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

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.