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arXiv:1510.03860 · Foundations of Physics

Higher-Order Interference in Extensions of Quantum Theory

Ciarán M. Lee, John H. Selby

Published Oct 22, 2016• 50 citations
arXivHTMLPDF

Abstract

Quantum interference, manifest in the two slit experiment, lies at the heart of several quantum computational speed-ups and provides a striking example of a quantum phenomenon with no classical counterpart. An intriguing feature of quantum interference arises in a variant of the standard two slit experiment, in which there are three, rather than two, slits. The interference pattern in this set-up can be written in terms of the two and one slit patterns obtained by blocking one, or more, of the slits. This is in stark contrast with the standard two slit experiment, where the interference pattern cannot be written as a sum of the one slit patterns. This was first noted by Rafael Sorkin, who raised the question of why quantum theory only exhibits irreducible interference in the two slit experiment. One approach to this problem is to compare the predictions of quantum theory to those of operationally-defined ‘foil’ theories, in the hope of determining whether theories that do exhibit higher-order interference suffer from pathological—or at least undesirable—features. In this paper two proposed extensions of quantum theory are considered: the theory of Density Cubes proposed by Dakić, Paterek and Brukner, which has been shown to exhibit irreducible interference in the three slit set-up, and the Quartic Quantum Theory of Życzkowski. The theory of Density Cubes will be shown to provide an advantage over quantum theory in a certain computational task and to posses a well-defined mechanism which leads to the emergence of quantum theory—analogous to the emergence of classical physics from quantum theory via decoherence. Despite this, the axioms used to define Density Cubes will be shown to be insufficient to uniquely characterise the theory. In comparison, Quartic Quantum Theory is a well-defined theory and we demonstrate that it exhibits irreducible interference to all orders. This feature of Życzkowski’s theory is argued not to be a genuine phenomenon, but to arise from an ambiguity in the current definition of higher-order interference in operationally-defined theories. Thus, to begin to understand why quantum theory is limited to a certain kind of interference, a new definition of higher-order interference is needed that is applicable to, and makes good operational sense in, arbitrary operationally-defined theories.

DOI
10.1007/s10701-016-0045-4

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Quantum interference, manifest in the two slit experiment, lies at the heart of several quantum computational speed-ups and provides a striking example of a quantum phenomenon with no classical counterpart. An intriguing feature of quantum interference arises in a variant of the standard two slit experiment, in which there are three, rather than two, slits. The interference pattern in this set-up can be written in terms of the two and one slit patterns obtained by blocking one, or more, of the slits. This is in stark contrast with the standard two slit experiment, where the interference pattern cannot be written as a sum of the one slit patterns. This was first noted by Rafael Sorkin, who raised the question of why quantum theory only exhibits irreducible interference in the two slit experiment. One approach to this problem is to compare the predictions of quantum theory to those of operationally-defined ‘foil’ theories, in the hope of determining whether theories that do exhibit higher-order interference suffer from pathological—or at least undesirable—features. In this paper two proposed extensions of quantum theory are considered: the theory of Density Cubes proposed by Dakić, Paterek and Brukner, which has been shown to exhibit irreducible interference in the three slit set-up, and the Quartic Quantum Theory of Życzkowski. The theory of Density Cubes will be shown to provide an advantage over quantum theory in a certain computational task and to posses a well-defined mechanism which leads to the emergence of quantum theory—analogous to the emergence of classical physics from quantum theory via decoherence. Despite this, the axioms used to define Density Cubes will be shown to be insufficient to uniquely characterise the theory. In comparison, Quartic Quantum Theory is a well-defined theory and we demonstrate that it exhibits irreducible interference to all orders. This feature of Życzkowski’s theory is argued not to be a genuine phenomenon, but to arise from an ambiguity in the current definition of higher-order interference in operationally-defined theories. Thus, to begin to understand why quantum theory is limited to a certain kind of interference, a new definition of higher-order interference is needed that is applicable to, and makes good operational sense in, arbitrary operationally-defined theories.
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.
The double-slit experiment is a direct demonstration of interference between quantum objects. Of course, interference is also observed in experiments using multiple slits. However, according to quantum theory, the interference patterns produced using such multi-slits can all be explained in terms of interference between pairs of slits [1] , [2] . Interestingly, there are proposed versions of quantum theory—so-called higher-order theories—wherein this is not possible [3] . In such theories the "third-order" interference (with three or more slits open) is genuinely different from second-order interference. Here we show that standard quantum mechanics actually allows for apparent high-order interference, if one considers nonlinear effects. Note, that another mechanism for this has also recently been shown based on so-called looped trajectories [4] – [6] .
Earth, Moon and Sun unite when they star together in the three-body problem, whose intricate plot still baffles us today. For some reason, the factorization of the two-body problem into two one-body problems does not, in general, cross the N=2 border. Is computational irreducibility responsible for this emergence of complexity, as Stephen Wolfram likes to think? We don't know. The introduction to this thesis in Chapter 1, however, makes it clear that the history of science is marked by intermittent encounters of sudden complexities when the number 2 is left behind. In Chapter 2, I present an experiment that is quite similar in spirit, for my colleagues and I observe three-photon interference without two-photon and single-photon interference. We had to overcome significant experimental challenges that are typical for most quantum interference experiments involving more than two photons. Next in line is the three-slit interference experiment. Again a deceptively simple extension of the famous double-slit experiment, we are faced with questions that are difficult to access experimentally: the existence of genuine three-slit interference was first denied and then affirmed, though no experiment has decided yet. My contribution to the study of this problem is outlined in Chapter 3, where I use symmetry of measurement settings in such interference experiments to theoretically derive higher-order interference terms. In Chapter 4, I take a step back in one sense, for we study a two-photon phenomenon, but we also leap forward and discover entirely new interference landscapes. Theoretically and experimentally, I demonstrate how to use a polarization-modulated lasers to go beyond the standard Hong-Ou-Mandel (HOM) dip, and generate both triangular and square wave HOM interference patterns. Two-photon interference is also subject of Chapter 5, but with an interesting twist. While laser HOM interference relies on two independent photons, here we endow the pair with the strongest known correlations, namely entanglement. More specifically, we entangle a polarization and a time-bin qubit and use this hybrid to assess the viability of a rather special interferometer for quantum communication purposes.
The double-slit experiment is the most direct demonstration of interference between individual quantum objects. Since similar experiments with more slits produce interference fringes reducible to a combination of double-slit patterns it is usually argued that quantum interference occurs between pairs of trajectories, compactly denoted as second-order interference. Here we show that quantum mechanics in fact allows for interference of arbitrarily high order. This occurs naturally when one considers multiple quantum objects interacting in the presence of a nonlinearity. We make this clear by treating a generalised M-slit interferometer using second-quantisation. We then present explicit experimentally-relevant examples both with photons interacting in nonlinear media and an interfering Bose-Einstein condensate with particle-particle interactions. These examples are all perfectly described by quantum theory, and yet exhibit higher-order interference.
Can quantum theory be seen as a special case of a more general probabilistic theory, as classical theory is a special case of the quantum one? We study here the class of generalized probabilistic theories defined by the order of interference they exhibit as proposed by Sorkin. A simple operational argument shows that the theories require higher-order tensors as a representation of physical states. For the third-order interference we derive an explicit theory of 'density cubes' and show that quantum theory, i.e. theory of density matrices, is naturally embedded in it. We derive the genuine non-quantum class of states and non-trivial dynamics for the case of a three-level system and show how one can construct the states of higher dimensions. Additionally to genuine third-order interference, the density cubes are shown to violate the Leggett–Garg inequality beyond the quantum Tsirelson bound for temporal correlations.
The existence of higher than pairwise quantum interference in the set-up, in which there are more than two slits, is currently under experimental investigation. However, it is still unclear what the confirmation of existence of such interference would mean for quantum theory -- whether that usual quantum mechanics is merely a limiting case of some more general theory or whether that some assumption of quantum theory taken as a fundamental one does not actually hold true. The present paper tries to understand why quantum theory is limited only to a certain kind of interference.