Hudson's theorem for finite-dimensional quantum systems
- 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, Vol. 47, Issue 12; Other Information: DOI: 10.1063/1.2393152; (c) 2006 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); ISSN 0022-2488
- Country of Publication:
- United States
- Language:
- English
Similar Records
Discrete Wigner functions and quantum computational speedup
Local equivalence, surface-code states, and matroids