Asymptotic Error Rates in Quantum Hypothesis Testing |
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Authors: | K M R Audenaert M Nussbaum A Szkoła F Verstraete |
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Institution: | 1. Institute for Mathematical Sciences, Imperial College London, 53 Prince’s Gate, London, SW7 2PG, UK 2. Dept. of Mathematics, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, UK 3. Department of Mathematics, Cornell University, Ithaca, NY, 14853, USA 4. Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103, Leipzig, Germany 5. Fakult?t für Physik, Universit?t Wien, Boltzmanngasse 5, 1090, Wien, Austria
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Abstract: | We consider the problem of discriminating between two different states of a finite quantum system in the setting of large
numbers of copies, and find a closed form expression for the asymptotic exponential rate at which the error probability tends
to zero. This leads to the identification of the quantum generalisation of the classical Chernoff distance, which is the corresponding
quantity in classical symmetric hypothesis testing.
The proof relies on two new techniques introduced by the authors, which are also well suited to tackle the corresponding problem
in asymmetric hypothesis testing, yielding the quantum generalisation of the classical Hoeffding bound. This has been done
by Hayashi and Nagaoka for the special case where the states have full support.
The goal of this paper is to present the proofs of these results in a unified way and in full generality, allowing hypothesis
states with different supports. From the quantum Hoeffding bound, we then easily derive quantum Stein’s Lemma and quantum
Sanov’s theorem. We give an in-depth treatment of the properties of the quantum Chernoff distance, and argue that it is a
natural distance measure on the set of density operators, with a clear operational meaning. |
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