A Quantum Version of Sanov's Theorem

Bjelaković, Igor; Deuschel, Jean–Dominique; Krüger, Tyll; Seiler, R.; Siegmund‐Schultze, Rainer; Szkola, Arleta

doi:10.1007/s00220-005-1426-2

Cited by 59 publications

(86 citation statements)

References 15 publications

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“…The authors of the present paper regrettably were not aware of this part of Hayashi's work during the preparation of [2].…”

mentioning

confidence: 76%

“…As already emphasized in [2], when passing from the classical to the quantum case, the universality mentioned above gets partially lost: there exists no longer a sequence of typical subspaces (of the underlying finite dimensional Hilbert spaces for the n-blocks of the system), which would work universally, whatever the reference states are. Consequently, speaking in the hypothesis testing terminology, for the alternative hypothesis only one process/state is admitted here.…”

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confidence: 99%

“…Universality with respect to the null hypothesis states is maintained, however. Also, in the quantum situation it is no longer possible to originate Sanov's theorem on the concept of empirical distributions (states), see [2], Chap. 4.…”

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confidence: 99%

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Typical Support and Sanov Large Deviations of Correlated States

Bjelaković

Deuschel

Krüger

et al. 2008

Commun. Math. Phys.

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Discrete stationary classical processes as well as quantum lattice states are asymptotically confined to their respective typical support, the exponential growth rate of which is given by the (maximal ergodic) entropy. In the iid case the distinguishability of typical supports can be asymptotically specified by means of the relative entropy, according to Sanov's theorem. We give an extension to the correlated case, referring to the newly introduced class of HP-states.

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“…The authors of the present paper regrettably were not aware of this part of Hayashi's work during the preparation of [2].…”

mentioning

confidence: 76%

mentioning

confidence: 99%

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Typical Support and Sanov Large Deviations of Correlated States

Bjelaković

Deuschel

Krüger

et al. 2008

Commun. Math. Phys.

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“…Moreover, we give an in-depth treatment of the properties of the quantum Chernoff distance in Sect. 4. More precisely, we show that it defines a distance measure between quantum states.…”

Section: Introductionmentioning

confidence: 85%

Asymptotic Error Rates in Quantum Hypothesis Testing

Audenaert

Nussbaum

Szkola

et al. 2008

Commun. Math. Phys.

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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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“…The method of irreducible decomposition provides the universal protocols in the quantum setting [14,16,[18][19][20][21][22]. However, even in the classical case, the universal channel coding requires the conditional type as well as the type [11].…”

Section: Introductionmentioning

confidence: 99%