On Bell's Theorem and Quantum-Mechanically Predicted Perfect Correlation

1. Introduction

This paper presents an explanation of the quantum-mechanically predicted perfect correlation between two spatially separated photons. The explanation is based on Einstein’s argument for defending his world view in the Einstein-Bohr debate on the conceptual foundations of quantum mechanics. The fundamental hypotheses underlying Einstein’s world view are locality (causal effects in the real world cannot propagate faster than light) and realism (physical objects in the real world possess definite properties even in the absence of measurement or observation). As indicated by a historical fact [1], Einstein did not write the famous Einstein, Podolsky, and Rosen (EPR) paper (namely, [2]). Podolsky wrote the EPR paper. The EPR argument developed in the EPR paper differs from Einstein’s argument. Einstein’s argument is based not only on his separability principle but also on his ensemble interpretation of $\psi$-functions [1]. Unfortunately, this fact is not widely known. Bohr’s argument is based on his framework of complementarity [3], which contradicts both the EPR argument and Einstein’s argument.

Bell intended to resolve the Einstein-Bohr debate. By resorting to local hidden-variable theories, Bell derived his inequality [4]. Bell’s inequality serves as the basis for his investigation and leads to the well-known Bell’s theorem that encourages inequalities similar to Bell’s inequality. One of such inequalities is the CHSH inequality [5]. The CHSH inequality was tested by experiments with single pairs of correlated photons. The experiments confirm the quantum-mechanically predicted perfect correlation between two spatially separated photons. By interpreting the correlation non-locally as the so-called entanglement, Bell’s theorem denies the existence of definite polarizations possessed by photons in the absence of measurement. The denial amounts to refuting Einstein’s argument. According to Bell’s theorem, Einstein’s argument seems to be wrong, and the Einstein-Bohr debate seems to have been resolved. Nowadays physicists are concerned with which hypothesis underlying Einstein’s argument is purportedly refuted by testing the CHSH inequality and seem to face the choice of rejecting locality or realism [6].

However, neither locality nor realism needs to be rejected. In the rest of this paper, Section 2 explains why Bell’s theorem is incorrect. Section 3 shows that the correlation is deterministic as explained by Einstein’s argument and presents a formal analysis supporting Einstein’s explanation. Section 4 discusses further Einstein’s explanation and concludes the paper.

2. Irrelevant CHSH Inequality and Biased Bell’s Theorem

According to Bell’s theorem, violating the CHSH inequality amounts to refuting Einstein’s argument in the Einstein-Bohr debate. But Bell’s theorem is incorrect for two reasons:

(i) The CHSH inequality is irrelevant to Einstein’s argument.

(ii) Bell’s theorem biases Bohr’s argument against Einstein’s argument.

When Bell started his investigation, he was a follower of Einstein and did not want to refute Einstein’s argument [7]. Regrettably, Bell erroneously regarded Einstein as a proponent of hidden variables [8] and insisted that he understood Einstein correctly (see [9], Appendix. Einstein and Hidden Variables). A conceptual connection between local hidden-variables and Einstein’s argument is the only way to link the former with the latter. The connection does not exist, for Einstein never endorsed any hidden-variable theory [8]. Consequently, the CHSH inequality is irrelevant to Einstein’s argument.

The derivation of the CHSH inequality involves four polarization correlation coefficients [10]. Calculations of the correlation coefficients are based on a $\psi$-function used to express a so-called entangled state. The $\psi$-function is a quantum superposition with two superposed components that constitute the polarization part of the corresponding state vector. The meaning of the quantum superposition is interpreted by current quantum theory. The quantum-mechanically calculated correlation coefficients yield following results about polarizations of two spatially separated photons to be detected in the corresponding experiment:

(a)

If nobody performs any measurement, neither of the photons have definite polarizations.

(b)

Once a polarization measurement is performed on one of the photons, the $\psi$-function collapses abruptly as a result of the reduction postulate in current quantum theory.

(c)

Polarizations of perfectly correlated photons are either $(+,+)$ or $(-,-)$ after the measurement.

Bohr’s argument already implies the above results. The results contradict Einstein’s argument. In particular, the reduction postulate is used to interpret not only the measurement-triggered collapse of the $\psi$-function but also the correlation between two spatially separated photons. The interpretation implies faster-than-light propagations of causal effects. According to Einstein’s argument, both the quantum superposition interpreted by current quantum theory and the reduction postulate are questionable. Bohr’s argument is not “free from spooky actions at a distance” [11]. Unlike Einstein, physicists nowadays take the legitimacy of the quantum superposition interpreted by current quantum theory for granted and consider the reduction postulate necessary to interpret the correlation [10].

The entangled state is also the basis for preparing single pairs of spatially separated photons used to test the CHSH inequality. Joint polarization measurements are performed on such photons in different pairs. Outcomes obtained by the measurements are random, but the random outcomes can be perfectly correlated [10]. An important issue in the Einstein-Bohr debate is whether the outcomes are inherently random or caused by the subjective state of ignorance. Where does the randomness come from? This question deserves a physically meaningful answer. But Bell’s theorem does not answer this question. However, the quantum superposition interpreted by current quantum theory implies quantum-mechanically calculated probabilities for the description of the outcomes. The description amounts to the assertion that the outcomes are inherently random. Einstein called such description “the fundamental dice-game” [11]. While Bohr’s framework of complementarity is controversial, Bell accepted Bohr’s argument. Thus, Bell’s theorem implies Bohr’s argument and contradicts Einstein’s argument. As shown above, Bell’s theorem ignores Einstein’s argument completely while implying Bohr’s argument. Therefore, Bell’s theorem is biased.

3. Deterministic Correlation

Bohr’s argument cannot resolve the Einstein-Bohr debate. It was hoped that the Einstein-Bohr debate might be resolved, if local hidden-variables introduced in the CHSH inequality could determine polarizations of perfectly correlated photons in different pairs and reproduce statistical predictions of quantum mechanics. While the CHSH inequality is irrelevant to Einstein’s argument as shown in the previous section, Bell’s theorem erroneously interprets the correlation non-locally as the so-called entanglement, which contradicts Einstein’s argument. The correlation can be explained reasonably by Einstein’s separability principle and his ensemble interpretation of $\psi$-functions. Einstein’s explanation is intuitively understandable: The correlation is deterministic.

To see the deterministic nature of the correlation, it is necessary to clarify the issue concerning the randomness observed in the experimental test of the CHSH inequality. Consider first a banal fact: In the real world, a photon can be detected at most once. As a consequence of the above fact, identical experimental conditions are needed to decide whether the observed randomness is an inherent property of the physical world or caused by the subjective state of ignorance. If photons in different pairs could be detected under identical conditions, Einstein’s argument might be refuted. However, in no sense can experimental physicists detect photons in different pairs under identical conditions, because they use continuous parameters to specify the experimental conditions. The continuous parameters include orientations of the polarizers for analyzing polarizations of detected photons. Orientations of the polarizers are given by unit vectors spanning the three-dimensional Euclidean space. While the vectors are well-defined mathematical entities, values of such abstract objects cannot be obtained by measurement in practice. Consequently, nobody knows values involved in specifying the experimental conditions. When photons in different pairs are detected, orientations of the corresponding polarizers are actually unknown and cannot be identical. Nevertheless, experimental physicists take values of continuous parameters for granted, as if they knew orientations of the polarizers, and as if the experimental conditions were identical for detecting photons in different pairs. Thus, outcomes obtained by joint polarization measurements performed on the photons are incorrectly interpreted as empirical evidence for an imaginary, inherently random physical world. However, the randomness is due to the subjective state of ignorance about unknown values of the vectors.

Clarifying the issue concerning the randomness paves the way toward explaining the correlation based on Einstein’s separability principle and his ensemble interpretation of $\psi$-functions. In the experimental test of the CHSH inequality, the entangled state is expressed by the $\psi$-function that is the quantum superposition interpreted by current quantum theory. The CHSH inequality is a numerical inequality and makes sense only for numbers. But $(+,+)$ and $(-,-)$ used to express polarizations of perfectly correlated photons in the correlation coefficients for deriving the CHSH inequality are not numbers. Thus, $(+,+)$ and $(-,-)$ are replaced by $(+1,+1)$ and $(-1,-1)$ in the correlation coefficients. Einstein argued against the quantum-mechanically interpreted $\psi$-function. The $\psi$-function interpreted by Einstein’s argument describes an ensemble. Members of this ensemble are single pairs consisting of perfectly correlated photons. The photons are spatially separated, thus satisfying Einstein’s separability principle. The separability principle says that any two spatially separated physical objects have their own separate real states. The states are independent of each other. In other words, the state of one object is independent of what happens to the state of the other object. The outcomes obtained by joint polarization measurements performed on spatially separated photons in different pairs of the ensemble are either $(+1,+1)$ or $(-1,-1)$. According to Einstein’s local-realist world view, polarizations of perfectly correlated photons are definite properties possessed by photons. The properties exist even in the absence of measurement. Such properties cannot be created by measurements.

Formally, denote by $\mathbb{N}$ the set of all positive integers. For each $n\in \mathbb{N}$, let ${\mathcal{P}}_n$ represent the n-th pair of the ensemble. Denote by ${\mathcal{V}}_n$ the unknown value of the vector used to specify orientations of the polarizers for detecting photons in ${\mathcal{P}}_n$. Write $D^{+}=(+1,+1)$ and $D^{-}=(-1,-1)$. Thus, $D_n\in \{D^{+},D^{-}\}$ represents the outcomes obtained by joint polarization measurement performed on photons in ${\mathcal{P}}_n$. In contrast to the unknown value ${\mathcal{V}}_n$, both $D^{+}$ and $D^{-}$ are attainable by measurements. According to Einstein’s argument based on his separability principle and his ensemble interpretation of $\psi$-functions, spatially separated photons in single pairs possess their autonomous real states of polarization (separability principle), and the pairs are members of the corresponding ensemble (ensemble interpretation of $\psi$-functions). Write

$$K=\{n_k\in \mathbb{N}:k=1,2,\dots \}$$

(1)

$$M=\{n_m\in \mathbb{N}:m=1,2,\dots \}$$

(2)

$$S=\{({\mathcal{V}}_n,D_n):n\in \mathbb{N}\}$$

(3)

$$S^{+}=\{({\mathcal{V}}_{n_k},D^{+}):k\in K\}$$

(4)

and

$$S^{-}=\{({\mathcal{V}}_{n_m},D^{-}):m\in M\}.$$

(5)

Hence

$$S^{+}\cup S^{-}=S.$$

(6)

Equation (6) follows immediately from the equations given before (6). The sets $S^{+}$ and $S^{-}$ are disjoint, namely,

$$S^{+}\cap S^{-}=\varnothing .$$

(7)

If (7) does not hold, there would be some $k\in K$ and some $m\in M$ such that ${\mathcal{V}}_{n_k}={\mathcal{V}}_{n_m}$ and $D^{+}=D^{-}$. This is absurd, for $D^{+}=(+1,+1)$ and $D^{-}=(-1,-1)$, but $(+1,+1)\ne (-1,-1)$. If ${\mathcal{V}}_{n_k}={\mathcal{V}}_{n_m}$ for each $k\in K$ and for each $m\in M$, i.e., if ${\mathcal{V}}_1,{\mathcal{V}}_2,\dots$ are all identical, then ${\mathcal{V}}_n=\mathcal{V}$ for each n, which implies

$$[\mathcal{V},(+1,+1\left)\right]=[\mathcal{V},(-1,-1\left)\right].$$

(8)

Equation (8) corresponds to detecting photons in different pairs under identical conditions. Thus, $(+1,+1)$ or $(-1,-1)$, which represent mutually exclusive properties belonging to photons in different pairs, are attached to an imaginary pair that does not exist in the real world. Nevertheless, experimental physicists do not consider $(+1,+1)=(-1,-1)$ absurd, for they insist that photons in different pairs can be detected under identical conditions.

Negating (7) not only leads to the absurdity but also contradicts Einstein’s argument based on his separability principle and his ensemble interpretation of $\psi$-functions for defending his local-realist world view. The absurdity can be avoided only by accepting Einstein’s argument. Testing the CHSH inequality cannot refute Einstein’s local-realist world view. Just like properties possessed by all physical objects in the real world, polarizations of perfectly correlated photons are definite and exist even in the absence of measurement. Outcomes such as $D^{+}=(+1,+1)$ or $D^{-}=(-1,-1)$ are merely reflections of such properties realized by measurements. The outcomes are unique. For example, if $+1$ is the polarization of one photon, the polarization of the other photon must also be $+1$. In other words, the polarizations of perfectly correlated photons in ${\mathcal{P}}_n$ for each $n\in \mathbb{N}$ are deterministically correlated. Local hidden-variables do not play any relevant role here.

4. Discussion and Conclusion

The quantum superposition interpreted by current quantum theory for describing the entangled state differs from the quantum superposition interpreted by Einstein’s ensemble interpretation of $\psi$-functions. In sharp contrast to the quantum superposition interpreted by current quantum theory, the quantum superposition interpreted by Einstein’s ensemble interpretation of $\psi$-functions uses disjunction (“or”) as the logical relation between the superposed components that represent outcomes obtained by joint polarization measurements performed on spatially separated photons in different pairs of the corresponding ensemble. Interpreted by current quantum theory, the quantum superposition uses conjunction (“and”) as the logical relation and attaches mutually exclusive properties that belong to photons in different pairs to an imaginary pair of photons. In general, quantum-mechanically calculated probabilities also appear in Einstein’s ensemble interpretation of $\psi$-functions, but the probabilities are used to describe the randomness caused by the subjective state of ignorance about unknown values of continuous parameters involved in specifying experimental conditions.

Interpreted by Einstein’s argument based on his separability principle and his ensemble interpretation of $\psi$-functions for defending his local-realist world view, the correlation between spatially separated photons in each single pair of the ensemble is deterministic. The correlation coefficients used to derive the CHSH inequality make little sense statistically. It should not be difficult to understand why results given by the CHSH inequality differ from statistical predictions of quantum mechanics. In addition, the deterministic correlation has nothing to do with faster-than-light propagations of casual effects, thus ruling out the measurement-triggered collapse of the $\psi$-function for describing the entangled state.

In conclusion, this paper presents an explanation of the quantum-mechanically predicted perfect correlation between spatially correlated photons. The explanation is based on Einstein’s separability principle and his ensemble interpretation of $\psi$-functions for defending his local-realist world view in the Einstein-Bohr debate on the conceptual foundations of quantum mechanics. It is shown that Bell’s theorem is incorrect. The correlation is deterministic. As indicated by the formal analysis presented in this paper, refuting Einstein’s argument leads to the absurdity that can be avoided only by accepting Einstein’s argument.