On Asymptotically Optimal Hypothesis Testing in Quantum Statistics

Холево, Александр Семенович

doi:10.1137/1123048

Cited by 90 publications

(89 citation statements)

References 3 publications

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“…The null hypothesis is then accepted or rejected according to the outcome of the measurement and the specified decision rule. The task of finding this optimal measurement is so fundamental that it was one of the first problems considered in the field of quantum information theory; it was solved in the one-copy case more than 30 years ago by Helstrom and Holevo [14,17]. We refer to the generalised ML-tests as Holevo-Helstrom tests.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Error Rates in Quantum Hypothesis Testing

Audenaert

Nussbaum

Szkola

et al. 2008

Commun. Math. Phys.

227

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: 99%

Asymptotic Error Rates in Quantum Hypothesis Testing

Audenaert

Nussbaum

Szkola

et al. 2008

Commun. Math. Phys.

227

242

View full text Add to dashboard Cite

show abstract

“…The problem is to discriminate two sources that output many identical copies of one out of two different quantum states and , and the question is to identify the exponent arising asymptotically when performing the optimal test to discriminate them. This task is so fundamental that it was probably the first problem ever considered in the field of quantum information theory; it was solved in the one-copy case more than 30 years ago [3,4]. In this Letter, we finally identify the asymptotic error exponent when the optimal strategy for discriminating the states is used.…”

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confidence: 99%

Discriminating States: The Quantum Chernoff Bound

et al. 2007

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We consider the problem of discriminating two different quantum states in the setting of asymptotically many copies, and determine the minimal probability of error. This leads to the identification of the quantum Chernoff bound, thereby solving a long-standing open problem. The bound reduces to the classical Chernoff bound when the quantum states under consideration commute. The quantum Chernoff bound is the natural symmetric distance measure between quantum states because of its clear operational meaning and because it does not seem to share some of the undesirable features of other distance measures.

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“…We will not further consider the critical point (π, π) because it does not yield a maximum of Eq. (15). In particular, for r 1 ≤ 0, r 2 ≥ 0, and x 1 ,…”

Section: Calculation Of Pminmentioning

confidence: 97%

“…Because the energy constraint is independent of θ 1 , θ 2 , it does not have to be taken into account to find the critical angles of the function in Eq. (15). We will not further consider the critical point (π, π) because it does not yield a maximum of Eq.…”

Section: Calculation Of Pminmentioning

confidence: 99%

Maximal trace distance between isoenergetic bosonic Gaussian states

Volkoff

2017

Journal of Mathematical Physics

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We locate the set of pairs $(\rho_{1},\rho_{2})$ of Gaussian states of a single mode electromagnetic field that exhibit maximal trace distance subject to the energy constraint $\langle a^{\dagger}a \rangle_{\rho_{1}}=\langle a^{\dagger}a \rangle_{\rho_{2}} = E$. Any such pair allows to achieve the minimum possible error in the task of binary distinguishability of two single mode, isoenergetic Gaussian quantum signals. In particular, we show that the logarithm of the minimal error probability for distinguishing two maximally trace distant, isoenergetic Gaussian states scales as $-E^{2}$, less than the achievable scaling of the minimal error probability for distinguishing, e.g., a pair of isoenergetic Heisenberg-Weyl coherent states with energy $E$ or a pair of isoenergetic quadrature squeezed states with energy $E$. For the case of a field consisting of $M>1$ modes, we locate the set of pairs of maximally trace distant isoenergetic, isocovariant Gaussian states. These results have basic applications in the theory of continuous variable quantum communications with Gaussian states of light.Comment: 11 pages, 2 figure

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On Asymptotically Optimal Hypothesis Testing in Quantum Statistics

Cited by 90 publications

References 3 publications

Asymptotic Error Rates in Quantum Hypothesis Testing

Asymptotic Error Rates in Quantum Hypothesis Testing

Discriminating States: The Quantum Chernoff Bound

Maximal trace distance between isoenergetic bosonic Gaussian states

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