# Using Measured Values in Bell’s Inequalities Entails at Least One Hypothesis in Addition to Local Realism

## Abstract

**:**

## 1. Introduction

_{CHSH}≡ |E(α,β) − E(α,β’)| + |E(α’,β’) +E(α’,β)| ≤ 2

^{ij}are the number of coincidences observed at the detectors, where i,j = + (−) label the transmitted (reflected) output of the analyzer and the first (second) index indicates the A (B) station. The |α,β on the right means that all the coincidences have been recorded with the analyzers set to the angle values {α,β}. Then, for example, C

^{+−}|0,π/8 is the number of coincidences recorded with a photon detected in the transmitted output of the analyzer at Station A and the reflected output at Station B, when the angle setting is 0 at A and π/8 at B. The predictions of quantum mechanics (QM) violate Equation (1). For example, for the entangled state of two particles |ϕ

^{+}〉 = (1/√2){|x

_{a},x

_{b}〉 + |y

_{a},y

_{b}〉}, if {α,β,α’,β’} are chosen to be equal to {0,π/8,π/4,3π/8}, then S

_{CHSH}= 2√2 > 2. This prediction (or others alike) has been experimentally confirmed under rigorous conditions [2,3,4,5,6,7,8,9,10]. The apparent conclusion is that locality and/or realism (in short: local realism, LR) are not valid in nature. This contradiction is a crucial issue, because LR is assumed not only in everyday life, but also in all scientific practice (excepting QM, of course). The contradiction should hence be solved in a fully satisfactory way in order to attain a coherent approach to the study of nature.

## 2. The Hypothesis Additional to LOCAL Realism

#### 2.1. Review of the Derivation of the CHSH Inequality

#### 2.2. The Additional Hypothesis

_{measured}in Equation (7) is an average over time. In practice, these two averages are implicitly assumed to be equal, but they may not be. The assumption stating that these two averages are equal is the well-known ergodic hypothesis. In other words, Bell’s inequalities are derived from assuming LR, but they can be used in the real world only if ergodicity (or a similar hypothesis, see later) is also assumed. The idea that non-ergodicity may be related with the solution of the QM vs. LR controversy is not new [13,14,15]. Yet, it must be warned that the meaning given to “non-ergodicity” in those early approaches was influenced by the proposal of an LR model based on a system of coupled oscillators (thus similar to the best known example of non-ergodicity: the Fermi–Pasta–Ulam system) [16]. This proposal was eventually shown to be reducible to a form of the time-coincidence loophole [17]. Instead, the meaning given to “non-ergodicity” here is more general.

^{6}), the numbers in the boxes are fatally averages over time, while the terms in the (theoretically derived) Bell’s inequality are averages over the ensemble of states of the hidden variable. Simply, the two averages are not necessarily equal. We cannot insert the numbers in the boxes into the derived Bell’s inequality (and to expect the result to be logically linked to the premises for deriving that inequality) unless we suppose the two averages (time and ensemble) are equal, which is the usual meaning of “ergodicity”.

^{ij}= 0. It is a zero-over-zero indeterminacy. Yet, counterfactual definiteness ensures that the rhs of Equation (12) does take some value (see later).

**A**≡ “I find Alice in the cafeteria” is 1, and 0 when I do not find her there. After many visits to the cafeteria, I measure the expectation value 〈

**A**〉 = 0.3. This is the available information. Now, let us consider the question, “What is the expectation value of

**A**when I don’t go to the cafeteria?” Assuming that Alice and the cafeteria have a well defined existence even when I do not go there (roughly speaking, if realism or counterfactual definiteness is assumed), and then slightly changing the definition of

**A**, from “I find” to “To find,” the expectation value of

**A**is some well defined number, say, q. However, the value of q cannot be known with the information available at this point. More information is needed regarding the behavior of Alice and the properties of the cafeteria when I am not observing them (that is, a “possible world” must be defined) to assign a numerical value to q. Defining this missing information means a hypothesis in addition to LR. If this additional hypothesis is not made (that is, if this missing information is not provided), then the values of the counterfactual terms in Equation (13) (which cannot be measured) remain undefined, and it is therefore impossible to know whether or not the results of an experiment (which measures only the factual terms) violate the inequality.

#### 2.3. The Conditions for Retrieving the Validity of Bell’s Inequalities

**A**〉 = 0.3). Each of the three counterfactual terms in the rhs in Equations (10) and (11) is then equal to the factual term; hence

_{CHSH}≤ 2.

_{1}, t

_{1}+ T] (say, when the variable is being observed) is equal to the time average recorded during any other interval [t

_{2}, t

_{2}+ T] (say, when the variable is not being observed) provided, of course, that T is sufficiently long. The relationship between the validity of the HDA and of a Markovian underlying dynamics seems intuitive, and deserves to be examined in detail elsewhere.

## 3. A Simple Model That Violates the Additional Hypotheses

#### 3.1. A Classical Mixture with Delay

^{+}〉 can be reproduced, regardless of the position of the analyzer in Station B (see Figure 1) by a statistical mixture of photon pairs polarized parallel and orthogonal to the analyzer in Station A, or

_{α}= ½ {|α〉〈α|

_{A}⊗|α〉〈α|

_{B}+ |α

_{⊥}〉〈α

_{⊥}|

_{A}⊗|α

_{⊥}〉〈α

_{⊥}|

_{B}}

_{α}is

^{++}(a,b,α) = ½ {cos

^{2}(a − α) × cos

^{2}(b − α) + sin

^{2}(a − α) × sin

^{2}(b − α)}

#### 3.2. The Static Case

_{CHSH}≈ 2√2. One of the three counterfactual terms (the one with the actual value of a) in the rhs of Equations (10) and (11) is equal to the factual time average, and the other two terms are zero. To reproduce these results, note that P

^{++}+ P

^{+−}= ½ and P

^{++}= P

^{−−}, so E(a,b,α) = 4 × P

^{++}(a,b,α) − 1, and use Equation (17). Hence, E(a,b) = E(a,b) = ±√2/2. Equation (14) is thus not fulfilled, and the usual CHSH inequality (Equation (1)) is not retrieved from Equation (13). In consequence, if values of measured E(a,b) are inserted into Equation (1), S

_{CHSH}> 2 (as was said before), but this result does not refute LR (because Equation (14) is not valid, so we are outside the gray set in Figure 2).

#### 3.3. The Random Variable Case

_{CHSH}and Δ is obtained after 2000τ with a coincidence rate of 500/τ and after discarding a transient of 200τ (total number of coincidences = 10

^{6}). The initial condition for a(t < 0) = 0. The rate of incident pairs is high enough to follow the variations of α(t) in detail; be aware that this condition is far from having been reached in any experiment. Due to fluctuations caused by the randomness of the jumps, the strict equality Δ = 0 is never obtained numerically. In what follows, I will say that Equation (14) holds (then we are inside the gray set in Figure 2) if Δ <<1 (Δ ≈ 0), and that it is violated (then we are outside the gray set) if Δ ≈ 1.

_{CHSH}> 2. In these cases, it is always found that Δ ≈ 1, that is, all the additional hypotheses are violated too. The inverse is not true: it is possible that Δ ≈ 1 and yet S

_{CHSH}< 2. In other words, the violation of the additional hypotheses is a condition necessary, but not sufficient, to retrieve the usual significance of Bell’s inequalities.

_{CHSH}and Δ as a function of the unknown parameter Γ are plotted for μτ = ¼. The vertical dotted lines at Γ ≈ 0.2 and Γ ≈ π/2 are the limits of the region where the CHSH inequality is violated. Note that, within this region, Δ ≥ 1. For Γ < 0.2, one gets S

_{CHSH}< 2, but it is still Δ ≈ 1. For Γ→0, Δ→0, and S

_{CHSH}→√2, which is the value produced by ρ

_{α}if there is no correlation between α(t) and a(t). At Γ ≈ π/2, S

_{CHSH}falls to √2 and Δ to zero abruptly. The origin of these results is understood from the inspection of the behavior of α(t), see Figure 4. For Γ = 1 (Figure 4a) α(t) rapidly adjusts to the changing values of a(t), the condition α(t) ≈ a(t) holds most of the time and hence S

_{CHSH}> 2. For Γ = 0.1 (Figure 4b) α(t) follows a(t) too slowly. It reaches the target values 0 and π/4 only rarely and, in consequence, S

_{CHSH}< 2. The fact that Δ ≈ 1 in these two cases is a non-trivial result. For Γ = 0.02 (Figure 4c), the tracking force is too weak, and α(t) makes a low amplitude zigzag around the middle value π/8. Therefore, α(t) and a(t) are mostly uncorrelated and S

_{CHSH}≈ √2. As α(t) is nearly constant, Equation (14) holds and Δ ≈ 0. If Γ ≥ π/2 (Γ = 2, Figure 4d) α(t) diverges exponentially, rotating faster and faster. The random jumps of a(t) make the evolution even wilder, reversing the rotation at random times. In consequence, α(t) loses any correlation with a(t). This explains why S

_{CHSH}falls abruptly to √2 at Γ ≈ π/2. The trajectory of α(t) in this case explores the whole phase space, apparently fulfilling the condition of “mixing” that makes the ergodic hypothesis valid. Thus, it is not surprising that Δ falls to zero abruptly at this point: ergodicity holds, so we are inside the gray set in Figure 2.

## 4. Summary and Discussion

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Scheme of a typical experiment to test Bell’s inequalities. The source S emits pairs of photons entangled in polarization towards Stations A and B separated by a distance L. At each station, the analyzer is set at some angle value. The expectation value E(α,β) is measured from the number of coincidences at each pair of outputs, recorded during some time interval (which can be continuous, or not). A photon detected in the transmitted (reflected) output of the analyzers is defined as a + (−) result.

**Figure 2.**Logical relationship between different additional hypotheses defining different “possible worlds.” For example, in a possible world where the ergodic hypothesis is valid, the homogeneous dynamics assumption (HDA)and Equation (14) are also valid. In this same possible world, the equality between factual and counterfactual results may be valid or not. The hypotheses in gray retrieve the usual significance of Bell’s inequalities. In the “other possible worlds” (some of them are discussed in [11]), there is no logical link between the observed violation of Bell’s inequalities and the validity of local realism (LR).

**Figure 3.**S

_{CHSH}and Δ as a function of Γ, μτ = ¼. The horizontal dotted line indicates the CHSH bound.

**Figure 4.**Evolution of the hidden variable α(t) for μτ = ¼ and several values of Γ, note the different vertical scales; (

**a**) for Γ = 1, α(t) promptly follows the random jumps of a(t) between 0 and π/4, S

_{CHSH}> 2 and Δ ≈ 1; (

**b**) for Γ = 0.1, α(t) follows a(t) slowly, rarely reaching 0 or π/4, Δ ≈ 1 but S

_{CHSH}< 2; (

**c**) for Γ = 0.02, α(t) zigzags around π/8 never reaching 0 or π/4, S

_{CHSH}≈ √2 and Δ ≈ 0; (

**d**) for Γ = 2, α(t) varies wildly, filling the whole phase space and losing correlation with a(t), S

_{CHSH}≈ √2 and Δ ≈ 0.

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Hnilo, A.A.
Using Measured Values in Bell’s Inequalities Entails at Least One Hypothesis in Addition to Local Realism. *Entropy* **2017**, *19*, 180.
https://doi.org/10.3390/e19040180

**AMA Style**

Hnilo AA.
Using Measured Values in Bell’s Inequalities Entails at Least One Hypothesis in Addition to Local Realism. *Entropy*. 2017; 19(4):180.
https://doi.org/10.3390/e19040180

**Chicago/Turabian Style**

Hnilo, Alejandro Andrés.
2017. "Using Measured Values in Bell’s Inequalities Entails at Least One Hypothesis in Addition to Local Realism" *Entropy* 19, no. 4: 180.
https://doi.org/10.3390/e19040180