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arXiv:1907.09419 · Physica Scripta

Contextuality and nonlocality of indistinguishable particles

Debajyoti Gangopadhyay, R Srikanth

Published Jun 26, 2019• 0 citations
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Abstract

Abstract Unlike in the case of distinguishable particles, the concept of entanglement—not to mention nonlocality—remains debated in the case of indistinguishable particles. Here, we show that certain existing all-versus-nothing type proofs of contextuality or nonlocality for distinguishable particles, based on a logical contradiction, may be carried over to indistinguishable particles.

DOI
10.1088/1402-4896/ab2c9c

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Abstract Unlike in the case of distinguishable particles, the concept of entanglement—not to mention nonlocality—remains debated in the case of indistinguishable particles. Here, we show that certain existing all-versus-nothing type proofs of contextuality or nonlocality for distinguishable particles, based on a logical contradiction, may be carried over to indistinguishable particles.
While entanglement in the case of distinguishable particles is clearly understood, in the case of indistinguishable particles, there is still a wide debate regarding basic conceptual issues. Here, I will address two such issues. First, the debate almost always rests on a differentiated treatment of the cases of distinguishability and indistinguishability, even with respect to the definition of the concept of entanglement. Second, the fact that symmetrized and antisymmetrized states of indistinguishable particles are non-factorizable leads to certain perplexities about the very nature of entanglement. In the present work, both issues will be addressed from the perspective of an ontology of properties, which finds its natural expression in the algebraic formalism of quantum mechanics. The final goal will be to show that the proposed ontology allows a unified treatment of entanglement in the distinguishability and indistinguishability cases and offers a straightforward way out to perplexities. This article is part of the theme issue 'Identity, individuality and indistinguishability in physics and mathematics'.
It is well known that in quantum mechanics we cannot always define consistently properties that are context independent.Many approaches exist to describe contextual properties, such as Contextuality by Default (CbD), sheaf theory, topos theory, and non-standard or signed probabilities.In this paper, we propose a treatment of contextual properties that is specific to quantum mechanics, as it relies on the relationship between contextuality and indistinguishability.In particular, we propose that if we assume the ontological thesis that quantum particles or properties can be indistinguishable yet different, no contradiction arising from a Kochen-Specker-type argument appears: when we repeat an experiment, we are in reality performing an experiment measuring a property that is indistinguishable from the first, but not the same.We will discuss how the consequences of this move may help us understand quantum contextuality.
It is well known that in quantum mechanics we cannot always define consistently properties that are context independent. Many approaches exist to describe contextual properties, such as Contextuality by Default (CbD), sheaf theory, topos theory, and non-standard or signed probabilities. In this paper, we propose a treatment of contextual properties that is specific to quantum mechanics, as it relies on the relationship between contextuality and indistinguishability. In particular, we propose that if we assume the ontological thesis that quantum particles or properties can be indistinguishable yet different, no contradiction arising from a Kochen–Specker-type argument appears: when we repeat an experiment, we are in reality performing an experiment measuring a property that is indistinguishable from the first, but not the same. We will discuss how the consequences of this move may help us understand quantum contextuality.
We study the role of entanglement and non-locality in quantum protocols that make use of systems of identical particles. Unlike in the case of distinguishable particles, the notions of entanglement and non-locality for systems whose constituents cannot be distinguished and singly addressed are still debated. We clarify why the only approach that avoids incongruities and paradoxes is the one based on the second quantization formalism, whereby it is the entanglement of the modes that can be populated by the particles that really matters and not the particles themselves. Indeed, by means of a metrological and of a teleportation protocol, we show that inconsistencies arise in formulations that force entanglement and non-locality to be properties of the identical particles rather than of the modes they can occupy. The reason resides in the fact that orthogonal modes can always be addressed while identical particles cannot.
Contextuality and nonlocality are nonclassical properties exhibited by quantum statistics whose implications profoundly impact both the foundations and applications of quantum theory. In this paper we provide some insights into logical contextuality and inequality-free proofs. The former can be understood as the possibility version of contextuality, while the latter refers to proofs of quantum contextuality and nonlocality that are not based on violations of some noncontextuality (or Bell) inequality. By ``possibilistic'' we mean a description in terms of possibilities for the outcomes, which are Boolean variables assuming value one when the corresponding probability is strictly larger than zero and zero otherwise. The present work aims to build a bridge between these two concepts from what we call possibilistic paradoxes, which are sets of possibilistic conditions whose occurrence implies contextuality and nonlocality. As the main result, we demonstrate the existence of possibilistic paradoxes whose occurrence is a necessary and sufficient condition for logical contextuality in a very important class of scenarios. Finally, we discuss some interesting consequences arising from the completeness of these possibilistic paradoxes.
Entanglement is well understood concept for distinguishable particles. However fundamental particles are inherently indistinguishable and when they occupy orthogonal modes, i.e. have orthogonal wave functions, they are effectively distinguishable. Thus there should exist a formulation for entanglement which works in the generic case, i.e. even when wave functions overlap. To achieve this, we consider R\`{e}nyi entropies which are measurable quantities. Outcomes of such measurements for a generic case are conjectured in terms of R\`{e}nyi entropies of spatial regions in a field theory. To leading order naive quantum mechanical answers are reproduced. Corrections depend on overlap of wave functions and exhibit novel features. In a generic case the Von Neumann entropy computed from these R\`{e}nyi entropies is argued to be a measure of entanglement of pure states and a formula for this is conjectured in terms of Von Neumann entropies of spatial regions in field theory. It seems, due to the corrections it is possible for the entanglement to exceed maximal quantum mechanical value, e.g. log2log 2log2 for a single qubit.