A proof of Bell’s inequality in quantum mechanics using causal interactions

Abstract

We give a simple proof of Bell’s inequality in quantum mechanics using theory from causal interaction, which, in conjunction with experiments, demonstrates that the local hidden variables assumption is false. The proof sheds light on relationships between the notion of causal interaction and interference between treatments.

Introduction

Neyman introduced a formal mathematical theory of counterfactual causation that now has become standard language in many quantitative disciplines, including statistics, epidemiology, philosophy, economics, sociology, and artificial intelligence, but not in physics. Several researchers in these disciplines (Frangakis et al., 2007; Pearl, 2009) have speculated that there exists a relationship between this counterfactual theory and quantum mechanics, but have not provided any substantive formal relation between the two. In this note, we show that theory concerning causal interaction, grounded in notions of counterfactuals, can be used to give a straightforward proof of a result in quantum physics, namely, Bell’s inequality. Our proof relies on recognizing that results on causal interaction (VanderWeele, 2010) can be used to empirically test for interference between treatments (VanderWeele et al., 2012). It should be stressed that a number of extremely short and elegant proofs of both Bell’s original inequality (and its generalizations) are already available in the physics literature (cf. Pitowsky, 1989; Kümmerer and Maassen, 1998; Gill et al., 2002; Gill, 2012). In fact some of these proofs are based on reasoning with counterfactuals (Gill et al., 2001). Our contribution is to explicitly show relations to the theory of causal interactions.

We motivate our proof with an exceedingly short history of the Bell Inequality that is elaborated upon later. A non-intuitive implication of quantum theory is that pairs of spin 1/2 particles (e.g., electrons) can be prepared in an entangled state with the following property. When the spins of both particles are measured along a common (spatial) axis, the measurement of one particle’s spin perfectly predicts the spin of the other; if the first particle’s spin is up, then the spin of the second must be down. One explanation would be that the measurement itself of the first particle determined the spin of the second, even if physically separated, perhaps, by many light years. This would mean that reality was not “local”; what occurred at one place would affect reality (i.e. the spin of the second electron) at another. However, Einstein believed in “local realism” and argued that the more plausible explanation was that both particles are carrying with them from their common source ‘hidden’ correlated spin outcomes which they will exhibit when measured (Einstein et al., 1935). He therefore argued for “local realism” and rejected the previous explanation. Bohr disagreed with Einstein and his “local realist” assumption. Neither Einstein nor Bohr apparently realized that the hypothesis of local realism was subject to empirical test. In 1964, John Bell showed that an empirical test was possible; he proved that if strict locality were true, there would be certain inequality relations between measurable quantities that must hold (Bell, 1964); quantum theory predicted that these inequalities must be violated. Experiments found Bell’s inequalities were indeed violated (though see discussion below for further comments). Einstein was wrong; local realism is false.

2. A Proof of Bell’s Inequality Using Causal Interactions

We now show how results on causal interaction can be used to produce an alternative proof of Bell’s theorem. Suppose we have two particles and can use devices to measure the spin of each, along any axis of our choosing. Let X₁ and X₂ be two “interventions” each taking values in {0, 1, 2}, where X₁ records the angle (i.e. axis in space) at which particle 1 is measured, and X₂ records the angle at which particle 2 is measured, and where 0, 1, 2 correspond to three particular angles. Let Y₁(x₁, x₂) be the binary spin (up= 1 or down = −1) of particle 1 and Y₂(x₁, x₂) be the spin for particle 2, when particle 1 is measured at angle x₁ and particle two is measured at x₂. In the language of the Neyman model Y_i(x₁, x₂) is the counterfactual response of particle i under the joint intervention (x₁, x₂). Let M(x₁, x₂) = 1{Y₁(x₁, x₂) = Y₂(x₁, x₂)} be an indicator function that the spins agree so that M(x₁, x₂) = 1 if the spins agree and M (x₁, x₂) = 0 if they disagree. Suppose that the particles are in a maximally entangled state (cf. Plenio and Virmani, 2007). Then, according to quantum mechanics of the 2 particle system, for i, j = 0, 1, 2, E[M(x₁ = i, x₂ = j)] = sin²(Δ_ij /2), where Δ_ij is the angle between angles i and j, and where the expectation is taken over repeated experiments (Ballentine, 1998). This result has been confirmed by experiments in which the angles of measurement were randomized (though see discussion below for further comments). Therefore in what follows we take {E[M(x₁, x₂)]; x₁ ∈ {0, 1, 2}, x₂ ∈ {0, 1, 2}} as known, based on the data from experiment. Since sin (0) = 0, M(i, i) = 0, i = 0, 1, 2, with probability 1 and, therefore, also Y₁(i, i) = −Y₂(i, i), with probability 1, as mentioned earlier.

We formalize the hypothesis of “local hidden variables” by the hypothesis that spin measured on one particle does not depend on the angle at which the other particle is measured. This can be stated as follows. For all angles (x₁, x₂), we have:

$$\begin{array}{l}{Y}_1(x_1,x_2)=Y_1(x_1)\\ Y_2(x_1,x_2)=Y_2(x_2).\end{array}$$ (H1)

In some of the experiments referenced above the times of the two measurements were sufficiently close and the separation of the particles sufficiently great that even a signal traveling at the speed of light could not inform one particle of the result of the other’s spin measurement. Therefore, refuting the hypothesis of “local hidden variables” implies reality is not local and therefore we can essentially treat the hypothesis of local hidden variable and local reality as the same; we return to this point in the discussion.

The hypothesis asserts both locality and reality. It asserts locality because the angle x₂ at which particle 2 is measured has no effect the spin Y₁(x₁) of particle 1. It asserts reality because the spin Y_i(x) of a particle measured along axis x is assumed to exist for every x, even though for each i, only one of the Y_i(x) is observed; the one corresponding to the axis along which particle was actually measured. All other Y_i(x) are missing data in the language of statisticians or, equivalently, hidden variables in the language of physicists. The counterfactuals Y_i(x) correspond exactly to what Einstein called “elements of reality”. In the language of counterfactual theory, the hypothesis of local reality is, by definition, the hypothesis of no interference between treatments. The following lemma is an immediate consequence of the above definition of ‘local hidden variables’. In the following a unit may be taken to be a pair of entangled particles.

Lemma 1

If for some particle i, Y_i(x₁, x₂) takes one value s ∈ {−1, 1} for exactly one level of (x₁, x₂) with x₁ ∈ {0, 1}, x₂ ∈ {0, 2} and Y_i(x₁, x₂) the value −s for the other three levels of (x₁, x₂) with x₁ ∈ {0, 1}, x₂ ∈ {0, 2}, then the hypothesis of ‘local hidden variables’ (H1) is false.

In the context of counterfactuals we would say that a particle i that satisfied the conditions of Lemma 1 would be said to exhibit a “causal interaction” (Vander-Weele, 2010) for the outcome Y_i, also referred to as “epistasis” (Bateson, 1909) or “compositional epistasis” (Philips, 2008; Cordell, 2009) in the genetics literature (cf. VanderWeele, 2010).

Theorem 1

If for x₁ ∈ {0, 1}, x₂ ∈ {0, 2}, we have that for some unit, M (x₁, x₂) = 1 if and only if (x₁, x₂) = (1, 2), then the supposition of Lemma 1 holds for either Y₁(x₁, x₂) or Y₂(x₁, x₂).

Proof

We have that M(1, 2) = 1 implies that Y₁(1, 2) = Y₂(1, 2) = s for some s ∈ {−1, 1}. We show that if Y₁(x₁, x₂) does not satisfy the hypothesis of Lemma 1 then Y₂(x₁, x₂) must. If Y₁(x₁, x₂) does not satisfy the hypothesis of Lemma 1 and Y₁(1, 2) = Y₂(1, 2) = s then either (a) the other 3 of the values of Y₁(x₁, x₂) are also s or (b) precisely one other value of Y₁(x₁, x₂) equals s and the other two are −s. In case (a), for x₁ ∈ {0, 1}, x₂ ∈ {0, 2}, since M (x₁, x₂) = 0 for (x₁, x₂) ≠ (1, 2), we have Y₂(x₁, x₂) equals −s for (x₁, x₂) ≠ (1, 2) and thus Y₂(x₁, x₂) satisfies the hypothesis of Lemma 1. In case (b), since M(x₁, x₂) = 0 for (x₁, x₂) ≠ (1, 2), we have Y₂(x₁, x₂) equals −s for precisely one value of (x₁, x₂), x₁ ∈ {0, 1}, x₂ ∈ {0, 2}, in which case Y₂(x₁, x₂) would again satisfy the hypothesis of Lemma 1.

A unit that satisfied the conditions of Theorem 1 would be said to exhibit a causal interaction for the outcome M. The next result is given in VanderWeele (2010) in the context of testing for a causal interaction in which an outcome occurs only if both of two treatments are present. The result below relates the empirical data E[M(x₁, x₂)] to the existence of a unit satisfying M(1, 2) = 1, M(0, 2) = M(1, 0) = M(0, 0) = 0. Within the counterfactual framework, this would constitute a causal interaction for the variable M. Since the proof of the result relating observed data E[M(x₁, x₂)] to units such that M(1, 2) = 1, M(0, 2) = M(1, 0) = M(0, 0) = 0 is essentially one line, we give it here also for completeness.

Theorem 2

If E[M(1, 2)] − E[M(0, 2)] − E[M(1, 0)] − E[M(0, 0)] > 0, then there must exist a unit with M(1, 2) = 1, M(0, 2) = M(1, 0) = M(0, 0) = 0.

Proof

By contradiction. Suppose there were no unit with M(1, 2) = 1, M(0, 2) = M(1, 0) = M(0, 0) = 0. Then, for all units, M(1, 2) = M(0, 2) = M(1, 0) = M(0, 0) = 0 which implies E[M(1, 2)] − E[M(0, 2)] − E[M(1, 0)] − E[M(0, 0)] ≤ 0, a contradiction.

An immediate corollary of Theorems 1 and 2 and Lemma 1 is then:

Corollary 1

If E[M(1, 2)] − E[M(0, 2)] − E[M(1, 0)] − E[M(0, 0)] > 0, then the hypothesis of ‘local hidden variables’is false.

This corollary is referred to as Bell’s theorem in the physics literature. Its premise is referred to as Bell’s inequality. As noted above, from the quantum mechanics of the 2-particle system, and confirmed by experiment, E[M(x₁ = i, x₁ = j)] = sin²(Δ_ij /2). Thus we have that:

$$E[M(1,2)]-E[M(0,2)]-E[M(1,0)]-E[M(0,0)]={sin}^2({\mathrm{\Delta }}_{12}/2)-{sin}^2({\mathrm{\Delta }}_{02}/2)-{sin}^2({\mathrm{\Delta }}_{10}/2)-0$$

From this it follows that the local hidden variables assumption is rejected if

$${sin}^2({\mathrm{\Delta }}_{12}/2)>{sin}^2({\mathrm{\Delta }}_{02}/2)+{sin}^2({\mathrm{\Delta }}_{10}/2)$$

but the angles 0, 1, 2 can be chosen arbitrarily, and thus easily chosen to satisfy this inequality. Thus the hypothesis of local hidden variables is false.

Note Lemma 1 and Theorem 1 do not require particle 1 to receive treatment (measurement axis) 2 or particle 2 to receive treatment 1. Further Lemma 1 and Theorem 1 remain true if we assume axis 1 and 2 are identical and thus everywhere replace “2”by “1”in the statements and proofs. The reason we assumed 3 directions {0, 1, 2} is due to the physics, not the mathematics; the physics requires it because, as noted earlier, quantum mechanics implies that the physical event M(1, 1) = 1 is not possible.

Note that although the connection to causal interactions is arguably novel, the actual algebra in the results above is in fact very similar to several proofs of Bell’s inequality already in the literature. We now walk through the logic that led us to realize that results on causal interaction can be used to test for interference between treatments and provide a proof of Bell’s inequality. From a counterfactual perspective, Bell’s Theorem implies that there is interference between treatments, i.e. the spin Y_i(x₁, x₂) of a particle i depends on treatment (measurement axis) of both particles. By regarding the two particles as a single unit, interference between treatments is recast as causal interaction between two treatments given to single unit. This recognition combined with experimental data on M(x₁, x₂) = 1{Y₁(x₁, x₂) = Y₂(x₁, x₂)} implied, by Theorem 2, the existence of a unit for which M(x₁, x₂) takes the value 1 if and only if x₁ = 1 and x₂ = 2, i.e. there is an epistatic interaction for M. By Lemma 1 and Theorem 1 this implies, for at least one of Y₁(x₁, x₂) or Y₂(x₁, x₂), there is epistatic interaction and thus dependence on both (x₁, x₂).

Interestingly, the same experiments that falsified the local hidden variable hypothesis, simultaneously confirmed that, as predicted by quantum mechanics, the marginal probability that the particle 1’s spin Y₁(x₁, x₂) is up is the same for all x₂. That is, from the perspective of the Neyman counterfactual model, although Bell’s theorem and experiment proves there exist units for which particle 2’s treatment has a causal effect on the the spin of particle 1, nonetheless, the expected (i.e. superpopulation average) causal effect is zero.

The prototypical Bell inequality, and accompanying experiment, has in recent years spawned a multitude of variations involving more than two particles, measurements with more than two outcomes, and more than two possible measurements at each location; see for instance Zohren, et al. (2010) for a striking version of “Bell” obtained simply by letting the number of outcomes be arbitrarily large. Popular inequalities and experiments are compared in terms of statistical efficiency by Van Dam et al. (2005). Other connections to statistics (missing data theory) and open problems are surveyed in Gill (2007).

3. Discussion

We claimed above that there were experimental results that violated Bell’s inequality and therefore ruled out local hidden variables. However, there remain several small possible loopholes. Perhaps the most important one of which is the following: in these experiments for every entangled pair that we measure we often fail to detect one of the two particles because the current experimental set-up is imperfect. The experimental results we noted above are actually the results conditional on both particles’ spins being measured. If those pairs were not representative of all pairs, that is, if the missing pairs are not missing at random, it is logically possible that the experimental results can be explained by local hidden variables where the values of Y₁(x₁) and Y₂(x₂) also determine the probability of the spin of both being observed. The results of experiments that close this loophole by observing a higher fraction of the pairs should be available within the next several years. Nearly all physicists believe that the results of these experiments will be precisely as predicted by quantum mechanics and thus violate Bell’s inequality.

Henceforth, we assume Bell’s inequality is violated and that we have therefore ruled out local hidden variables. We now return to the question of whether this rules out local reality. As noted above, experiments have been conducted such that the times of the two measurements were sufficiently close and the separation of the particles sufficiently great that even a signal traveling at the speed of light could not inform one particle of the result of the other’s spin measurement. Therefore ruling out local hidden variable effectively rules out local reality.

Since the hypothesis of local reality is false, we conclude that for some particle pairs the angle at which particle 1 is measured has a causal effect on the spin of particle 2. Note, even under the alternative, we have assumed that Y₁(x₁, x₂) exists for all (x₁, x₂). Thus our assumption of ‘reality’ remains; the hypothesis that “reality” is local has been rejected. However, quantum mechanics is generally assumed to be irreducibly stochastic. We could have accommodated this assumption by positing stochastic counterfactuals p₁(x₁, x₂) and p₂(x₁, x₂) defined for all (x₁, x₂) with the measured spin Y_i(x₁, x₂) being the realization of a Bernoulli random with success probability p_i(x₁, x₂). That is, we could assume that the elements of reality are the counterfactual probabilities p_i(x₁, x₂). Our hypothesis of stochastic locality is then p₁(x₁, x₂) = p₁(x₁) and p₂(x₁, x₂) = p₂(x₂). The proof given above, again combined with the experimental results, can be used to reject this hypothesis by using a coupling argument as in VanderWeele and Robins (2012).

A perhaps more radical point of view is the one that is often attributed to the Copenhagen school: the mathematical theory of quantum mechanics successfully predicts the results of experiments, without positing any “elements of reality” (counter-factuals), even the above non-local stochastic ones. The Copenhagen standpoint says that there is nothing “behind” quantum mechanics. Bell’s Theorem only gives us locality problems if (a) we insist on the “existence” of counterfactual outcomes of measurements which we did not actually do and also (b) we insist on placing them in time and space alongside of the actual outcomes of the actually performed measurements. In other words, we only have “non-locality” in a mathematical picture of the world which we built ourselves, and only after we have added into it, alongside of the things which actually happen (located in space-time when and where they happen), also things which only have a theoretical existence. At the level of observable phenomena, quantum mechanics doesn’t predict non-locality. It does not allow “action at a distance” owing to the fact that for each particle, the marginal probabilty of the spin being ‘up’ does not depend on the treatment (measurement direction) of the second particle. Thus, there is no conflict with any relativistic principles (principles of causality), but quantum mechanics does predict phenomena which would be impossible under classical physics. According to this view attributed to the Copenhagen School, the question of the existence of “elements of reality” is not a scientific question, as it is not subject to empirical test and our most successful scientific theory, quantum mechanics, has no need of them. This is appealing to physicists because it restores locality in the following sense. To become entangled two particles must interact and this interaction, even in the laws of quantum mechanics, occurs locally. Entanglement leads to correlated measurements. Once entangled, these correlations will persist irrespective of the particles’ separation as described earlier. However, following the Copenhagen school, to say counterfactuals Y_i(x₁, x₂) do not exist is to say that the question of whether the measurement axis of the spin of particle 1 had an effect on the spin on particle 2 cannot be asked; not every event has a cause. In all physical theories prior to quantum theory, it was possible to imagine, alongside the actual measurements of actual experiments, what would have been observed had we done something differently (i.e. counterfactuals), while still preserving locality. This is not possible with quantum mechanics. In summary, Bell’s inequality (and its experimental support) show that the Copenhagen standpoint of abandoning counter-factuals is not only possible, it is also necessary to take this standpoint if we want to retain “locality” as a fundamental part of our world picture.