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.
Highlights
1.1 Overview of ResultsThe present paper investigates two proposed extensions of quantum theory from the point of view of their interference behaviour
The two theories which shall be investigated are: the theory of Density Cubes proposed by Dakic, Paterek and Brukner, which has been shown to exhibit third-order interference in the three slit set-up [1], and the Quartic Quantum Theory [2] of Zyczkowski
There are no genuinely new features resulting from considering three slits instead of two. This is in stark contrast with the existence of second-order, i.e. quantum-like, interference, for which there exists a two-slit experiment whose interference pattern cannot be written as a sum of the one slit patterns obtained by blocking each one of the slits
Summary
The present paper investigates two proposed extensions of quantum theory from the point of view of their interference behaviour. This investigation clarifies the impact of these two generalised theories to ongoing experimental tests for “higher-order interference” and explores potential information-theoretic consequences of post-quantum interference in concrete theories. In particular it highlights an ambiguity in the current definition of higher-order interference. The two theories which shall be investigated are: the theory of Density Cubes proposed by Dakic, Paterek and Brukner, which has been shown to exhibit third-order interference in the three slit set-up [1], and the Quartic Quantum Theory [2] of Zyczkowski.
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