On the Meaning of Local Realism

Abstract

We present a pragmatic analysis of the different meanings assigned to the term “local realism” in the context of the empirical violations of Bell-type inequalities since its inception in the late 1970s. We point out that most of them are inappropriate and arise from a deeply ingrained prejudice that originated in the celebrated 1935 paper by Einstein-Podolski-Rosen. We highlight the correct connotation that arises once we discard unnecessary metaphysics.

Appendices

Appendix A: Apparatuses’ Influence on Measurements

We can include the influence of the measuring devices in the derivation of the Bell inequality. Their explicit inclusion makes it more evident that the Bell inequality cannot falsify preexisting values. On the contrary, we can surmise the inclusion of such influences is according to the orthodox Copenhagen dictum that observation creates such values. Of course, all about preexistence is mere speculation, but the explicit inclusion of apparatuses’ hidden variables may help us get rid of such prejudices.

In 1971 Bell showed [27] how to include the uncontrollable influences of the measuring devices in the derivation of his inequality. That is according to Bohr’s views:

Indeed the finite interaction between object and measuring agencies conditioned by the very existence of the quantum of action entails – because of the impossibility of controlling the reaction of the object on the measuring instruments if these are to serve their purpose – the necessity of a final renunciation of the classical ideal of causality and a radical revision of our attitude towards the problem of physical reality. [25]

The argument is so trivial that Bell just explained, “If we average first over these instruments variables”; without giving further details [27]. Reference [51] contains a more thorough explanation.

Here we present a slightly different approach. We start with the expression for the joint probability of a stochastic local non-conspiratorial hidden variable model

$$\begin{aligned} P(A,B\mid x,y)=\int _{{{\varvec{\Lambda }}}} P(A\mid x,\lambda ) P(B\mid y,\lambda ) P(\lambda )\,d\lambda \end{aligned}$$

(A1)

In (A1), $A,B\in \{-1,1\}$ are the results, and x, y are the measurement settings. The hidden variables $\lambda$ are the local common causes, and $\varvec{\Lambda }$ is the space of these variables. We include the apparatuses’ effects assuming hidden variables $\xi \in \varvec{\Lambda }_1$ for Alice’s measuring device and $\eta \in \varvec{\Lambda }_2$ for Bob’s. After the inclusion of these variables, the probabilities for the measurement results respectively become

$$\begin{aligned} P(A\mid x,\lambda ,\xi )\quad \text {and}\quad P(B\mid y,\lambda ,\eta ) \end{aligned}$$

The $\lambda$ represents common causes lying at the causal past of the measuring events while $\xi$ and $\eta$ describe the local instruments. Therefore all these variables are independent of each other and are independently distributed. Putting $P_\lambda =P(\lambda ),\,P_\xi (x)=P(\xi ,x),\,P_\eta (y)=P(\eta ,y)$

$$\begin{aligned} P(A,B\mid x,y)=\int _{\varvec{\Lambda }}\!P_\lambda d\lambda \int _{\varvec{\Lambda }_1}P_\xi (x) d\xi \int _{\varvec{\Lambda }_2}\,P_\eta (y) d\eta P(A\mid x,\lambda ,\xi ) P(B\mid y,\lambda ,\eta ) \end{aligned}$$

(A2)

The fact that $P(\lambda )$ is independent of x and y is a consequence of the non-conspiratorial hypothesis. Integrating first over the instruments variables $\xi$ and $\eta$

$$\begin{aligned} \int _{\varvec{\Lambda }_1} P_\xi (x) P(A\mid x,\lambda ,\xi )\,d\xi= & {} \overline{P}(A\mid x,\lambda ) \end{aligned}$$

(A3)

$$\begin{aligned} \int _{\varvec{\Lambda }_2} P_\eta (y) P(B\mid y,\lambda ,\eta )\,d\eta= & {} \overline{P}(B\mid y,\lambda ) \end{aligned}$$

(A4)

Taking (A3) and (A4) into (A2) we have

$$\begin{aligned} P(A,B\mid x,y)=\int _{\varvec{\Lambda }} \overline{P}(A\mid x,\lambda ) \overline{P}(B\mid y,\lambda ) P_\lambda \,d\lambda \end{aligned}$$

(A5)

Since $\overline{P}(A\mid x,\lambda )$ and $\overline{P}(B\mid y,\lambda )$ range between 0 and 1, the derivation of the Bell inequality goes through as usual.

Appendix B: On Stapp’s Counterfactual Arguments

To discuss the differences between Stapp’s counterfactual inequality and the Bell inequality, we abandon our neutrality regarding the LSI and HVLSI interpretations and embrace the latter. Quantum nonlocality should be proved by different means, just as Einstein and Bell argued. We cannot justify the reasons for our interpretation here, but Ref. [49] develops the argument why we sustain that Bell did not interpret his inequality as proof of quantum nonlocality.

Stapp admired the Bell theorem. In 1975, he famously said

Bell’s theorem is the most profound discovery of science. [52]

Indeed, proving the metaphysical speculation (quantum mechanics completion) that would allow a classical down-to-earth interpretation of quantum mechanics is accessible to direct experimental tests is not a minor feat.

However, he sensed that the presence of hidden variables was an undesired characteristic for claiming quantum nonlocality. He was aware of the difference between the proof of quantum nonlocality and a no-go theorem for local hidden variables.

Stapp’s program to prove quantum nonlocality is quite different from Bell’s. Remarkably, much of the physical community has missed that Bell did not base his quantum nonlocality arguments on his inequality. Bell and Einstein’s arguments for quantum nonlocality are indeed very similar [49]. The purpose of the Bell theorem, and the Bell inequality, is to prove that we cannot supplement quantum mechanics with additional variables to render a local theory. In the third line of his 1964 introduction, Bell wrote:

These additional variables were to restore to the theory causality and locality. [28]

But he did not say such variables were supposed to prove the nonlocal character of orthodox quantum mechanics. Assuming that quantum mechanics is nonlocal, Bell proved we cannot turn it local by adding hidden variables. From a logical and conceptual stance, that is light-years away from proving quantum nonlocality.

On the other hand, Stapp also was largely misunderstood. Stapp intended a theoretical argument for quantum nonlocality, while the Bell theorem is a directly falsifiable no-go theorem for local hidden variables.

From a historical and conceptual point of view, it is relevant to follow Stapp’s arguments because the failure to recognize the different nature between Stapp’s and Bell’s inequalities resulted in widespread confusion when conflating Stapp’s and Bell’s methods.

Stapp began his long crusade to prove quantum nonlocality in 1971. From 1971 to 1997, he used an inequality without hidden variables [39, 53, 54]. Then from 1997 to 2004 [55–57], Stapp turned to modal logic. Finally, in 2004, he recognized the problems with his modal logic approach and devised a reasoning based on Hardy’s 1993 paper [58].

Stapp’s method in the period from 1971 to 1997 is particularly relevant. It still influences the formulation of the Bell theorem to this day, notwithstanding the different nature of the Bell inequality.

Here we shall review Stapp’s reasoning during that period and remark on the differences between his inequality and Bell’s.

1.1 Appendix B.1: Counterfactual Derivation Without Hidden Variables

Stapp assumed a Bell-like experimental scenario. As usual, Alice and Bob each receive a particle in the singlet-state

$$\begin{aligned} \mid \psi \rangle =\frac{1}{\sqrt{2}}(\mid +\rangle \mid -\rangle - \mid -\rangle \mid +\rangle ) \end{aligned}$$

(B6)

After measuring in one of two possible directions $i\in \{1,2\}$, Alice finds the result $A_i\in \{-1,+1\}$. Analogously for Bob, $B_k\in \{-1,+1\}$, $k\in \{1,2\}$.

Similarly to the case of the Clauser, Horne, Shimony,Holt (CHSH) [59] form of the Bell inequality, Stapp considers the four possible experiments and their eight possible results

$$\begin{aligned} A_1B_1,\,A'_1B_2,\,A_2B'_1,\,A'_2B'_2 \end{aligned}$$

(B7)

Stapp interprets those results according to the following counterfactual rule:

Of these eight numbers only two can be compared directly to experiment. The other six correspond to the three alternative experiments that could have been performed but were not. [60]

The counterfactual rule plus the locality condition impose some restrictions on the possible values of (B7). Suppose $A_1B_1$ are the measured values Alice and Bob find, then had Bob measured in direction 2 instead of 1, he would have found a different result, say $B_2$. Locality demands Alice’s previous result $A_1$ cannot change because of Bob’s different choice. So, the second term $A'_1B_2$ in (B7) becomes $A_1B_2$.

In a similar way, from the actual result $A_1B_1$, we can infer that if it was Alice who decided to change her measurement direction to 2, Bob’s result would not have changed and we find for the third term that $A_2B'_1=A_2B_1$. Finally, had both Alice and Bob changed their settings, they would have found the same values as if the other one did not change hers finding that $A'_2B'_2=A_2B_2$.

Thus, the counterfactual interpretation plus the locality condition reduces the eight possible numbers in (B7) to only four

$$\begin{aligned} A_1B_1,\,A_1B_2,\,A_2B_1,\,A_2B_2 \end{aligned}$$

(B8)

Note that nowhere in the reasoning do we require the physical preexistence of not performed experiments results. Let us assume a series of actual experiments performed with settings (1, 1)

$$\begin{aligned} lim \frac{1}{N}\sum _k A^{(k)}_1 B^{(k)}_1 = E^*(1,1) \end{aligned}$$

(B9)

Based on the results of the actual experiments as given by (B9), Stapp reasoned about what would have happened in the other three different series of hypothetical experiments, given the locality constraint (B8) on the possible results in each run k of the experiment

$$\begin{aligned} lim \frac{1}{N}\sum _k A^{(k)}_1 B^{(k)}_2= & {} E^*(1,2) \end{aligned}$$

(B10)

$$\begin{aligned} lim \frac{1}{N}\sum _k A^{(k)}_2 B^{(k)}_1= & {} E^*(2,1) \end{aligned}$$

(B11)

$$\begin{aligned} lim \frac{1}{N}\sum _k A^{(k)}_2 B^{(k)}_2= & {} E^*(2,2) \end{aligned}$$

(B12)

The only necessary assumptions to write down (B10), (B11), and (B12) are locality and the possibility of those experiments or that they were a real possibility. There is no need for determinism or the infamous counterfactual definiteness hypothesis. Indeed we do not need the physical existence of the counterfactual results, much less the simultaneous existence of those experiments. Neither do we need the no conspiracy hypothesis or statistical independence. That hypothesis only is necessary in the presence of hidden variables.

Note the irrelevance of the odd ideas usually invoked, such as incompatible experiments and simultaneous existence of conjugate magnitudes. The four series of experiments (B9), (B10), (B11), and (B12) are not supposed to exists simultaneously. They are independent possibilities; if ever, only one eventually takes place. There is nothing in those experiments that is inconsistent with quantum mechanics. The best proof of that is that quantum mechanics also makes precise predictions for those series of experiments, say $E(1,1),\,E(1,2),\,E(2,1),\,E(2,2)$.

Stapp proved that for certain settings, the values imposed by locality on the statistical correlations $E^*(1,1),\,E^*(1,2),\,E^*(2,1),\,E^*(2,2)$ are incompatible with the quantum mechanical predictions $E(1,1),\,E(1,2),\,E(2,1),\,E(2,2)$ of those correlations. For instance, in [60] and [53], he showed that incompatibility arriving at the Stapp inequality

$$\begin{aligned} \sqrt{2}\le 1 \end{aligned}$$

(B13)

Stapp’s argument is simple and direct, albeit he received many criticisms [10, 61–63]. In most cases, he responded to those criticisms by trying to explain misinterpretations and unnecessary complications introduced by such misunderstandings [64–67]. Even very eminent physicists [10] seem to have missed the simple reasoning implied by (B13). It is about the mathematical incompatibility of the predictions for different series of experiments, experiments that are not supposed to have meshed together in incompatible or inconsistent forms. In our opinion, Stapp’s argument, as we reproduce it here, is unassailable. Probably the only weak point is that it relies on counterfactual reasoning:

Although philosophers contend that counterfactual concepts pervade science, and are needed for science, the significance of results based on the use of counterfactuals remains somewhat shaky in the minds of most quantum physicists. [68]

Indeed, sometimes it is claimed that we cannot apply counterfactual reasoning to quantum mechanics. But that is just an unjustified prejudice, at least for the case that Stapp considered. Quantum mechanics makes precise predictions for all four independent series of experiments irrespective of whether they are real or hypothetical, so the factual or counterfactual nature of such experiments cannot invalidate the logical inference obtained from them:

Quantum mechanical statistical predictions are incompatible with locality.

However, we shall never be able to directly falsify the predictions (B10), (B11), and (B12). Indeed, we do not even know the values of $A^{(k)}_2$ and $B^{(k)}_2,\,k\in \{1,\ldots ,N\}$. But whatever those values, quantum mechanics statistical predictions are incompatible with the locality constraints. Thus, although quantum mechanics is no-signaling, there is a clear sense in which it implies a sort of nonlocality. It is controversial whether it is a sort of nonlocality requiring action-at-a-distance inconsistent with relativistic causality. It is not our intention to delve into that problem.

1.2 Appendix B.2: Comparison of Stapp’s and Bell’s Formulations

From (B9), (B10), (B11), and (B12) we can also arrive at a Bell-CHSH inequality

$$\begin{aligned} \mid \langle A_1B_1\rangle -\langle A_1B_2\rangle +\langle A_2B_1\rangle +\langle A_2B_2\rangle \mid \le 2 \end{aligned}$$

(B14)

To the best of our knowledge, the first to apply Stapp’s counterfactual method to obtain a Bell-CHSH inequality was Eberhard⁴ in 1976 [69].

Thus, we have two different formulations of the Bell-CHSH inequality based on quite different hypothesis. The original Bell-CHSH inequality as derived by Bell and by Clauser, Hall, Shimony, and Holt is based on:

• (1) Hidden variables.

• (2) Local causality or the screening off condition.

• (3) Statistical independence.

It is relevant to note that the hidden variables approach does not use counterfactual reasoning [42]. As originally formulated, it is a no-go theorem for local hidden variables. The LSI interpretational current vigorously resists this interpretation of the Bell-CHSH inequality. According to them, the list of hypotheses should reduce to 2) and 3) because hidden variables are a consequence of locality [5–9]. In our opinion, Bell wisely avoided that approach. He preferred to argue for quantum nonlocality differently and only introduced his inequality to prove that a local completion is not possible unless we violate statistical independence.

On the other hand, if we use Stapp’s counterfactual method to derive the Bell-CHSH inequality, the interpretation is quite different. Stapp’s reasoning allows us to arrive at (B14) assuming only locality and is directly applicable to quantum mechanics.

However, from an empirical point of view, the counterfactual (B14) is not very useful. While the hidden variable approach is a constraint on every four series of actual experiments, therefore is directly falsifiable, the counterfactual Bell-CHSH inequality only establishes the eventual existence of four such series satisfying it. So, the counterfactual Bell-CHSH inequality reduces to a not falsifiable thought experiment that proves quantum nonlocality.

Stapp’s formulation is not very informative for those who accept quantum nonlocality for different reasons, such as Bell and Einstein. For them, the relevant significance of the Bell inequality is that we cannot fix quantum nonlocality and a local completion is impossible unless we accept superdeterminism.