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AIP conference proceedings

Topological effects of Aharonov-Bohm type and the role of the quantum entanglement

Hasan Özgür Çıldıroğlu, A. U. Yilmazer

Published Jan 1, 2018• 4 citations
Publisher

Abstract

The state of two or more particle systems in quantum mechanics is generally expressed as linear superpositions of tensor products of discrete states; and entanglement is connected to the superpositions that cannot be expressed by a single tensor product. On the other hand a discussion of nonlocality is always present in quantum entanglement and furthermore this nonlocality may be revealed via entanglement in the cases of the topological effects in quantum mechanics. In this work, aiming a more systematic and a complementary framework we focus on a comparison of the topological effects of Aharonov-Bohm type whose roots seem to be common. In this context, we will discuss the roles of three topological effects on an entangled quantum mechanical system.

DOI
10.1063/1.5078917

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The state of two or more particle systems in quantum mechanics is generally expressed as linear superpositions of tensor products of discrete states; and entanglement is connected to the superpositions that cannot be expressed by a single tensor product. On the other hand a discussion of nonlocality is always present in quantum entanglement and furthermore this nonlocality may be revealed via entanglement in the cases of the topological effects in quantum mechanics. In this work, aiming a more systematic and a complementary framework we focus on a comparison of the topological effects of Aharonov-Bohm type whose roots seem to be common. In this context, we will discuss the roles of three topological effects on an entangled quantum mechanical system.
Since its discovery in 1959 the Aharonov-Bohm effect has continuously been the cause for controversial discussions of various topics in modern physics, e.g. the reality of gauge potentials, topological effects and nonlocalities. In the present paper we juxtapose the two rival interpretations of the Aharonov-Bohm effect. We show that the conception of nonlocality encountered in the Aharonov-Bohm effect is closely related to the nonseparability which is common in quantum mechanics albeit distinct from it due to its topological nature. We propose a third alternative interpretation based on the loop space of holonomies which serves to solve some of the problems and we trace back the topological nonlocality and thereby the Aharonov-Bohm effect to their quantum mechanical origin. All three discussed interpretations are, of course, empirically equivalent. In fact, they present us with an instructive case study for the thesis of theory underdetermination by empirical data.
The Aharonov-Bohm effect is a quantum mechanical effect, that is, has no classical counterpart. The effect was predicted in 1959 in a seminal paper of Y. Aharonov and D. Bohm [AB59] in which they demonstrated that a beam of electrons is affected by the existence of the electric/magnetic field even though electrons travel through field-free regions. Aharonov and Bohm carried out two hypothetical experiments to support their claim that potentials are more fundamental than fields and they are respon-sible of the effect. Since then, the debate has arisen around whether potentials are mathematical tools or fundamental entities in physics. Different arguments have been set to explain the results predicted by Aharonov and Bohm and experimentally confirmed. Amongst these arguments, the first argument adopted by Aharonov and Bohm was that potentials are physically significant. Many have claimed that fields do have non-local features, i.e. action at a distance. Others have claimed that topological effects may interpret the effect in which potentials are modeled as connections in higher-dimensional fiber bun-dle geometries. The most recent argument has been proposed by Vaidman [Vai12] who claimed that the the composite system is represented by one state, an entangled state, and due to the electromagnetic interactions part of this state is changed, hence, the total state. In the present essay, I will discuss the latter argument as well as reviewing some other arguments. i
The two main features of the Aharonov-Bohm effect are the topological dependence of accumulated phase on the winding number around the magnetic fluxon, and nonlocality-local observations at any intermediate point along the trajectories are not affected by the fluxon. The latter property is usually regarded as exclusive to quantum mechanics. Here we show that both the topological and nonlocal features of the Aharonov-Bohm effect can be manifested in a classical model that incorporates random noise. The model also suggests new types of multiparticle topological nonlocal effects which have no quantum analog.
Although the Aharonov-Bohm and related effects are familiar in solid-state and high-energy physics, the nonlocality of these effects has been questioned. Here we show that the Aharonov-Bohm effect has two very different aspects. One aspect is instantaneous and nonlocal; the other aspect, which depends on entanglement, unfolds continuously over time. While local, gauge-invariant variables may occasionally suffice for explaining the continuous aspect, we argue that they cannot explain the instantaneous aspect. Thus the Aharonov-Bohm effect is, in general, nonlocal.
At first sight the Aharonov-Bohm effect appears nonlocal, though not in the way EPR/ Bell correlations are generally acknowledged to be nonlocal. This paper applies an analysis of nonlocality to the Aharonov-Bohm effect to show that its peculiarities may be blamed either on a failure of a principle of local action or on a failure of a principle of separability. Different interpretations of quantum mechanics disagree on how blame should be allocated. The parallel between the Aharonov-Bohm effect and violations of Bell inequalities turns out to be so close that a balanced assessment of the nature and significance of quantum nonlocality requires a detailed study of both effects.