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arXiv:1306.3700 · Physical Review Letters

Contextuality in Bosonic Bunching

Paweł Kurzyński, Akihito Soeda, Jayne E. Thompson, Dagomir Kaszlikowski

Published Jan 14, 2014• 6 citations
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Abstract

In the classical probability theory a sum of probabilities of three pairwise exclusive events is always bounded by one. This is also true in quantum mechanics if these events are represented by pairwise orthogonal projectors. However, this might not be true if the three events refer to a system of indistinguishable particles. We show that one can find three pairwise exclusive events for a system of three bosonic particles whose corresponding probabilities sum to 3/2. This can be done under assumptions of realism and noncontextuality, i.e., that it is possible to assign outcomes to events before measurements are performed and in a way that does not depend on a particular measurement setup. The root of this phenomenon comes from the fact that for indistinguishable particles there are events that can be deduced to be exclusive under the aforementioned assumptions, but at the same time are complementary because the corresponding projectors are not orthogonal.

DOI
10.1103/physrevlett.112.020403

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In the classical probability theory a sum of probabilities of three pairwise exclusive events is always bounded by one. This is also true in quantum mechanics if these events are represented by pairwise orthogonal projectors. However, this might not be true if the three events refer to a system of indistinguishable particles. We show that one can find three pairwise exclusive events for a system of three bosonic particles whose corresponding probabilities sum to 3/2. This can be done under assumptions of realism and noncontextuality, i.e., that it is possible to assign outcomes to events before measurements are performed and in a way that does not depend on a particular measurement setup. The root of this phenomenon comes from the fact that for indistinguishable particles there are events that can be deduced to be exclusive under the aforementioned assumptions, but at the same time are complementary because the corresponding projectors are not orthogonal.
We show that violation of Klyachko-Can-Binicioglu-Shumovsky [Phys. Rev. Lett. {\bf 101}, 020403 (2008)] pentagram-like inequality can exceed 5\sqrt{5}5​ provided that exclusive events do not have to be comeasurable and that one uses bosonic systems which exhibit bunching effects. We also show that in this case one can find three pairwise exclusive events whose sum of probabilities is 3/2.
The presence of contextuality in quantum theory was first highlighted by Bell, Kochen and Specker, who discovered that for quantum systems of three or more dimensions, measurements could not be viewed as deterministically revealing pre-existing properties of the system. More precisely, no model can assign deterministic outcomes to the projectors of a quantum measurement in a way that depends only on the projector and not the context (the full set of projectors) in which it appeared, despite the fact that the Born rule probabilities associated with projectors are independent of the context. A more general, operational definition of contextuality introduced by Spekkens, which we will term 'probabilistic contextuality', drops the assumption of determinism and allows for operations other than measurements to be considered contextual. Even two-dimensional quantum mechanics can be shown to be contextual under this generalised notion. Probabilistic noncontextuality represents the postulate that elements of an operational theory that cannot be distinguished from each other based on the statistics of arbitrarily many repeated experiments (they give rise to the same operational probabilities) are ontologically identical. In this paper, we introduce a framework that enables us to distinguish between different noncontextuality assumptions in terms of the relationships between the ontological representations of objects in the theory given a certain relation between their operational representations. This framework can be used to motivate and define a 'possibilistic' analogue, encapsulating the idea that elements of an operational theory that cannot be unambiguously distinguished operationally can also not be unambiguously distinguished ontologically. We then prove that possibilistic noncontextuality is equivalent to an alternative notion of noncontextuality proposed by Hardy. Finally, we demonstrate that these weaker noncontextuality assumptions are sufficient to prove alternative versions of known 'no-go' theorems that constrain ψ -epistemic models for quantum mechanics.
Quantum theory has the intriguing feature that is inconsistent with noncontextual hidden variable models, for which the outcome of a measurement does not depend on which other compatible measurements are being performed concurrently. While various proofs of such contextual behavior of quantum systems have been established, relatively little is known concerning the possibility to demonstrate this intriguing feature for indistinguishable particles. Here, we show in a simple and systematic manner that with projective measurements alone, it is possible to demonstrate quantum contextuality for such systems of arbitrary Hilbert space dimensions, including those corresponding to a qubit. Our demonstration is applicable to a single fermion as well as multiple fermions, and thus also a composite boson formed from an even number of fermions. In addition, our approach gives a clear demonstration of the intimate connection between complementarity and contextuality, two seemingly unrelated aspects of quantum theory.
Quantum theory has the intriguing feature that is inconsistent with noncontextual hidden variable models, for which the outcome of a measurement does not depend on which other compatible measurements are being performed concurrently. While various proofs of such contextual behavior of quantum systems have been established, relatively little is known concerning the possibility to demonstrate this intriguing feature for indistinguishable particles. Here, we show in a simple and systematic manner that with projective measurements alone, it is possible to demonstrate quantum contextuality for such systems of arbitrary Hilbert space dimensions, including those corresponding to a qubit. Our demonstration is applicable to a single fermion as well as multiple fermions, and thus also a composite boson formed from an even number of fermions. In addition, our approach gives a clear demonstration of the intimate connection between complementarity and contextuality, two seemingly unrelated aspects of quantum theory.
In a recent article entitled "A simple explanation of the quantum violation of a fundamental inequality," Cabello proposes a condition on a class of probabilistic models that, he claims, gives the same bound on contextuality for the KCBS inequality as quantum mechanics, and also rules out PR-box nonlocality. He conjectures that the condition will also reproduce quantum limits on contextuality in other scenarios. Here we show that the proposed principle is actually too weak to derive these results. Cabello has implicitly assumed in the proofs that if all pairs in a set of events are pairwise exclusive (so that their probabilities must sum to less than 1), the set can itself be considered exclusive. Perhaps surprisingly, this is not in general true in the general probabilistic framework under discussion. We call this hidden assumption "Consistent Exclusivity" or CE. With this extra assumption Cabello's proofs are sound. Furthermore, it is established that CE holds in quantum mechanics, providing a reasonable and simple new principle that may characterise quantum non-contextuality in many scenarios.
There exist quantum phenomena that cannot be explained by noncontextual hidden-variable theories, yet the majority of them requires measurements that are performed on a single quantum system at a time. This fact constrains the phenomenon of contextuality to the microscopic domain. It is therefore natural to ask if quantum contextuality can be observed in measurements on collections of particles. Since particles in nature are identical, one can expect that such contextuality would be linked to bosonic and fermionic properties. Analysis of quantum contextuality in such scenarios would broaden our understanding of nonclassical effects in composite systems and perhaps would give us a hint on how to observe quantum phenomena in the macroscopic world. In this work I propose a generalization of quantum contextuality to the case of many identical particles. I show that a type of contextuality exhibited by a collection of particles (state dependent, state independent, or noncontextual) depends on their type and their number. I also discuss further properties of this generalization and identify major open questions.