How well can we approximate a quantum channel output state using a codebook with a certain size? In this work, we study the so-called quantum soft covering problem, which is using a random codebook to approximate the target output state of a quantum channel. We establish a one-shot exponential bound on the expected trace-norm distance between the codebook-induced state and the true state. When using an independently and identically distributed random codebook with a rate above the quantum mutual information, we prove that the trace distances decay exponentially with error exponents determined by the Legendre transform of the quantum sandwiched Rényi information. As a result, it implies a tight bound on the information leakage to Eavesdroppers in private communication over wiretap quantum channels. Our proof technique is to establish a novel matrix concentration inequality by using interpolation of noncommutative L_p space. This may have applications elsewhere.
This talk is based on [IEEE TIT 70(5), 2024].
