Nonsignaling Deterministic Models for Nonlocal Correlations have to be Uncomputable

Abstract

Quantum mechanics postulates random outcomes. However, a model making the same output predictions but in a deterministic manner would be, in principle, experimentally indistinguishable from quantum theory. In this work we consider such models in the context of non-locality on a device independent scenario. That is, we study pairs of non-local boxes that produce their outputs deterministically. It is known that, for these boxes to be non-local, at least one of the boxes' output has to depend on the other party's input via some kind of hidden signaling. We prove that, if the deterministic mechanism is also algorithmic, there is a protocol which, with the sole knowledge of any upper bound on the time complexity of such algorithm, extracts that hidden signaling and uses it for the communication of information. Bell nonlocality [1] makes us choose between determinism and the non-signaling principle [2]. That is, if one wants to account in a deterministic manner for the non-local correlations that quantum mechanics predicts and which we are now almost certain [3][4][5] that Nature exhibits, one must allow for the existence of some kind of signaling mechanism that links distant measurement choices and outcomes. But, since quantum correlations are non-signaling, such signaling mechanism must be restricted to the so-called hidden variables, and not reach the phenomenological level. Known examples of deterministic non-local theories violating the non-signaling principle (also referred to as parameter independence [6]) at the hidden-variable level are: the hidden variable model with communication of Toner and Bacon [7] and, more prominently, Bohmian mechanics [8]. For those models that use classical communication to mimic nonlocality, one can in fact study the amount of communication needed (see, for example, [9-11]).A reasonable feature that one would expect of any physical model is that it is computable [12]. This means that, in principle, one should be able to write a computer program that given a description of an experiment (that is, the measurement choices and the state of the system) outputs the model's outcomes predictions (these being probabilities in the case of quantum mechanics).Our main result is that, on the contrary, deterministic models of non-local correlations need to be uncomputable if we want to prevent those correlations from being signaling. In other words, we show that if the deterministic model is computable, the hidden signaling mechanism used to exhibit non-locality can be extracted at the observation level and used for the communication of information. More specifically, we give a protocol to perform one-way communication between two observers holding computable non-local boxes.There are a few previous results on this direction. First, this result has a flavour similar to [13]. However, we obtain our result in a device-independent scenario, that is, without assuming quantum mechanics, and provide an explicit communication protocol. Second, in [14,15] it is shown that some non-local boxes fed with...

Randomness? What Randomness?

“…40 We only defined 1-randomness for outcome sequences of fair coin flips, but the definition can be extended to other measures, see Downey & Hirschfeldt (2010), §6.12 and references therein. 41 In this light I draw attention to an important result of Senno (2017), Theorem 3.2.7, see also Bendersky et al (2017), which is entirely consistent with the above analysis: If the functions f1 and g are computable (within a computable time bound), then Alice and Bob can signal superluminally. In other words, where mathematically speaking averaging the hidden state over the probability measure µ ψ suffices to guarantee (surface) signal-locality even in a theory without hidden signal-locality (Valantini, 2002), if this averaging is done by sampling Λ in a long run of repeated measurements, then this sampling must at least be incomputable.…”

“…• Neither of these classical meanings is at all identical with the dominant usage from medieval times to the early 20th century, which was exemplified by Spinoza, who claimed that not only miracles, but also circumstances that have concurred by chance are reducible to ignorance of the true causes of phenomena, for which ultimately the will of God ('the sanctuary of ignorance') is invoked as a placeholder. 8 Thus Spinozist randomness lies in the absence of full knowledge of the entire causal chain of events.…”

“…And similarly, but highlighting the alleged negativity of the concept even more: 8 See Ethics, Part I, Appendix. 9 See also Lüthy & Palmerino (2016), §2.7 for part of the following analysis.…”

Epistemic Horizons and the Foundations of Quantum Mechanics

“…In seeming contradiction to the above result, there exist theories that do attribute definite values of properties to quantum systems, nevertheless reproducing the full array of quantum mechanical predictions, such as the de Broglie-Bohm theory [56,57]. However, the results of Yurtsever [14], Bendersky et al [15], and Calude and Svozil [16] already imply that any theory capable of reproducing quantum mechanics must be noncomputable.…”

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