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Mathematical Proceedings of the Cambridge Philosophical Society

An elementary proof of Gleason's theorem

Roger M. Cooke, Michaël Keane, William Moran

Published Jul 1, 1985• 115 citations
Publisher

Abstract

Abstract Gleason's theorem characterizes the totally additive measures on the closed sub-spaces of a separable real or complex Hilbert space of dimension greater than two. This paper presents an elementary proof of Gleason's theorem which is accessible to undergraduates having completed a first course in real analysis.
DOI
10.1017/s0305004100063313

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Letters to Joseph Renes

Christopher A. Fuchs

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.

Cambridge University Press eBooks•Jan 6, 2011•0 citations

Gleason’s theorem for composite systems

Markus Frembs, Andreas Döring

Abstract Gleason’s theorem (Gleason 1957 J. Math. Mech. 6 885) is an important result in the foundations of quantum mechanics, where it justifies the Born rule as a mathematical consequence of the quantum formalism. Formally, it presents a key insight into the projective geometry of Hilbert spaces, showing that finitely additive measures on the projection lattice extend to positive linear functionals on the algebra of bounded operators . Over many years, and by the effort of various authors, the theorem has been broadened in its scope from type I to arbitrary von Neumann algebras (without type I 2 factors). Here, we prove a generalisation of Gleason’s theorem to composite systems. To this end, we strengthen the original result in two ways: first, we extend its scope to dilations in the sense of Naimark (1943 Dokl. Akad. Sci. SSSR 41 359) and Stinespring (1955 Proc. Am. Math. Soc. 6 211) and second, we require consistency with respect to dynamical correspondences on the respective (local) algebras in the composition (Alfsen and Shultz 1998 Commun. Math. Phys. 194 87). We show that neither of these conditions changes the result in the single system case, yet both are necessary to obtain a generalisation to bipartite systems.

arXiv:2205.00493•Journal of Physics A Mathematical and Theoretical•Sep 20, 2023•3 citations

Quantum theory and measures on Hilbert space

Henry Krips

It is shown that merely the decision to represent the fundamental events of quantum theory by vectors in a Hilvert space uniquely determines the form of any probability measure over the set of events. The proof proceeds by showing that any nonnegative meausre on the vectors of a Hilbert space, for which the sum of the measures on the vectors of any complete orthonormal set is 1, takes the usual form damanded by quantum theory. This result represents a considerable strengthening of one of the consequences of Gleason’s theorem.

Journal of Mathematical Physics•May 1, 1977•1 citation

A mechanistic macroscopic physical entity with a three-dimensional Hilbert space description

Diederik Aerts, Bob Coecke, Bart D’Hooghe +1 more

It is sometimes stated that Gleason's theorem prevents the construction of hidden-variable models for quantum entities described in a more than two-dimensional Hilbert space. In this paper however we explicitly construct a classical (macroscopical) system that can be represented in a three-dimensional real Hilbert space, the probability structure appearing as the result of a lack of knowledge about the measurement context. We briefly discuss Gleason's theorem from this point of view.

arXiv:quant-ph/0111074•Helvetica Physica Acta (Switzerland)•Sep 1, 1997•10 citations

Mathematical Questions Related to Non-Existence of Hidden Variables

Mirko Navara, Luigi Accardi, Guillaume Adenier +5 more

The famous Gleason’s Theorem gives a characterization of states on lattices of subspaces of Hilbert spaces. The attempts to simplify its proof have led to easy proofs of some consequences, mainly the non‐existence of hidden variables (dispersion‐free states). Here we simplify some of them. We also formulate related open problems concerning spaces with rational coordinates and group‐valued measures.

AIP conference proceedings•Jan 1, 2009•1 citation

Why qubits are exceptional in Gleason's theorem: a Bloch-space perspective

Massimiliano Sassoli de Bianchi

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.

quant-pharXiv:2603.07745•Mar 10, 2026•0 citations
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 (Gleason 1957 J. Math. Mech. 6 885) is an important result in the foundations of quantum mechanics, where it justifies the Born rule as a mathematical consequence of the quantum formalism. Formally, it presents a key insight into the projective geometry of Hilbert spaces, showing that finitely additive measures on the projection lattice extend to positive linear functionals on the algebra of bounded operators . Over many years, and by the effort of various authors, the theorem has been broadened in its scope from type I to arbitrary von Neumann algebras (without type I 2 factors). Here, we prove a generalisation of Gleason’s theorem to composite systems. To this end, we strengthen the original result in two ways: first, we extend its scope to dilations in the sense of Naimark (1943 Dokl. Akad. Sci. SSSR 41 359) and Stinespring (1955 Proc. Am. Math. Soc. 6 211) and second, we require consistency with respect to dynamical correspondences on the respective (local) algebras in the composition (Alfsen and Shultz 1998 Commun. Math. Phys. 194 87). We show that neither of these conditions changes the result in the single system case, yet both are necessary to obtain a generalisation to bipartite systems.
It is shown that merely the decision to represent the fundamental events of quantum theory by vectors in a Hilvert space uniquely determines the form of any probability measure over the set of events. The proof proceeds by showing that any nonnegative meausre on the vectors of a Hilbert space, for which the sum of the measures on the vectors of any complete orthonormal set is 1, takes the usual form damanded by quantum theory. This result represents a considerable strengthening of one of the consequences of Gleason’s theorem.
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.