User-friendly confidence regions for quantum state tomography

TL;DR

This paper introduces a new method for constructing confidence regions in quantum state tomography that are efficient, versatile, and easy to report, improving practical state estimation.

Contribution

It provides a novel confidence region construction with optimal sample efficiency, broad applicability, and simple ellipsoid representation, based on a vector Bernstein inequality.

Findings

Achieves asymptotically optimal sample cost.

Applicable to any measurement scheme.

Provides easy-to-interpret ellipsoid confidence regions.

Abstract

Quantum state tomography is the standard technique for reconstructing a quantum state from experimental data. In the regime of finite statistics, experimental data cannot give perfect information about the quantum state. A common way to express this limited knowledge is by providing confidence regions in the state space. Though other confidence regions were previously proposed, they are either too wasteful to be of practical interest, cannot easily be applied to general measurement schemes, or are too difficult to report. Here we construct confidence regions that solve these issues, as they have an asymptotically optimal sample cost and good performance for realistic parameters, are applicable to any measurement scheme, and can be described by an ellipsoid in the space of Hermitian operators. Our construction relies on a vector Bernstein inequality and bounds with high probability the Hilbert-Schmidt norm error of sums of multinomial samples transformed by linear maps.