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arXiv:1212.4344 · Physical Review Letters

Bell Inequality Violations Under Reasonable and Under Weak Hypotheses

Edson de Faria, Charles P. Tresser

Published Jun 28, 2013• 2 citations
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Abstract

Given a sequence of pairs (${p}_{i}$, ${\overline{p}}_{i}$) of spin-$1/2$ particles in the singlet state, assume that Alice measures the normalized projections ${a}_{i}$ of the spins of the ${p}_{i}$'s along vector $\mathbit{a}$ while Bob measures the normalized projections ${b}_{i}$ of the spins of the ${\overline{p}}_{i}$'s along vector $\mathbit{b}$. Then quantum mechanics (QM) lets one evaluate the correlation $⟨a,b⟩$ as $\ensuremath{-}\mathrm{cos}({\ensuremath{\theta}}_{\mathbit{a}}\ensuremath{-}{\ensuremath{\theta}}_{\mathbit{b}})$ where ${\ensuremath{\theta}}_{\mathbit{v}}$ is the angle between the vector $\mathbit{v}$ and a reference vector chosen once and for all and in a fixed plane. Assuming classical microscopic realism (CMR) there exist also normalized projection pairs (${a}_{i}^{\ensuremath{'}}$, ${b}_{i}^{\ensuremath{'}}$) of the spins of the pairs (${p}_{i}$, ${\overline{p}}_{i}$) along (${\mathbit{a}}^{\ensuremath{'}}$, ${\mathbit{b}}^{\ensuremath{'}}$) so that $⟨{a}^{\ensuremath{'}},{b}^{\ensuremath{'}}⟩=\ensuremath{-}\mathrm{cos}({\ensuremath{\theta}}_{{\mathbit{a}}^{\ensuremath{'}}}\ensuremath{-}{\ensuremath{\theta}}_{{\mathbit{b}}^{\ensuremath{'}}})$. Since all projections are in ${\ensuremath{-}1,1}$, $|⟨c,d⟩+⟨c,e⟩|+|⟨f,d⟩\ensuremath{-}⟨f,e⟩|\ensuremath{\le}2$ for $c$, $d$, $e$, and $f$ in ${a,b,{a}^{\ensuremath{'}},{b}^{\ensuremath{'}}}$. Assuming locality (the impossibility of any effect of an event on another event when said events are spatially separated) beside QM and CMR, Bell's theory lets one deduce various violations of this inequality at some choices of quadruplets $Q\ensuremath{\equiv}(\mathbit{a},\mathbit{b},{\mathbit{a}}^{\ensuremath{'}},{\mathbit{b}}^{\ensuremath{'}})$. Our main result is the existence of such $Q$'s where at least one of the above inequalities is violated if one only assumes QM, CMR, and some very mild further hypotheses that only concern the behavior of correlations that appear in these inequalities near special $Q$'s.

DOI
10.1103/physrevlett.110.260409

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Given a sequence of pairs (pi{p}_{i}pi​, p‾i{\overline{p}}_{i}p​i​) of spin-1/21/21/2 particles in the singlet state, assume that Alice measures the normalized projections ai{a}_{i}ai​ of the spins of the pi{p}_{i}pi​'s along vector \mathbita\mathbit{a}\mathbita while Bob measures the normalized projections bi{b}_{i}bi​ of the spins of the p‾i{\overline{p}}_{i}p​i​'s along vector \mathbitb\mathbit{b}\mathbitb. Then quantum mechanics (QM) lets one evaluate the correlation ⟨a,b⟩⟨a,b⟩⟨a,b⟩ as \ensuremath−cos(\ensuremathθ\mathbita\ensuremath−\ensuremathθ\mathbitb)\ensuremath{-}\mathrm{cos}({\ensuremath{\theta}}_{\mathbit{a}}\ensuremath{-}{\ensuremath{\theta}}_{\mathbit{b}})\ensuremath−cos(\ensuremathθ\mathbita​\ensuremath−\ensuremathθ\mathbitb​) where \ensuremathθ\mathbitv{\ensuremath{\theta}}_{\mathbit{v}}\ensuremathθ\mathbitv​ is the angle between the vector \mathbitv\mathbit{v}\mathbitv and a reference vector chosen once and for all and in a fixed plane. Assuming classical microscopic realism (CMR) there exist also normalized projection pairs (ai\ensuremath′{a}_{i}^{\ensuremath{'}}ai\ensuremath′​, bi\ensuremath′{b}_{i}^{\ensuremath{'}}bi\ensuremath′​) of the spins of the pairs (pi{p}_{i}pi​, p‾i{\overline{p}}_{i}p​i​) along (\mathbita\ensuremath′{\mathbit{a}}^{\ensuremath{'}}\mathbita\ensuremath′, \mathbitb\ensuremath′{\mathbit{b}}^{\ensuremath{'}}\mathbitb\ensuremath′) so that ⟨a\ensuremath′,b\ensuremath′⟩=\ensuremath−cos(\ensuremathθ\mathbita\ensuremath′\ensuremath−\ensuremathθ\mathbitb\ensuremath′)⟨{a}^{\ensuremath{'}},{b}^{\ensuremath{'}}⟩=\ensuremath{-}\mathrm{cos}({\ensuremath{\theta}}_{{\mathbit{a}}^{\ensuremath{'}}}\ensuremath{-}{\ensuremath{\theta}}_{{\mathbit{b}}^{\ensuremath{'}}})⟨a\ensuremath′,b\ensuremath′⟩=\ensuremath−cos(\ensuremathθ\mathbita\ensuremath′​\ensuremath−\ensuremathθ\mathbitb\ensuremath′​). Since all projections are in \ensuremath−1,1{\ensuremath{-}1,1}\ensuremath−1,1, ∣⟨c,d⟩+⟨c,e⟩∣+∣⟨f,d⟩\ensuremath−⟨f,e⟩∣\ensuremath≤2|⟨c,d⟩+⟨c,e⟩|+|⟨f,d⟩\ensuremath{-}⟨f,e⟩|\ensuremath{\le}2∣⟨c,d⟩+⟨c,e⟩∣+∣⟨f,d⟩\ensuremath−⟨f,e⟩∣\ensuremath≤2 for ccc, ddd, eee, and fff in a,b,a\ensuremath′,b\ensuremath′{a,b,{a}^{\ensuremath{'}},{b}^{\ensuremath{'}}}a,b,a\ensuremath′,b\ensuremath′. Assuming locality (the impossibility of any effect of an event on another event when said events are spatially separated) beside QM and CMR, Bell's theory lets one deduce various violations of this inequality at some choices of quadruplets Q\ensuremath≡(\mathbita,\mathbitb,\mathbita\ensuremath′,\mathbitb\ensuremath′)Q\ensuremath{\equiv}(\mathbit{a},\mathbit{b},{\mathbit{a}}^{\ensuremath{'}},{\mathbit{b}}^{\ensuremath{'}})Q\ensuremath≡(\mathbita,\mathbitb,\mathbita\ensuremath′,\mathbitb\ensuremath′). Our main result is the existence of such QQQ's where at least one of the above inequalities is violated if one only assumes QM, CMR, and some very mild further hypotheses that only concern the behavior of correlations that appear in these inequalities near special QQQ's.
We revisit the computation of correlations of spin projections onto unit vectors for spin-1/2 particles in Quantum Mechanics. We then choose one of the Boole inequalities that, as we recall, must be obeyed by collections of sequences of normalized spin-1/2 projections onto unit vectors that belong to, say, the plane that is orthogonal to the classical trajectory or to the three dimensional space. Next we define the concept of being a-p for i-indexed sequences of spin-1/2 particles: it turns out that a sequence of spin-1/2 particles is a-p if and only if the particles behave as if they had indeed been prepared (in the usual sense) to have definite spins +1 or -1 along the axis a, the sequence of signs being pre-determined. The a-p property is formulated in a way abstract enough to let one generalize it and we define the concept of being a-p and b-p. However, using our chosen Boole inequality we prove that no particle sequence can be both a-p and b-p when a and b are not co-linear. We next consider two modifications of Quantum Mechanics: - The first one by assuming both Weak Realism, the weakest form of Realism needed to develop any main theorem in Bell's Theory, and Locality. - The second one by assuming Weak Realism and an hypothesis that turns out to be none else than (relativistic) Causality extended to cover as well the observable values that make sense only if one assumes Weak Realism. In both cases we arrive at the conclusion that in the EPRB setting where successive pairs of spin-1/2 particles are prepared in the so-called singlet state, one of these particles sequences is both a-p and b-p and a and b are the opposite of the vectors respectively used to project the spin of the other particle and to get the Realism-based alternate value for a projection. Thus the usual discussions of Bell's inequalities under such hypotheses do not even make physical sense.
Some new, powerful, and very simple Bell inequalities are derived. They establish that for two spin-sss particles in the singlet state, the quantum-theoretic Einstein-Podolsky-Rosen correlations for spin measurements along all pairs of directions from a set of NNN axes are incompatible with local realism for any set of distinct coplanar axes when N=3N=3N=3, and for any set of distinct axes whatever when N=4N=4N=4, with the possible exception of sets restricted by one of the constraints $\stackrel{^}{a}\ifmmode\pm\else\textpm\fi{}\stackrel{^}{b}\ifmmode\pm\else\textpm\fi{}\stackrel{^}{c}\ifmmode\pm\else\textpm\fi{}\stackrel{^}{d}=0$. These results hold for any value of the spin.
While it is widely agreed that Bell's theorem is an important result in the foundations of quantum physics, there is much disagreement about what exactly Bell's theorem shows. It is agreed that Bell derived a contradiction with experimental facts from some list of assumptions, thus showing that at least one of the assumptions must be wrong; but there is disagreement about what the assumptions were that went into the argument. In this paper, I make a few points in order to help clarify the situation. Introduction Different authors have expressed very different views about what Bell's theorem shows about physics. The disagreement concerns particularly the question of which assumptions go into the argument. Since we have to give up one of the assumptions leading to the empirically violated Bell inequality, knowing what the assumptions were is crucial. For example, if author X believes that the argument requires only one assumption, A 1 , while author Y believes that it requires two, A 1 and A 2 , then X will conclude that A 1 ¬ must be abandoned, while Y will conclude that either A 1 or A 2 must abandoned, so A 1 may well be true in our world if A 2 is false. In this paper, I consider several assumptions that have been mentioned in connection with Bell's theorem, and I look into their roles in the proof of Bell's theorem. While I do not say anything here that has never been said before, I hope that my remarks can nevertheless be helpful to some readers. For the sake of definiteness, the version of the relevant experiment that I will consider involves two spin-1\2 particles, initially in the singlet spin state ψ = 2 −1/2 (|↑↓ ⟩ − |↓↑ ⟩). Two widely separated experimenters, Alice and Bob, have one particle transported to each of them. At spacelike separation, they each choose a direction in space, represented by the unit vectors a and b , and carry out a quantum measurement of the spin observable in this direction on their particle, ( a · σ ) ⊗ I and I ⊗ ( b · σ ), with I the 2 × 2 unit matrix and σ the triple of Paulimatrices. The possible outcomes, A and B , are±1 on each side. Bell's theorem is the statement that under the assumptions we are about to discuss, Bell's inequality holds; quantum theory predicts violations of Bell's inequality, and experiments have confirmed these predictions.
We review the computation of correlations of successive projections of the spin onto axes for spin-1/2 particles and EPRB pairs (in the Singlet state). We assume forms of Realism (at least as general as the Predictive Hidden Variables in the classical Bell's Theory), to review one particle correlations before we further assume Locality and analyze EPRB pairs. To our surprise, we find that the Abstract Bell Inequalities with three or four axes cannot have physical meaning i.e., two spin projections cannot belong to one particle or some correlations that one would need cannot be evaluated. A recent version of Bell's Theory without Locality turns out to be unaffected by our results that refocus the open issues on "realism" rather than "local realism", inviting us to leave locality alone so to speak.
The geometric conditions for violation of Bell's inequality, in its original form dealing with a pair of spin one-half particles formed in the singlet spin state, are discussed. The parameters x1,x2{x}_{1},\phantom{\rule{0.3em}{0ex}}{x}_{2}x1​,x2​, and x3{x}_{3}x3​ are introduced as the cosines of angles ${\ensuremath{\theta}}_{ab}=\ensuremath{\angle}(\mathbf{a},\mathbf{b})$, ${\ensuremath{\theta}}_{ac}=\ensuremath{\angle}(\mathbf{a},\mathbf{c})$, and ${\ensuremath{\theta}}_{bc}=\ensuremath{\angle}(\mathbf{b},\mathbf{c})$ between the unit vectors a,b\mathbf{a},\phantom{\rule{0.3em}{0ex}}\mathbf{b}a,b, and c\mathbf{c}c defining the orientation of the corresponding Stern-Gerlach magnets. A Monte Carlo experiment shows that in order to obtain violation, the three parameters must belong to a definite region in the parametric space that encompasses 1∕31∕31∕3 of all possible triplets of unit vectors generated randomly from the isotropic distribution in the ordinary space. By introducing a measure of violation $D\ensuremath{\equiv}\ensuremath{\mid}{x}_{1}\ensuremath{-}{x}_{2}\ensuremath{\mid}+{x}_{3}\ensuremath{-}1$, it is found that $0\ensuremath{\leqslant}D\ensuremath{\leqslant}1∕2$ and that the maximal violation D=1∕2D=1∕2D=1∕2 is obtained only in two isolated cases, when the relevant angles are ${\ensuremath{\theta}}_{ab}=2\ensuremath{\pi}∕3$, ${\ensuremath{\theta}}_{ac}={\ensuremath{\theta}}_{bc}=\ensuremath{\pi}∕3$, or when ${\ensuremath{\theta}}_{ab}={\ensuremath{\theta}}_{bc}=\ensuremath{\pi}∕3$, ${\ensuremath{\theta}}_{ac}=2\ensuremath{\pi}∕3$. In both cases the unit vectors a,b\mathbf{a},\phantom{\rule{0.3em}{0ex}}\mathbf{b}a,b, and c\mathbf{c}c are coplanar.
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.