Discrete Wigner functions and quantum computational speedup

Abstract

Gibbons et al. [Phys. Rev. A 70, 062101 (2004)] have recently defined a class of discrete Wigner functions W to represent quantum states in a finite Hilbert space dimension d. I characterize the set C d of states having non-negative W simultaneously in all definitions of W in this class. For d ≤ 5 I show C d is the convex hull of stabilizer states. This supports the conjecture that negativity of W is necessary for exponential speedup in pure-state quantum computation.

Mutually Unbiased Bases and the Complementarity Polytope

“…Note that this observation provides the formal proof that the faces really are faces, that is that they consist of points belonging to the boundary of the polytope. Note also that this very way of characterizing the facets of the polytope was used by Galvão [10], although he expressed it a little differently-his statement is that the discrete Wigner function, defined by Wootters [8], vanishes on facets of the polytope.…”

Classicality in discrete Wigner functions

“…Following [20], let us now characterize the set of states having non-negative discrete Wigner functions W simultaneously in all definitions proposed by Gibbons et al [11] for power-of-prime dimension d.…”

“…Our first result is a complete characterization of the set of quantum states having non-negative discrete Wigner functions W . This is done by proving a conjecture presented by one of us in [20] (a related discussion in a somewhat different context, using concepts in high-dimensional geometry appeared in [21,22]). Our proof is elementary and constructive, and shows that the only pure states with non-negative W are stabilizer states, i.e.…”

Extended reduction criterion and lattice states

“…By means of the extended reduction map, we shall associate most of them with specific properties of the corresponding lattice states like that of being separable, NPT-entangled or PPT-entangled. Because of this, besides enriching the phenomenology of entangled states, the class of lattice states may also turn out to be a useful arena for the approach based on the discrete Wigner functions [13,14,15,16] The plan of the paper is as follows: we shall first introduce the lattice states and their presently known properties; then, we shall improve their classification by using the extended reduction criterion; finally, we shall completely characterize some subclasses of them and discuss which ones need stronger entanglement witnesses than those available so far.…”