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Undergraduate texts in physics

Entanglement

Ray LaPierre

Published Jan 1, 2022• 0 citations
Publisher

Abstract

There exist multiphoton states that cannot be expressed as a product of individual photon states. These states are called entangled states. Einstein used hidden variables in an attempt to explain the probabilities of quantum mechanics. John Bell proposed a test, using entangled states, showing that quantum mechanics cannot be explained by hidden variables.

DOI
10.1007/978-3-031-12432-7_8

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There exist multiphoton states that cannot be expressed as a product of individual photon states. These states are called entangled states. Einstein used hidden variables in an attempt to explain the probabilities of quantum mechanics. John Bell proposed a test, using entangled states, showing that quantum mechanics cannot be explained by hidden variables.
Hidden variable theories, as encouraged by the EPR paper, that proliferated in the 1950s, were taken quite seriously in some academic quarters, despite the existence of the 1932 von Neumann ‘proof’ that supposedly showed that these hidden theories were incompatible with the stochasticity of quantum mechanics. In 1964, John Bell, a particle theorist, reexamined the von Neumann proof, found it ‘wanting’, and offered his own proof that reaffirmed that hidden variable theories were incompatible with quantum mechanics. Bell added a twist: his proof was in the form a neat inequality theorem that was easy to comprehend and became an instant success. Bell's theorem led to a new generation of experimentalists to rediscover quantum entanglement, and to conduct these experiments in the visible spectrum, to test for evidence of hidden variable theories. The results were negative thus reaffirming the validity of quantum mechanics and its indeterminacy. In this chapter the premises of Bell's theorem are explained as well as its link to the probabilities of quantum entanglement.
Quantum mechanics makes surprising predictions that defy our classical understanding of the world.Experiments done on entangled particles-particles with linked properties-prove that actions on one particle can affect the other one no matter the distance.In response, physicists came up with local hidden variable theories in an attempt to understand this quantum phenomena through a classical lens.John Bell formulated a way to test whether nature is dictated by local hidden variable theories or quantum mechanics, and since then it has been the goal of numerous physicists to experimentally test the theorem and improve upon the results.This paper describes the history of these experiments, their pros and cons, their advancements, and the potential applications that they will enable.
EPR showed that two particles emitted from a source can be entangled by a shared wavefunction where two non-commuting observables (position, momentum) can be simultaneously real, leading to a contradiction with quantum mechanics (two non-commuting variables can not be simultaneously real). John Bell derived an inequality where any local hidden variables prediction is bounded and quantum mechanics can violate the inequality. Bell tests on correlated photon pairs showed a clear violation of the Bell inequality and agreement with quantum mechanics. This study revealed that the Bell inequality holds at any individual instances in a given system but it does not hold over the entire population. The Bell inequality is not an incompatibility criterion for local hidden variables vs quantum mechanics rather it is a criterion for an individual nature vs population dynamics. The nonlocality prediction given by wavefunction is a system approximation and local causality (photon polarization) provides a complete description of the Bell tests. The paper concludes with great satisfaction that the measurement of two correlated photons at a distance was a local cause (photon-filter interaction did not violate the special theory of relativity).
Now that we have a few more concepts under our belts, I would like to go back and revisit in some more detail just what John Bell proved in 1964 about entangled states. He considered a simple quantum situation like the entangled ions in Alice and Bob’s pocket watches and then set about to compare the predictions of quantum theory for such a setup to the predictions of a classical hidden variable theory that had all of Einstein’s constraints: reality, locality, and certainty. Remember-there was already the Bohm hidden variable theory, well known in 1964, which did predict the same outcome as quantum theory, but this Bohm theory doesn’t count because it violates Einstein’s locality condition. (Einstein thumbs his nose at the Bohmians.) Bell’s goal was to see if he could construct a local classical hidden variable theory that would agree with all the predictions of quantum theory and satisfy Einstein’s three conditions. If such a hidden variable theory could be found, then it would be tempting to just replace quantum theory with that hidden variable theory and all the spookiness would just disappear. But in order for this project to succeed, the classical hidden variable theory had to obey all three of Einstein’s reality, locality, and certainty conditions and reproduce all the predictions of quantum theory. at was just too tall of an order to ask of any local hidden variable theory, Bell discovered. Something, somewhere, had to give.
From its seemingly non-intuitive and puzzling nature, most evident in numerous EPR-like gedanken experiments to its almost ubiquitous presence in quantum technologies, entanglement is at the heart of modern quantum physics. First introduced by Erwin Schrödinger nearly a century ago, entanglement has remained one of the most fascinating ideas that came out of quantum mechanics. Here, we attempt to explain what makes entanglement fundamentally different from any classical phenomenon. To this end, we start with a historical overview of entanglement and discuss several hidden variables models that were conceived to provide a classical explanation and demystify quantum entanglement. We discuss some inequalities and bounds that are violated by quantum states thereby falsifying the existence of some of the classical hidden variables theories. We also discuss some exciting manifestations of entanglement, such as N00N states and the non-separable single particle states. We conclude by discussing some contemporary results regarding quantum correlations and present a future outlook for the research of quantum entanglement.
Bell derived the given inequalities on the basis of one rather forceful assumption that was supposed to hold in the hidden variable theory. However, this assumption has been so strong that it has corresponded only to the classical physics; any dependence on spin orientations of individual objects (i.e., on quantum characteristics) having been practically excluded. And consequently, the hidden-variable theory has not been refused by corresponding experimental data. On the other side, Einstein's criticism of Copenhagen quantum mechanics based on ontological approach refusing nonlocality has been entitled as a much higher limit value corresponds to Bell's probability combination in such a case. And there is not any reason to believe in direct interaction between matter objects at greater distances as shown recently also by K. Hess on the basis of another approach.