Asymptotic Error Rates in Quantum Hypothesis Testing
Koenraad Audenaert, Michael Nussbaum, Arleta Szkola, and Frank Verstraete
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Submission date: 06. Sep. 2007
published in: Communications in mathematical physics, 279 (2008) 1, p. 251-283
DOI number (of the published article): 10.1007/s00220-008-0417-5
MSC-Numbers: 81-xx, 62-xx, 60-xx
Keywords and phrases: quantum hypothesis testing, quantum Baysian error rates, quantum Chernoff distance, quantum Hoeffding bound, quantum Hoeffding bound, quantum Sanov theorem
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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 specified 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, thereby solving a long standing open problem. The proof relies on a new trace inequality for pairs of positive operators as well as on a special mapping from pairs of density operators to pairs of probability distributions. These two new techniques have been introduced in [quant-ph/0610027] and [quant-ph/0607216], respectively. They are also well suited to prove the quantum generalisation of the Hoeffding bound, which is a modification of the Chernoff distance and specifies the optimal achievable asymptotic error rate in the context of asymmetric hypothesis testing. This has been done subsequently by Hayashi [quant-ph/0611013] and Nagaoka [quant-ph/0611289] for the special case where both hypotheses have full support. Moreover, quantum Stein's Lemma and quantum Sanov's theorem may be derived directly from quantum Hoeffding bound combining it with a result obtained recently in [math/0703772]. Actually, the goal of this paper is to present the proofs of the above mentioned results in a unified way and in full generality (allowing hypothetic states with different supports) using mainly the techniques from [quant-ph/0607216] and [quant-ph/0610027]. Additionally, we give an in-depth treatment of the properties of the quantum Chernoff distance. We argue that due to its clear operational meaning it is a natural distance measure on the set of density operators, although it is not a metric.