Minimax quantum tomography: the ultimate bounds on accuracy

TL;DR

This paper develops the first minimax estimators for quantum state tomography under relative entropy risk, revealing fundamental limits on accuracy and proposing a practical alternative with improved average performance.

Contribution

It introduces the first minimax estimators for quantum tomography and analyzes their worst-case scaling, highlighting sampling mismatch issues and proposing a more accurate, computationally feasible alternative.

Findings

Minimax risk scales as O(1/√N), worse than classical O(1/N).

Sampling mismatch causes bias in minimax estimators.

Proposed alternative estimator has better average accuracy with similar worst-case behavior.

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.