Amortized Channel Divergence for Asymptotic Quantum Channel Discrimination

TL;DR

This paper extends the understanding of quantum channel discrimination, showing that adaptive strategies do not improve the asymptotic exponential error rate for classical-quantum channels, using the concept of amortized distinguishability.

Contribution

It establishes the strong Stein's lemma for classical-quantum channels and provides bounds on adaptive protocol power in asymptotic quantum channel discrimination.

Findings

Adaptive strategies do not improve the exponential error rate for classical channels asymptotically.

The strong Stein's lemma is proven for classical-quantum channels.

Various bounds are given on the effectiveness of adaptive protocols in quantum channel discrimination.

Abstract

It is well known that for the discrimination of classical and quantum channels in the finite, non-asymptotic regime, adaptive strategies can give an advantage over non-adaptive strategies. However, Hayashi [IEEE Trans. Inf. Theory 55(8), 3807 (2009)] showed that in the asymptotic regime, the exponential error rate for the discrimination of classical channels is not improved in the adaptive setting. We extend this result in several ways. First, we establish the strong Stein's lemma for classical-quantum channels by showing that asymptotically the exponential error rate for classical-quantum channel discrimination is not improved by adaptive strategies. Second, we recover many other classes of channels for which adaptive strategies do not lead to an asymptotic advantage. Third, we give various converse bounds on the power of adaptive protocols for general asymptotic quantum channel discrimination. Intriguingly, it remains open whether adaptive protocols can improve the exponential error rate for quantum channel discrimination in the asymmetric Stein setting. Our proofs are based on the concept of amortized distinguishability of quantum channels, which we analyse using data-processing inequalities.