On the Colbeck–Renner theorem

Abstract

In three papers Colbeck and Renner (Nature Communications 2:411, (2011); Phys. Rev. Lett. 108, 150402 (2012); arXiv:1208.4123) argued that "no alternative theory compatible with quantum theory and satisfying the freedom of choice assumption can give improved predictions." We give a more precise version of the formulation and proof of this remarkable claim. Our proof broadly follows theirs, which relies on physically well motivated axioms, but to fill in some crucial details certain technical assumptions have had to be added, whose physical status seems somewhat obscure.

Completely real? A critical note on the claims by Colbeck and Renner

“…In this paper I argue that thus far no solid satisfactory proof has been presented to support this claim. Building on the earlier work of Leifer (2014), Landsman (2015) and Leegwater (2016), I present and prove two results that only partially support this claim. I then discuss the arguments by Colbeck, Renner and Leegwater concerning how these results are to generalize to the full claim.…”

“…Thus there is little hope for finding an easy proof for Claim 1 based on Claim 2. My focus is therefore on Claim 1 alone, which Landsman (2015) has criticized but which was endorsed by Leegwater (2016) after he clarified some of the technicalities behind the proof. I shall argue that a rigorous proof for Claim 1 is currently lacking.…”

“…Since then, Leifer (2014), relying on (Colbeck and Renner, 2017), presented a rigorous proof for the second claim by Colbeck and Renner that does not rely on the validity of the completeness claim. Landsman (2015) gave a critical assessment of the completeness claim, arguing that on top of these explicit assumptions, the proof for ψcompleteness relies on no less than four further rather technical assumptions. A far more friendly conclusion was reached by Leegwater (2016) who gave a thorough reworking of Colbeck and Renner's original proof.…”

“…To this end a formal statement of their two claims in terms of the ontic models framework is given in section 3 after motivating the use of this framework in section 2. In section 3 it is also shown that the completeness claim is logically stronger than the ψ-ontology claim, thus showing that the issues of Landsman (2015) are not simply resolved by reworking the proof of Leifer (2014). The general strategy for proving the completeness claim is to start with proving it for the special case of two systems in a maximally entangled state.…”

Non-locality and quasiclassical reality in Kent’s formulation of relativistic quantum theory

“…Colbeck and Renner [18] (made rigorous by Landsman [19]) show that, under some apparently natural extra assumptions: any theory that supplements orthodox quantum theory, in the sense of recovering orthodox Born-rule probabilities by probabilistic averaging over "hidden variables", must violate either:…”