Reading Club: A resource approach to the Stein’s Lemma for channel discrimination

Reading Club: A resource approach to the Stein’s Lemma for channel discrimination

Seminar author:Andreas Winter

Event date and time:02/12/2020 12:00:pm

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The quantum Stein’s Lemma gives the asymptotic error for simple i.i.d. hypothesis testing between two states, where the type-I  error is bounded away from 1, and the type-II error is exponentially small. The exponent is given by the quantum relative entropy between the states. The corresponding question for quantum channels (cptp maps) a priori has two answers, depending on whether we allow adaptive strategies to discriminate the channels or not. In a series of recent papers, Berta et al. (1808.01498), Wang/Wilde (1907.06306) and Fang et al. (1909.05826) have solved both questions and shown that the answers are the same: the Stein exponent for asymmetric channel hypothesis testing, i.e. discriminating between n copies of a channel N and n copies of a channel M, using adaptive strategies is given by the amortized channel divergence, and this quantity equals the regularized plain channel divergence, which is the optimal exponent for parallel strategies. The first part relies on a beautiful resource theory, going back to Matsumoto (1010.1030) in the state setting, where the objects are _pairs_ of quantum channels. The second part is surprisingly not operational, but based on chain rule relations for the relative entropy.