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arXiv:2302.07818

A new estimation of the quantum Chernoff bound

Mohsen Kian, Dinh, Trung Hoa, Moslehian, Mohammad Sal, Osaka, Hiroyuki

Published Feb 15, 2023• 0 citations
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Abstract

Relating to finding possible upper bounds for the probability of error for discriminating between two quantum states, it is well-known that \begin{align*} \mathrm{tr}(A+B) - \mathrm{tr}|A-B|\leq 2\, \mathrm{tr}\big(f(A)g(B)\big) \end{align*} holds for every positive-valued matrix monotone function $f$, where $g(x)=x/f(x)$, and all positive definite matrices $A$ and $B$. In this paper, we introduce a new class of functions that satisfy the above inequality. As a consequence, we derive a novel estimation of the quantum Chernoff bound. Additionally, we characterize matrix decreasing functions and establish matrix Powers-Störmer type inequalities for perspective functions.

DOI
10.48550/arxiv.2302.07818

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Relating to finding possible upper bounds for the probability of error for discriminating between two quantum states, it is well-known that \begin{align*} \mathrm{tr}(A+B) - \mathrm{tr}|A-B|\leq 2, \mathrm{tr}\big(f(A)g(B)\big) \end{align*} holds for every positive-valued matrix monotone function fff, where g(x)=x/f(x)g(x)=x/f(x)g(x)=x/f(x), and all positive definite matrices AAA and BBB. In this paper, we introduce a new class of functions that satisfy the above inequality. As a consequence, we derive a novel estimation of the quantum Chernoff bound. Additionally, we characterize matrix decreasing functions and establish matrix Powers-Störmer type inequalities for perspective functions.
Determining the closeness of two quantum states with minimal error probability is a central problem in quantum information theory. The optimal exponents characterizing the error probability in the symmetric and asymmetric settings are given by the quantum Chernoff bound (QCB) and the quantum Hoeffding bound (QHB), respectively. In this work, we introduce the first algorithms for estimating the QCB and QHB of two unknown states. We propose a unitary encoding for diagonal density matrix based on the copy matrix, which enables unbiased estimation of nonlinear matrix functions via quantum singular value transformation (QSVT). For general density matrices, we combine QSVT with the Hadamard test to estimate both QCB and QHB. Numerical simulations demonstrate that the algorithm converges to the optimal value within a limited number of iterations. Furthermore, we establish the first integration of QSVT with convex optimization, yielding a hybrid quantum-classical framework for quantum estimation.
We consider the problem of detecting the true quantum state among r possible ones, based of measurements performed on n copies of a finite-dimensional quantum system. A special case is the problem of discriminating between r probability measures on a finite sample space, using n i.i.d. observations. In this classical setting, it is known that the averaged error probability decreases exponentially with exponent given by the worst case binary Chernoff bound between any possible pair of the r probability measures. Define analogously the multiple quantum Chernoff bound, considering all possible pairs of states. Recently, it has been shown that this asymptotic error bound is attainable in the case of r pure states, and that it is unimprovable in general. Here we extend the attainability result to a larger class of r-tuples of states which are possibly mixed, but pairwise linearly independent. We also construct a quantum detector which universally attains the multiple quantum Chernoff bound up to a factor 1/3.
We consider the problem of detecting the true quantum state among r possible ones, based of measurements performed on n copies of a finite-dimensional quantum system. A special case is the problem of discriminating between r probability measures on a finite sample space, using n i.i.d. observations. In this classical setting, it is known that the averaged error probability decreases exponentially with exponent given by the worst case binary Chernoff bound between any possible pair of the r probability measures. Define analogously the multiple quantum Chernoff bound, considering all possible pairs of states. Recently, it has been shown that this asymptotic error bound is attainable in the case of r pure states, and that it is unimprovable in general. Here we extend the attainability result to a larger class of r-tuples of states which are possibly mixed, but pairwise linearly independent. We also construct a quantum detector which universally attains the multiple quantum Chernoff bound up to a factor 1/3.
The quantum Chernoff bound is a famous result about discriminating two different states in a setting where large numbers of copies are available, which gives the analytic asymptotic rate at which the minimal error probability decays to zero exponentially. It is however a challenging task to calculate the quantum Chernoff bound exactly in practical scenarios. In this paper, from the viewpoint of differential geometry, we demonstrate a remarkable link between the quantum Chernoff bound and Wigner-Yanase skew information. As a result, the quantum Chernoff bound can be estimated efficiently by virtue of the skew information. We present several examples to illustrate the efficiency of estimation about the quantum Chernoff bound via Wigner-Yanase skew information.
Some properties and applications of the new quantum fff-divergences
Recently, a new definition for quantum fff-divergences was introduced based on an integral representation. These divergences have shown remarkable properties, for example when investigating contraction coefficients under noisy channels. At the same time, many properties well known for other definitions have remained elusive for the new quantum fff-divergence because of its unusual representation. In this work, we investigate alternative ways of expressing these quantum fff-divergences. We leverage these expressions to prove new properties of these fff-divergences and demonstrate some applications. In particular, we give a new proof of the achievability of the quantum Chernoff bound by establishing a strengthening of an inequality by Audenaert et al. We also establish inequalities between some previously known Renyi divergences and the new Renyi divergence. We further investigate some monotonicity and convexity properties of the new fff-divergences, and prove inequalities between these divergences for various functions.