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Journal of Physics G Nuclear and Particle Physics

Joint probabilities for an aligned spin-1 system

A. R. Usha Devi, Swarnamala Sirsi, G Devi

Published Nov 1, 1994• 5 citations
Publisher

Abstract

Following Margenau and Hill (1961) we adduce arguments to show that an aligned spin-1 system (for e.g. 2 H or 14 N nuclei in external electric quadrupole fields) can be viewed in terms of bivariate probability distributions obtained by identifying the associated characteristic function.
DOI
10.1088/0954-3899/20/11/014

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In contrast to the canonically conjugate variates q, p representing the position and momentum of a particle in the phase space distributions, the three Cartesian components, Jx, Jy, Jz of a spin-j system constitute the mutually non-commuting variates in the quasi-probabilistic spin distributions. It can be shown that a univariate spin distribution is never squeezed and one needs to look into either bivariate or trivariate distributions for signatures of squeezing. Several such distributions result if one considers different characteristic functions or moments based on various correspondence rules. As an example, discrete probability distribution for an arbitrary spin-1 assembly is constructed using Wigner-Weyl and Margenau-Hill correspondence rules. It is also shown that a trivariate spin-1 assembly resulting from the exposure of nucleus with non-zero quadrupole moment to combined electric quadrupole field and dipole magnetic field exhibits squeezing in certain cases.
Measurement incompatibility and the negativity of quasi-probability distributions both demonstrate the signature of nonclassicality. However, their relations largely remained qualitative, and no explicit operational connection has been established. We provide an operational link between measurement incompatibility and the Margenau-Hill(MH) quasi-probability distribution associated with two dichotomic observables. { We first derive the joint measurability (measurement compatibility) condition for any pair of dichotomic observables in arbitrary finite dimension ddd.} We then rigorously prove that for any pair of unsharp dichotomic measurements in dimension ddd, the positivity of the MH distribution is equivalent to joint measurability \emph{i.e.}, the MH distribution is positive \emph{if and only if} the measurements are jointly measurable. We further introduce the MH-like quasi-probability distribution for nnn dichotomic unsharp observables in arbitrary finite dimension and derive a sufficient condition of joint-measurability when the observables are mutually anticommuting. Finally, we propose an interferometric setup, inspired by quantum-switch architectures, that directly reconstructs MH quasi-probability which in turn provides a test of incompatibility.
A class of signed joint probability measures for n arbitrary quantum observables is derived and studied based on quasicharacteristic functions with symmetrized operator orderings of Margenau-Hill type. It is shown that the Wigner distribution associated with these observables can be rigorously approximated by such measures. These measures are given by affine combinations of Dirac delta distributions supported over the finite spectral range of the quantum observables and give the correct probability marginals when coarse-grained along any principal axis. We specialize to bivariate quasiprobability distributions for the spin measurements of spin- 1 / 2 particles and derive their closed-form expressions. As a side result, we point out a connection between the convergence of these particle approximations and the Mehler-Heine theorem. Finally, we interpret the supports of these quasiprobability distributions in terms of repeated thought experiments.