Physical Review A

Geometric conditions for violation of Bell’s inequality

Istok P. Mendaš

Published Mar 24, 2005• 2 citations

Abstract

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 ${x}_{1},\phantom{\rule{0.3em}{0ex}}{x}_{2}$, and ${x}_{3}$ 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 $\mathbf{a},\phantom{\rule{0.3em}{0ex}}\mathbf{b}$, and $\mathbf{c}$ 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 $1∕3$ 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 $D=1∕2$ 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 $\mathbf{a},\phantom{\rule{0.3em}{0ex}}\mathbf{b}$, and $\mathbf{c}$ are coplanar.

DOI: 10.1103/physreva.71.034103

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Physical Review A

Geometric conditions for violation of Bell’s inequality

Istok P. Mendaš

Published Mar 24, 2005• 2 citations

Publisher

Abstract

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

{x}_{1},\phantom{\rule{0.3em}{0ex}}{x}_{2}

, and

{x}_{3}

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

\mathbf{a},\phantom{\rule{0.3em}{0ex}}\mathbf{b}

, and

\mathbf{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∕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∕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

\mathbf{a},\phantom{\rule{0.3em}{0ex}}\mathbf{b}

, and

\mathbf{c}

are coplanar.

DOI: 10.1103/physreva.71.034103

Related papers

Analytic quantification of the singlet nonlocality for the first Bell inequality

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Recently an alternative way to quantifying the Bell nonlocality has been proposed [E. A. Fonseca and F. Parisio, Phys. Rev. A 92, 030101(R) (2015)]. In this work we further develop this concept, the volume of violation, and analytically calculate its value for the spin-singlet state with respect to the settings of the first Bell inequality. These settings correspond to three directions in space, or three arbitrary points on the unit sphere. It is shown that the triples of directions that lead to violations in local causality correspond to $1/3$ of all possible configurations. From the perspective of quantum communication, this means that two distant parties that were able to align their measurements in one direction only (the remaining direction in each site being random) have a probability of about $33.3%$ to certify their entanglement.Recently an alternative way to quantifying the Bell nonlocality has been proposed [E. A. Fonseca and F. Parisio, Phys. Rev. A 92, 030101(R) (2015)]. In this work we further develop this concept, the volume of violation, and analytically calculate its value for the spin-singlet state with respect to the settings of the first Bell inequality. These settings correspond to three directions in space, or three arbitrary points on the unit sphere. It is shown that the triples of directions that lead to violations in local causality correspond to $1/3$ of all possible configurations. From the perspective of quantum communication, this means that two distant parties that were able to align their measurements in one direction only (the remaining direction in each site being random) have a probability of about $33.3%$ to certify their entanglement.

arXiv:1511.01027Physical Review AMar 2, 20165 citations

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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.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.

arXiv:1212.4344Physical Review LettersJun 28, 20132 citations

The Assumptions of Bell's Proof

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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.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.

arXiv:1501.04168Cambridge University Press eBooksAug 31, 201612 citations

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Foundations of PhysicsMar 14, 20241 citation

Two-particle entanglement as a property of three-particle entangled states

Jose Luis Cereceda

In a recent article [Phys. Rev. A 54, 1793 (1996)] Krenn and Zeilinger investigated the conditioned two-particle correlations for the subensemble of data obtained by selecting the results of the spin measurements by two observers 1 and 2 with respect to the result found in the corresponding measurement by a third observer. In this paper we write out explicitly the condition required in order for the selected results of observers 1 and 2 to violate Bell's inequality for general measurement directions $e{\ensuremath{\rightarrow}}_{1},$ $e{\ensuremath{\rightarrow}}_{1}^{\ensuremath{'}},$ $e{\ensuremath{\rightarrow}}_{2},$ $e{\ensuremath{\rightarrow}}_{2}^{\ensuremath{'}},$ and $e{\ensuremath{\rightarrow}}_{3}.$ It is shown that there are infinitely many sets of directions giving the maximum level of violation. Further, we extend the analysis by the authors to the class of triorthogonal states $|\ensuremath{\Psi}〉{=c}_{1}|{z}_{1}〉|{z}_{2}〉|{z}_{3}〉{+c}_{2}|\ensuremath{-}{z}_{1}〉|\ensuremath{-}{z}_{2}〉|\ensuremath{-}{z}_{3}〉.$ It is found that a maximal violation of Bell's inequality occurs provided the corresponding three-particle state yields a direct (``all or nothing'') nonlocality contradiction.In a recent article [Phys. Rev. A 54, 1793 (1996)] Krenn and Zeilinger investigated the conditioned two-particle correlations for the subensemble of data obtained by selecting the results of the spin measurements by two observers 1 and 2 with respect to the result found in the corresponding measurement by a third observer. In this paper we write out explicitly the condition required in order for the selected results of observers 1 and 2 to violate Bell's inequality for general measurement directions $e{\ensuremath{\rightarrow}}_{1},$ $e{\ensuremath{\rightarrow}}_{1}^{\ensuremath{'}},$ $e{\ensuremath{\rightarrow}}_{2},$ $e{\ensuremath{\rightarrow}}_{2}^{\ensuremath{'}},$ and $e{\ensuremath{\rightarrow}}_{3}.$ It is shown that there are infinitely many sets of directions giving the maximum level of violation. Further, we extend the analysis by the authors to the class of triorthogonal states $|\ensuremath{\Psi}〉{=c}_{1}|{z}_{1}〉|{z}_{2}〉|{z}_{3}〉{+c}_{2}|\ensuremath{-}{z}_{1}〉|\ensuremath{-}{z}_{2}〉|\ensuremath{-}{z}_{3}〉.$ It is found that a maximal violation of Bell's inequality occurs provided the corresponding three-particle state yields a direct (``all or nothing'') nonlocality contradiction.

arXiv:quant-ph/9905095Physical Review ASep 1, 19972 citations

Bell Inequalities with a Range of Violation that Does Not Diminish as the Spin Becomes Arbitrarily Large

Anupam Garg, N. David Mermin

Some new, powerful, and very simple Bell inequalities are derived. They establish that for two spin- $s$ particles in the singlet state, the quantum-theoretic Einstein-Podolsky-Rosen correlations for spin measurements along all pairs of directions from a set of $N$ axes are incompatible with local realism for any set of distinct coplanar axes when $N=3$ , and for any set of distinct axes whatever when $N=4$ , 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.Some new, powerful, and very simple Bell inequalities are derived. They establish that for two spin-$s$ particles in the singlet state, the quantum-theoretic Einstein-Podolsky-Rosen correlations for spin measurements along all pairs of directions from a set of $N$ axes are incompatible with local realism for any set of distinct coplanar axes when $N=3$, and for any set of distinct axes whatever when $N=4$, 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.

Physical Review LettersSep 27, 198292 citations

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

{x}_{1},\phantom{\rule{0.3em}{0ex}}{x}_{2}

, and

{x}_{3}

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

\mathbf{a},\phantom{\rule{0.3em}{0ex}}\mathbf{b}

, and

\mathbf{c}

1∕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∕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

\mathbf{a},\phantom{\rule{0.3em}{0ex}}\mathbf{b}

, and

\mathbf{c}

are coplanar.

1/3

of all possible configurations. From the perspective of quantum communication, this means that two distant parties that were able to align their measurements in one direction only (the remaining direction in each site being random) have a probability of about

33.3%

to certify their entanglement.

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.

Some new, powerful, and very simple Bell inequalities are derived. They establish that for two spin-

s

particles in the singlet state, the quantum-theoretic Einstein-Podolsky-Rosen correlations for spin measurements along all pairs of directions from a set of

N

axes are incompatible with local realism for any set of distinct coplanar axes when

N=3

, and for any set of distinct axes whatever when

N=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.