# Classical (Local and Contextual) Probability Model for Bohm–Bell Type Experiments: No-Signaling as Independence of Random Variables

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## Abstract

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## 1. Introduction

#### 1.1. Towards CP-Representation

#### 1.2. No-Go Statements

- probabilities→states,
- random variables→ Hermitian operators,

- There is one jpd for all observables of the experiment.
- There is a deterministic hidden-variables model for the experiment.

#### 1.3. Can Experimental Violation of Bell Type Inequalities Be Checked in the Absence of Classical Probabilistic Representation?

#### 1.4. Conditional Probability Approach

#### 1.5. CP-Representations in the Presence of Signaling

## 2. Bohm–Bell Type Experiment: Traditional Description

#### 2.1. Description of (Four) Observables

#### 2.2. Classical Probability Model (BCHSH) for the Bohm–Bell Experiment: Four Random Variables

#### 2.3. BCHSH-Rule for Correspondence between Observational and Classical Probabilities

#### 2.4. Missed Component of Experimental Arrangement

**A randomness condition:**The inputs that we give to Alice and Bob to select experimental settings must be random. By this, we mean that Alice and Bob cannot predict the inputs that they will receive and thus adapt their strategy to the future values of the inputs.

## 3. Bohm–Bell Type Experiments: Taking into Account Random Generators

#### 3.1. Description of (Six) Observables

#### 3.2. Complete CP-Model: Six Random Variables

- ${a}_{i}=0$ (with probability one), if the i-setting was not selected, i.e., ${r}_{A}\ne i;$
- ${b}_{j}=0$ (with probability one), if the j-setting was not selected, i.e., ${r}_{B}\ne j.$

#### 3.3. Constraints on Joint Probabilities Implied by Matching Condition

#### 3.4. Correspondence between Observational and Classical Conditional Probabilities

#### 3.5. Violation of the CHSH-Inequality by Conditional Correlations

#### 3.6. Construction of jpd from Observational Probabilities

## 4. (No-)Signaling

#### 4.1. No-Signaling in Quantum Physics

#### 4.2. No-Signaling as a Condition of Independence of Random Variables

#### 4.3. Interpretation of No-Signaling: From Random Variables to Observables

#### 4.4. (No-) Signaling in Experiments in Quantum Physics and Psychology

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

- RVs are context-independent, i.e., the C-index can be omitted:$$\langle f(A,B,\dots ,K)\rangle ={\int}_{\mathsf{\Lambda}}f(a\left(\lambda \right),b\left(\lambda \right),\dots ,k\left(\lambda \right))d{P}_{C}\left(\lambda \right),$$
- 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.

- 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:$$$\langle f(A,B,\dots ,K)\rangle ={\int}_{\mathsf{\Lambda}}f({a}_{C}\left(\lambda \right),{b}_{C}\left(\lambda \right),\dots ,{k}_{C}\left(\lambda \right))dP\left(\lambda \right).$$
- The model is nonlocally contextual.

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.

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.

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 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.

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

#### 5.1. How Can This Happen?

## 6. Conclusions

- It demystifies the probabilistic structure of quantum mechanics, namely, the representation of probabilities by complex amplitudes and observables by Hermitian operators:
- It justifies the use of CP-based mathematical statistics for analysis of data from quantum experiments.
- It shows the possibility to describe the experimental schemes of the Bohm–Bell type with the aid of local contextual hidden-variables models.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

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Khrennikov, A.; Alodjants, A.
Classical (Local and Contextual) Probability Model for Bohm–Bell Type Experiments: No-Signaling as Independence of Random Variables. *Entropy* **2019**, *21*, 157.
https://doi.org/10.3390/e21020157

**AMA Style**

Khrennikov A, Alodjants A.
Classical (Local and Contextual) Probability Model for Bohm–Bell Type Experiments: No-Signaling as Independence of Random Variables. *Entropy*. 2019; 21(2):157.
https://doi.org/10.3390/e21020157

**Chicago/Turabian Style**

Khrennikov, Andrei, and Alexander Alodjants.
2019. "Classical (Local and Contextual) Probability Model for Bohm–Bell Type Experiments: No-Signaling as Independence of Random Variables" *Entropy* 21, no. 2: 157.
https://doi.org/10.3390/e21020157