Optimal error intervals for properties of the quantum state

TL;DR

This paper introduces optimal error intervals for directly estimating specific properties of quantum states from data, providing a statistically rigorous measure of accuracy that is computationally feasible and illustrated with qubit examples.

Contribution

It develops a method for constructing optimal error intervals for quantum state properties, bypassing full state estimation and enabling more efficient and meaningful property estimation.

Findings

Optimal error intervals maximize likelihood for data and specified size.

The method is applicable to single-qubit and two-qubit systems.

It includes an iterative algorithm for reliable marginal likelihood computation.

Abstract

Quantum state estimation aims at determining the quantum state from observed data. Estimating the full state can require considerable efforts, but one is often only interested in a few properties of the state, such as the fidelity with a target state, or the degree of correlation for a specified bipartite structure. Rather than first estimating the state, one can, and should, estimate those quantities of interest directly from the data. We propose the use of optimal error intervals as a meaningful way of stating the accuracy of the estimated property values. Optimal error intervals are analogs of the optimal error regions for state estimation [New J. Phys. 15, 123026 (2013)]. They are optimal in two ways: They have the largest likelihood for the observed data and the pre-chosen size, and are the smallest for the pre-chosen probability of containing the true value. As in the state situation, such optimal error intervals admit a simple description in terms of the marginal likelihood for the data for the properties of interest. Here, we present the concept and construction of optimal error intervals, report on an iterative algorithm for reliable computation of the marginal likelihood (a quantity difficult to calculate reliably), explain how plausible intervals --- a notion of evidence provided by the data --- are related to our optimal error intervals, and illustrate our methods with single-qubit and two-qubit examples.