Discrete Wigner function

This work is intended as an attempt to present two essentially different constructions of harmonic oscillator states in a FD Hilbert space. We propose some new definitions of the states and find their explicit forms in the Fock representation. For the convenience of the reader, we also bring together several known FD quantum-optical states, thus making our exposition more self-contained. We shall discuss FD coherent states, FD phase coherent states, FD displaced number states, FD Schrodinger cats, and FD squeezed vacuum. We shall show some intriguing properties of the states with the help of the discrete Wigner function. 

The phase states can also be used in construction of a discrete Wigner function as will be described in Section III. The FD annihilation and creation operators in are defined by... 

Figure 1. Examples of discrete Wigner function on a torus in 19-dimensional Hilbert space i s = 18) (a) vacuum 0) (b) single-photon number state 1) (c) FD preferred phase state ( phase vacuum ) 0o)(5) (d) FD coherent state, a) ss a) (e) FD displaced number state, oc, 1)(j) oc, 1)(j) (0 FD phase coherent state, P, 0o)(j) P,0o)(5), with equal displacement parameters, 7. — 7. — P — P — 0.5 and 0o = 0. The darker is a region, the higher is the value of the Wigner function.

For a better understanding of the discrete Wigner function, let us recall its close correspondence to the discrete Pegg-Bamett phase distribution [26]... 

Although many properties of the phase CS are known by now, for their better understanding it is very useful to analyze graphs of their quasidistributions. The discrete Wigner function, as defined by Wootters [55] (see also Ref. 57), takes the following form for s > 1... 

An alternative way for describing the discrete Wigner function is the one given in terms of the discrete phase-point operator, given by [26] ... 

U. Leonhardt, Quantum-state tomography and discrete Wigner function, Phys. Rev. Lett. 74 (1995) 4101. 

U. Leonhardt, Discrete Wigner function and quantum-state tomography, Phys. Rev. A S3 (1996) 2998. 

Figure 5.15 Generic scattering circuit implemented in Ref. [29] to measure the discrete Wigner function. Adapted with permission from [29].

As a last topic in this chapter, we review the work of Miquel et al. [29] who implemented a quantum circuit to measure the discrete Wigner function of a two-qubit system through NMR. In Chapter 3 it was made a brief introduction to Wigner functions and the quantum processing of information in phase space. 

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