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Fortschritte der Physik

Stochastic Simulations and Discreteness of Light

Radim Filip

Published Oct 1, 2001• 0 citations
Publisher

Abstract

The disability of stochastic simulations of a quantum system using continuous variables to describe the discreteness of light is illustrated on the example of the nonlinear interaction. In this case, the well-known occurrence of divergent trajectories in the simulations is suppressed and correct results can be achieved only if the discrete character is artificially implemented into the simulation process.

DOI
10.1002/1521-3978(200110)49:10/11<949::aid-prop949>3.0.co;2-#

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The disability of stochastic simulations of a quantum system using continuous variables to describe the discreteness of light is illustrated on the example of the nonlinear interaction. In this case, the well-known occurrence of divergent trajectories in the simulations is suppressed and correct results can be achieved only if the discrete character is artificially implemented into the simulation process.
The stochastic simulations of quantum evolution are introduced, alternatively, to the Ito-stochastic simulations, for the normal, symmetrical and antinormal ordering in the double-dimensional phase space. The nonclassical effects in the linearized parametric amplifier can be very well simulated in contrast to the collapses-revivals effect in a nonlinear version, which is stimulated by the discreteness of light. The reason is that the stochastic simulations use the continuous variables, whereas the quantum noise in nonlinear systems cannot generally be treated continuously.
The stochastic simulations of quantum evolution are introduced, alternatively to the Ito-stochastic simulations, for the normal, symmetrical and antinormal ordering in the double dimensional phase-space. The nonclassical effects in the linearized parametric amplifier can be very well simulated in contrast to the collapses-revivals effect in the nonlinear version of amplifier. It is illustrated that the stochastic simulation method using the continuous variables cannot be used if the discreteness of light has pronounced influence on the dynamics.
Most of the fundamental characteristics of quantum mechanics, such as nonlocality and contextuality, are manifest in discrete, finite-dimensional systems. However, many quantum information tasks that exploit these properties cannot be directly adapted to continuous variable systems. To access these quantum features, continuous quantum variables can be made discrete by binning together their different values, resulting in observables with a finite number, ddd, of outcomes. While direct measurement indeed confirms their manifestly discrete character, here we employ a salient feature of quantum physics known as mutual unbiasedness to show that such coarse-grained observables are in a sense neither continuous nor discrete. Depending on ddd, the observables can reproduce either the discrete or the continuous behavior, or neither. To illustrate these results, we present an example for the construction of such measurements and employ it in an optical experiment confirming the existence of four mutually unbiased measurements with d=3d=3d=3 outcomes in a continuous variable system, surpassing the number of mutually unbiased continuous variable observables.
By exploiting the complexity intrinsic to quantum dynamics, quantum technologies promise a host of computational advantages. One such advantage lies in the field of stochastic modeling, where it has been shown that quantum stochastic simulators can operate with a lower memory overhead than their best classical counterparts. This advantage is particularly pronounced for continuous-time stochastic processes; however, the corresponding quantum stochastic simulators heretofore prescribed operate only on a quasi-continuous-time basis and suffer an ever-increasing circuit complexity with increasing temporal resolution. Here, by establishing a correspondence with quantum trajectories (a method for modeling open quantum systems), we show how truly continuous-time quantum stochastic simulators can be embedded in such open quantum systems, bridging this gap and obviating previous constraints. We further show how such an embedding can be made for discrete-time stochastic processes, which manifest as jump-only trajectories, and discuss how viewing the correspondence in the reverse direction provides a means of studying structural complexity in quantum systems themselves. Published by the American Physical Society 2024