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arXiv:1206.1487 · Journal of Physics A Mathematical and Theoretical

Symmetric discrete coherent states forn-qubits

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

Published May 30, 2012• 12 citations
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Abstract

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'.

DOI
10.1088/1751-8113/45/24/244014

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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'.
We consider the phase space for nnn identical qudits (each one of dimension ddd, with ddd a primer number) as a grid of ${d}^{n}\ifmmode\times\else\texttimes\fi{}{d}^{n}$ 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.
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.
The coherent states for a quantum particle on a Möbius strip are constructed and their relation to the natural phase space for fermionic fields is shown. The explicit comparison of the obtained states with previous works where the cylinder quantization was used and the spin-1/2 was introduced by hand is given, and the relation between the geometrical phase space, constraints and projection operators is analyzed and discussed. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.
Nonlinear fermions of degree n ( n -fermions) are introduced as particles with creation and annihilation operators obeying the simple nonlinear anticommutation relation AA † + A † n A n = 1. The ( n + 1)th-order nilpotency of these operators follows from the existence of unique A -vacuum. Supposing appropriate ( n + 1)th-order nilpotent para-Grassmann variables and integration rules the sets of n -fermion number states, 'right' and 'left' ladder operator coherent states (CS) and displacement-operator-like CS are constructed. The ( n + 1) × ( n + 1) matrix realization of the related para-Grassmann algebra is provided. General ( n + 1)th-order nilpotent ladder operators of finite-dimensional systems are expressed as polynomials in terms of n -fermion operators. Overcomplete sets of (normalized) 'right' and 'left' eigenstates of such general ladder operators are constructed and their properties are briefly discussed. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.