Measuring Quantum Entropy

Abstract

The entropy of a quantum system is a measure of its randomness, and has applications in measuring quantum entanglement. We study the problem of measuring the von Neumann entropy, $S(\rho )$, and Rényi entropy, $S_{\alpha }(\rho )$ of an unknown mixed quantum state $\rho$ in $d$ dimensions, given access to independent copies of $\rho$. We provide an algorithm with copy complexity $O(d^{2/\alpha })$ for estimating $S_{\alpha }(\rho )$ for $\alpha <1$, and copy complexity $O(d^2)$ for estimating $S(\rho )$, and $S_{\alpha }(\rho )$ for non-integral $\alpha >1$. These bounds are at least quadratic in $d$, which is the order dependence on the number of copies required for learning the entire state $\rho$. For integral $\alpha >1$, on the other hand, we provide an algorithm for estimating $S_{\alpha }(\rho )$ with a sub-quadratic copy complexity of $O(d^{2-2/\alpha })$. We characterize the copy complexity for integral $\alpha >1$ up to constant factors by providing matching lower bounds. For other values of $\alpha$, and the von Neumann entropy, we show lower bounds on the algorithm that achieves the upper bound. This shows that we either need new algorithms for better upper bounds, or better lower bounds to tighten the results. For non-integral $\alpha$, and the von Neumann entropy, we consider the well known Empirical Young Diagram (EYD) algorithm, which is the analogue of empirical plug-in estimator in classical distribution estimation. As a corollary, we strengthen a lower bound on the copy complexity of the EYD algorithm for learning the maximally mixed state by showing that the lower bound holds with exponential probability (which was previously known to hold with a constant probability). For integral $\alpha >1$, we provide new concentration results of certain polynomials that arise in Kerov algebra of Young diagrams.

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• Multivariate trace estimation in constant quantum depth
• Quantum simulation of in-medium QCD jets: momentum broadening, gluon production, and entropy growth
• Estimating Quantum Mutual Information Through a Quantum Neural Network
• Entanglement theory with limited computational resources
• Linear programming with unitary-equivariant constraints
• The topology of data hides in quantum thermal states
• Time-Efficient Quantum Entropy Estimator via Samplizer
• Quantum Lower Bounds by Sample-to-Query Lifting
• Measuring quantum relative entropy with finite-size effect
• Disentangling quantum neural networks for unified estimation of quantum entropies and distance measures
• Testing identity of collections of quantum states: sample complexity analysis
• The Cramér-Rao approach and global quantum estimation of bosonic states
• Entropy density benchmarking of near-term quantum circuits
• Variational quantum simulation of the quantum critical regime
• Performance Guarantees for Quantum Neural Estimation of Entropies