Strong converse exponents for a quantum channel discrimination problem and quantum-feedback-assisted communication

We study the difficulty of discriminating between an arbitrary quantum channel and a “replacer” channel that discards its input and replaces it with a fixed state.1 The results obtained here generalize those known in the theory of quantum hypothesis testing for binary state discrimination. We show that, in this particular setting, the most general adaptive discrimination strategies provide no asymptotic advantage over non-adaptive tensor-power strategies. This conclusion follows by proving a quantum Stein’s lemma for this channel discrimination setting, showing that a constant bound on the Type I error leads to the Type II error decreasing to zero exponentially quickly at a rate determined by the maximum relative entropy registered between the channels. The strong converse part of the lemma states that any attempt to make the Type II error decay to zero at a rate faster than the channel relative entropy implies that the Type I error necessarily converges to one. We then refine this latter result by identifying the optimal strong converse exponent for this task. As a consequence of these results, we can establish a strong converse theorem for the quantum-feedback-assisted capacity of a channel, sharpening a result due to Bowen. Furthermore, our channel discrimination result demonstrates the asymptotic optimality of a non-adaptive tensor-power strategy in the setting of quantum illumination, as was used in prior work on the topic. The sandwiched Rényi relative entropy, [16, 27], is a key tool in our analysis. Background: Quantum channel discrimination is a natural extension of a basic problem in quantum hypothesis testing, that of distinguishing between the possible states of a quantum system. In an i.i.d. binary state discrimination problem, the discriminator is provided with n quantum systems in the state ρ⊗n or σ⊗n, and the task is to apply a binary measurement {Qn, I−Qn} to these n systems, with 0 ≤ Qn ≤ I⊗n. One is then concerned with two kinds of error probabilities: αn(Qn) ≡ Tr {(I⊗n −Qn)ρ} , the probability of incorrectly rejecting the null hypothesis, the Type I error, and βn(Qn) ≡ Tr {Qnσ} , the probability of incorrectly rejecting the alternative hypothesis, the Type II error. One studies the asymptotic behaviour of αn and βn as n → ∞, expecting there to be a trade-off between minimising αn and minimising βn. In quantum channel discrimination, we have a quantum channel with input system A and output system B, and we are given that the channel is described by either the completely positive trace-