Locality, Causality, and Realism in the Derivation of Bell's Inequality

Abstract. For several years, a significant disagreement has persisted between (mainly) philosophers, and (mainly) physicists concerning the consequences of the empirical violation of Bell's inequality. On the one hand, it is claimed that empirical violations of Bell's inequality show us that the world is “non-local,” which might give rise to a serious conflict with the special theory of relativity. On the other hand, several authors maintain that this conflict can be avoided by giving up a “realistic” interpretation of quantum mechanics. I use a recent derivation of a Bell-type inequality from Reichenbach's principle of the common cause to explicate the different notions of “non-locality” and “realism” involved in the debate, and to assess the adequacy of the different claims. My two main conclusions are: First, if “realism” is understood as the existence of sufficient conditions for the measurement outcomes in the experimental setups under consideration, giving it up will not allow us to maintain locality. Second, however, I will argue that there is, in fact, a plausible notion of “realism” which we could reject in order to save locality. Instead of challenging the special theory of relativity, this option impugns common notions of causality.

1 From a Dilemma to a “Monolemma”

The empirical violation of Bell's inequality forces us to deny at least one of the premises from which it is derived.¹ There is, however, a significant disagreement as to what assumptions about the physical world or our theories of it are at stake.

On the one hand, people (mainly physicists) maintain that it is an open question whether we should admit that there are non-local interactions in the world, and thus challenge the special theory of relativity,² or whether we should, rather, give up a certain notion of realism:

The experimental violation of mathematical relations known as Bell's inequalities sounded the death knell of Einstein's idea of “local realism,” in quantum mechanics. But which concept, locality or realism, is the problem? Aspect 2007, 866

On the other hand, (mainly) philosophers urge that this stance is untenable because even if we dispense with our cherished notions of realism we have to accept non-local interactions in the world. Their argument is the following.

One can derive Bell's inequality from essentially the following two assumptions:

The outcome of a measurement at one wing in an Einstein-Podolsky-Rosen (EPR) experiment³ does not causally depend on the outcome or the measurement operations at the other wing.

The outcomes of the measurements in an EPR experiment are co-determined by some events which are themselves causally independent of the measurement operations.

The first of these two assumptions comprises locality conditions, for which one can make strong supportive arguments based on the theory of special relativity. The second assumption postulates something akin to what Einstein called, in the EPR paper, “elements of reality.” The existence or not of elements of reality, and the properties that they instantiate should be, moreover, independent from the actions of the observer, according to the second assumption.

Let us, in line with this “zeroth” characterization, abbreviate the two set of assumptions by “ ” and “ ,” respectively. Then, the derivation of Bell's inequality can be represented as the following entailment. Let us call it, for the purposes of the present discussion, “Bell's theorem”:

( & ) Bell's Inequality.

Because of the empirical violations of the consequence of this entailment, i. e., of Bell's inequality, at least one of the two sets of premises has to be false. If the derivation is minimal in the sense that dropping one of the premisses will lead to an invalid argument, then the argument gives no reason to give up more than one of the two problematic sets of premises.

However, according to several authors, the derivation from the above two premises is not minimal, because, according to them, Realism is a necessary condition for Locality, i. e.

.

They take it that Einstein, Podolsky, and Rosen have established that claim:

Many believe that because Bell starts by assuming the world conforms to what is called local realism, he therefore proved that either locality or realism is violated. Thus, the world could be local if it violates “realism.” But this idea overlooks—or misunderstands—that the original “EPR” argument of Albert Einstein, Boris Podolsky and Nathan Rosen rules out the possibility of quantum locality without the realism Bell uses. Albert and Galchen 2009, 31

If Albert and Galchen are right and Realism is, indeed, a necessary condition for Locality, “Bell's theorem” gives rise to a derivation of Bell's inequality from the assumption of Locality alone:

Bell's Inequality.

This is also the conclusion that Maudlin draws:

So experiments verifying the violation of Bell's inequality would doom locality tout court. Maudlin 2002, 20

Thus, according to Albert, Galchen, and Maudlin, we cannot avoid the challenges, which a violation of Locality most likely poses to the theory of special relativity, by giving-up Realism alone.

2 Details of the Derivation of Bell's Inequality

In order to assess the adequacy of the divergent claims by, on the one hand, the physicist Aspect and, on the other hand, Albert, Galchen, and Maudlin, I will sketch, in the following, a more detailed and explicit derivation of a Bell-type inequality, which is based on work by Graßhoff et.al. 2005. This derivation is minimal with respect to an important class of other derivations. For example, the authors do not require different correlations to have the same common cause. I will subdivide the derivation into two steps so as to be able to compare it more readily with Maudlin's, and Albert and Galchen's arguments to the effect that Bell's inequality derives from locality conditions alone.

I will set aside some auxiliary assumptions. This is, of course, always a pragmatic instead of a principled decision and depends on where one wants to put the focus of the investigation. Most of the auxiliary assumptions are made explicit by Graßhoff et.al. 2005, but even there the authors did, for instance, not address, in any detailed manner, the question as to how to interpret the probabilities which are used in the derivation. I will bracket out this question here, too.

The first step of the derivation proceeds from three principal assumptions to an intermediate conclusion. The assumptions involve perfect correlations, causal independence of correlated outcomes, and a variant of Reichenbach's 1956 principle of the common cause:

• PCORR: Upon parallel measurements, the outcomes in an EPR experiment are perfectly anti-correlated.

• C-OI: The measurement outcomes at one wing are not causally relevant for the outcomes at the other wing.

• SCR: If two types of events and are correlated and neither is causally relevant for nor vice-versa then there exists a third type of event which statistically “screens-off” from :

From these three assumptions follows (given the auxiliary assumptions characterizing the experimental setup and a suitable interpretation of the probabilities) that there are sufficient conditions for the correlated effects.⁴ More precisely, for instance, setting up both the apparatus on the left and on the right to measure the spin of the particle relative to direction no. 1, , together with a certain type of screening-off event, , is a sufficient condition for the outcome of the measurement at the left wing being , and also for the outcome at the right wing being . In a similar way, we arrive at the following four conditional statements: ⁵

Since we assume that

• C-P:: the measurement operations on the particle at the distant wing of the experimental setup are not causally relevant to the outcome of measurements on the particle in question,

the above conditionals should be true even without the statements, in the antecedent, that the measurement apparatus at the distant wing is set up in a particular way. That is, for instance, we can discard from the sufficient condition for , or from the sufficient condition for . This manipulation of the antecedent, is an application of the regularity theory of causality as developed by Mackie 1974, and elaborated by, e.g., Baumgartner 2008, and Graßhoff and May 2001. According to these theories of causality, if, for instance,

would cease to be valid when is discarded, would, in fact, be causally relevant for given certain additional conditions.⁶

Although I here performed the transition from the conditionals which contain both measurement settings to the conditionals which contain only one of them using a theory of causal regularities, the essential assumption involved in the transition is again screening-off. For, under suitable interpretations of the probabilities, the above manipulation of the conditionals is tantamount to reading C-PI as the claim that, e.g., screens off from , i. e.,

or, equivalently,

Thus, if we do not take the distant measurement operations to be causally relevant for the nearby outcome and assume an interpretation of causal irrelevance which either implicitly or explicitly is a screening-off condition, we end up with the following sufficient conditions for the outcomes:

• SUFF::

We can graphically represent these relations by way of causal graphs (see Figure 1).⁷

From these conditionals, we can derive the relative frequencies of the outcomes in terms of the relative frequencies of the screener-offs and the relative frequencies of the measurement operations. If we add the condition that the screener-offs are statistically independent from the measurement operations, we can even express the frequencies of the measurement outcomes in terms of only the screener-offs. That is, if we assume

• NO-CONS: The conjunction of the measurement operations and the conjunction of the screener-offs are statistically independent from each other:⁸

we have

Because the right-hand side of the last equation contains only terms which also occur in the right-hand side of the previous two equations, the relative frequency which is expressed by the last equation cannot be larger than the sum of the other two frequencies, i. e.,

• BELL: .

This is a version of Bell's inequality, which is violated significantly in the experiments which measure the relative frequencies to which the inequality refers.⁹ Therefore, we must conclude that at least one of the assumptions from which the inequality has been derived must be false.

3 Locality and Reality Conditions

According to my sketch of the detailed derivation of Bell's inequality, we need the assumptions of (i) perfect correlation, causal independence of the outcomes on each other (ii) and on the distant measurement operations (iii), (iv) the principle of the common cause, and (v) the statistical independence of the common causes on the measurement operations. That is, the following entailment holds:

PCORR & C-OI & C-PI & SCR & NO-CONS BELL

We performed the derivation in two steps, the first being the derivation of sufficient conditions for the measurement outcomes (SUFF), from the perfect correlations between causally independent outcomes (PCORR, C-OI), using a variant of the principle of the common cause (SCR):

PCORR & C-OI & C-PI & SCR SUFF

Using the intermediate conclusion SUFF, we proceeded, in a second step, to the derivation of Bell's inequality (BELL) from the causal independence of the outcomes on the distant measurement operations (C-PI), and the statistical independence of the screener-offs and the measurements operations:

C-PI & NO-CONS & SUFF BELL

This entailment shows us that we cannot maintain SUFF without giving up at least one of C-PI or NO-CONS. If SUFF, i. e. the existence of sufficient conditions for the outcomes, is our notion of “realism” any “realistic” account of the quantum correlations in question will have to feature a causal dependence of the outcomes on the distant measurement operations or a statistical dependence between the measurement operations and the “elements of reality,” understood as the screener-offs of the perfect correlations:

However, the first step of our derivation shows us that even if we are willing to give up “realism” in the sense of sufficient conditions for the outcomes, we still have to give up either C-OI, C-PI or SCR, or deny that the empirical evidence lends itself to the description as perfect correlations of the outcomes with parallel measurement settings.

Therefore, giving up realism (understood as SUFF) does not help us as much as it might seem at first sight in avoiding the possible conflict between quantum mechanical phenomena, such as the EPR experiment, and the special theory of relativity. When considering the advantages of giving up “realism,” we should not only take into account the costs of maintaining it but also the costs that we nonetheless have to carry if we do give it up.

Understood along these lines, Maudlin's and Albert's and Galchen's critical remarks are to the point. However, the way they phrase the critique has several unattractive consequences.

If they maintain that the derivation of Bell's inequality is a reductio argument against locality conditions alone, they must subsume all of C-PI, NO-CONS, C-OI, SCR, and PCORR under their notion of locality. Also, if they regard the first step of the derivation as a variant of EPR's argument, which derives Realism from Locality, and the second step, “Bell's theorem” in their terms, as a derivation of Bell's inequality from Locality and Realism, the notion of locality in the first and the second step of the derivation is not necessarily the same. In the first step, “Locality” must entail the causal independence of the measurement outcomes and the version of the principle of the common cause:¹⁰

LOC_EPR C-OI & C-PI & SCR

In the second step, “Locality” must entail the causal independence of the outcomes on the distant measurement operations and the statistical independence of the measurement operations and the screener-offs:

LOC_BELL C-PI & NO-CONS.

In both steps, they must identify “Realism” with SUFF:

Realism SUFF.

These consequences are unattractive mainly because assumptions such as NO-CONS or SCR have to be classified as “locality” conditions. Yet, NO-CONS, to begin with, is a condition relating event types which are instantiated by time-like separated events, whereas I take the usual referents of locality conditions to be event types which are instantiated by space-like separated events.¹¹

Also SCR is not a particular statement about space-like separated events. Rather, it applies to any two types of events and says that there cannot be a correlation between them when they do not cause each other or when there is no screener-off of the correlation.

Of course, from a logical point of view, one can define whatever one pleases as long as the definitions are consistent. But in order to relate the discussion about the consequences of the empirical violation of Bell's inequality to the more general concepts of “locality” and “realism” we are well advised to link the premises of the derivation as closely as we possibly can to the meaning these terms have in their typical use. And on this measure, there is a much better alternative than the broad notion of locality which includes virtually any premise of Bell's inequality.

The alternative is to identify NO-CONS and SCR with “Realism,” i. e.,

Realism NO-CONS & SCR,

and C-OI and C-PI with “Locality”:

Locality C-OI & C-PI.

Like this, on the one hand, the locality conditions state particular cases of the prohibition that an event type be causally relevant for another event type when their coinciding instances are space-like separated. On the other hand, the realism conditions spell out the intuition “that there has to be something” accounting for our observations, and that this “something” is there, independent of our eventual measurement operations.

4 Thinking About Local, “Non-Realistic” Models

The previous analysis opens up the possibility, contra Maudlin, Albert, and Galchen, of maintaining locality and giving up realism instead. Local, “non-realistic” models would satisfy both C-PI and C-OI, but not both NO-CONS and SCR.

Models which violate only NO-CONS will have to address delicate, though perhaps not unresolvable, issues involving backward in time causation,¹² or a hitherto unknown class of causal relations between the measurement operations and the events which are associated with the production of quantum mechanical singlet states.

Models which violate SCR are the other local but non-realistic option. These models will have to allow correlations between events which do not cause each other and which do not have a screener-off. However, whether such correlations can exist, is a question which has to do with our notion of causality, and the answer to this question is logically independent from the requirement that space-like separated events do not cause each other.

Acknowledgments

Research for this article was supported by the Swiss National Science Foundation under the grant no. 105211-129730 (principal investigator Gerd Graßhoff). A generous postdoctoral fellowship (Sept. 2011 – Apr. 2012) from the University of Pittsburgh's Center for Philosophy of Science gave me a particularly encouraging opportunity to write up these results.

References

Footnotes