[Paper Review] Putting probabilities first. How Hilbert space generates and constrains them

Michael Janas, Michael E. Cuffaro+1 more|arXiv (Cornell University)|Oct 23, 2019

Quantum Mechanics and Applications138 references4 citations

TL;DR

This paper uses correlation arrays and geometric representations in Hilbert space to show how quantum correlations—particularly those from the singlet state of spin-1/2 particles—are constrained by the elliptope, a mathematical structure derived from the Born rule and inner products. It argues that quantum probabilities are primary, generated by Hilbert space structure, and that Bell-type nonlocality is not a flaw but a feature of quantum mechanics, with local hidden-variable models constrained by raffle simulations that form polytopes inside the elliptope.

ABSTRACT

We use Bub's (2016) correlation arrays and Pitowksy's (1989b) correlation polytopes to analyze an experimental setup due to Mermin (1981) for measurements on the singlet state of a pair of spin-$\frac12$ particles. The class of correlations allowed by quantum mechanics in this setup is represented by an elliptope inscribed in a non-signaling cube. The class of correlations allowed by local hidden-variable theories is represented by a tetrahedron inscribed in this elliptope. We extend this analysis to pairs of particles of arbitrary spin. The class of correlations allowed by quantum mechanics is still represented by the elliptope; the subclass of those allowed by local hidden-variable theories by polyhedra with increasing numbers of vertices and facets that get closer and closer to the elliptope. We use these results to advocate for an interpretation of quantum mechanics like Bub's. 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 equation for the elliptope derived in this paper is a general constraint on correlation coefficients that can be found in older literature on statistics and probability theory. Yule (1896) already stated the constraint. De Finetti (1937) already gave it a geometrical interpretation.

Motivation & Objective

• To clarify the kinematical constraints on quantum correlations using a geometric framework based on correlation arrays.
• To demonstrate that quantum correlations in the singlet state of spin-1/2 particles are confined within the elliptope, a convex set defined by the Born rule and inner products.
• To investigate whether these quantum correlations can be simulated by classical raffles with pre-printed outcomes, revealing a class of local hidden-variable models.
• To generalize the analysis to higher-spin singlet states, showing that the quantum correlation set remains the elliptope while classical simulations converge toward it as spin increases.
• To advocate for Bubism—an information-theoretic interpretation of quantum mechanics—where probabilities and expectation values are primary, derived from Hilbert space structure.

Proposed method

• Represent quantum correlations using correlation arrays derived from the singlet state of spin-1/2 particles, mapping them into a non-signaling cube.
• Geometrically represent the set of quantum correlations as the elliptope, defined by the Tsirelson bound and the Born rule, using inner products in Hilbert space.
• Model local hidden-variable theories via raffles with tickets pre-printed with outcomes for all measurement settings, simulating classical correlations.
• Show that raffle-simulable correlations form a tetrahedron (in the spin-1/2 case) or polytopes with increasing complexity for higher spins, contained within the elliptope.
• Use Wigner d-matrices to generalize the formalism to higher-spin systems, preserving non-signaling and anti-correlation structure.
• Trace the historical roots of the elliptope constraint back to Yule (1897) and de Finetti (1937), showing its foundational role in probability theory.

Experimental results

Research questions

• RQ1 How do Hilbert space inner products generate and constrain the set of possible quantum correlations in the singlet state?
• RQ2 What is the geometric relationship between quantum correlations (elliptope) and classical correlations simulated by raffles (polytopes) in the spin-1/2 case?
• RQ3 How do the constraints on classical raffle simulations of quantum correlations change as the spin of the particles increases?
• RQ4 Why are the Bell-type nonlocal correlations of quantum mechanics not a flaw but a necessary feature of the theory’s probabilistic structure?
• RQ5 To what extent can the elliptope be seen as a universal constraint on correlation coefficients, independent of quantum mechanics?

Key findings

• The set of quantum correlations for the spin-1/2 singlet state is geometrically represented by the elliptope, a convex set defined by the Tsirelson bound and the Born rule.
• Classical raffle simulations of these correlations produce a tetrahedron contained within the elliptope, demonstrating that not all quantum correlations can be explained by local hidden variables.
• For higher-spin singlet states, the quantum correlation set remains the elliptope, while raffle-simulable correlations form polytopes with more vertices and facets that converge toward the elliptope as spin increases.
• The elliptope equation is a general constraint on correlation coefficients, predating quantum mechanics and already identified by Yule (1897) and de Finetti (1937).
• The paper establishes that the 'small' and 'big' measurement problems are not defects but features of quantum mechanics, consistent with the informational interpretation of Bubism.
• Hilbert space does not represent physical states directly but encodes families of probability distributions over observables, with the structure of the space imposing kinematical constraints analogous to spacetime geometry.