Typical Support and Sanov Large Deviations of Correlated States

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

doi:10.1007/s00220-008-0440-6

Cited by 22 publications

(40 citation statements)

References 30 publications

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“…This may be seen as a quantum generalisation of Sanov's theorem. In a recent paper [5] an extension of this result to the case where the hypotheses correspond to sources emitting correlated (not necessarily i.i.d.) classical or quantum data has been given.…”

Section: Introductionmentioning

confidence: 89%

“…As this holds for any η > 0, we find that β * R ( ) ≥ S(ρ σ ). With β * R ( ) ≥ S(ρ σ ) the two hypotheses associated to the pair of density operators (ρ, σ ) satisfy the HP-condition in the terminology of the paper [5]. Thus Proposition 1 in [5] implies β * R ( ) = S(ρ σ ).…”

Section: Quantum Stein's Lemma and Quantum Version Of Sanov's Theoremmentioning

confidence: 98%

“…Here we use the quantum HBCL Theorem to prove that the relative entropy S(ρ σ ) is an achievable error rate limit and deduce optimality of this bound from Proposition 1 in [5].…”

Section: Quantum Stein's Lemma and Quantum Version Of Sanov's Theoremmentioning

confidence: 99%

“…Secondly, we present a complete proof of the quantum Hoeffding bound in a unified way. Moreover, we derive quantum Stein's Lemma as well as quantum Sanov's Theorem from the quantum Hoeffding bound combined with the mentioned equivalence relation proved in [5].…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic Error Rates in Quantum Hypothesis Testing

Audenaert

Nussbaum

Szkola

et al. 2008

Commun. Math. Phys.

226

242

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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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Section: Introductionmentioning

confidence: 89%

Section: Quantum Stein's Lemma and Quantum Version Of Sanov's Theoremmentioning

confidence: 98%

“…Here we use the quantum HBCL Theorem to prove that the relative entropy S(ρ σ ) is an achievable error rate limit and deduce optimality of this bound from Proposition 1 in [5].…”

Section: Quantum Stein's Lemma and Quantum Version Of Sanov's Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Asymptotic Error Rates in Quantum Hypothesis Testing

Audenaert

Nussbaum

Szkola

et al. 2008

Commun. Math. Phys.

226

242

View full text Add to dashboard Cite

show abstract

“…An advantage of this approach, in addition to its divergence-optimality base, is that it permits the possibility of flexible families of distributions that need not be Gaussian in nature. For discussions relative to the flexible family of distributions, under given values of moments and indirect noisy sample observations, see [3][4][5][6][7].…”

Section: Introductionmentioning

confidence: 99%

Implications of the Cressie-Read Family of Additive Divergences for Information Recovery

Judge

Mittelhammer

2012

Entropy

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Abstract:To address the unknown nature of probability-sampling models, in this paper we use information theoretic concepts and the Cressie-Read (CR) family of information divergence measures to produce a flexible family of probability distributions, likelihood functions, estimators, and inference procedures. The usual case in statistical modeling is that the noisy indirect data are observed and known and the sampling model-error distribution-probability space, consistent with the data, is unknown. To address the unknown sampling process underlying the data, we consider a convex combination of two or more estimators derived from members of the flexible CR family of divergence measures and optimize that combination to select an estimator that minimizes expected quadratic loss. Sampling experiments are used to illustrate the finite sample properties of the resulting estimator and the nature of the recovered sampling distribution.

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A Generalization of Quantum Stein’s Lemma

Brandão

Plenio

2010

Commun. Math. Phys.

104

185

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Given many independent and identically-distributed (i.i.d.) copies of a quantum system described either by the state ρ or σ (called null and alternative hypotheses, respectively), what is the optimal measurement to learn the identity of the true state? In asymmetric hypothesis testing one is interested in minimizing the probability of mistakenly identifying ρ instead of σ, while requiring that the probability that σ is identified in the place of ρ is bounded by a small fixed number. Quantum Stein's Lemma identifies the asymptotic exponential rate at which the specified error probability tends to zero as the quantum relative entropy of ρ and σ.We present a generalization of quantum Stein's Lemma to the situation in which the alternative hypothesis is formed by a family of states, which can moreover be non-i.i.d.. We consider sets of states which satisfy a few natural properties, the most important being the closedness under permutations of the copies. We then determine the error rate function in a very similar fashion to quantum Stein's Lemma, in terms of the quantum relative entropy.Our result has two applications to entanglement theory. First it gives an operational meaning to an entanglement measure known as regularized relative entropy of entanglement. Second, it shows that this measure is faithful, being strictly positive on every entangled state. This implies, in particular, that whenever a multipartite state can be asymptotically converted into another entangled state by local operations and classical communication, the rate of conversion must be non-zero. Therefore, the operational definition of multipartite entanglement is equivalent to its mathematical definition.

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Typical Support and Sanov Large Deviations of Correlated States

Cited by 22 publications

References 30 publications

Asymptotic Error Rates in Quantum Hypothesis Testing

Asymptotic Error Rates in Quantum Hypothesis Testing

Implications of the Cressie-Read Family of Additive Divergences for Information Recovery

A Generalization of Quantum Stein’s Lemma

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