1 research outputs found
Hudson's Theorem for finite-dimensional quantum systems
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 counter-example. 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.Comment: 17 pages, 3 figures; References updated. Title changed to match
published version. See also quant-ph/070200