Measuring quantum relative entropy with finite-size effect

Abstract

We study the estimation of relative entropy $D(\rho \Vert \sigma )$ when $\sigma$ is known. We show that the Cramér-Rao type bound equals the relative varentropy. Our estimator attains the Cramér-Rao type bound when the dimension $d$ is fixed. It also achieves the sample complexity $O(d^2)$ when the dimension $d$ increases. This sample complexity is optimal when $\sigma$ is the completely mixed state. Also, it has time complexity $O(d^6 \polylog d)$. Our proposed estimator unifiedly works under both settings.

References in corpus (31)

• A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks
• Exponential decreasing rate of leaked information in universal random privacy amplification
• Verifiable measurement-only blind quantum computing with stabilizer testing
• Multi-parameter estimation beyond Quantum Fisher Information
• Sample-optimal tomography of quantum states
• Efficient Quantum Circuits for Schur and Clebsch-Gordan Transforms
• Estimating the spectrum of a density operator
• Second-order asymptotics for quantum hypothesis testing
• The Spectra of Density Operators and the Kronecker Coefficients of the Symmetric Group
• Optimal verification of entangled states with local measurements
• Optimal sequence of quantum measurements in the sense of Stein's lemma in quantum hypothesis testing
• Correlation Detection and an Operational Interpretation of the Renyi Mutual Information
• Efficient Verification of Pure Quantum States in the Adversarial Scenario
• Quantum universal variable-length source coding
• Uncertainty Relation Revisited from Quantum Estimation Theory
• Quantum Semiparametric Estimation
• Comparison between the Cramer-Rao and the mini-max approaches in quantum channel estimation
• A quantum version of Sanov's theorem
• On Nonzero Kronecker Coefficients and their Consequences for Spectra
• Quantum state estimation with nuisance parameters
• Exponents of quantum fixed-length pure state source coding
• Group theoretical study of LOCC-detection of maximally entangled state using hypothesis testing
• An efficient high dimensional quantum Schur transform
• Measuring Quantum Entropy
• One from many: Estimating a function of many parameters
• Hypothesis Testing on Invariant Subspaces of the Symmetric Group, Part I - Quantum Sanov's Theorem and Arbitrarily Varying Sources
• Robust and efficient verification of graph states in blind measurement-based quantum computation
• Fast Quantum Algorithms for Trace Distance Estimation
• Communication Cost of Quantum Processes
• Quantum chi-squared tomography and mutual information testing
• Efficient qubit phase estimation using adaptive measurements

Cited by in corpus (2)

• Time-Efficient Quantum Entropy Estimator via Samplizer
• The Cramér-Rao approach and global quantum estimation of bosonic states