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Physical Review A

Bell’s inequality violation with four entangled fermions

Shahen Hacyan

Published Oct 1, 1997• 2 citations
Publisher

Abstract

A simple scheme for a Bell-inequality experiment employing four spin-$$$\frac{1}{2}$ particles is presented. Two pairs of fermions, say, ${P}_{1}\ensuremath{-}{P}_{2}$ and ${P}_{3}\ensuremath{-}{P}_{4},$ are emitted in different directions; then, particles ${P}_{1}\ensuremath{-}{P}_{3}$ and ${P}_{2}\ensuremath{-}{P}_{4}$ meet at two causally disconnected regions where their phases are shifted. The result is a correlation between the spin states of distant pairs of particles that violates Bell's inequality. The required phase shifts could be manipulated, for instance, with magnetic fields.

DOI
10.1103/physreva.56.r2489

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A simple scheme for a Bell-inequality experiment employing four spin-$$$\frac{1}{2}particlesispresented.Twopairsoffermions,say,particles is presented. Two pairs of fermions, say,particlesispresented.Twopairsoffermions,say,{P}{1}\ensuremath{-}{P}{2}andandand{P}{3}\ensuremath{-}{P}{4},areemittedindifferentdirections;then,particlesare emitted in different directions; then, particlesareemittedindifferentdirections;then,particles{P}{1}\ensuremath{-}{P}{3}andandand{P}{2}\ensuremath{-}{P}{4}$ meet at two causally disconnected regions where their phases are shifted. The result is a correlation between the spin states of distant pairs of particles that violates Bell's inequality. The required phase shifts could be manipulated, for instance, with magnetic fields.
Using real fields instead of complex ones, it was recently claimed, that all fermions are made of pairs of coupled fields (strings) with an internal tension related to mutual attraction forces, related to Planck’s constant. Quantum mechanics is described with real fields and real operators. Schrodinger and Dirac equations then are solved. The solution to Dirac equation gives four, real, 2-vectors solutions ψ 1 =( U 1 D 1 ) ψ 2 =( U 2 D 2 ) ψ 3 =( U 3 D 3 ) ψ 4 =( U 4 D 4 ) where ( ψ 1 , ψ 4 ) are coupled via linear combinations to yield spin-up and spin-down fermions. Likewise, ( ψ 2 , ψ 3 ) are coupled via linear combinations to represent spin-up and spin-down anti-fermions. For an incoming entangled pair of fermions, the combined solution is Ψ in = c 1 ψ 1 + c 4 ψ 4 where c 1 and c 4 are some hidden variables. By applying a magnetic field in +Z and +x the theoretical results of a triple Stern-Gerlach experiment are predicted correctly. Then, by repeating Bell’s and Mermin Gedanken experiment with three magnetic filters σ θ , at three different inclination angles θ , the violation of Bell’s inequality is proven. It is shown that all fermions are in a mixed state of spins and the ratio between spin-up to spin-down depends on the hidden variables.
With Bell's inequalities one has a formal expression to show how essentially all local theories of natural phenomena that are formulated within the framework of realism may be tested using a simple experimental arrangement. For the case of entangled pairs of spin-1/2 particles we propose an alternative measurement setup which is consistent to the necessary assumptions corresponding to the derivation of the Bell inequalities. We find that the Bell inequalities are never violated with respect to our suggested measurement process.
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In this paper, the Lorentz transformation of entangled Bell states seen by a moving observer is studied. The calculated Bell observable for four joint measurements turns out to give a universal value, $〈\^a\ensuremath{\bigotimes}\mathrm{b\ifmmode \hat{}\else \^{}\fi{}}〉+〈\^a\ensuremath{\bigotimes}{b}^{\ensuremath{'}}〉+〈{a}^{\ensuremath{'}}\ensuremath{\bigotimes}\mathrm{b\ifmmode \hat{}\else \^{}\fi{}}〉\ensuremath{-}〈{a}^{\ensuremath{'}}\ensuremath{\bigotimes}{b}^{\ensuremath{'}}〉=(2/\sqrt{2\ensuremath{-}{\ensuremath{\beta}}^{2}})(1+\sqrt{1\ensuremath{-}{\ensuremath{\beta}}^{2}}),$ where $\^a,\mathrm{b\ifmmode \hat{}\else \^{}\fi{}}$ are the relativistic spin observables derived from the Pauli-Lubanski pseudovector and $\ensuremath{\beta}=(v/c).$ We found that the degree of violation of the Bell's inequality is decreasing with increasing velocity of the observer and Bell's inequality is satisfied in the ultrarelativistic limit where the boost speed reaches the speed of light.
Bell-inequality violation is one of the most widely known manifestations of entanglement in quantum mechanics; indicating that experiments on physically separated quantum mechanical systems cannot be given a local realistic description. However, despite the importance of Bell inequalities, it is not known in general how to determine whether a given entangled state will violate a Bell inequality. This is because one can choose to make many different measurements on a quantum system to test any given Bell inequality and the optimization over measurements is a high-dimensional variational problem. In order to better understand this problem we present algorithms that provide, for a given quantum state and Bell inequality, both a lower bound and an upper bound on the maximal violation of the inequality. In many cases these bounds determine measurements that would demonstrate violation of the Bell inequality or provide a bound that rules out the possibility of a violation. Both bounds apply techniques from convex optimization and the methodology for creating upper bounds allows them to be systematically improved. Examples are given to illustrate how these algorithms can be used to conclude definitively if some quantum states violate a given Bell inequality.
Previous work on Bell's inequality realized in the laboratory has used entangled photons. Here we describe how entangled atoms can violate Bell's inequality, and how these violations can be measured with a very high detection efficiency. We first discuss a simple scheme based on two-level atoms inside a cavity to prepare the entangled state. We then discuss a scheme using three-level atoms, which requires a parameter regime much easier to access experimentally using current technology. As opposed to other schemes, our proposal relies on the presence of finite decay rates and its implementation should therefore be much less demanding.