Hudson's theorem for finite-dimensional quantum systems

Journal Article · Fri Dec 15 04:00:00 EST 2006 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.2393152· OSTI ID:20861553

Gross, D ^[1]

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1. Institute for Mathematical Sciences, Imperial College London, London SW7 2BW, United Kingdom and QOLS, Blackett Laboratory, Imperial College London, London SW7 2BW (United Kingdom)

We show that, on a Hilbert space of odd dimension, the only pure states to possess a non-negative Wigner function are stabilizer states. The Clifford group is identified as the set of unitary operations which preserve positivity. The result can be seen as a discrete version of Hudson's theorem. Hudson established that for continuous variable systems, the Wigner function of a pure state has no negative values if and only if the state is Gaussian. Turning to mixed states, it might be surmised that only convex combinations of stabilizer states give rise to non-negative Wigner distributions. We refute this conjecture by means of a counterexample. Further, we give an axiomatic characterization which completely fixes the definition of the Wigner function and compare two approaches to stabilizer states for Hilbert spaces of prime-power dimensions. In the course of the discussion, we derive explicit formulas for the number of stabilizer codes defined on such systems.

• OSTI ID: 20861553
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 12 Vol. 47; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
FOURIER TRANSFORMATION
FUNCTIONS
GROUP THEORY
HILBERT SPACE
MIXED STATE
QUANTUM MECHANICS
WIGNER DISTRIBUTION