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.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.

arXiv:0907.3845Physical Review AOct 27, 200934 citations

Coherent states in Finite Quantum Mechanics

Gregory G. Athanasiu, Emmanuel G. Floratos

Nuclear Physics BAug 1, 199435 citations

Wigner function and the successive measurement of position and momentum

Pier A. Mello, M. Revzen

Wigner function is a ``quasidistribution'' that provides a representation of the state of a quantum mechanical system in the phase space of position and momentum. In this paper we find a relation between the Wigner function and appropriate measurements involving the system's position and momentum which generalize the von Neumann model of measurement. We introduce two probes coupled successively in time to projectors associated with the system's position and momentum. We show that one can relate the Wigner function to the Kirkwood joint quasidistribution of position and momentum, the latter, in turn, being a particular case of successive measurements. We first consider the case of a quantum mechanical system described in a continuous Hilbert space and then turn to the case of a discrete, finite-dimensional Hilbert space.Wigner function is a ``quasidistribution'' that provides a representation of the state of a quantum mechanical system in the phase space of position and momentum. In this paper we find a relation between the Wigner function and appropriate measurements involving the system's position and momentum which generalize the von Neumann model of measurement. We introduce two probes coupled successively in time to projectors associated with the system's position and momentum. We show that one can relate the Wigner function to the Kirkwood joint quasidistribution of position and momentum, the latter, in turn, being a particular case of successive measurements. We first consider the case of a quantum mechanical system described in a continuous Hilbert space and then turn to the case of a discrete, finite-dimensional Hilbert space.

arXiv:1307.2877Physical Review AJan 13, 201411 citations

On the coherent states, displacement operators and quasidistributions associated with deformed quantum oscillators

Paolo Aniello, Vladimir I. Man’ko, Giuseppe Marmo +2 more

The Wigner (W), Husimi-Kano (Q) and Glauber-Sudarshan (P) quasidistributions are generalized to f-deformed quasidistributions which extend the parametric family of s-ordered quasidistributions of Cahill and Glauber. The deformation procedure is obtained via a canonical nonisometric transform of the displacement operators which preserves the form of the standard creation-annihilation commutation relation, hence the Heisenberg-Weyl algebra, but changes the scalar product in the Hilbert space of the oscillator states. A whole class of new resolutions of the identity is introduced. The time evolution equation for the new generalized quasidistributions is derived.The Wigner (W), Husimi-Kano (Q) and Glauber-Sudarshan (P) quasidistributions are generalized to f-deformed quasidistributions which extend the parametric family of s-ordered quasidistributions of Cahill and Glauber. The deformation procedure is obtained via a canonical nonisometric transform of the displacement operators which preserves the form of the standard creation-annihilation commutation relation, hence the Heisenberg-Weyl algebra, but changes the scalar product in the Hilbert space of the oscillator states. A whole class of new resolutions of the identity is introduced. The time evolution equation for the new generalized quasidistributions is derived.

Journal of Optics B Quantum and Semiclassical OpticsNov 15, 200060 citations

Quantum Systems in Finite-Dimensional Hilbert Spaces

Gianluca Stefanucci

Springer eBooksJan 1, 20240 citations

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.