Notes on the Qubit Phase Space and Discrete Symplectic Structures

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.

12 pages

References in corpus (4)

• Classicality in discrete Wigner functions
• Discrete Wigner functions and quantum computational speedup
• Wigner Functions and Separability for Finite Systems
• Picturing Qubits in Phase Space

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• Minimal sets of dequantizers and quantizers for finite-dimensional quantum systems
• Unitary and Non-Unitary Matrices as a Source of Different Bases of Operators Acting on Hilbert Spaces
• Transition probabilities and transition rates in discrete phase space