We introduce quantum states associated with single phase space points in the Wigner formalism for finite-dimensional spaces. We consider both continuous and discrete Wigner functions. This analysis provides a procedure for a direct practical observation of the Wigner functions for states and transformations without inversion formulas.

Physical Review AMay 20, 200416 citations

Wigner transform for continuous and for discrete finite-dimensional Hilbert spaces

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arXiv:2402.09922

Interpreting symplectic linear transformations in a two-qubit phase space

William K. Wootters

Published Feb 15, 2024• 0 citations

arXiv HTML PDF

Abstract

d \times d

phase space when the Hilbert-space dimension

d

is odd. It does not hold for a

d \times d

phase space if the dimension is even. Here we show, though, that a generalized version of this correspondence does apply in the case of a two-qubit phase space. In this case, a symplectic linear permutation of the points of the phase space, together with a certain reinterpretation of the Wigner function, is equivalent to a unitary transformation.

DOI: 10.48550/arxiv.2402.09922

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d \times d

phase space when the Hilbert-space dimension

d

is odd. It does not hold for a

d \times d

We study the Wigner Transform (WT) in the case of an infinite-dimensional continuous Hilbert space and a discrete Hilbert space of finite prime-number dimensionality

N

. In the discrete case we define a family of WT's as a function of a phase parameter. It is only for a specific value of the parameter that all the properties we have examined, including the matrix elements of the phase-space operator

P

and the relation between Wigner's function and Kirkwood's distribution, have a parallel in both types of Hilbert space. This close parallel was found constructing the two WT's in terms of a complete set of operators having similar properties, in the sense that the position and momentum operators appear disentangled in the two situations. We give a geometric interpretation of the properties involving the phase-space operator in the discrete case.