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Journal of Physics A Mathematical and Theoretical

Isotropic and squeezed fluctuations in ann-qubit system

C Muñoz, A. B. Klimov

Published Dec 23, 2011• 1 citation
Publisher

Abstract

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.

DOI
10.1088/1751-8113/45/4/045301

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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.
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.
Abstract Collective spin operators for symmetric multi-quDit (namely identical D -level atom) systems generate a U( D ) symmetry. We explore generalizations to arbitrary D of SU(2)-spin coherent states and their adaptation to parity (multi-component Schrödinger cats), together with multi-mode extensions of NOON states. We write level, one- and two-quDit reduced density matrices of symmetric N -quDit states, expressed in the last two cases in terms of collective U( D )-spin operator expectation values. Then, we evaluate level and particle entanglement for symmetric multi-quDit states with linear and von Neumann entropies of the corresponding reduced density matrices. In particular, we analyze the numerical and variational ground state of Lipkin–Meshkov–Glick models of 3-level identical atoms. We also propose an extension of the concept of SU(2)-spin squeezing to SU( D ) and relate it to pairwise D -level atom entanglement. Squeezing parameters and entanglement entropies are good markers that characterize the different quantum phases, and their corresponding critical points, that take place in these interacting D -level atom models.
Approximating a continuous superposition by a discrete one leads to an understanding of the role of quantum interference in generating squeezed coherent states, and in its quantum properties. It was shown that discrete superposition with a large enough number of coherent states with appropriate coefficients can have practically the same features as the squeezed coherent state. Dependent on the parameters W, r and 8, different numbers of coherent states are required for the interference fringe to form a Wigner function resembling that of the squeezed vacuum state and to reduce the deviation below a certain value. For the case of rotating the direction of squeezing, about 20 states were required to lower the deviation to 0.05%, while 640 states would be necessary when a unit displacement along the imaginary axis is included. When only a few coherent states are superimposed, the resulting state changes chaotically. In the case when the approximated squeezed coherent state was displaced along the imaginary axis an interesting Schrodinger cat-like state was found with about 20 coherent states on the real axis. At this point a significant peak occurs in the variances of the quadratures. It turned out that non-classical states can be effectively produced by a discrete superposition of coherent states with the coefficients derived from the one-dimensional representation of the given state.
We investigate the presence of spin- and planar-squeezing in generalized superpositions of atomic (or spin) coherent states. Spin-squeezing has been shown to be a useful tool in determining the presence of entanglement in multipartite systems, such as collections of two-level atoms, as well as being an indication of reduced projection noise and sub-shot-noise-limited phase uncertainty in Ramsey spectroscopy, suitable for measuring phases ϕ ∼ 0 . On the other hand, planar-squeezed states display reduced projection noise in two directions simultaneously and have been shown to lead to enhanced metrological precision in measuring phases without the need for explicit prior knowledge of the phase value. In this paper, we show that the generalized superposition state can be parametrized to display both spin-squeezing along all orthogonal axes and planar-squeezing along all orthogonal planes for all values of J > 1 / 2 . We close with an application of the maximally spin- and planar-squeezed states to quantum metrology.