Fractional angular momentum and quasi-probability densities for angular degrees of freedom

In the present study, we consider quasi-probability densities W[θ, p] and W1/2[θ, p] for pure states depending on two classical parameters θ and p, where −π ≤ θ ≤ π and p can take any real number. For integer values of p, the corresponding marginal distributions W[p] and W1/2[p] are positive and in accordance with Born’s rule in quantum mechanics for an angular momentum observable L. For the state ψ(θ) in Skagerstam and Rekdal [J. Opt. 26, 095201 (2024); arXiv:2312.16535v1 [quant-ph]], which can have half-integer angular momentum expectation values, negative values of W[θ, p] and W1/2[θ, p] can reveal non-classical features of ψ(θ) but in an ambiguous manner. In line with Franke-Arnold et al. [New J. Phys. 6, 103 (2004)], it is shown that experimental data of the uncertainties Δθ and ΔL can be sufficient to reveal quantum-mechanical features of such states ψ(θ) without necessarily making use of quasi-probability densities.