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Asymptotic error rates in quantum hypothesis testing

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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.
Keywords
RELATIVE ENTROPY, PROBABILITY

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Chicago
Audenaert, KMR, M Nussbaum, A Szkola, and Frank Verstraete. 2008. “Asymptotic Error Rates in Quantum Hypothesis Testing.” Communications in Mathematical Physics 279 (1): 251–283.
APA
Audenaert, K., Nussbaum, M., Szkola, A., & Verstraete, F. (2008). Asymptotic error rates in quantum hypothesis testing. COMMUNICATIONS IN MATHEMATICAL PHYSICS, 279(1), 251–283.
Vancouver
1.
Audenaert K, Nussbaum M, Szkola A, Verstraete F. Asymptotic error rates in quantum hypothesis testing. COMMUNICATIONS IN MATHEMATICAL PHYSICS. 2008;279(1):251–83.
MLA
Audenaert, KMR et al. “Asymptotic Error Rates in Quantum Hypothesis Testing.” COMMUNICATIONS IN MATHEMATICAL PHYSICS 279.1 (2008): 251–283. Print.
@article{8589268,
  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.},
  author       = {Audenaert, KMR and Nussbaum, M and Szkola, A and Verstraete, Frank},
  issn         = {0010-3616},
  journal      = {COMMUNICATIONS IN MATHEMATICAL PHYSICS},
  language     = {eng},
  number       = {1},
  pages        = {251--283},
  title        = {Asymptotic error rates in quantum hypothesis testing},
  url          = {http://dx.doi.org/10.1007/s00220-008-0417-5},
  volume       = {279},
  year         = {2008},
}

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