aqora

The problem of dispersion-free probabilities in Gleason-type theorems for a two-dimensional Hilbert space

Arkady Bolotin

arXiv:1806.01436Quantum Studies Mathematics and FoundationsDec 4, 20180 citations

Generalized Complex Spherical Harmonics, Frame Functions, and Gleason Theorem

Valter Moretti, Davide Pastorello

arXiv:1205.4504Annales Henri PoincaréNov 16, 20129 citations

Infinite and finite Gleason’s theorems and the logic of indeterminacy

Itamar Pitowsky

In the first half of the paper I prove Gleason’s lemma: Every non-negative frame function on the set of rays in R3 is continuous. This is the central and most difficult part of Gleason’s theorem. The proof is a reconstruction of Gleason’s idea in terms of orthogonality graphs. The result is a demonstration that this theorem is actually combinatorial in nature. It depends only on a finite graph structure. In the second half of the paper I use the graph construction to obtain results about probability distributions (non-negative frame functions with weight one) on finite sets of rays. For example, given any two distinct nonorthogonal rays a and b, I construct a finite set of rays Γ that contains them, and has the following property: No probability distribution on Γ assigns both a and b a truth value (probability zero or one) unless they are both false. Thus the principle of indeterminacy turns into a theorem of propositional quantum logic (or partial Boolean algebras).In the first half of the paper I prove Gleason’s lemma: Every non-negative frame function on the set of rays in R3 is continuous. This is the central and most difficult part of Gleason’s theorem. The proof is a reconstruction of Gleason’s idea in terms of orthogonality graphs. The result is a demonstration that this theorem is actually combinatorial in nature. It depends only on a finite graph structure. In the second half of the paper I use the graph construction to obtain results about probability distributions (non-negative frame functions with weight one) on finite sets of rays. For example, given any two distinct nonorthogonal rays a and b, I construct a finite set of rays Γ that contains them, and has the following property: No probability distribution on Γ assigns both a and b a truth value (probability zero or one) unless they are both false. Thus the principle of indeterminacy turns into a theorem of propositional quantum logic (or partial Boolean algebras).

Journal of Mathematical PhysicsJan 1, 199864 citations

Gleason measures and quantum comparative probability

W. Ochs

Lecture notes in mathematicsJan 1, 19859 citations

Gleason-Type Theorems from Cauchy’s Functional Equation

Victoria J Wright, Stefan Weigert

Gleason-type theorems derive the density operator and the Born rule formalism of quantum theory from the measurement postulate, by considering additive functions which assign probabilities to measurement outcomes. Additivity is also the defining property of solutions to Cauchy’s functional equation. This observation suggests an alternative proof of the strongest known Gleason-type theorem, based on techniques used to solve functional equations.

arXiv:1905.12751Foundations of PhysicsJun 1, 20198 citations

Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements

Carlton M. Caves, Christopher A. Fuchs, Manne, K +1 more

We prove a Gleason-type theorem for the quantum probability rule using frame functions defined on positive-operator-valued measures (POVMs), as opposed to the restricted class of orthogonal projection-valued measures used in the original theorem. The advantage of this method is that it works for two-dimensional quantum systems (qubits) and even for vector spaces over rational fields--settings where the standard theorem fails. Furthermore, unlike the method necessary for proving the original result, the present one is rather elementary. In the case of a qubit, we investigate similar results for frame functions defined upon various restricted classes of POVMs. For the so-called trine measurements, the standard quantum probability rule is again recovered.We prove a Gleason-type theorem for the quantum probability rule using frame functions defined on positive-operator-valued measures (POVMs), as opposed to the restricted class of orthogonal projection-valued measures used in the original theorem. The advantage of this method is that it works for two-dimensional quantum systems (qubits) and even for vector spaces over rational fields--settings where the standard theorem fails. Furthermore, unlike the method necessary for proving the original result, the present one is rather elementary. In the case of a qubit, we investigate similar results for frame functions defined upon various restricted classes of POVMs. For the so-called trine measurements, the standard quantum probability rule is again recovered.

Jun 26, 200377 citations

Probability Distributions and Gleason’s Theorem

Abstract

Related papers

Probability Distributions and Gleason’s Theorem

Abstract

Related papers