Enhanced quantum parameter estimation based on the Hardy paradox

TL;DR

This paper explores how the Hardy paradox and quantum weak values can be used to enhance phase estimation in quantum metrology, revealing the conditions under which this enhancement is most effective.

Contribution

It introduces a novel post-selected quantum metrology framework inspired by the Hardy paradox, linking weak values to phase estimation enhancement.

Findings

Enhanced sensitivity depends on the anomalous weak value.

Efficiency decreases when expectation values differ from the inverse of the weak value.

A detailed understanding of weak and expectation values is crucial for optimization.

Abstract

Statistical paradoxes such as the Hardy paradox and the enhancement of phase estimation via post-selection both draw upon the same non-classical features of quantum statistics described by non-positive quasi-probabilities. In this paper, we introduce a post-selected quantum metrology scenario where the initial state, the dynamics associated with the phase shift, and the post-selection are all inspired by the Hardy paradox. Specifically, we identify an anomalous weak value that is characteristic of both the Hardy paradox and the potential enhancement of sensitivity by the post-selection. We find that the efficiency of the enhancement is reduced when the expectation value associated with the anomalous weak value is different from the inverse of this value. We conclude that the relation between enhanced phase estimation and the Hardy paradox requires a detailed understanding of the relation between weak values and expectation values.