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arXiv:1407.0884 · Physical Review A

Asymmetric quantum hypothesis testing with Gaussian states

Gaetana Spedalieri, Samuel L. Braunstein

Published Nov 6, 2014• 32 citations
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Abstract

We consider the asymmetric formulation of quantum hypothesis testing, where two quantum hypotheses have different associated costs. In this problem, the aim is to minimize the probability of false negatives and the optimal performance is provided by the quantum Hoeffding bound. After a brief review of these notions, we show how this bound can be simplified for pure states. We then provide a general recipe for its computation in the case of multimode Gaussian states, also showing its connection with other easier-to-compute lower bounds. In particular, we provide analytical formulas and numerical results for important classes of one- and two-mode Gaussian states.

DOI
10.1103/physreva.90.052307

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We consider the asymmetric formulation of quantum hypothesis testing, where two quantum hypotheses have different associated costs. In this problem, the aim is to minimize the probability of false negatives and the optimal performance is provided by the quantum Hoeffding bound. After a brief review of these notions, we show how this bound can be simplified for pure states. We then provide a general recipe for its computation in the case of multimode Gaussian states, also showing its connection with other easier-to-compute lower bounds. In particular, we provide analytical formulas and numerical results for important classes of one- and two-mode Gaussian states.
In this thesis we investigate the theory of quantum hypothesis testing and its potential applications for the new area of quantum technologies. We first consider the asymmetric formulation of quantum hypothesis testing where the aim is to minimize the probability of false negatives and the main tool is provided by the quantum Hoeffding bound. In this context we provide a general recipe for computing this bound in the most important scenario for continuous variable quantum information, that of Gaussian states. We then study both asymmetric and symmetric quantum hypothesis testing in the context of quantum channel discrimination. Here we show how the use of quantum-correlated light can enhance the detection of small variations of transmissivity in a sample of photodegrabable material, while a classical source of light either cannot retrieve information or would destroy the sample. This non-invasive quantum technique might be useful to realize in-vivo and real-time probing of very fragile biological samples, such as DNA or RNA. We also show that the same principle can be exploited to build next-generation memories for the confidential storage of confidential data, where information can be read only by well-tailored sources of entangled light.
An upper bound on simple quantum hypothesis testing in the asymmetric setting is shown using a useful inequality by Audenaert et al. [Phys. Rev. Lett. 98, 160501 (2007)] which was originally invented for symmetric setting. Using this upper bound, we obtain the Hoeffding bound, which is identical with the classical counterpart if the hypotheses, composed of two density operators, are mutually commutative. Its attainability has been a long-standing open problem. Further, using this bound, we obtain a better exponential upper bound of the average error probability of classical-quantum channel coding.
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.
Quantum hypothesis testing is one of the most basic tasks in quantum information theory and has fundamental links with quantum communication and estimation theory. In this paper, we establish a formula that characterizes the decay rate of the minimal type-II error probability in a quantum hypothesis test of two Gaussian states given a fixed constraint on the type-I error probability. This formula is a direct function of the mean vectors and covariance matrices of the quantum Gaussian states in question. We give an application to quantum illumination, which is the task of determining whether there is a low-reflectivity object embedded in a target region with a bright thermal-noise bath. For the asymmetric-error setting, we find that a quantum illumination transmitter can achieve an error probability exponent stronger than a coherent-state transmitter of the same mean photon number, and furthermore, that it requires far fewer trials to do so. This occurs when the background thermal noise is either low or bright, which means that a quantum advantage is even easier to witness than in the symmetric-error setting because it occurs for a larger range of parameters. Going forward from here, we expect our formula to have applications in settings well beyond those considered in this paper, especially to quantum communication tasks involving quantum Gaussian channels.
This paper is a summary of [1].We formulate some problem on quantum hypothesis testing for special Gaussian states, which naturally appear in photonic systems.Moreover, we introduce an optimal test for each quantum hypothesis testing problem.Optimal tests are constructed by using group symmetry of Gaussian states and some results in classical hypothesis testing. 1 Gaussian state We introduce Gaussian states on H=L2(R)\mathcal{H}=L^{2}(\mathbb{R})H=L2(R) , which represents a single-mode bosonic system and is spanned by sets {∣k⟩}k∈Z≥0\{|k\rangle\}_{k\in \mathbb{Z}_{\geq 0}}{∣k⟩}k∈Z≥0​​ of the k-th Hermitian functions ∣k⟩.|k\rangle.∣k⟩.AAA most typical example of a single-mode bosonic system is the one-mode photonic system, in which the vector ∣k⟩|k\rangle∣k⟩ is called the kkk -photon number state because it is regarded as the state corresponding to kkk photons.Then, the set MN={∣k⟩⟨k∣}k∈Z>0M_{N}=\{|k\rangle\langle k|\}_{k\in \mathbb{Z}>0}MN​={∣k⟩⟨k∣}k∈Z>0​ , the number measurement, forms a positive-operator-valued measure (POVM).When the vector ∣ξ−)∈L2(R)(ξ∈C)|\xi^{-})\in L^{2}(\mathbb{R})(\xi\in \mathbb{C})∣ξ−)∈L2(R)(ξ∈C) is defined by ∣ξ):=e−⌋⋆2∑k=0∞ξkk!∣k⟩,| \xi):=e^{-\rfloor}\star^{2}\sum_{k=0}^{\infty}\frac{\xi^{k}}{\sqrt{k!}}|k\rangle,∣ξ):=e−⌋⋆2∑k=0∞​k!​ξk​∣k⟩,