Wigner distributions for finite state systems without redundant phase point operators

Abstract

We set up Wigner distributions for $N$ state quantum systems following a Dirac inspired approach. In contrast to much of the work on this case, requiring a $2N\times 2N$ phase space, particularly when $N$ is even, our approach is uniformly based on an $N\times N$ phase space grid and thereby avoids the necessity of having to invoke a `quadrupled' phase space and hence the attendant redundance. Both $N$ odd and even cases are analysed in detail and it is found that there are striking differences between the two. While the $N$ odd case permits full implementation of the marginals property, the even case does so only in a restricted sense. This has the consequence that in the even case one is led to several equally good definitions of the Wigner distributions as opposed to the odd case where the choice turns out to be unique.

Latex, 14 pages

References in corpus (8)

• Hudson's Theorem for finite-dimensional quantum systems
• Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations
• Framed Hilbert space: hanging the quasi-probability pictures of quantum theory
• Quasiprobability distribution functions for periodic phase-spaces: I. Theoretical Aspects
• Discrete phase-space structure of $n$-qubit mutually unbiased bases
• Spectra of phase point operators in odd prime dimensions and the extended Clifford group
• The isotropic lines of Z_{d}^{2}
• Semi-classical Scar functions in phase space

Cited by in corpus (9)

• Weak mutually unbiased bases
• Coherent states in projected Hilbert spaces
• Aligned SICs and embedded tight frames in even dimensions
• Classical codes in quantum state space
• Counting stabiliser codes for arbitrary dimension
• Transition probabilities and transition rates in discrete phase space
• Optimal quantum tomography with constrained elementary measurements arising from unitary bases
• A quantum prediction as a collection of epistemically restricted classical predictions
• Grand Unification of All Discrete Wigner Functions on $d\times d$ Phase Space