Notes on the Qubit Phase Space and Discrete Symplectic Structures

TL;DR

This paper explores the structure of discrete phase spaces for qubits and finite-dimensional systems, focusing on the Moyal product, non-commutative geometry, and the comparison between quantum and classical dynamics.

Contribution

It provides explicit expressions for the Moyal bracket in discrete phase spaces and demonstrates its role as the only consistent symplectic structure, extending analysis to prime dimensions.

Findings

The classical Poisson bracket fails to satisfy the Jacobi identity in discrete settings.

The Moyal bracket acts as the unique consistent symplectic structure in discrete phase spaces.

Explicit formulas for the Moyal bracket in qubit and prime-dimensional systems.

Abstract

We start from Wootter's construction of discrete phase spaces and Wigner functions for qubits and more generally for finite dimensional Hilbert spaces. We look at this framework from a non-commutative space perspective and we focus on the Moyal product and the differential calculus on the discrete phase spaces. In particular, the qubit phase space provides the simplest example of a four-point non-commutative phase space. We give an explicit expression of the Moyal bracket as a differential operator. We then compare the quantum dynamics encoded by the Moyal bracket to the classical dynamics: we show that the classical Poisson bracket does not satisfy the Jacobi identity thus leaving the Moyal bracket as the only consistent symplectic structure. We finally generalizes our analysis to Hilbert spaces of prime dimensions d and their associated d*d phase spaces.