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arXiv:0907.4967

Defining the local part of a hidden variable model: a comment

Roger Colbeck, Renato Renner

Published Jul 28, 2009• 2 citations
arXivHTMLPDF

Abstract

In [Physical Review Letters 101, 050403 (2008)], we showed that quantum theory cannot be explained by a hidden variable model with a non-trivial local part. The purpose of this comment is to clarify our notion of local part, which seems to have caused some confusion in the recent literature. This notion is based on Bell's and demands that local hidden variables are physical, the idea being that, if discovered, they would not contradict basic physical principles. We explain why the recent supposed "counterexamples" that have appeared are not counterexamples to our theorem--in fact they are based on a definition of local hidden variables which would allow signalling and is therefore not physical.

DOI
10.48550/arxiv.0907.4967

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In [Physical Review Letters 101, 050403 (2008)], we showed that quantum theory cannot be explained by a hidden variable model with a non-trivial local part. The purpose of this comment is to clarify our notion of local part, which seems to have caused some confusion in the recent literature. This notion is based on Bell's and demands that local hidden variables are physical, the idea being that, if discovered, they would not contradict basic physical principles. We explain why the recent supposed "counterexamples" that have appeared are not counterexamples to our theorem--in fact they are based on a definition of local hidden variables which would allow signalling and is therefore not physical.
Bell's theorem is 50 years old. Still there is a controversy about its implications. Much of it has its roots in confusion regarding the premises from which the theorem can be derived. Some claim that a derivation of Bell's inequalities requires just locality assumption, and nothing more. Violations of the inequalities are then interpreted as ``nonlocality'' or ``quantum nonlocality''. We show that such claims are unfounded and that every derivation of Bell's inequalities requires a premise -- in addition to locality and freedom of choice -- which is either assumed tacitly, or unconsciously, or is embedded in a single compound condition (like Bell's ``local causality''). The premise is equivalent to the assumption of existence of additional variables which do not appear in the quantum formalism (in form of determinism, or joint probability for outcomes of all conceivable measurements, or ``additional causes`'', or ``hidden variables'', ``complete description of the state'' or counterfactual definiteness, etc.). A certain irony is that perhaps the main message of violation of Bell's inequalities is that our notion of locality should be based on an operationally well-defined no-signalling condition, rather than on local causality.
It was shown by Bell that no local hidden variable model is compatible with quantum mechanics. If, instead, one permits the hidden variables to be entirely nonlocal, then any quantum mechanical predictions can be recovered. In this Letter, we consider general hidden variable models which can have both local and nonlocal parts. We show the existence of (experimentally verifiable) quantum correlations that are incompatible with any hidden variable model having a nontrivial local part, such as the model proposed by Leggett.
The conceptually puzzling features of quantum mechanics as a statistical theory all have their source in the impossibility of relating the probability assignments defined by the quantum state to distributions over determinate values of the physical magnitudes. While some interpretations of quantum mechanics, as well as “hidden variable” modifications of the theory, have proposed several constructions for assigning values to magnitudes (beyond the value assignments defined by the quantum state alone), a variety of theorems exist which impose severe restrictions on such theoretical constructions. This note will deal with Bell's proof (in [2]), that any hidden variable theory satisfying a physically reasonable locality condition is characterized by an inequality which is inconsistent with the quantum statistics. Bell's result has initiated a series of experiments designed to refute the entire class of local hidden variable theories. We show that Bell's inequality actually characterizes a feature of hidden variable theories which is much weaker than locality in the sense considered physically motivated.
As clarified in preceding chapters, conventional “quantum mechanics” is not a complete theory. This fact was pointed out by Einstein-Podolsky-Rosen (1935). In connection with the completeness problem, the locality of hidden-variable theories of quantum mechanics was suspected. It seems that Bell’s discussion (1964) based on his inequality has convinced people that any hidden-variable theory becomes necessarily incompatible with quantum mechanics if it satisfies the locality. But, as will be shown, Bell’s conclusion (1964) is not correct.