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Open Physics

formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

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Volume 15, Issue 1


Volume 13 (2015)

Nonrecurrence and Bell-like inequalities

Douglas G. Danforth
Published Online: 2017-12-29 | DOI: https://doi.org/10.1515/phys-2017-0089


The general class, Λ, of Bell hidden variables is composed of two subclasses ΛR and ΛN such that ΛRΛN = Λ and ΛRΛN = {}. The class ΛN is very large and contains random variables whose domain is the continuum, the reals. There are an uncountable infinite number of reals. Every instance of a real random variable is unique. The probability of two instances being equal is zero, exactly zero. ΛN induces sample independence. All correlations are context dependent but not in the usual sense. There is no “spooky action at a distance”. Random variables, belonging to ΛN, are independent from one experiment to the next. The existence of the class ΛN makes it impossible to derive any of the standard Bell inequalities used to define quantum entanglement.

Keywords: Bell; inequalities; nonrecurrent; variables; quantum

PACS: 03.65.Ud

1 Introduction

John S. Bell [1] derived an inequality claiming it holds for all local hidden variable models of quantum mechanics (of the singlet state). Bell’s formulation is incomplete. It does not hold for all possible hidden variables even though the class, Λ, of his hidden variables is general. Bell writes

Let this more complete specification be effected by means of parameters λ. It is a matter of indifference in the following whether λ denotes a single variable or a set, or even a set of functions, and whether the variables are discrete or continuous. However, we write as if λ were a single continuous parameter.

Bell does not make use of the properties of continuous hidden variables. Instances of random variables belonging to the continuum (reals) [2] do not repeat and are members of ΛN. Every instance of a real random variable is unique. The probability of two instances being equal is zero, exactly zero [3].

His correlations restrict Λ to a subset ΛR consisting of hidden variables that repeat under different measurement device orientations. That implies Bell’s inequality does not govern the behavior of correlations derived from nonrecurrent hidden variables, ΛN. This suggests Bell’s formulation is not correct for nonrecurrent hidden variables.

Consider experiments. We write the sample average with a bar over variables x (instances of a random variable X) and note the sample average approaches the theoretical expected value for large sample size N (law of large numbers1) x¯N=1Nn=0N1xnlimNx¯N=X

When the random variable is a function of nonrecurrent hidden variables the sample average of the kth experiment is written AB¯k,N=1Nn=0N1Aak,λknBbk,λkn

Due to nonrecurrence each instance has its own unique hidden variable λkn specific to the kth experiment and nth occurrence. An experiment labeled k has measurement device orientations (a⃗k,b⃗k) called the configuration.

In vector notation Akn=Aak,λknBkn=Bbk,λkn

The sample average is then the inner product of the two vectors AB¯k,N=1NAkBk

The probability density ρN(λ) specifies hidden variables are nonrecurrent. To take into account experiment independence write the correlation as rk=dλρNλAkak,λBkbk,λ

Each experiment, k, has its own random variables Ak and Bk satisfying AiAj=δi,j

(and similarly for B). The Ai are independent of the Aj. Their means are zero and hence their correlation is zero. This formulation is local. Each function is dependent only on its local orientation and the hidden variable, λ.

It appears this formulation is context sensitive, and it is but not in the usually sense. There is no “spooky action at a distance”. Alice’s data at the time of recording is not a function of Bob’s orientation and Bob’s data at the time of recording is not a function of Alice’s orientation. But note, when the data are correlated they are brought to a common point and the joint orientations are revealed. An experiment consists of the set of data for which the configuration (joint orientations) is constant. For example, with the CHSH [4] experiments the configurations consist of

Table 1

CHSH configurations

When Alice collects her data she only knows her own orientations a⃗,a⃗′ and similarly for Bob’s orientations b⃗, b⃗′. It is the job of the correlator to segment Alice’s and Bob’s data into sequences of constant configuration, Kk. That makes the sequences correlation context sensitive, but the data are unchanged by segmentation only the partitions are created. Hence the label k for the random variables in this formulation reflects segmentation and not data dependency. The data in each segment are independent of every other segment when the hidden variables belong to ΛN.

2 Inequalities

The existence of the class ΛN makes it impossible to derive any of the standard inequalities: Bell [1], CHSH [4], CH [5] or the GHZ [6] constraint, used to define quantum entanglement [7, 8].

Quantum entanglement is a physical phenomenon that occurs when pairs or groups of particles are generated or interact in ways such that the quantum state of each particle cannot be described independently of the others, even when the particles are separated by a large distance—instead, a quantum state must be described for the system as a whole.

The class ΛN generates new predictions as specified in the Table 2 comparing the inequalities and their equivalent nonrecurrent form. The derivation of each form is presented below.

Table 2

Recurrent vs. nonrecurrent constraints

2.1 Bell’s inequality

The Bell inequality is composed of 3 correlations. In Bell’s notation the r3 correlation is r3 = P(b⃗,c⃗). That correlation never occurs for the class ΛN because P(b⃗,c⃗) is obtained by assuming the A of A(a⃗,λ)B(b⃗,λ) is the same as the A for A(a⃗,λ′)B(c⃗,λ′). That assumption is false for the class ΛN since each configuration (a, b) and (a, c) happens at a different time and hence have different hidden variables. The A’s do not multiply to 1 and P(b⃗,c⃗) does not occur. The nonrecurrent inequality holds for all r1 and r2. It places no constraint on those correlations.

John S Bell’s original inequality [1] is modified for nonrecurrent hidden variables as follows. We write rk for the correlation of the kth experiment rk=1Nn=0N1Aak,λknBbk,λkn

where each λkn is unique. Following Bell the difference of two such correlations is written rirj=1Nn=0N1Aai,λinBbi,λin1Nn=0N1Aaj,λjnBbi,λjnrirj=1Nn=0N1Aai,λinBbi,λinAaj,λjnBbi,λjn

which in the limit of large N becomes rirj=AiaiBibiAjajBjbj

As with Bell factor that expression rirj=AiaiBibi1AiaiBibiAjajBjbj

and take the absolute value using AiaiBibi=1

to obtain the inequality rirj1AiaiBibiAjajBjbj

which is rirj1AiaiBibiAjajBjbj,

Now, unlike Bell, the product Aai,λinAaj,λjn

for ai=aj

is not equal to 1 since, for hidden variables belonging to ΛN, ijλinλjn.

That fact is the crucial difference between Bell’s ΛR and ΛN. One can say, in general, that Bell’s assumption A(a⃗,λin)A(a⃗,λjn) = 1 does not hold for hidden variables belonging to ΛN.

For ΛN different samples are independence and so the expectation of the product equals to the product of the expectations rirj1AiaiBibiAjajBjbj,

or rirj1rirj,

and rirj+rirj1.

ΛN leads to an inequality that differs from Bell’s. It places no constraints on ri and rj. They hold for all −1 ≤ ri ≤ 1 and −1 ≤ rj ≤ 1. To show that, define maxij = MAX(ri, rj) and minij = MIN(ri, rj) and note that for all ri and rj 01+minij,01maxij,

and hence 01+minij1maxij.

Expand that product to give maxijminij+maxijminij1

which is equivalent to rirj+rirj1.

Hence the ΛN inequality is true for all ri and rj.

Bell’s procedure does not bound the correlations formed for nonrecurrent hidden variables.

2.2 CHSH inequality

The CHSH form arises from a double application of the Bell form. It places no constraints on the correlations r1, r2, r3, r4.

The same steps as with Bell can be performed for the derivation of the CHSH [4] inequality which is written as r1r2+r3+r42

Start from r1r2+r3+r4=A1B1A2B2+A3B3+A4B4A1B11A1B1A2B2+A3B31+A3B3A4B41A1B1A2B2+1+A3B3A4B41A1B1A2B2+1+A3B3A4B41A1B1A2B2+1+A3B3A4B42r1r2+r3r4

so r1r2+r3+r4+r1r2r3r42

We have already shown that for all r1 and r2 r1r2+r1r21

so correspondingly r3r4+r3r41

Hence that inequality holds for all r1, r2, r3, and r4. No constraints are placed on those correlations.

2.3 CH inequality

The CH inequality again assumes the hidden variables recur under different configurations. When the expressions are written taking into account the independence of each configuration the upper bound increases to 1. Experiments that violate the upper bound of 0 do not violate the upper bound of 1 and hence a local model is not ruled out by 0 violation.

Clauser [5] in his “Bells theorem: experimental tests and implications” writes

Following our discussion of §3.3, we assumed that, given λ, a and b, the probabilities p1(λ, a) and p2(λ, b) are independent. Thus we write the probabilities of detecting both components as p12(λ,a,b)=p1(λ,a)p2(λ,b)(3.15)

The ensemble average probabilities of equations (3.14) are then given by: p1(a)=Λp1λ,adρp2(b)=Λp2λ,bdρp12(a,b)=Λp12λ,a,bdρ(3.16)

To proceed, CH introduce the following lemma, the proof of which may be found in their paper: if x, x′, y, y′, X, Y are real numbers such that 0 ≤ x, x′ ≤ X and 0 ≤ y, y′ ≤ Y, then the inequality XYxyxy+xy+xyYxXy0(3.17)

holds. Inequality (3.17) and equation (3.15) yield: 1p12(λ,a,b)p12(λ,a,b)+p12(λ,a,b)+p12(λ,a,b)p1(λ,a)p2(λ,b)0(3.18)

Integrating inequality (3.18) over λ with distribution ρ, and using equation (3.16), one obtains the result: 1p12(a,b)p12(a,b)+p12(a,b)+p12(a,b)p1(a)p2(b)0(3)

Implicit in the derivation of (3) is the assumption the same λ can appear with different configurations (a, b), (a, b′), (a′, b), and (a′, b′). That assumption is false for nonrecurrent random variables. Each configuration has its own unique set of hidden variables. Hence equation (3.18) does not hold for those hidden variables. That equation lacks generality. If λ appears in p12(λ, a, b) it does not appear in any of the other terms. For nonrecurrent hidden variables the context in (3.15) must be specified. The CH recurrent form (3.17) has 4 variables: x, x′, y, y′. The corresponding nonrecurrent form has 8 variables: 0xiX,0yiY,i=1,2,3,4

We repeat (3.17) with the recurrent “kernel” specified as KR=xyxy+xy+xyYxXy

and then write the corresponding nonrecurrent “kernel” KN=x1y1x2y2+x3y3+x4y4Yx3Xy1

The choice of -Yx3 -Xy1 is arbitrary. They could just as well have been set to -Yx5 -Xy5 in a fifth experiment but we seek the least upper bound and assume the x and y are taken from the experiments as written. Doing so increases the constraints on the variables. Regroup the nonrecurrent kernel and write KN=x1Xy1x2y2+x3y3Y+x4y4

Each term in that expression is independent of every other term. As such the max of KN is obtained by maximizing each term separately. MAXKN=MAXx1Xy1+MAXx2y2+MAXx3y3Y+MAXx4y4MAXKN=0+0+0+XY=XY

The minimum is likewise determined [MINKN=MINx1Xy1+MINx2y2+MINx3y3Y+MINx4y4MINKN=XYXYXY+0=3XY

Setting XY = 1 leads to 3KN1

as compared to 1KR0

The upper bound for experimental results is 1 and not 0. Hence any experiment that exceeds 0 does not imply nonlocality but rather that the hidden variables are nonrecurrent (and local).

A recent paper by Giustina et al. [9] purports to violate the CH-Eberhard upper bound of 0 and hence simultaneously close several loopholes. The nonrecurrent upper bound is 1 for CH and hence there is no difficulty explaining those experimental results with a local model. The CH derivation is invalidated by nonrecurrent hidden variables. In general the CH inequality is false.

2.4 GHZ constraint

The GHZ constraint again assumes the hidden variables recur under different configurations. When the independence of those configurations is taken into account the GHZ contradiction does not occur and the EPR [10] program is maintained.

The GHZ [6] paper derives a condition that does not use inequalities. They introduce the expressions Ifφ1+φ2φ3φ4=0,thenEψn1,n2,n3,n4=1(10a) Ifφ1+φ2φ3φ4=π,thenEψn1,n2,n3,n4=1(10b)

and then list 4 premises

  1. Perfect correlations: With four Stern-Gerlach analyzers set at angles satisfying the conditions of either (10a) or (10b), knowledge of the outcomes for any three particles enables a prediction with certainty of the outcome for the fourth.

  2. Locality: Since at the time of measurement the four particles are arbitrarily far apart, they presumably do not interact, and hence no real change can take place in any one of them in consequence of what is done to the other three.

  3. Reality: same as in Sec. 2.

  4. Completeness: Same as in Sec. 2.

We now reproduce their salient arguments with the corresponding nonrecurrent form. Consider the meaning for ΛN. Each case is a different configuration. As such each case must be tagged with a different hidden variable. For notation simplicity we write Ai rather than Aλi. We also suppress the arguments of the functions since they are recoverable from the configuration table (see Table 3 below).

Table 3

GHZ Configurations

They state Ifφ1+φ2φ3φ4=0,thenAγ(φ1)Bγ(φ2)Cγφ3Dγφ4=1(11a) Ifφ1+φ2φ3φ4=π,thenAγφ1Bγφ2Cγφ3Dγφ4=1(11b)

Let us now consider some implications of just one of (11a) and (11b), say, the first. Four instances of (11a) are Aγ0Bγ0Cγ0Dγ0=1(12a) A1B1C1D1=1(N12a) AγϕBγ0CγϕDγ0=1(12b) A2B2C2D2=1(N12b) AγϕBγ0Cγ0Dγϕ=1(12c) A3B3C3D3=1(N12c) Aγ2ϕBγ0CγϕDγϕ=1(12d) A4B4C4D4=1(N12d)

The configurations are specified by the angles used in (12a-12d)

From Eqs. (12a) and (12b) we obtain Aγ0Cγ0=AγϕCγϕ,(13a) A1B1C1D1=A2B2C2D2(13Na)

There is no cancellation of factors because the hidden variables are different. The B1(0) and B2(0) do not cancel nor do the D1(0) and D2(0).

and from Eqs. (12a) and (12c) we obtain AγφDγφ=Aγ0Dγ0.(13b) A1B1C1D1=A3B3C3D3(N13b)

A consequence of these is Cγφ/Dγφ=Cγ0/Dγ0,(14a) A2A3B2B3C2/D3=C3/D2(N14a)

which can be rewritten as CγφDγφ=Cγ0Dγ0,(14b) A2A3B2B3C2D3=C3D2(N14b)

because Dγ (φ) ± 1 and hence equals its inverse, and the same for Dγ(0). We then obtain from (12d) and (14b) Aγ2φBγ0Cγ0Dγ0=1(15) A4B3C3D2A2A3B2B4C2C4D3D4=1(N15)

which in combination with Eq. (12a) yields Aγ2φ=Aγ0=constforallφ.(16) A42ϕ=A10fϕ,fϕ=A2ϕA3ϕB10B20B30B40C10C2ϕC30C4ϕD10D20D3ϕD4ϕ.(N16)

Equation (16) is a quite surprising preliminary result. By itself, this equation is not mathematically contradictory, but physically it is very troublesome: For if Aγ (φ) is intended as EPR’s program suggest, to represent an intrinsic spin quantity, then Aγ (0) and Aγ (π) would be expected to have opposite signs. The trouble becomes manifest, and an actual contradiction emerges, when we use (11b)–which until now has not been brought into play–to obtain Aγθ+πBγ0CγθDγ0=1(17) A5θ+πB50C5θD50=1(N17)

which in combination with Eq. (12b) yields Aγθ+π=Aγθ(18) A5θ+π=A2θgθ,gθ=B20B50C2θC5θD20D50.(N18)

This result confirms the sign change that we anticipated on physical grounds in EPR’s program, but it also contradicts the earlier result of Eq. (16) (let φ = π/2, θ = 0). We have thus brought to the surface an inconsistency hidden in premises (i)-(iv).

Using the suggested values for φ we obtain Aγπ=Aγ0(16) A4π=A10fπ(N16) Aγπ=Aγ0(17) A5π=A20g0(N17)

We immediately see that A4(π) can not be set equal to A5(π). Those functions occur under different experimental configurations K4=π,0,π/2,π/2K5=π,0,0,0

and hence, for ΛN, they have different hidden values. In general they are not equal. Because they are not equal there is no contradiction. Because there is no contradiction the premises (i)-(iv) hold. Because those premises hold the EPR hidden variable program is maintained. Quantum theories based on hidden variables remain possible.

3 Conclusion

We have presented four situations based on the existence of the hidden variable class ΛN which are capable of violating the well-known inequalities and condition. Experimental violation does not discriminate between local hidden variable models and standard Hilbert space based quantum mechanics. The state of the local hidden variable model takes one and only one value at each point in time. In contrast the state of standard quantum mechanics takes all possible values at each point in time. Leonard Susskind mentioned in a recent lecture that Richard Feynman said “Hilbert space is so damn big!”. We now see that such excess is unnecessary. Real local single valued hidden variables can violate Bell’s inequalities. Complex nonlocal multivalued quantum mechanics can violate Bell’s inequality. Inequality violation does not determine which model is correct.

The literature is vast on the implications of inequality violation. Much of that literature is made suspect by the hidden variable class ΛN.

The abstract of the EPR paper [10] says

In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false. One is thus led to conclude that the description of reality given by a wave function is not complete.

John S. Bell states in his paper [1]:

The paradox of Einstein, Podolsky and Rosen [10] was advanced as an argument that quantum mechanics could not be a complete theory but should be supplemented by additional variables. These additional variables were to restore the theory causality and locality.

We have presented evidence that Bell’s formulation fails for nonrecurrent hidden variables. As such Einstein’s program of completing quantum mechanics with hidden variables remains viable.


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  • 1

    Wikipedia states “A ‘law of large numbers’ is one of several theorems expressing the idea that as the number of trials of a random process increases, the percentage difference between the expected and actual values goes to zero.” 

About the article

Received: 2017-09-12

Revised: 2017-11-05

Accepted: 2017-11-06

Published Online: 2017-12-29

Citation Information: Open Physics, Volume 15, Issue 1, Pages 762–768, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2017-0089.

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© 2017 Douglas G. Danforth. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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