Enhanced quantum parameter estimation based on the Hardy paradox

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.

Figures (3)

• Figure 1: Post-selected QFI $I_\mathrm{select}$ and contextual probability $P(a,a|\phi_0)$ as a function of the free parameter $|\langle 0 | a \rangle|^2$ characterizing the Hardy paradox. The dotted lines indicate the location of the maximal values.
• Figure 2: Conversion efficiency $\eta$ as a function of the free parameter $|\langle 0 | a \rangle|^2$. The conversion efficiency drops sharply just below $|\langle 0 | a \rangle|^2=0.75$, indicating that contextuality cannot be accessed for enhanced parameter estimation when $|\langle 0 | a \rangle|^2$ is above that value.
• Figure 3: Ratio between QFI $I_\mathrm{select}$ enhanced by post-selection $\hat{\Pi}$ and QFI $I_0$ without post-selection. The value of this ratio measures the enhancement of the QFI induced by the post-selection. Post-selection enhances QFI for $I_\mathrm{select}/I_\mathrm{0}>1$, observed in the range of $[0.1505,0.7384]$ for the free parameter $|\langle0|a\rangle|^2$.