Beating noise with abstention in state estimation

TL;DR

This paper demonstrates that allowing abstentions in quantum state estimation significantly improves fidelity under noisy measurements, especially for large numbers of qubits, by effectively reducing noise impact.

Contribution

It introduces an optimal abstention-based protocol for noisy quantum state estimation and derives analytical expressions for fidelity and abstention probability asymptotically.

Findings

Abstention improves average fidelity in noisy quantum state estimation.

Fidelity gain decreases exponentially with the number of qubits, characterized by a relative entropy.

Estimation with abstention is especially beneficial when noise increases with system size.

Abstract

We address the problem of estimating pure qubit states with non-ideal (noisy) measurements in the multiple-copy scenario, where the data consists of a number N of identically prepared qubits. We show that the average fidelity of the estimates can increase significantly if the estimation protocol allows for inconclusive answers, or abstentions. We present the optimal such protocol and compute its fidelity for a given probability of abstention. The improvement over standard estimation, without abstention, can be viewed as an effective noise reduction. These and other results are exemplified for small values of N. For asymptotically large N, we derive analytical expressions of the fidelity and the probability of abstention, and show that for a fixed fidelity gain the latter decreases with N at an exponential rate given by a Kulback-Leibler (relative) entropy. As a byproduct, we obtain an asymptotic expression in terms of this very entropy of the probability that a system of N qubits, all prepared in the same state, has a given total angular momentum. We also discuss an extreme situation where noise increases with N and where estimation with abstention provides a most significant improvement as compared to the standard approach.