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Entropy 2017, 19(6), 253; doi:10.3390/e19060253

Ruling out Higher-Order Interference from Purity Principles

1
Centre for the Mathematics of Quantum Theory (QMATH), Department of Mathematical Sciences, University of Copenhagen, DK-2100 Copenhagen, Denmark
2
Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131, USA
3
Department of Physics, University College London, London WC1E 6BT, UK
4
Department of Computer Science, University of Oxford, Oxford OX1 3QD, UK
5
Department of Physics, Imperial College London, London SW7 2AZ, UK
*
Authors to whom correspondence should be addressed.
Academic Editors: Giacomo Mauro D’Ariano and Paolo Perinotti
Received: 21 April 2017 / Revised: 21 May 2017 / Accepted: 22 May 2017 / Published: 1 June 2017
(This article belongs to the Special Issue Quantum Information and Foundations)
View Full-Text   |   Download PDF [383 KB, uploaded 3 June 2017]

Abstract

As first noted by Rafael Sorkin, there is a limit to quantum interference. The interference pattern formed in a multi-slit experiment is a function of the interference patterns formed between pairs of slits; there are no genuinely new features resulting from considering three slits instead of two. Sorkin has introduced a hierarchy of mathematically conceivable higher-order interference behaviours, where classical theory lies at the first level of this hierarchy and quantum theory theory at the second. Informally, the order in this hierarchy corresponds to the number of slits on which the interference pattern has an irreducible dependence. Many authors have wondered why quantum interference is limited to the second level of this hierarchy. Does the existence of higher-order interference violate some natural physical principle that we believe should be fundamental? In the current work we show that such principles can be found which limit interference behaviour to second-order, or “quantum-like”, interference, but that do not restrict us to the entire quantum formalism. We work within the operational framework of generalised probabilistic theories, and prove that any theory satisfying Causality, Purity Preservation, Pure Sharpness, and Purification—four principles that formalise the fundamental character of purity in nature—exhibits at most second-order interference. Hence these theories are, at least conceptually, very “close” to quantum theory. Along the way we show that systems in such theories correspond to Euclidean Jordan algebras. Hence, they are self-dual and, moreover, multi-slit experiments in such theories are described by pure projectors. View Full-Text
Keywords: higher-order interference; generalised probabilistic theories; Euclidean Jordan algebras higher-order interference; generalised probabilistic theories; Euclidean Jordan algebras
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Barnum, H.; Lee, C.M.; Scandolo, C.M.; Selby, J.H. Ruling out Higher-Order Interference from Purity Principles. Entropy 2017, 19, 253.

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