## Abstract

Non-locality, or quantum-non-locality, are buzzwords in the community of quantum foundation and information scientists, which purportedly describe the implications of Bell’s theorem. When such phrases are treated seriously, that is it is claimed that Bell’s theorem reveals non-locality as an inherent trait of the quantum description of the micro-world, this leads to logical contradictions, which will be discussed here. In fact, Bell’s theorem, understood as violation of Bell inequalities by quantum predictions, is consistent with Bohr’s notion of complementarity. Thus, if it points to anything, then it is rather the significance of the principle of Bohr, but even this is not a clear implication. Non-locality is a necessary consequence of Bell’s theorem only if we reject complementarity by adopting some form of realism, be it additional hidden variables, additional hidden causes, etc., or counterfactual definiteness. The essay contains two largely independent parts. The first one is addressed to any reader interested in the topic. The second, discussing the notion of local causality, is addressed to people working in the field.

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## Notes

- 1.
The phrases in brackets indicate the text of EPR transformed in such a way so that it fits the considered three-particle example. Their

*Q*(position) is now horizontal component of the spin, and*P*(momentum) is the vertical one. - 2.
The mathematical formalism of quantum mechanics reflects complementarity of pairs “observables”. If this is the case say for observables

*A*and*B*, e.g. describing two different components of spin, then they “do not commute”. This is turn means that in the formal quantum description “operators” associated which the two observables we have the following property: \(AB\ne BA\). Of course complementarity can occur in various degrees. We have perfect complementarity when experiments measuring B give completely random results for quantum systems prepared in any state, which was prepared by measuring A and selecting only systems which gave the same result of this measurement. For example, photons which are selected by a polarization analyzer which allows only linearly polarized photons to pass through it, would upon subsequent measurement of circular polarization give fully random results. Either clockwise or anti-clockwise polarized photons would appear, with equal probabilities. like in a coin toss. - 3.
Such an approach accepts so called “counterfactual” statements or conditionals. Such statements contain an “if” clause which describes a situation which in fact did not occur: e.g., “If EPR knew the results of the GHZ paper, they would not have written their 1935 work”.

- 4.
EPR forgot that if a new notion is to be introduced to a theory, then it must checked whether it is consistent with

*all*predictions of the theory... - 5.
“Free will” is usually not a challenged assumption, thus we assume it to hold throughout the discussion.

- 6.
Note already here, that \(\lambda \)’s do not appear in quantum mechanics, thus they are

*hidden variables. Basically this could already end the discussion, as hidden variables are a program of completing quantum mechanics, just like the aim of EPR. As a matter of fact elements of reality are indeed hidden variables.* - 7.
Some authors reserve the phrase Bell’s second theorem to his independent derivation of the impossibility of non-contextual hidden variables.

- 8.
Of course there is a full mathematical equivalence between local causal theories and stochastic local hidden variable theories of Clauser and Horne. I shall argue that additionally there is no conceptual difference.

- 9.

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Żukowski, M. (2017). Bell’s Theorem Tells Us *Not* What Quantum Mechanics *Is*, but What Quantum Mechanics *Is Not*
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In: Bertlmann, R., Zeilinger, A. (eds) Quantum [Un]Speakables II. The Frontiers Collection. Springer, Cham. https://doi.org/10.1007/978-3-319-38987-5_10

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