Are there physical, probabilistic or information-theoretic principles which characterize the quantum probabilities and distinguish them from the classical case as well as from other probability theories, or which reveal why quantum mechanics requires its very special mathematical formalism? The paper identifies the fundamental absence of third-order interference as such a principle of `quantumness'. Considering three-slit experiments, the concept of third-order interference was originally introduced by Sorkin in 1994.
Are there physical, probabilistic or information-theoretic principles which characterize the quantum probabilities and distinguish them from the classical case as well as from other probability theories, or which reveal why quantum mechanics requires its very special mathematical formalism? The paper identifies the fundamental absence of third-order interference as such a principle of `quantumness'. Considering three-slit experiments, the concept of third-order interference was originally introduced by Sorkin in 1994.
Quantum probabilities differ from classical ones in many ways, e.g. by violating the well-known Bell and Clauser–Horne–Shimony–Holt inequalities or another simple inequality due to R Wright. The latter has recently regained attention because of its equivalence to a novel noncontextual inequality by Klyachko et al . On the other hand, quantum probabilities still obey many limitations which need not hold in more general probabilistic theories (super quantum probabilities). Wright, Popescu and Rohrlich identified states which are included in such theories, but impossible in quantum mechanics, and they showed this using the Hilbert space formalism. Recently, Fritz et al and Cabello detected that the impossibility of these states can be derived from very general principles (local orthogonality and global exclusive disjunction, respectively) without using Hilbert space techniques. In the paper, an alternative derivation from rather different physical principles will be presented. These are a reasonable calculus of conditional probability (i.e. a model for the quantum measurement process) and the absence of third-order interference. The concept of third-order interference was introduced by Sorkin, who also recognized its impossibility in quantum mechanics.
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.
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.
Most quantum logics do not allow for a reasonable calculus of conditional probability. However, those ones which do so provide a very general and rich mathematical structure, including classical probabilities, quantum mechanics, and Jordan algebras. This structure exhibits some similarities with Alfsen and Shultz's noncommutative spectral theory, but these two mathematical approaches are not identical. Barnum, Emerson, and Ududec adapted the concept of higher-order interference, introduced by Sorkin in 1994, into a general probabilistic framework. Their adaption is used here to reveal a close link between the existence of the Jordan product and the nonexistence of interference of third or higher order in those quantum logics which entail a reasonable calculus of conditional probability. The complete characterization of the Jordan algebraic structure requires the following three further postulates: a Hahn-Jordan decomposition property for the states, a polynomial functional calculus for the observables, and the positivity of the square of an observable. While classical probabilities are characterized by the absence of any kind of interference, the absence of interference of third (and higher) order thus characterizes a probability calculus which comes close to quantum mechanics but still includes the exceptional Jordan algebras.
More general probability sum-rules for describing interference found in quantum mechanics (QM) were formulated by Sorkin in a hierarchy of such rules. The additivity of classical measure theory corresponds to the second sum-rule. QM violates this rule, but satisfies the third and higher sum-rules. This evokes the question of whether there are physical principles that forbid their violation. We show that in a theory that is indistinguishable from quantum mechanics in first and second-order interferences, the violation of higher sum-rules allows for superluminal signaling, essentially because probability measures can be contextual in such theories.
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] .