Minimax quantum tomography: the ultimate bounds on accuracy

Abstract

A minimax estimator has the minimum possible error ("risk") in the worst case. We construct the first minimax estimators for quantum state tomography with relative entropy risk. The minimax risk of non-adaptive tomography scales as $O(1/\sqrt{N})$, in contrast to that of classical probability estimation which is $O(1/N)$. We trace this deficiency to sampling mismatch: future observations that determine risk may come from a different sample space than the past data that determine the estimate. This makes minimax estimators very biased, and we propose a computationally tractable alternative with similar behavior in the worst case, but superior accuracy on most states.

References in corpus (4)

• Experimental Quantum State Tomography of Optical Fields and Ultrafast Statistical Sampling
• The Quantum Chernoff Bound
• Quantum Tomography via Compressed Sensing: Error Bounds, Sample Complexity, and Efficient Estimators
• Quantum state estimation with informationally overcomplete measurements

Cited by in corpus (12)

• Multi-parameter estimation beyond Quantum Fisher Information
• Sample-optimal tomography of quantum states
• Optimal verification of entangled states with local measurements
• A comparative study of estimation methods in quantum tomography
• Capacity estimation and verification of quantum channels with arbitrarily correlated errors
• Behavior of the maximum likelihood in quantum state tomography
• Quantum metrology in the finite-sample regime
• Quantum chi-squared tomography and mutual information testing
• Maximum a posteriori estimation of quantum states
• Discrimination of correlated and entangling quantum channels with selective process tomography
• Minimax estimation of qubit states with Bures risk
• Adaptive tomography of qubits: Purity versus statistical fluctuations