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Optimal Min-Max Tests for Quantum Hypothesis Testing Problems on Gaussian States (Theory and application of statistical inference in quantum theory)

Wataru Kumagai

Published May 1, 2013• 0 citations
OpenAlex

Abstract

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 $\mathcal{H}=L^{2}(\mathbb{R})$ , which represents a single-mode bosonic system and is spanned by sets $\{|k\rangle\}_{k\in \mathbb{Z}_{\geq 0}}$ of the k-th Hermitian functions $|k\rangle.$$A$ most typical example of a single-mode bosonic system is the one-mode photonic system, in which the vector $|k\rangle$ is called the $k$ -photon number state because it is regarded as the state corresponding to $k$ photons.Then, the set $M_{N}=\{|k\rangle\langle k|\}_{k\in \mathbb{Z}>0}$ , the number measurement, forms a positive-operator-valued measure (POVM).When the vector $|\xi^{-})\in L^{2}(\mathbb{R})(\xi\in \mathbb{C})$ is defined by $| \xi):=e^{-\rfloor}\star^{2}\sum_{k=0}^{\infty}\frac{\xi^{k}}{\sqrt{k!}}|k\rangle,$

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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\rangle.$$A 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⟩,
Efficiently testing whether a quantum state possesses a given structure is both a fundamental and a practical task in quantum information. Among the most important structured families are Gaussian states, which underpin quantum optics and many-body physics while defining paradigmatic regimes of efficient classical simulation. Despite the central role of Gaussian states, optimal tests of Gaussianity have remained unknown. Here we give optimal and robust algorithms for testing fermionic and bosonic Gaussian states, with sample complexity independent of system size, particle number, and energy. Remarkably, the protocols implement experimentally feasible measurements associated with known exact symmetry characterizations of Gaussianity, using Bell sampling on two copies for fermions and passive interferometry with photon counting on three copies for bosons. We further obtain tolerant testers requiring joint measurements on a larger, but still constant, number of copies. At the heart of our analysis is a unified approach---which we call the gradient-flow method---that applies across all structured families considered here and underlies our optimal testers: representation theory turns approximate satisfaction of the defining symmetry into quantitative control of the gradient of the rejection probability, allowing a continuous descent from the input state to the target family whose length bounds their distance.
Gaussian states are widely regarded as one of the most relevant classes of continuous-variable (CV) quantum states, as they naturally arise in physical systems and play a key role in quantum technologies. This motivates a fundamental question: given copies of an unknown CV state, how can we efficiently test whether it is Gaussian? We address this problem from the perspective of representation theory and quantum learning theory, characterizing the sample complexity of Gaussianity testing as a function of the number of modes. For pure states, we prove that just a constant number of copies is sufficient to decide whether the state is exactly Gaussian. We then extend this to the tolerant setting, showing that a polynomial number of copies suffices to distinguish states that are close to Gaussian from those that are far. In contrast, we establish that testing Gaussianity of general mixed states necessarily requires exponentially many copies, thereby identifying a fundamental limitation in testing CV systems. Our approach relies on rotation-invariant symmetries of Gaussian states together with the recently introduced toolbox of CV trace-distance bounds.
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.
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.
Bosonic and fermionic Gaussian quantum states are families of quantum states that play a central role in quantum physics and quantum information processing. We study the problem of testing whether a given state is either Gaussian, or has bounded overlap with any Gaussian state. Previous analyses of this testing problem led to tests that require a number of copies of the state that scales with the number of modes (the system size). In this work we show that a constant number of copies of the state suffices. We derive these results based on the symmetries of identical copies of Gaussian states, and by bounding the distance to the set of Gaussian states through a gradient flow and a spectral bound studied previously for the Kac random walk on a sphere.
Quantum fidelity is a measure to quantify the closeness between two quantum states. In an operational sense, it is defined as the minimal overlap between the probability distributions of measurement outcomes and the minimum is taken over all possible positive-operator valued measures (POVMs). Quantum fidelity has been investigated in various scientific fields, but the identification of associated optimal measurements has often been overlooked despite its great importance both for fundamental interest and practical purposes. We find here the optimal POVMs for quantum fidelity between multimode Gaussian states in a closed analytical form. Our general finding is applied for selected single-mode Gaussian states of particular interest and we identify three types of optimal measurements: an excitation-number-resolving detection, a projection onto the eigenbasis of operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}+\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}$, and a quadrature variable detection, each of which corresponds to distinct types of single-mode Gaussian states. We also show the equivalence between optimal measurements for quantum fidelity and those for quantum parameter estimation when two arbitrary states are infinitesimally close. It is applied for simplifying the derivations of quantum Fisher information and the associated optimal measurements, exemplified by displacement, phase, squeezing, and loss parameter estimation using Gaussian states.