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What can we learn from Bell test experiments about locality and realism?

Bell test experiments are empirical constructs in demonstrating violation of Bell’s inequality in quantum systems and proving that local realism can never reproduce all predictions of quantum mechanics (Wikipedia, Bell’s theorem). Since the initial publication of Bell’s paper (Bell, 1964), various versions of Bell test experiment has been proposed and conducted. However, imperfect implementations of the conceptual framework of the experiment suffer from loopholes that undermine its credibility. Is there a practically feasible loophole-free implementation of Bell test experiment that can sentence the death of local realism? If local realism is false, then which one is problematic, locality or realism? This paper explores what we can (and cannot) learn from Bell test experiments about locality and realism by answering these two questions. Under reasonable metaphysical and statistical assumptions, a recent experiment (Hensen, 2015) provides a good loophole-free demonstration of violation of Bell’s inequality. Meanwhile, it is not trivial to understand what an ideal Bell test experiment can tell about locality and realism separately.

Introduction

We start by introducing two central terms: locality and realism. According to special relativity, causal influence cannot travel faster than the speed of light. An object cannot be affected by things beyond its immediate surroundings; more precisely, it can only be affected by events within its backward light cone. This is the principle of locality. Locality was proposed in the classical physics era, and is conformed by both special relativity and general relativity (Wikipedia, Principle of locality).

Realism is an ontological view, often claimed regarding a specific (family of) object. Realism has a weaker interpretation (the entity version) and a stronger interpretation (the fact version). In the entity version, realism on an object is the view that the object enjoys mind-independent existence. The fact version of realism, in addition to agreeing with the entity version, further claims that the physical properties of an object exist independently of human mind. It follows that any property of an object must have an objective definite value, no matter we measure it or not (Weaver, 2016).

Hidden variable theory is an interpretation of quantum mechanics under the philosophy of realism (as well as determinism). It claims that quantum mechanics is incomplete, that physical system cannot be completely described by quantum state, and that there are “hidden” variables – real physical entities inaccessible for observation – governing the seemingly random behavior of quantum observables. Local hidden variable theory is built on hidden variable theory with the additional constraint of locality.

Local hidden variable theory is suggested by Einstein, Podolsky, and Rosen (Einstein et al, 1935) as complementing the theory of quantum mechanics. Starting from the assumption of local realism, they found that an entangled pair of particles must either violate Heisenberg’s uncertainty principle (which is deemed impossible) or violate the principle of locality, which is known as the EPR paradox. It was believed by EPR that this paradox shows that quantum mechanics theory is incomplete, and they propose to introduce hidden variables for the remedy. However, the distinction between the non-local quantum mechanics theory and the local hidden variable theory was not quantifiable until Bell proposed an inequality by which the local hidden variable theory must follow (Bell, 1964). If the inequality is observed to be violated, then local hidden variable theory must not be true. Note that “local realism” and “local hidden variable theory” are synonyms and are often used interchangeably.

Loopholes and Complications in Bell Test

Bell’s inequality is constructed from a conceptual framework involving a series of conditions. If a Bell test experiment fails to strictly implement this framework, violation of Bell’s inequality in this particular experiment is not a valid proof of the violation itself. This is known as loophole in Bell test experiments. Loopholes undermine a Bell test experiment’s credibility in refuting against local realism. So far, the loopholes of utmost concern are the detection loophole and the communication loophole.

The detection loophole stands in the way of linking the experimentally determined frequency and the theoretical probability. The probability of particles in a specific state is derived from the frequency of measuring particles in that state, and then plugged into the Bell’s inequality. If the detector only captures some, but not all, of the particles (it has a low efficiency/fidelity), then it better has fair sampling, that is, it has the same successful rate on detecting particles in different states. It is conceivable that this fair sampling assumption may be violated by the conspiracy of hidden variables, leading to a violation of Bell’s inequality with values from observed inaccurate frequencies while there is no violation with the underlying probabilities. The detection loophole can be addressed by using high-efficiency detectors on appropriate particle-property, avoiding having to make the fair sampling assumption.

The communication loophole arises when certain independence between the two entangling parties is not guaranteed. Bell’s inequality builds upon two independence conditions: remote outcome independence – that the outcome of measurement on one party gives no information about the outcome of measurement on the other party, and remote context independence – that the outcome of measurement on one party is independent of the type of measurement performed on the other party (Bell’s Theorem, SEP part 2). If these independence conditions are not strictly met, Bell’s inequality cannot be said to be violated by the experiment. The closure of communication loophole is typically attempted and achieved by space-like separating the detections on two parties (Bell’s Theorem, SEP part 5). (Aspect et al, 1982) proposes the method of random basis-selection with rapid alternation, which is often accompanied by taking the two parties kilometers apart to create space-like separated detections.

Although efforts on closing the loopholes in Bell test have never ceased since the 1970s, the detection loophole and the communication loophole had never been addressed simultaneously in a single experiment until recent years (Hensen et al, 2015). Potential reasons include that scholars were exploring the best way to address each loophole, and that there is not so much interest in perfecting the proof of Bell’s theorem as in putting the theorem into applications. However, we identify a deeper reason for the difficulty of combining the remedies of the two loopholes.

There is a fundamental incompatibility between the closure of detection loophole and the closure of communication loophole in practice. In order to ensure the space-like separation of the two measurement events, detections must be made within a certain time window governed by special relativity. Meanwhile, with decreased size of detection time window comes decreased efficiency of detectors, which in turn introduces the requirement for the fair sampling assumption; if the fair sampling assumption is not proved to hold in the experiment, the detection loophole will emerge. On one hand, if space-like separation is guaranteed to be met, one risks lowering the fidelity of detection beyond a lower bound threshold that eliminates the fair sampling assumption. On the other hand, if the fidelity of detection is elevated to provide statistical significance for closure of detection loophole, the detection time window may need to be expanded beyond the range guaranteeing space-like separation. This circumstance is somewhat analogous to the uncertainty principle, in which case with increased confidence of one variable comes decreased confidence of the other.

Fortunately, there are two characteristics distinguishing the incompatibility of loopholes closure from the incompatibility of observables. First, the closure of communication loophole is a dichotomous variable, because it is good as long as the time window size is kept below a certain value. This can be exploited by setting the time window size as large as (a small fraction below) that threshold to both comply to the closure of communication loophole and attain greater confidence on the closure of detection loophole. Second, there is no theoretical implication on how “incompatible” these two closures of loophole are, as is in the uncertainty principle. The extent of incompatibility is dependent on specific experimental setting. To illustrate, an experiment using photon pair and entanglement of polarization may give a 70% detector efficiency with time window size on the threshold, while using electron pair and entanglement of spin may give a 90% detector efficiency with time window size on the threshold (the figures are fictitious, just for sake of illustration). Therefore, proper experimental setting should be wisely selected to achieve optimal confidence of loophole-free.

In (Hensen et al, 2015), an experiment claimed to close both the detection loophole and the communication loophole altogether is conducted. The quantum system involved is electron pair spin; efficient electron spin detection closes the detection loophole by making the fair sampling assumption unnecessary; fast detection, random basis-selection and placing the entangled electrons in two afar labs create space-like separation that closes the communication loophole. With time window size maximized under the constraint of space-like separation, a detector fidelity of over 96% is achieved, and Bell’s inequality is claimed to be violated with 2.1 sigma confidence. Simultaneously published were wo other papers (Shalm et al, 2015) (Giustina et al, 2015) also claiming to have closed all loopholes with photon polarization systems.

One defect in this experiment is that the random numbers governing basis selection is not perfectly random – they are produced by a pair of space-like separated pseudorandom number generators. Even using quantum random number generators, the problem still exists because these generators could still be operating under hidden variables. The way out is to use human free will as random number generators in future experiments, which in turn introduces the question of whether human beings have free will. But after all, scientific research has to build on some basic framework or assumptions of metaphysics, so this issue will be beyond the scope of our topic.

With the limitation imposed by the uncertainty principle in quantum systems, it seems that we have to rely on statistical tools in closing both loopholes (i.e. to convey our confidence that the detection loophole is closed, since there is no detector of quantum particles with 100% fidelity in practice). In fact, statistics has become a prevalent tool in studying modern physics (think of LIGO’s announcement of finding gravitational wave with 5.1 sigmas confidence (Abbott et al, 2016), so there is no problem in the methodological aspect. Therefore, wisely designed and conducted, a Bell test experiment is a legitimate proof that local realism is an incorrect philosophical guidance for modeling quantum systems.

Implications of Ideal Bell Test

Violation of the Bell’s inequality nullifies local hidden variable theory as a candidate interpretation of quantum mechanics; that is, at least one of locality and realism is false in the quantum world. By Boolean logic, if A and B is false, then either A or B, or both, is false; by propositional logic, if a false premise C is derived from premises A and B, then at least one of these premises is false. We are interested to know which one(s) is false. Unfortunately, even if an ideal, loophole-free Bell test experiment is performed, its result does not show whether non-locality, anti-realism, or both, is the correct picture. This limit has to do with the basic construct of local realism and Bell’s inequality.

At the first glance, Bell test experiment seems capable to distinguish between cases of non-local realism, local anti-realism, and non-local anti-realism. The method to achieve this is to remove either or both of the local realism assumptions, and check if the resulting Bell’s inequality agrees with experimental results. If the non-local realism version of Bell’s inequality is incompatible with observations, then the realism assumption must be false; likewise, if the local anti-realism version of Bell’s inequality is incompatible with observations, then the locality assumption must be false. A special remark that if the non-local realism version of Bell’s inequality is compatible with observations, it is underdetermined whether realism is true (because nature might still be non-local and anti-realism); likewise, if the local anti-realism version of Bell’s inequality is compatible with observations, it is underdetermined whether locality is true.

However, this method is fallacious in that it does not capture how Bell’s inequality is built from the assumptions of locality and realism. It is not the naïve case that the conjunction of locality and realism leading to a certain bound of Bell’s inequality, while say the conjunction of locality and anti-realism leading to an equality with somewhat relaxed bound. Instead, locality and realism are indispensable assumptions in the construction of Bell’s inequality itself. In the conceptual framework of Bell’s inequality, the system of two parties is assumed to be described by a complete state function / vector. Note that this assumption implies realism, since in the case of anti-realism, a physical system will have mind-dependent properties and thus can never be characterized by a complete state in isolation. Then, on the way of constructing Bell’s inequality, a set of probabilities are defined, and a set of equations with regard to these probabilities are assumed to hold in accordance to locality. Armed with these assumptions, we can derive Bell’s inequality in the form most commonly used to demonstrate violation to it:

-2  <A1B1> + <A2B1> + <A1B2> - <A2B2>  2

where A1 and A2 are observables of one party of the system, and B1 and B2 are observables of the other party. As predicted by quantum mechanics, the maximally entangled bi-particle state with certain choice of observables can boost the expression in the middle up to ~2.828. If one of locality and realism is absent from the conceptual framework of Bell’s inequality, there would be no such thing as Bell’s inequality at all! There is not counterpart of Bell’s inequality that corresponds to the assumptions of non-local realism, local anti-realism, or non-local anti-realism (actually, (Gröblacher et al, 2007) attempted to test non-local realism with a constructed inequality analogous to Bell’s, but it was later criticized for conceptual fallacies (Tausk, 2008; Zukowski, 2008)), and people may need to search for alternative methods and design more experiments to distinguish them (or there is no way at all to make the distinction if two theories predict the exactly same observable outcomes).

Here I will discuss and refute some scholar’s view that Bell’s theorem settles the debate between anti-realism and non-locality. Some believe that quantum non-locality is incompatible with the locality axiom as postulated by special theory of relativity, which is deemed the most heavily verified theory of physics. This is actually confusing the meaning of “locality” in the context of quantum mechanics and that of relativity. In relativity, locality means that no information can be transferred exceeding the speed of light, or in short, no signaling. This by nature is causation. In contrast, non-locality in quantum mechanics only refers to instantaneous correlation between two physical systems that are space-like separated. As indicated by the “no signaling theorem” (Nielsen, 2010, p. 3), It cannot be used to transfer information because the collapse of state is a random process. Therefore, the non-local behavior shown in quantum systems does not violate locality in relativity. On the other hand, (Gisin, 2012) suspects the necessity of a realism assumption in the construction of Bell’s inequality, thereby arguing that Bell test experiments give an outright refutation of locality. In his argument, the usage of realism only exists in the distinction of whether the output of measurement is real, rather than in the derivation of Bell’s inequality. This is not true. Bell’s inequality is built upon the well-definition of the probability of the value of measurement outcome on one side, conditional on that value of measurement on the other side and their respective choice of measurement basis. However, such probability is not well-defined if the quantum system of two parties cannot be described completely by a state vector / function, which is what realism suggests. Although the dependency of Bell’s inequality on the realism assumption is quite difficult to grasp, realism, together with locality, consists of essential foundation of Bell’s inequality. With this dual dependency, Bell test experiments is incapable of determining the truthfulness of non-locality.

Beyond the local hidden variable theory, there are numerous interpretations of quantum mechanics that hold different (combinations of) views on locality and realism. Non-local hidden variable theory is in favor of non-local realism, and is endorsed by the de Broglie-Bohm theory; the predominant Copenhagen interpretation, on the other hand, believes in non-locality and anti-realism. A comprehensive chart on the comparison of various interpretations can be found under the Wikipedia page “Interpretations of quantum mechanics”.

There are many competing interpretations of quantum mechanics, and Bell test experiment is definitely not capable of indicating which is the correct one. It only proves that the combination of locality and realism is wrong, and that’s it. No information about locality and realism in isolation can be extracted from this experiment. However, this is already significantly striking.

Conclusion

We analyzed what we can learn from Bell test experiments on locality and realism in two steps. First, we explored whether we can empirically know that the conjunction of locality and realism is false by implementing a loophole-free Bell test experiment, and our answer is yes. Although we have discovered intrinsic mechanisms that backfire the simultaneous closure of all loopholes, certain technical characteristics of carefully chosen experimental setup can still make loophole-free possible. Then, we further explored if we can learn whether it is wrong with reality or it is wrong with locality with the methodology of Bell test, and our answer to this question is no. Both locality and realism are essential assumptions in the construction of Bell’s inequality, whose prediction is violated by empirical data on quantum systems, and thus making the Bell test alone incapable of ruling out either locality or realism. Experiments attempting to make the distinction between the remaining alternatives – non-local realism, local anti-realism, and non-local anti-realism – in a Bell’s fashion (inequality violation) have not been reported successful. Different interpretations of quantum mechanics hold different beliefs on locality, realism, and many other physical and metaphysical issues, and physics and the philosophy of science will continue their endeavor to the truth of the microscopic world.

[Note: my original critical thinking is largely reflected in “Loopholes and Complications in Bell Test” paragraphs 5-6, 8-9, and “Implications of Ideal Bell Test” paragraphs 1-4.]

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