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Physica Scripta

Coherent, isotropic and squeezed states in anN-qubit system

A. B. Klimov, Carlos A Muñoz

Published Feb 11, 2013• 18 citations
Publisher

Abstract

We discuss how to construct a sensible set of discrete coherent states for an N-qubit system and how to use them as a tool for the analysis and representation of states in the discrete phase space. Considering N qubits as a single system of dimension 2N, we analyze the concept of squeezing as a reduction of isotropic fluctuations below the minimum achievable level for factorized (but non-necessary symmetric under permutations) qubit states.

DOI
10.1088/0031-8949/87/03/038110

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We discuss how to construct a sensible set of discrete coherent states for an N-qubit system and how to use them as a tool for the analysis and representation of states in the discrete phase space. Considering N qubits as a single system of dimension 2N, we analyze the concept of squeezing as a reduction of isotropic fluctuations below the minimum achievable level for factorized (but non-necessary symmetric under permutations) qubit states.
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.
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 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'.
A general definition of squeezed states is proposed and its main features are illustrated through a discussion of the standard optical coherent states represented by 'Gaussian pure states'. The set-up involves representations of groups on Hilbert spaces over homogeneous spaces of the group, and relies on the construction of a square integrable (coherent state) group representation modulo a subgroup. This construction depends upon a choice of a Borel section which has a certain permissible arbitrariness in its selection; this freedom is attributable to a squeezing of the defining coherent states of the representation, and corresponds in this way to a sort of gauging.