Superposing compass states for asymptotic isotropic sub-Planck phase-space sensitivity

TL;DR

This paper introduces generalized compass states formed by superposing multiple compass states, demonstrating that as the number of states increases, the sensitivity to phase-space displacement becomes isotropic, enhancing quantum measurement precision.

Contribution

The paper develops a new class of superposed compass states and proves their isotropic phase-space sensitivity in the limit of infinite superpositions.

Findings

Derived Wigner functions for generalized compass states

Provided approximate overlaps with displaced states

Showed isotropic sensitivity as the number of superpositions approaches infinity

Abstract

Compass states deliver sub-Planck phase-space structure in the sense that sensitivity to phase-space displacement is superior to the sensitivity of displacing the vacuum state in any direction, but this sensitivity is anisotropic: better sensitivity for some directions of phase-space displacement vs others. Here we introduce generalised compass states as superpositions of $n$ compass states, with each oriented by $\frac\pi{2n}$ with respect to its predecessor. Specifically, we derive Wigner functions for these generalised compass states and approximate closed-form expressions for overlaps between generalised compass states and their displaced counterparts. Furthermore, we show that generalised compass states, in the limit $n\to\infty$, display isotropic sensitivity to phase-space displacement in any direction.