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IEEE International Symposium on Information Theory (ISIT)

Limit Distribution for Quantum Relative Entropy

Sreejith Sreekumar, Mario Berta

Published Jul 7, 2024• 2 citations
Publisher

Abstract

Estimation of quantum relative entropy is a fundamental statistical task in quantum information theory, physics, and beyond. While several estimators of the same have been proposed in the literature along with their computational complexities explored, a limit distribution theory which characterizes the asymptotic fluctuations of the estimation error is still premature. As our main contribution, we characterize these asymptotic distributions in terms of Fréchet derivatives of elementary operator-valued functions. We achieve this by leveraging an operator version of Taylor's theorem and identifying the regularity conditions needed. As an application of our results, we consider an estimator of quantum relative entropy based on Pauli tomography of quantum states and show that the resulting asymptotic distribution is a centered normal, with its variance characterized in terms of the Pauli operators and states. We utilize the knowledge of the aforementioned limit distribution to obtain asymptotic performance guarantees for a multi-hypothesis testing problem.

DOI
10.1109/isit57864.2024.10619162

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Estimation of quantum relative entropy is a fundamental statistical task in quantum information theory, physics, and beyond. While several estimators of the same have been proposed in the literature along with their computational complexities explored, a limit distribution theory which characterizes the asymptotic fluctuations of the estimation error is still premature. As our main contribution, we characterize these asymptotic distributions in terms of Fréchet derivatives of elementary operator-valued functions. We achieve this by leveraging an operator version of Taylor's theorem and identifying the regularity conditions needed. As an application of our results, we consider an estimator of quantum relative entropy based on Pauli tomography of quantum states and show that the resulting asymptotic distribution is a centered normal, with its variance characterized in terms of the Pauli operators and states. We utilize the knowledge of the aforementioned limit distribution to obtain asymptotic performance guarantees for a multi-hypothesis testing problem.
Estimation of quantum relative entropy and its Rényi generalizations is a fundamental statistical task in quantum information theory, physics, and beyond. While several estimators of these divergences have been proposed in the literature along with their computational complexities explored, a limit distribution theory which characterizes the asymptotic fluctuations of the estimation error is still premature. As our main contribution, we characterize these asymptotic distributions in terms of Fréchet derivatives of elementary operator-valued functions. We achieve this by leveraging an operator version of Taylor’s theorem and identifying the regularity conditions needed. As an application of our results, we consider an estimator of quantum relative entropy based on Pauli tomography of quantum states and show that the resulting asymptotic distribution is a centered normal, with its variance characterized in terms of the Pauli operators and states. We utilize the knowledge of the aforementioned limit distribution to obtain asymptotic performance guarantees for a multi-hypothesis testing problem.
We present our recent results on limit theorems for quantum entropies and their applications in quantum information theory. They include the full extensions of the fundamental asymptotic equipartition property (AEP) to ergodic quantum informations sources and the ergodic version of Stein's Lemma on quantum hypothesis testing. These limit theorems are then used to obtain a data compression theorem for ergodic quantum sources and a quantum version of Sanov's theorem on large deviations for uncorrelated quantum sources. Moreover, Stein's lemma clearifies the operational interpretation of the quantum Kullback-Leibler distance (relative entropy), and this operational understanding of this quantity can be employed to give a new intuitive proof of the monotonicity of this statistical distance measure relying on information-theoretic methods.
We establish a general quantum fluctuation-response inequality that bounds the difference in expectation values of an observable between two quantum states via the quantum relative entropy. For observables with bounded spectra, we further strengthen this result by exploiting the sub-Gaussian property, and explicitly relate our bound to the sub-Gaussian norm of the observable. This allows us to derive a novel non-asymptotic bound on the sum of statistical errors in quantum hypothesis testing, which complements and can be more informative than existing bounds. We also demonstrate the versatility of our results through problems such as thermodynamic hypothesis testing and quantum speed limits in physics, and generalizability of algorithms in machine learning.
Abstract We extend quantum Stein’s lemma in asymmetric quantum hypothesis testing to composite null and alternative hypotheses. As our main result, we show that the asymptotic error exponent for testing convex combinations of quantum states ρ⊗n\rho ^{\otimes n}ρ⊗n ρ ⊗ n against convex combinations of quantum states σ⊗n\sigma ^{\otimes n}σ⊗n σ ⊗ n can be written as a regularized quantum relative entropy formula. We prove that in general such a regularization is needed but also discuss various settings where our formula as well as extensions thereof become single-letter. This includes an operational interpretation of the relative entropy of coherence in terms of hypothesis testing. For our proof, we start from the composite Stein’s lemma for classical probability distributions and lift the result to the non-commutative setting by using elementary properties of quantum entropy. Finally, our findings also imply an improved recoverability lower bound on the conditional quantum mutual information in terms of the regularized quantum relative entropy—featuring an explicit and universal recovery map.
Estimating quantum entropies and divergences is an important problem in quantum physics, information theory, and machine learning. Quantum neural estimators (QNEs), which utilize a hybrid classical-quantum architecture, have recently emerged as an appealing computational framework for estimating these measures. Such estimators combine classical neural networks with parametrized quantum circuits, and their deployment typically entails tedious tuning of hyperparameters controlling the sample size, network architecture, and circuit topology. This work initiates the study of formal guarantees for QNEs of measured (Rényi) relative entropies in the form of non-asymptotic error risk bounds. We further establish exponential tail bounds showing that the error is sub-Gaussian and thus sharply concentrates about the ground truth value. For an appropriate sub-class of density operator pairs on a space of dimension ddd with bounded Thompson metric, our theory establishes a copy complexity of O(∣Θ(U)∣d/ε2)O(|Θ(\mathcal{U})|d/ε^2)O(∣Θ(U)∣d/ε2) for QNE with a quantum circuit parameter set Θ(U)Θ(\mathcal{U})Θ(U), which has minimax optimal dependence on the accuracy εεε. Additionally, if the density operator pairs are permutation invariant, we improve the dimension dependence above to O(∣Θ(U)∣polylog(d)/ε2)O(|Θ(\mathcal{U})|\mathrm{polylog}(d)/ε^2)O(∣Θ(U)∣polylog(d)/ε2). Our theory aims to facilitate principled implementation of QNEs for measured relative entropies and guide hyperparameter tuning in practice.
The density matrix is a positive semidefinite operator of trace 1 characterizing the state of a quantum system. We consider the inverse problem to reconstruct such density matrices from indirect measurements, also known as quantum state tomography. To solve such inverse problems in high or infinite dimensional settings, we study variational regularization using the quantum relative entropy as penalty functional. Quantum relative entropy is an analog of the well-known maximum entropy functional with compositions of functions replaced by the spectral functional calculus. The main aim of this paper is to establish the regularizing property of this scheme. As a crucial intermediate step, we establish lower semi-compactness of the penalty functional with respect to the weak-∗*∗-topology. Moreover, we compute the subgradient, proximal operator, and conjugate functional of the quantum relative entropy on finite dimensional spaces. This enables us to apply iterative algorithms from convex optimization to solve the regularized problems numerically. To show the validity and practical value of our results, we apply our theory to the examples of Photon-Induced Near-field Electron Microscopy (PINEM) and to optical homodyne tomography.