Error Exponent and Strong Converse for Quantum Soft Covering

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, which uses a pairwise-independent random codebook to approximate the target output state of a quantum channel. We establish a one-shot error exponent bound and a one-shot strong converse bound on the approximation error measured in terms of the expected trace distance between the codebook-induced state and the true channel output state. When using independent and identically-distributed random codebook with a rate above the quantum mutual information
$I(X{\,:\,}B)_{\rho} $
, we prove that the error decays exponentially with its error exponent expressed by the sandwiched Rényi information. On the other hand, when the rate of the codebook size is below the quantum mutual information, the error converges to one exponentially fast. Similar results are obtained using a random constant composition codebook, whereas the sandwiched Augustin information gives the error exponent. In addition to the above large deviation analysis, our results also hold in the moderate deviation regime.