Abstract
The 1964 theorem of John Bell shows that no model that reproduces the predictions of quantum mechanics can simultaneously satisfy the assumptions of locality and determinism. On the other hand, the assumptions of signal locality plus predictability are also sufficient to derive Bell inequalities. This simple theorem, previously noted but published only relatively recently by Masanes, Acin and Gisin, has fundamental implications not entirely appreciated. Firstly, nothing can be concluded about the ontological assumptions of locality or determinism independently of each other—it is possible to reproduce quantum mechanics with deterministic models that violate locality as well as indeterministic models that satisfy locality. On the other hand, the operational assumption of signal locality is an empirically testable (and well-tested) consequence of relativity. Thus Bell inequality violations imply that we can trust that some events are fundamentally unpredictable, even if we cannot trust that they are indeterministic. This result grounds the quantum-mechanical prohibition of arbitrarily accurate predictions on the assumption of no superluminal signalling, regardless of any postulates of quantum mechanics. It also sheds a new light on an early stage of the historical debate between Einstein and Bohr.
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Notes
Our usage of ‘locality’ here is the same as that of Bell in 1964: “that the result of a measurement on one system be unaffected by operations on a distant system”. This assumption is sometimes referred to as “parameter independence” and is strictly weaker than the assumption of “local causality”, which was later introduced by Bell [5] to show that determinism does not need to be assumed, and that no locally causal model (deterministic or otherwise) can reproduce quantum mechanics. See also Sect. 4.
Besides, Valentini’s main result seems to be flawed. He seems to have only proven the weaker result that there exist distributions over the hidden variables which would allow signalling, as the reader may be convinced by analysing the first equation on p. 276 of [21].
After a first version of this work was posted on the arXiv (arXiv:0911.2504v1), another arXiv post (arXiv:0911.3427v1, eventually published in Nature [15]) underscored the importance of the distinction by proposing a scheme to generate random numbers certified by violation of a Bell inequality. Those results are however somewhat distinct from the present one, in that to derive bounds on the randomness of the output of a Bell experiment, those authors assumed the validity of the laws of quantum mechanics. Here no such assumption is made (but consequently no bound is given on the randomness or unpredictability of the output).
For the physicist trained to be suspicious of philosophical terms, note that in this context the term ‘metaphysics’ does not refer to mysticism, but to the study of formal and empirical properties of physical theories themselves. (Experimental) metaphysics is to physics as metamathematics is to mathematics. It includes the study of sets of physical theories which fail to represent observations, where this analysis can be illuminating in understanding those that do not.
Note that considering the possibility that further variables exist is not the same as assuming that they exist; there is no “hidden-variable assumption” in Bell’s theorem.
Indeed it is a corollary of the present result that in any deterministic model that reproduces quantum theory the ontic variables must be necessarily unknowable.
We remind the reader that this corresponds to Bell’s definition of locality introduced in 1964 [3]. There are instances in less formal publications, for example Ref. [4], where Bell used the term “locality” more loosely to mean the property which quantum mechanics lacked, as revealed by his theorem. However these are very much the exception, and from 1976 on, Bell almost invariably used the term “local causality” for this property.
There is another useful sense of determinism which needs to be distinguished from the one we are using here. Quantum mechanics can be said to be deterministic in the sense that for a closed system the state at a later time is determined through unitary evolution by the state at an initial time. However, operational quantum mechanics is not deterministic in the sense used in this paper, since of course a system undergoing a measurement interaction is no longer a closed system.
Although, of course, this would have no implication for Einstein’s later attacks on the completeness of quantum theory. Of course all that we have said about the Bohr-Einstein debates accepts that Bohr’s narrative of the events is accurate. It has been argued by Howard [13] and by Bacciagaluppi and Valentini [1] that Bohr misunderstood Einstein’s arguments, and that actually Einstein was arguing for incompleteness (based on nonseparability), rather than incorrectness, the whole time.
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Acknowledgements
We are grateful to Rob Spekkens for useful feedback and encouragement. This work was partly supported by the Australian Research Council Discovery grant DP0984863 and Discovery Early-Career Researcher Award DE120100559.
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Cavalcanti, E.G., Wiseman, H.M. Bell Nonlocality, Signal Locality and Unpredictability (or What Bohr Could Have Told Einstein at Solvay Had He Known About Bell Experiments). Found Phys 42, 1329–1338 (2012). https://doi.org/10.1007/s10701-012-9669-1
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DOI: https://doi.org/10.1007/s10701-012-9669-1