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In the paper ''Quantum non-locality as an axiom'' (Popescu, S. & Rohrlich, D. Found Phys (1994) 24: 379. https://doi.org/10.1007/BF02058098), it is stated that the conventional formalism of quantum mechanics has indeterminism as an axiom and non-locality as a theorem. Contrary to the standard approach, in this paper the authors start with two axioms (relativistic causality and non-locality) and proceed on to deduce that quantum theory has to be indeterministic.

My question is this: what theorem did the authors have in mind when they mentioned that non-locality is a theorem and indeterminism is an axiom? In other words how to I deduce non-locality from indeterminism? As far as I know , non-locality originates primarily from the way symmetric and anti-symmetric wavefunctions are constructed, to account for the indistinguishable nature of quantum particles.

Qmechanic
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In the following I am using the arXiv version of the paper.

The word theorem is not used once in the paper. The word axiom never refers to indeterminism (a word used only once at the end of the paper). Therefore we can not state that the authors mentions "that non-locality is a theorem and indeterminism is an axiom". This appears only in the abstract of the 1994 version linked in the question (I have no access to the paper itself, please correct me if it is significatively different), where it may be a rhetorical abuse of langage.

Non-locality in quantum mechanics manifests itself via non-local correlations: because EPR correlations cannot be explained by hidden variables, which has been experimentally proven by Alain Aspect and many other teams in the form of observed violations of Bell's inequalities (CHSH in the paper), we can consider them as non-local. The relevant theorem here is Bell's theorem, which states that "no physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics."

Indeterminism in quantum mechanics appears in certain expositions where it is considered that two laws are competing for the evolution in time of a quantum system: a deterministic one when a system is not measured (the unitary Schrödinger equation), and a non-deterministic one (collapse/reduction) upon measurement. Since measurement outcomes are only predicted via probabilities (Born rule) one can say that quantum mechanics is not deterministic. Note that many people still claim that QM is deterministic and consider that the measurement problem is to be solved by decoherence approaches where everything happens within the unitary evolution; the jury is still out.

Now for the articulation of non-locality and determinism: it is because measurement outcomes are probabilistic that we would need hidden variables to explain EPR correlations. Since they are experimentally ruled out, we are left with distant correlations that without hidden variables are by definition non-local: what we call hidden variables here is whatever would locally move along the entangled parts of a quantum system.

In that sense indeterminism would be an experimental fact (not an axiom), and non-locality a logical consequence in the context of realist interpretations of quantum mechanics.

Stéphane Rollandin
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Suppose we send a single photon to a half-silvered diagonal mirror, so that the photon can be detected later at two different places, A and B, each with probability 1/2. If we require energy to be not just conserved on the average but exactly conserved, in accord with the 1926 Bothe-Geiger experiment, then there must be a perfect anticorrelation between detection at A and detection at B. A and B can be spacelike in relation to one another, so the correlation demonstrates a pretty serious type of non-locality.