DR

This paper investigates the quantum soft covering problem, establishing how the trace distance between the induced and target states decays exponentially depending on the codebook rate relative to the quantum mutual information, with results applicable in large and moderate deviation regimes.

Contribution

It provides the first analysis of error exponents and strong converse properties for quantum soft covering, linking decay rates to sandwiched Rényi and Augustin information measures.

Findings

Expected trace distance decays exponentially with rate above quantum mutual information

Trace distance converges to one exponentially fast below the mutual information

Results extend to moderate deviation regimes with asymptotic vanishing trace distance

Abstract

How well can we approximate a quantum channel output state using a random codebook with a certain size? In this work, we study the quantum soft covering problem. Namely, we use a random codebook with codewords independently sampled from a prior distribution and send it through a classical-quantum channel to approximate the target state. When using a random codebook sampled from an independent and identically distributed prior with a rate above the quantum mutual information, we show that the expected trace distance between the codebook-induced state and the target state decays with exponent given by the sandwiched R\'enyi information. On the other hand, when the rate of the codebook size is below the quantum mutual information, the trace distance converges to one exponentially fast. We obtain similar results when using a random constant composition codebook, whereas the sandwiched Augustin information expresses the error exponent. In addition to the above large deviation analysis, our results also hold in the moderate deviation regime. That is, we show that even when the rate of the codebook size approaches the quantum mutual information moderately quickly, the trace distance still vanishes asymptotically.