Discrete phase space based on finite fields

Journal Article · Wed Dec 01 04:00:00 EST 2004 · Physical Review. A

DOI:https://doi.org/10.1103/PhysRevA.70.062101· OSTI ID:20649890

Gibbons, Kathleen S ^[1]; Hoffman, Matthew J ^[1]; Wootters, William K ^[1]

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1. Department of Physics, Williams College, Williamstown, Massachusetts 01267 (United States)

The original Wigner function provides a way of representing in phase space the quantum states of systems with continuous degrees of freedom. Wigner functions have also been developed for discrete quantum systems, one popular version being defined on a 2Nx2N discrete phase space for a system with N orthogonal states. Here we investigate an alternative class of discrete Wigner functions, in which the field of real numbers that labels the axes of continuous phase space is replaced by a finite field having N elements. There exists such a field if and only if N is a power of a prime; so our formulation can be applied directly only to systems for which the state-space dimension takes such a value. Though this condition may seem limiting, we note that any quantum computer based on qubits meets the condition and can thus be accommodated within our scheme. The geometry of our NxN phase space also leads naturally to a method of constructing a complete set of N+1 mutually unbiased bases for the state space.

• OSTI ID: 20649890
• Journal Information: Physical Review. A, Journal Name: Physical Review. A Journal Issue: 6 Vol. 70; ISSN 1050-2947; ISSN PLRAAN
• Country of Publication: United States
• Language: English

Related Subjects

74 ATOMIC AND MOLECULAR PHYSICS
DEGREES OF FREEDOM
FUNCTIONS
GEOMETRY
INFORMATION THEORY
PHASE SPACE
QUANTUM MECHANICS
WIGNER DISTRIBUTION