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arXiv:0907.3845 · Physical Review A

Discrete coherent and squeezed states of many-qudit systems

A. B. Klimov, Carlos A Muñoz, Luis Lorenzo Sánchez-Soto

Published Oct 27, 2009• 34 citations
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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.

DOI
10.1103/physreva.80.043836

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We consider the phase space for nnn identical qudits (each one of dimension ddd, with ddd a primer number) as a grid of dn\ifmmode×\else\texttimes\fidn{d}^{n}\ifmmode\times\else\texttimes\fi{}{d}^{n}dn\ifmmode×\else\texttimes\fidn points and use the finite Galois field GF(dn)\text{GF}({d}^{n})GF(dn) to label the corresponding axes. The associated displacement operators permit to define sss-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.
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We show that 2 n discrete coherent states of an n -qubit system, generated by application of the discrete displacement operators to a symmetric fiducial state, have isotropic fluctuations, with n /2 ⩽ 〈Δ S 2 〉 ⩽ n , in a specific tangent plane, which in general is not orthogonal to the mean spin direction. This allows us to use them as reference states to define a discrete squeezing for non-symmetric n -qubit states. Examples of states with reduced fluctuations, obtained after application of XOR gates to correlate (partially entangle) qubits, are analyzed.
For a qudit, a quantum system on a finite dimensional Hilbert space, we construct minimum uncertainty states and a non-negative quasi probability distribution. They are the discrete version of the coherent states and the Husimi function in the continuous quantum mechanics. The two bases related by the discrete Fourier transformation are identified with the position and momentum bases. We prove an inequality concerning uncertainties of the position and momentum of a qudit. The minimum uncertainty states are the ones which saturate the inequality. Like the coherent states, they are not mutually orthogonal but comprise an overcomplete set in the state space. As the dimension of the Hilbert space goes to infinity, these states approach the coherent states. Using the minimum uncertain states, we construct a non-negative quasi probability distribution for a qudit. Its marginal distributions are smeared out, but we show that there does not exist a non-negative quasi probability distribution with sharper marginal properties if the translational covariance is assumed.
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We put forward a method of constructing discrete coherent states for n qubits. After establishing appropriate displacement operators, the coherent states appear as displaced versions of a fiducial vector that is fixed by imposing a number of natural symmetry requirements on its Q -function. Using these coherent states, we establish a partial order in the discrete phase space, which allows us to picture some n -qubit states as apparent distributions. We also analyze correlations in terms of sums of squared Q -functions. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.
Using the Pegg-Barnett phase operator formalism we have introduced the coherent phase state and the displaced phase state in a finite dimension. Using the definition of phase creation and annihilation operators we have constructed a displacement operator which yields coherent phase states and displaced phase states within a finite-dimensional basis and numerically evaluated observables as a function of the size of the basis set. We have shown that, when the dimension of the state space is much larger than the average excitation of phase quanta, then these states are the coherent phase state and displaced phase state of a harmonic oscillator which has an infinite number of eigenstates and exhibits phase squeezing up to 100%. The coherent phase state is a minimum-uncertainty state with respect to the phase quadrature operators but not the minimum-uncertainty state with respect to number and phase operators. The properties of coherent phase states for a two-state system are also investigated. It is interesting to realize that these coherent phase states and displaced phase states are exactly the same as the phase vacuum state and different phase states respectively for the single-mode light field case.