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).
The basic ingredients of the ‘‘consistent histories’’ approach to a generalized quantum theory are ‘‘histories’’ and decoherence functionals. The main aim of this program is to find and to study the behavior of consistent sets associated with a particular decoherence functional d. In its recent formulation by Isham [‘‘Quantum logic and the histories approach to quantum theory,’’ J. Math. Phys. 35, 2157–2185 (1994)] it is natural to identify the space 𝒰𝒫 of propositions about histories with an orthoalgebra or lattice. When 𝒰𝒫 is given by the lattice of projectors 𝒫(𝒱) in some Hilbert space 𝒱, consistent sets correspond to certain partitions of the unit operator in 𝒱 into mutually orthogonal projectors {α1,α2,...}, such that the function d(α,α) is a probability distribution on the boolean algebra generated by {α1,α2,...}. Using the classification theorem for decoherence functionals proven in Isham et al. [‘‘The classification of decoherence functionals: An analog of Gleason’s Theorem,’’ J. Math. Phys. 35, 6360–6370 (1994)] we show that in the case where 𝒱 is some separable Hilbert space there exists for each partition of the unit operator into a set of mutually orthogonal projectors, and for any probability distribution p(α) on the corresponding boolean algebra, decoherence functionals d with respect to which this set is consistent and which are such that for the probability functions d(α,α)=p(α) holds.
Strong O-valued contextuality: ruling out discrete nondeterministic alternatives to quantum theory
Gleason's theorem identifies the Born rule via non-contextuality over an infinite continuous lattice of projectors, while its corollary the Kochen-Specker (KS) theorem rules out the specific class of deterministic ({0,1}-valued) noncontextual models using finite sets of projectors. Gleason's theorem also indicates the existence of KS-type finite vector constructions to rule out other discrete nondeterministic alternatives to quantum theory beyond the {0,1} case. Here, we construct finite measurement configurations to rule out noncontextual empirical models with outcome probabilities drawn from an arbitrary finite subset O⊂[0,1]. We do this by two means: (i) constructing a family of experimentally feasible state-dependent Hardy-type tests, and (ii) proving a generalized KS theorem for a broad class of O that includes prior results as special cases. In the sheaf-theoretic framework of Abramsky and Brandenburger, a hierarchy of probabilistic-possibilistic-strong contextuality has been established quantifying contextuality as a resource. We extend this framework by introducing strong O-valued contextuality, showing that quantum theory evades global sections of all finite-valued presheaves. We also discuss the implications of the result on finite many-valued logics as viable ontological models for quantum theory and for contextuality-based (semi)-device-independent protocols.
June 2000, “References for Gleason” Some relevant references. The important ones are Gleason's paper, Cooke et al .'s paper, and Pitowsky's paper. R. Cooke, M. Keane, and W. Moran, “An Elementary Proof of Gleason's Theorem,” Math. Proc. Camb. Phil. Soc. 98 , 117–128 (1981). A. M. Gleason, “Measures on the Closed Subspaces of a Hilbert Space,” J. Math. Mech. 6 , 885–894 (1957). I. Pitowsky, “George Boole's ‘Conditions of Possible Experience’ and the Quantum Puzzle,” Brit. J. Phil. Sci. 45 , 95–125 (1994). I. Pitowsky, “Infinite and Finite Gleason's Theorems and the Logic of Indeterminacy,” J. Math. Phys. 39 , 218–228 (1998). F. Richman and D. Bridges, “A Constructive Proof of Gleason's Theorem,” J. Func. Anal. 162 , 287–312 (1999). December 2000, “New Ideas and Old Men” Thanks for the new ideas. Let me try to make a few comments. I'm in a smoky hotel bar right now; I wonder how this is affecting my laptop's health? I know it's wreaking havoc on mine. I'm still in Vienna, by the way. I leave Tuesday morning. Unfortunately I haven't gotten an ounce written of what I had hoped to. Renesism 1: This got me thinking about this stuff, and it seems that in some sense what we're gathering information about, when we make measurements, is the ability to predict future measurements . You haven't read my paper with Kurt Jacobs yet, have you? [See quant-ph/0009101.] Come on, at least read the Introduction and Footnote 27.
Abstract Gleason’s theorem identifies the Born rule via non-contextuality over an infinite continuous lattice of projectors, while its corollary the Kochen-Specker (KS) theorem rules out the specific class of deterministic ({0, 1}-valued) noncontextual models using finite sets of projectors. Gleason’s theorem also indicates the existence of KS-type finite vector constructions to rule out other discrete nondeterministic alternatives to quantum theory beyond the {0, 1} case. Here, we construct finite measurement configurations to rule out noncontextual empirical models with outcome probabilities drawn from an arbitrary finite subset O ⊂ [0, 1]. We do this by two means: (i) constructing a family of experimentally feasible state-dependent Hardy-type tests, and (ii) proving a generalized KS theorem for a broad class of O that includes prior results as special cases. In the sheaf-theoretic framework of Abramsky and Brandenburger, a hierarchy of probabilistic-possibilistic-strong contex-tuality has been established quantifying contextuality as a resource. We extend this framework by introducing strong O-valued contextuality, showing that quantum theory evades global sections of all finite-valued presheaves. We also discuss the implications of the result on finite many-valued logics as viable ontological models for quantum theory and for contextuality-based (semi)-device-independent protocols.
Of what are experimental quantum propositions primary bearers? As it is widely accepted in the modern literature, rather than being bearers of truth and falsity, these entities are bearers of probability values. Consequently, their truth values can be regarded as no more than degenerate probabilities (i.e., ones that have only the values 0 and 1). The mathematical motivation for precedence of probabilistic semantics over propositional semantic for the logic of experimental quantum propositions is Gleason's theorem. It proves that the theory of probability measures on closed linear subspaces of a Hilbert space (which represent experimental quantum propositions) does not admit any probability measure having only the values 0 and 1. -- By contrast, in the present paper, it is proclaimed that experimental propositions about quantum systems are primary bearers of truth values. As this paper demonstrates, algebraic properties of separable Hilbert spaces of finite dimension equal or greater than 2 do not allow in valuations (that is, truth assignments) which are dispersion-free, i.e., total functions from the set of atomic propositions to the set of two objects, true and false. Providing a probability function can be interpreted as a measure of the (un)certainty in the assignment of truth values, the fact that valuations cannot be dispersion-free gives rise to probabilistic semantics for the logic of experimental quantum propositions.
Gleason's theorem is often cited as establishing the Born rule from the structure of Hilbert space, yet its original proof is mathematically sophisticated and rarely accessible to physicists. In this article we present a simple route to the dimensional distinction at the heart of Gleason's result, using the generalized Bloch representation of quantum states. More precisely, we consider a restricted but instructive class of probability assignments obtained by replacing the linear dependence of the Born expression on a Bloch-space scalar product with a real function f. For qubits, infinitely many continuous non-Born assignments of this form satisfy normalization for every projective measurement. In dimension greater than 3, by contrast, the geometry of the measurement simplex imposes additional relations among the relevant scalar products. Requiring normalization for arbitrary states then leads to a Cauchy-type functional equation, which, together with boundedness, forces f to be linear. Our argument provides a complementary geometric illustration of why the freedom present in dimension two disappears in higher dimensions, reaching this conclusion directly from the geometry of the measurement simplices and normalization, without explicitly invoking the intertwining of projectors across different orthogonal resolutions of the identity.