Putting probabilities first. How Hilbert space generates and constrains them

Michael Janas, Michael E. Cuffaro, Michel Janssen

Research output: Contribution to journalArticlepeer-review

Abstract

We use correlation arrays, the workhorse of Bub’s (2016) Bananaworld, to analyze the correlations found in an experimental setup due to Mermin (1981) for measurements on the singlet state of a pair of spin- 12 particles. Adopting an approach pioneered by Pitowsky (1989b) and promoted in Bananaworld, we geometrically represent the class of correlations allowed by quantum mechanics in this setup as an elliptope in a non-signaling cube. To determine which of these quantum correlations are allowed by local hidden-variable theories, we investigate which ones we can simulate using raffles with baskets of tickets that have the outcomes for all combinations of measurement settings printed on them. The class of correlations found this way can be represented geometrically by a tetrahedron contained within the elliptope. We use the same Bub-Pitowsky framework to analyze a generalization of the Mermin setup for measurements on the singlet state of two particles with higher spin. The class of correlations allowed by quantum mechanics in this case is still represented by the elliptope; the subclass of those whose main features can be simulated with our raffles can be represented by polyhedra that, with increasing spin, have more and more vertices and facets and get closer and closer to the elliptope. We use these results to advocate for Bubism (not to be confused with QBism), an interpretation of quantum mechanics along the lines of Bananaworld. Probabilities and expectation values are primary in this interpretation. They are determined by inner products of vectors in Hilbert space. Such vectors do not themselves represent what is real in the quantum world. They encode families of probability distributions over values of different sets of observables. As in classical theory, these values ultimately represent what is real in the quantum world. Hilbert space puts constraints on possible combinations of such values, just as Minkowski space-time puts constraints on possible spatio-temporal constellations of events. Illustrating how generic such constraints are, the constraint derived in this paper, the equation for the elliptope, is a general constraint on correlation coefficients that can be found in older literature on statistics and probability theory. Yule (1897) already stated the constraint. De Finetti (1937) already gave it a geometrical interpretation sharing important features with its interpretation in Hilbert space.

Original languageEnglish (US)
JournalUnknown Journal
StatePublished - Oct 23 2019
Externally publishedYes

Keywords

  • Bell inequalities
  • Born rule
  • Bubism
  • Correlation arrays
  • Correlation polytopes
  • Kinematical constraints
  • Measurement
  • Principle theories
  • Tsirelson bound

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