## 5. Hidden-Variables Models: Noncontextual versus Contextual, Local versus Nonlocal

Hidden variables were introduced in the line with von Neumann’s no-go theorem as representing dispersion free states, see, e.g., Bell [

15,

16,

79] and especially Gudder [

80,

81,

82]. Each value

${\lambda}_{0}$ of hidden variable

$\lambda $ determines uniquely the values of all observables. Thus, the observables can be mathematically represented as functions of

$\lambda .$ Such hidden-variables models are known as

deterministic. Mathematically, they are represented by Kolmogorov probability spaces [

11], triples of the form

$(\mathsf{\Lambda},\mathcal{F},P).$ Here,

$\mathsf{\Lambda}$ is the set of hidden variables,

$\mathcal{F}$ is a

$\sigma $-algebra of subsets of

$\mathsf{\Lambda},$ and

P is a probability measure on

$\mathcal{F}.$ Observables are represented by RVs, (measurable) functions on

$\mathsf{\Lambda}.$ The ranges of values of observables and corresponding RVs should coincide (see Mermin [

78] for details). Average of observable

A which is represented by RV

a is given by

More generally, for any set of compatible (jointly measurable) observables

$A,B,\dots ,K$ and any (bounded and measurable) function

$f,$ average of the operator-function

$f(A,B,\dots ,K)$ is given by

Such hidden-variables models are known as

noncontextual. Model

${\mathcal{M}}_{BCHSH}$ explored by Bell and Clauser, Horne, Shimony, and Holt (see

Section 2.2) is a (deterministic) noncontextual model.

It is well known that Bell argued that “the result of an observation may reasonably depend not only upon the state of the system (including the hidden variables) but also on the complete disposition of the apparatus” [

79]. Shimony [

83] stressed that this is the first statement about contextuality (although Bell did not use this terminology). Hidden-variables models of such type are known as contextual. In fact, Bell’s statement is closely coupled with Bohr’s emphasis of the role of experimental arrangement. However, Bohr considered quantum mechanics as a complete theory. The state of a system is given by wave function

$\psi $ and there is no need in supplementary parameters

$\lambda .$ (We remind that the name “contextualistic” was introduced by Shimony [

84] and a shortening to “contextual” was performed by Beltrametti and Cassinelli [

85].) Shimony made the Bohr–Bell statement concrete on the role of experimental arrangement as follows [

83]:

“John Stewart Bell (1928-90) gave a new lease on life to the program of hidden variables by proposing contextuality. In the physical example just considered, the complete state $\lambda $ in a contextual hidden variables model would indeed ascribe an antecedent element of physical reality to each squared spin component ${s}_{n}^{2}$ but in a complex manner: the outcome of the measurement of ${s}_{n}^{2}$ is a function ${s}_{n}^{2}(\lambda ,C)$ of the hidden variable $\lambda $ and the context $C,$ which is the set of quantities measured along with ${s}_{n}^{2}.$ ... a minimum constraint on the context C is that it consists of quantities that are quantum mechanically compatible, which is represented by self-adjoint operators which commute with each other...”

For a contextual model, average’s representation (

31) is modified as follows:

where context

C is determined by the set of compatible observables

$C=\{A,B,\dots ,K\}$ which are represented by RVs

${a}_{C},{b}_{C},\dots ,{k}_{C}.$ We continue citation of Shimony [

83]:

“Another reasonable constraint on C of great conceptual importance was proposed by Bell when the system of interest consists of two or more spatially separated parts, and the physical quantity of interest A concerns one of these parts. C should not include quantities whose measurements are events with space-like separation from the measurement of $A,$ since there would be a violation of relativistic locality if those measurements affected the outcome of the measurement of $A.$”

Thus, we have two types of contextuality, local and nonlocal. This local versus nonlocal structure of contextual models with hidden variables is not so much emphasized in modern studies on contextuality. Quantum contextuality is identified with a nonlocal one.

Model

${\mathcal{M}}_{KH}$ is a contextual hidden-variables model. We point out that, by writing paper [

31], its author was unaware about original works on contextual hidden-variable models (Gudder [

80,

81,

82], Bell [

79], Shimony [

83], Mermin [

78]). This lack of knowledge led to the statement: “We emphasize that our construction of the classical probability space for the EPR–Bohm–Bell experiment cannot be used to support the hidden variable approach to the quantum phenomena. The classical random parameter involved in our considerations cannot be identified with the hidden variable which is used the Bell-type considerations.” This statement was a consequence of the very restricted picture of hidden-variables models borrowed from the original Bell paper [

15] (see also [

26]). Our model has three distinguishing features:

RVs are context-independent, i.e., the

C-index can be omitted:

Contextual probabilities $\left\{{P}_{C}\right\}$ can be selected as conditional probabilities with respect to a single probability measure $P:{P}_{C}\left(E\right)=P\left(E\right|C).$ (In particular, contexts have the set-representation and conditional probability is given by Bayes’ formula.)

The model is locally contextual.

This is the good place to mention the hidden-variables interpretation of CP-model ${\mathcal{M}}_{DZ}$:

RVs are context-dependent, i.e., the C-index cannot be omitted.

Instead of a family of contextual probabilities

$\left\{{P}_{C}\right\},$ one can proceed with a single probability measure

$P:$The model is nonlocally contextual.

The rest of this section is devoted to analysis of the locality issue. This issue is very complex and it is not basic for the present paper which is devoted to the analysis of the possibility of construction of the CP-representation of quantum probabilities. Therefore, the coming analysis cannot be considered as complete. We come back to it in one of the further publications.

In his seminal paper [

16], Bell used the following definition of locality:

Now we make the hypothesis [68], and it seems one at least worth considering, that if the two measure- ments are made at places remote from one another the orientation of one magnet does not influence the result obtained with the other.

This definition matches with Einstein’s viewpoint on locality, see Bell’s citation [

16] of Einstein [

68]:

But on one supposition we should, in my opinion, absolutely hold fast: the real factual situation of the system $S1$ is independent of what is done with the system $S2,$ which is spatially separated from the former.

Bell concluded his article [

16] with the following statement:

In a theory in which parameters are added to quantum mechanics to determine the results of individual measurements, without changing the statistical predictions, there must be a mechanism whereby the setting of one measuring device can influence the reading of another instrument, however remote. Moreover, the signal involved must propagate instantaneously, so that such a theory could not be Lorentz invariant.

One of the problems with treatment of the locality issue in the Bell-framework is that space-time is absent in Bell’s mathematical formalization (see [

86,

87] for a discussion). In the following consideration, we shall ignore this problem (consideration of locality without using a mathematical model based on Minkovsky’s space-time, cf. [

88].

In the hidden-variables framework, Bell formalized the notion of locality (locality hypothesis) as follows. To make our notation closer to Bell’s notation, denote by

$a(i,j,\lambda )$ and

$b(j,i,\lambda )$ RVs corresponding to measurement of observables

${A}_{i}$ and

${B}_{j},$ respectively, under selection of settings

${r}_{a}=i$ and

${r}_{b}=j.$ The locality hypothesis is that RV

$a(i,j,\lambda )$ does not depend on the index

j and RV

$b(j,i,\lambda )$ does not depend on the index

i (see [

67], p. 65, Equations (2) and (3)).

Let us analyze the locality issue for CP-model

${\mathcal{M}}_{\mathrm{KH}}.$ Consider realization of this model based on the space of hidden variables

$\mathsf{\Lambda}$ defined in Equation (

13). For reader’s convenience, we list these 16 points once again:

where

${\alpha}_{i},{\beta}_{j}=\pm 1.$ RVs are functions defined on

$\mathsf{\Lambda}.$ For

$\alpha ,\beta =\pm 1,$ the following equalities hold:

and

Thus, the values of RV

${a}_{1}$ representing observable

${A}_{1}$ do not depend on the values of RV

${r}_{b}$ ruling selection of experimental settings for

${S}_{2}$ nor on the values of

b-RVs representing

B-observables (“the real factual situation of system

${S}_{1}$ is independent from what is done with system

${S}_{2}^{\u2033})$.

The CP-model ${\mathcal{M}}_{\mathrm{KH}}$ is local in the Einstein–Bell sense.

Model ${\mathcal{M}}_{\mathrm{KH}}$ is locally contextual. It is contextual because the values of RV ${a}_{i}$ depend on outcomes of RV ${r}_{a}$ representing observable ${R}_{a}$ compatible with observable ${A}_{i}.$

The considered locality condition is the analog of

local causality or the locality condition as considered by Bell [

79]. Typically, these conditions are formulated in the probabilistic terms for stochastic hidden-variables models. The condition of locality is formulated as the

probability factorization condition [

79]. Hidden-variables model

${\mathcal{M}}_{KH}$ is deterministic. However, to proceed closer to the standard framework, we shall also use the probabilistic terminology. We remark that, for a deterministic model, all conditional probabilities are equal either to zero or to one. In our notation, the factorization condition can be written as follows (for

$\alpha ,\beta =\pm 1)$:

or

Let

$i=1,j=1$ and let

$\lambda =(\alpha ,0,\beta ,0,1,1).$ Then,

Thus, for this

$\lambda ,$In the same way, we consider all cases of matching the indexes of ${a}_{i}$ and ${b}_{j}$ with the last digits of $\lambda $:

$i=1,j=2$ and $\lambda =(\alpha ,0,0,\beta ,1,2),$

$i=2,j=1$ and $\lambda =(0,\alpha ,\beta ,0,2,1),$

$i=2,j=2,$ and $\lambda =(0,\alpha ,0,\beta ,2,2).$

The conditional probabilities on the right-hand and left-hand sides of Equation (

37) also equal one.

Now, consider mismatching the indexes of RVs

${a}_{i}$ and

${b}_{j}$ with the last digits of

$\lambda .$ For example, let

$i=1,j=1$ and

$\lambda =(\alpha ,0,0,{\beta}^{\prime},1,2).$ Here,

We extend the definition of conditioning to the case such that both nominator and denominator equal zero. In such a case, we set conditional probability to zero. Thus,

Thus, the factorization condition trivially holds as

$0=0.$One may think that this (natural) regularization of conditional probability is the root of violation of Bell’s theorem. This is not the case. Even regularized conditional probability

$P(\alpha ,\beta |{r}_{a}=i,{r}_{b}=j,\lambda )$ provides the right representation:

The main issue is the correspondence rule coupling probabilities of model

${\mathcal{M}}_{KH}$ with observational probabilities. The probability on the right-hand side of Equality (

39) does not coincide with the observational probability. The latter equals

$P({a}_{i}=\alpha ,{b}_{j}=\beta |{r}_{a}=i,{r}_{b}=j).$#### 5.1. How Can This Happen?

We pointed to the possibility to violate Bell’s type inequalities in the local contextual framework:

By rejection of the BCHSH-rule for coupling observational probabilities with CP-probability on the space of hidden variables.