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arXiv:1010.4020

The Consistency of Quantum Mechanics Implies its Non-Determinism

Iégor Reznikoff

Published Oct 19, 2010• 0 citations
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Abstract

In a previous paper (arXiv:1008.3661v1[quant-ph] 21 Aug 2010), we have given a purely logical proof of the Conway and Kochen Free Will theorem in QM: the freedom of the observer implies the freedom of the observed particle. Here we show that the hypothesis of the observer's freedom is not necessary: the assumption of the (informal) consistency (non contradiction) of QM implies its non-determinism relative to physical events (the freedom of observed particles).

DOI
10.48550/arxiv.1010.4020

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In a previous paper (arXiv:1008.3661v1[quant-ph] 21 Aug 2010), we have given a purely logical proof of the Conway and Kochen Free Will theorem in QM: the freedom of the observer implies the freedom of the observed particle. Here we show that the hypothesis of the observer's freedom is not necessary: the assumption of the (informal) consistency (non contradiction) of QM implies its non-determinism relative to physical events (the freedom of observed particles).
I begin with description of quantum entanglement and its relevance to the EPR-type experiments, which played a decisive role in rejecting determinism and hidden-variable theories. Thereby I prove the closely-related Kochen-Specker paradox, on which also the main object of my paper, the recently published Conway and Kochen’s Free Will Theorem, is founded. It is being shown there, on the basis of three physical axioms, following from the special theory of relativity and quantum mechanics, that if the choice for a particular type of spin 1 experiment is not a function of the information accessible to the experimenters, then its outcome is equally not a function of the information accessible to the particles. Therefor the response of the particles can be seen as free. In conclusion I mention some critical remarks on the theorem, expressed in subsequent responses to the original article.
Although the concept of an observing device with memory is very simple and definable in terms of algorithmic machine, it is not compatible with a predictive theory involving this concept. This is sufficient to show that Physics as a whole cannot be deterministic; moreover, using the Conway-Kochen Free Will Theorem [1] and the simple logical proof of it we gave in [2], we show, without any need of free will or observer's freedom, that neither Quantum Mechanics nor any extension of this theory can be deterministic.
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The title refers to the Free Will Theorem by Conway and Kochen whose flashy formulation is: if experimenters possess free will, then so do particles. In more modest terms, the theorem says that individual pairs of spacelike separated particles cannot be described by deterministic systems provided their mixture is the same for all choices of measurement settings. We reformulate and generalize the Free Will Theorem theorem in terms of systems of random variables, and show that the proof is based on two observations: (1) some compound systems are contextual (non-local), and (2) any deterministic system with spacelike separated components is non-signaling. The contradiction between the two is obtained by showing that a mixture of non-signaling deterministic systems, if they exist, is always noncontextual. The "experimenters' free will" (independence) assumption is not needed for the proof: it is made redundant by the assumption (1) above, critical for the proof. We next argue that the reason why an individual pair of particles is not described by a deterministic system is more elementary than in the Free Will Theorem. A system, contextual or not and deterministic or not, includes several choices of settings, each of which can be factually used without changing the system. An individual pair of particles can only afford a single realization of random variables for a single choice of settings. With this conceptualization, the "free will of experimenters" cannot be even meaningfully formulated, and the choice between the determinism and "free will of particles" becomes arbitrary and inconsequential.