Classicality in discrete Wigner functions

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 Hilbert space with finite dimension. We show that the only pure states having non-negative W for all such functions are stabilizer states, as conjectured by one of us [Phys. Rev. A 71, 042302 (2005)]. We also show that the unitaries preserving non-negativity of W for all definitions of W form a subgroup of the Clifford group. This means pure states with non-negative W and their associated unitary dynamics are classical in the sense of admitting an efficient classical simulation scheme using the stabilizer formalism.

Operational quasiprobabilities for qudits

“…[13,14]. The maximal degree of nonclassicality is N max = ( √ 2 − 1)/4, as in the case of the two complementary observables, which is obtained by ρ ∈ {(±1,±1,0)/ √ 2,(±1,0,±1)/ √ 2,(0,±1,±1)/ √ 2}.…”

“…This incommensurability makes it difficult to interpret the nonclassicality of a quasiprobability distribution. This problem remains unsolved in the approaches of generalizing quasiprobability functions to discrete systems [13,14]. On the other hand, consider a joint probability distribution in the sequence of measuring p first and x later [15]:…”

Quantum Fourier transform and tomographic Rényi entropic inequalities

“…It was widely discussed in connection with constructing the Wigner function for a finite Hilbert space and in quantum cryptography [27], [38]- [41].…”

Wigner formalism for a particle on an infinite lattice: dynamics and spin

“…In the finite-dimensional case and for odd dimensions, Gross showed [50] that the only pure states with positive WFs are stabilizer states. The presence of negative values in the WF has been, in this case, connected to a quantum resource related to a possible quantum speedup [28,51] or the nonsimulability of certain quantum computations involving states with nonpositive WFs [7,31].…”