Continuous phase-space methods on discrete phase spaces

TL;DR

This paper connects discrete quasiprobability distributions with continuous $SU(N)$ distributions using phase-point operators, and applies this to improve the treatment of diverging trajectories in quantum dynamics.

Contribution

It introduces a method to derive discrete quasiprobability distributions from continuous $SU(N)$ distributions via phase-point operators, bridging discrete and continuous phase-space formalisms.

Findings

Discrete quasiprobability distributions can be obtained as discretizations of continuous $SU(N)$ distributions.

Discretization of the positive-P function helps address diverging trajectories in quantum simulations.

Semiclassical approximation effectively describes long-time dynamics of the transverse-field Ising chain.

Abstract

We show that discrete quasiprobability distributions defined via the discrete Heisenberg-Weyl group can be obtained as discretizations of the continuous $SU(N)$ quasiprobability distributions. This is done by identifying the phase-point operators with the continuous quantisation kernels evaluated at special points of the phase space. As an application we discuss the positive-P function and show that its discretization can be used to treat the problem of diverging trajectories. We study the dissipative long-range transverse-field Ising chain and show that the long-time dynamics of local observables is well described by a semiclassical approximation of the interactions.