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Journal of Modern Optics

Robust quantum state engineering using coherent states on a truncated von Neumann lattice

Lampros K. Stergioulas, Apostol Vourdas

Published Jun 1, 1998• 6 citations
Publisher

Abstract

Quantum states are constructed as a superposition of a few coherent states on a truncated von Neumann lattice. It is shown that the reconstruction is robust, in the sense that random noise in the coefficients affects the constructed state weakly. Numerical results for squeezed states and number eigenstates demonstrate that the technique is very accurate.

DOI
10.1080/09500349808230906

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Quantum states are constructed as a superposition of a few coherent states on a truncated von Neumann lattice. It is shown that the reconstruction is robust, in the sense that random noise in the coefficients affects the constructed state weakly. Numerical results for squeezed states and number eigenstates demonstrate that the technique is very accurate.
Arbitrary quantum states can be expanded with a very good accuracy in terms of coherent states on a truncated von Neumann lattice. Optimization techniques are used to further reduce the size of the basis, whilst still preserving the same degree of accuracy. Various p -norms are adopted as accuracy measures, which reflect different practical needs. Numerical examples demonstrate that a significant reduction in the size of the basis can be achieved with optimization techniques, without substantial loss of accuracy.
An orthonormal basis of modified coherent states is constructed. Each member of the basis is an infinite sum of coherent states on a von Neumann lattice. A single state is assigned to each unit cell of area h (Planck constant) in the phase plane. The uncertainties of the coordinate x and the square of the momentum p 2 for these states are shown to be similar to those for the usual coherent states. Expansions in the newly established set are discussed and it is shown that any function in the kq -representation can be written as a sum of two fixed kq -functions. Approximate commuting operators for x and p 2 are defined on a lattice in phase plane according to von Neumann's prescription.
An efficient method of quantum state engineering based on discrete superpositions of coherent states is presented. A systematic method is developed for obtaining optimized superpositions from the one-dimensional representation of the desired state. It is shown that superposition of even a small number of coherent states put along a straight line or on a circle in phase space can approximate nonclassical field states with a high degree of accuracy. An experimental scheme is proposed for generating equidistant coherent-state superpositions on a circle with arbitrary coefficients. \textcopyright{} 1996 The American Physical Society.
We significantly extend recently developed methods to faithfully reconstruct unknown quantum states that are approximately low-rank, using only a few measurement settings. Our new method is general enough to allow for measurements from a continuous family, and is also applicable to continuous-variable states. As a technical result, this work generalizes quantum compressed sensing to the situation where the measured observables are taken from a so-called tight frame (rather than an orthonormal basis) --- hence covering most realistic measurement scenarios. As an application, we discuss the reconstruction of quantum states of light from homodyne detection and other types of measurements, and we present simulations that show the advantage of the proposed compressed sensing technique over present methods. Finally, we introduce a method to construct a certificate which guarantees the success of the reconstruction with no assumption on the state, and we show how slightly more measurements give rise to "universal" state reconstruction that is highly robust to noise.