John S. Bell  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)  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 .
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)
When the random variable is a function of nonrecurrent hidden variables the sample average of the kth experiment is written
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
The sample average is then the inner product of the two vectors
The probability density ρN(λ) specifies hidden variables are nonrecurrent. To take into account experiment independence write the correlation as
Each experiment, k, has its own random variables Ak and Bk satisfying
(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  experiments the configurations consist of
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.
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.
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  is modified for nonrecurrent hidden variables as follows. We write rk for the correlation of the kth experiment
where each λkn is unique. Following Bell the difference of two such correlations is written
which in the limit of large N becomes
As with Bell factor that expression
and take the absolute value using
to obtain the inequality
Now, unlike Bell, the product
is not equal to 1 since, for hidden variables belonging to ΛN,
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
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
Expand that product to give
which is equivalent to
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  inequality which is written as
We have already shown that for all r1 and r2
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  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 (3.15)
The ensemble average probabilities of equations (3.14) are then given by: (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 (3.17)
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:
We repeat (3.17) with the recurrent “kernel” specified as
and then write the corresponding nonrecurrent “kernel”
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
Each term in that expression is independent of every other term. As such the max of KN is obtained by maximizing each term separately.
The minimum is likewise determined
Setting XY = 1 leads to
as compared to
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.  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  program is maintained.
The GHZ  paper derives a condition that does not use inequalities. They introduce the expressions (10a) (10b)
and then list 4 premises
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.
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.
Reality: same as in Sec. 2.
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).
They state (11a) (11b)
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).
A consequence of these is (14a) (N14a)
which can be rewritten as (14b) (N14b)
which in combination with Eq. (12a) yields (16) (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 (17) (N17)
which in combination with Eq. (12b) yields (18) (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 (16) (N16) (17) (N17)
We immediately see that A4(π) can not be set equal to A5(π). Those functions occur under different experimental configurations
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.
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  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 :
The paradox of Einstein, Podolsky and Rosen  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.
Rudin W., Principles of Mathematical Analysis, McGraw Hill, 1964. Google Scholar
Feller W., An Introduction To Probability Theory and its Applications, Vol II, 1971. Google Scholar
Clauser J. F., Shimony A., Bell’s Theorem: experimental tests and implications, 1978, 1895, in Rep. Prog. Phys. 1978 41 1881-1927, The Instittue of Physics. Google Scholar
Wikipedia, Quantum Entanglement, https://en.wikipedia.org/wiki/Quantum_entanglement
Plenio M. B., Virmani S., An introduction to entanglement measures, https://arxiv.org/abs/quant-ph/0504163v3
Giustina M., Marijn A., Versteegh M., Wengerowsky S, Handsteiner J., Hochrainer A., et al., Significant-Loophole-Free Test of Bell’s Theorem with Entangled Photons, Phys. Rev. Lett., 2015, 115, 250401. CrossrefWeb of ScienceGoogle Scholar
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Published Online: 2017-12-29