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Physical Review A

Parametrized discrete phase-space functions

Tomáš Opatrný, D.‐G. Welsch, Vladimír Bužek

Published Jun 1, 1996• 44 citations
Publisher

Abstract

Using discrete displacement-operator expansion, s-parametrized phase-space functions associated with the operators in a finite-dimensional Hilbert space are introduced and their properties are studied. In particular, the phase-space functions associated with the density operator can be regarded as quasidistributions whose properties are similar to those of the well-known quasidistributions in the continuous phase space. So the Q function (s=-1) is non-negative and can be measured directly in particular experiments, whereas the P function (s=1) corresponds to the diagonal form of the density operator in an overcomplete basis. Except for the W function (s=0), the introduction of discrete phase-space functions requires the choice of a special reference state. We finally present a simple model for measuring the discrete Q function. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.53.3822

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Using discrete displacement-operator expansion, s-parametrized phase-space functions associated with the operators in a finite-dimensional Hilbert space are introduced and their properties are studied. In particular, the phase-space functions associated with the density operator can be regarded as quasidistributions whose properties are similar to those of the well-known quasidistributions in the continuous phase space. So the Q function (s=-1) is non-negative and can be measured directly in particular experiments, whereas the P function (s=1) corresponds to the diagonal form of the density operator in an overcomplete basis. Except for the W function (s=0), the introduction of discrete phase-space functions requires the choice of a special reference state. We finally present a simple model for measuring the discrete Q function. \textcopyright{} 1996 The American Physical Society.
We present a Q function of a state of a quantum-mechanical system in a finite-dimensional Hilbert space. This discrete Q function is defined with the help of the W\'odkiewicz concept of propensities, i.e., we define the Q function as a discrete convolution of two Wigner functions based on Wootter's formalism, one of the state itself and one of the filter state. The discrete Q function takes nonnegative values in all ``points'' of the discrete phase space and is normalized and it is possible to reconstruct from it the density operator of the state under consideration. We analyze Q-function graphs for several states of interest.
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 NNN. 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 PPP 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.
By means of a mod(N)\mathrm{mod}(N)mod(N)-invariant operator basis, sss-parametrized phase-space functions associated with bounded operators in a finite-dimensional Hilbert space are introduced in the context of the extended Cahill-Glauber formalism, and their properties are discussed in details. The discrete Glauber-Sudarshan, Wigner, and Husimi functions emerge from this formalism as specific cases of sss-parametrized phase-space functions where, in particular, a hierarchical process among them is promptly established. In addition, a phase-space description of quantum tomography and quantum teleportation is presented and new results are obtained.
The evolution of the discrete Wigner function is formally similar to a probabilistic process, but the transition probabilities, like the discrete Wigner function itself, can be negative. We investigate these transition probabilities, as well as the transition rates for a continuous process, aiming particularly to give simple criteria for deciding when a set of such quantities corresponds to a legitimate quantum process. We also show how the transition rates for any Hamiltonian evolution can be worked out by expanding the Hamiltonian as a linear combination of displacement operators in the discrete phase space.
A discussion of discrete Wigner functions in phase space related to mutually unbiased bases is presented. This approach requires mathematical assumptions, which limits it to systems with density matrices defined on complex Hilbert spaces of dimension p n where p is a prime number. With this limitation, it is possible to define a phase space and Wigner functions in close analogy to the continuous case. That is, we use a phase space that is a direct sum of n two-dimensional vector spaces each containing p 2 points. This is in contrast to the more usual choice of a two-dimensional phase space containing p 2 n points. A useful aspect of this approach is that we can relate complete separability of density matrices and their Wigner functions in a natural way. We discuss this in detail for bipartite systems and present the generalization to arbitrary numbers of subsystems when p is odd. Special attention is required for two qubits ( p = 2) and our technique fails to establish the separability property for more than two qubits. Finally, we give a brief discussion of Hamiltonian dynamics in the language developed in the paper.