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Optica Acta International Journal of Optics

Limits to Concentration by Passive Means

Ian M. Bassett

Published Dec 1, 1985• 2 citations
Publisher

Abstract

Liouville's theorem and its optical counterparts are extended into the wave domain, whereby some upper bounds, representing absolute limits on the performance of passive systems, are obtained. The equality of Liouville's theorem is degraded in general to an inequality, although the limiting equality sign is sometimes realizable. The main emphasis is on optics, and the main methods those of quantum optics.

DOI
10.1080/713821675

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Liouville's theorem and its optical counterparts are extended into the wave domain, whereby some upper bounds, representing absolute limits on the performance of passive systems, are obtained. The equality of Liouville's theorem is degraded in general to an inequality, although the limiting equality sign is sometimes realizable. The main emphasis is on optics, and the main methods those of quantum optics.
Quantum objects, sometimes called quantons, often display a characteristic feature referred to as wave-particle duality (WPD). Lately, this and other quantum traits have been subjected to intensive research, mainly motivated by the development of quantum information science. As a consequence, the scopes of some concepts have been extended, and it has been realized that they are not in the exclusive domain of quantum physics. This is particularly clear in optics, where qubits may show up as Jones vectors and WPD has its counterpart as wave-ray duality. WPD was originally addressed by focusing on a single qubit, which was afterwards supplemented with a second one playing the role of a path-marker in an interferometer setup. Fringe contrast, a sign of wave-like behavior, was proved to be diminished in connection with the effectiveness of the marker, the inducer of particle-like behavior. Going from bipartite to tripartite states is a natural and necessary step towards better understanding of WPD. This step is what we have accomplished in this work. We report some constraints ruling WPD for tripartite systems, as well as their experimental display with single photons.
In linear optics, photons are scattered in a network through passive optical elements including beamsplitters and phase shifters, leading to many intriguing applications in physics, such as Mach-Zehnder interferometry, Hong-Ou-Mandel effect, and tests of fundamental quantum mechanics. Here we present a general analytic expression governing the upper limit of the transition amplitudes in sampling bosons, through all realizable linear optics. Apart from boson sampling, this transition bound results in many other interesting applications, including behaviors of Bose-Einstein Condensates (BEC) in optical networks, counterparts of Hong-Ou-Mandel effects for multiple photons, and approximating permanents of matrices. Also, this general bound implies the existence of a polynomial-time randomized algorithm for estimating transition amplitudes of bosons, which represents a solution to an open problem raised by Aaronson and Hance in 2012.