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.
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.
This dissertation investigates relative entropies, also called generalized divergences, and how they can be used to characterize information-theoretic tasks in quantum information theory. The main goal is to further refine characterizations of the optimal rates for quantum source coding, state redistribution, and measurement compression with quantum side information via second order asymptotic expansions and strong converse theorems. The dissertation consists of a mathematical and an information-theoretic part. In the mathematical part, we focus on the
α-sandwiched Rényi divergence (
α-SRD). We first investigate the limit
α→0 to determine whether this recovers the well-known
0-Rényi relative divergence. We then prove various new results for entropic quantities derived from the
α-SRD, including dimension bounds and useful bounds in terms of the fidelity between two quantum states. Furthermore, we derive a necessary and sufficient algebraic condition for equality in the data processing inequality (viz. monotonicity under quantum operations) for the
α-SRD, and give applications to entropic bounds. In the information-theoretic part, we first derive the second order asymptotics of visible quantum source coding using a mixed source. For the achievability part, we develop universal quantum source codes achieving a given second order rate for a memoryless source. As a corollary of the main result, we obtain the second order asymptotics of quantum source coding using a single memoryless source. We then prove strong converse theorems for state redistribution (with or without feedback) and measurement compression with quantum side information. The key ingredients in proving these theorems are the aforementioned fidelity bounds on Rényi entropic quantities derived from the
α-SRD.
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.
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
d with bounded Thompson metric, our theory establishes a copy complexity of
O(∣Θ(U)∣d/ε2) for QNE with a quantum circuit parameter set
Θ(U), which has minimax optimal dependence on the accuracy
ε. Additionally, if the density operator pairs are permutation invariant, we improve the dimension dependence above to
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.
Quantum generalizations of the Fisher information are important in quantum information science, with applications in high energy and condensed matter physics and in quantum estimation theory, machine learning, and optimization. One can derive a quantum generalization of the Fisher information matrix in a natural way as the Hessian matrix arising in a Taylor expansion of a smooth divergence. Such an approach is appealing for quantum information theorists, given the ubiquity of divergences in quantum information theory. In contrast to the classical case, there is not a unique quantum generalization of the Fisher information matrix, similar to how there is not a unique quantum generalization of the relative entropy or the Rényi relative entropy. In this paper, I derive information matrices arising from the log-Euclidean,
α-
z, and geometric Rényi relative entropies, with the main technical tool for doing so being the method of divided differences for calculating matrix derivatives. Interestingly, for all non-negative values of the Rényi parameter
α, the log-Euclidean Rényi relative entropy leads to the Kubo-Mori information matrix, and the geometric Rényi relative entropy leads to the right-logarithmic derivative Fisher information matrix. Thus, the resulting information matrices obey the data-processing inequality for all non-negative values of the Rényi parameter
α even though the original quantities do not. Additionally, I derive and establish basic properties of
α-
z information matrices resulting from the
α-
z Rényi relative entropies. For parameterized thermal states and time-evolved states, I establish formulas for their
α-
z information matrices and hybrid quantum-classical algorithms for estimating them, with applications in quantum Boltzmann machine learning.
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.