Abstract
We consider the phase space for $n$ identical qudits (each one of dimension $d$, with $d$ a primer number) as a grid of ${d}^{n}\ifmmode\times\else\texttimes\fi{}{d}^{n}$ points and use the finite Galois field $\text{GF}({d}^{n})$ to label the corresponding axes. The associated displacement operators permit to define $s$-parametrized quasidistributions on this grid, with properties analogous to their continuous counterparts. These displacements allow also for the construction of finite coherent states, once a fiducial state is fixed. We take this reference as one eigenstate of the discrete Fourier transform and study the factorization properties of the resulting coherent states. We extend these ideas to include discrete squeezed states, and show their intriguing relation with entangled states of different qudits.
Highlights
The concept of phase-space representation of quantum mechanics, introduced in the pioneering works of Weyl [1], Wigner [2], and Moyal [3], is a very useful and enlightening approach that sheds light on the correspondence between quantum and classical worlds.Numerous applications of the phase-space methods to physical problems have been extensively discussed in the last decades [4, 5, 6, 7]
Even if this picture is quite popular, especially when applied to qubits, one can rightly argue that there is a lot of information redundancy there and that the phase space should be a grid of points, as one could expect for a truly discrete system
The strategy we adopt to deal with this problem is to look for eigenstates of the discrete Fourier transform [60]. They have a very distinguishable behavior that is at the basis of the remarkable properties of coherent states
Summary
The concept of phase-space representation of quantum mechanics, introduced in the pioneering works of Weyl [1], Wigner [2], and Moyal [3], is a very useful and enlightening approach that sheds light on the correspondence between quantum and classical worlds. The resulting Wigner function, naturally related to the SU(2) dynamical group, has been further studied by a number of authors [44, 45, 46, 47], has been applied to some problems in quantum optics [48, 49] and extended to more general groups [50] These Wigner functions are not defined in a discrete phase space. The strategy we adopt to deal with this problem is to look for eigenstates of the discrete Fourier transform [60] For continuous variables, they have a very distinguishable behavior that is at the basis of the remarkable properties of coherent states. The program developed in this paper can be seen as a handy toolbox for the phase-space analysis of many-qudit systems, which should be of interest to a large interdisciplinary community working in these topics
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