Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1808.08091 · Journal of Physics A Mathematical and Theoretical

A Gleason-type theorem for qubits based on mixtures of projective measurements

Victoria J Wright, Stefan Weigert

Published Dec 18, 2018• 18 citations
arXivHTMLPDF

Abstract

Abstract We derive Born’s rule and the density-operator formalism for quantum systems with Hilbert spaces of dimension two or larger. Our extension of Gleason’s theorem only relies upon the consistent assignment of probabilities to the outcomes of projective measurements and their classical mixtures. This assumption is significantly weaker than those required for existing Gleason-type theorems valid in dimension two.

DOI
10.1088/1751-8121/aaf93d

Related papers

A Gleason-type theorem for qubits based on mixtures of projective measurements

Stefan Weigert, Victoria J Wright

We derive Born's rule and the density-operator formalism for quantum systems with Hilbert spaces of dimension two or larger. Our extension of Gleason's theorem only relies upon the consistent assignment of probabilities to the outcomes of projective measurements and their classical mixtures. This assumption is significantly weaker than those required for existing Gleason-type theorems valid in dimension two.

•Jan 10, 2019•28 citations

Gleason's Theorem for a Qubit as Part of a Composite System

Vincenzo Fiorentino, Stefan Weigert

We extend Gleason's theorem to the two-dimensional Hilbert space of a qubit by invoking the standard axiom that describes composite quantum systems. The tensor-product structure allows us to derive density matrices and Born's rule for $d=2$ from a simple requirement: the probabilities assigned to measurement outcomes must not depend on whether a system is considered on its own or as a subsystem of a larger one. In line with Gleason's original theorem, our approach assigns probabilities only to projection-valued measures, while other known extensions rely on considering more general classes of measurements. This extension of Gleason's theorem to two-dimensional systems is shown to remain valid for some foil theories of quantum theory.

quant-pharXiv:2511.15607•Physical Review A•Nov 20, 2025•0 citations

Comment on "Gleason-Type Theorem for Projective Measurements, Including Qubits" by F. De Zela

Michael J. W. Hall

It has recently been claimed by De Zela that Gleason's theorem, for probability measures on the lattice of projection operators, can be extended to qubits by adding assumptions related to continuity and the existence of 'eigenstates'. This amounts to a claim of the derivation of Born's rule for Hermitian qubit observables. I point out a simple counterexample, and the flaw in De Zela's derivation (these are equally applicable to the repetition of the derivation given in a recent Reply). I also briefly discuss a valid extension to qubits given by Busch.

arXiv:1611.00613•Nov 2, 2016•3 citations

Born Statistics from Decoherence and Contraction Dynamics on Projective Hilbert Space

J. Rosa

We derive the Born rule as a consequence of two physical processes: environment-induced decoherence, which selects the pointer-basis decomposition of the density matrix, and a contraction map on projective Hilbert space, which assigns each pointer state to a definite outcome basin. The contraction map partitions projective space into basins of attraction whose boundaries are determined by the measurement projectors. We prove that the decoherence-selected decomposition of the reduced density matrix, combined with the basin assignment, reproduces the Born probabilities p_i = ⟨ψ|Π_i|ψ⟩ exactly for projective measurements in any finite dimension, including dimension 2 where Gleason's theorem does not apply. For generalized measurements (POVMs), the result extends via Naimark dilation. The derivation resolves the "probability problem" of decoherence-based approaches: standard decoherence produces the correct diagonal density matrix but leaves open why diagonal entries should be interpreted as outcome probabilities. Our answer is geometric — each diagonal component corresponds to a pointer state lying deep inside a definite basin of the contraction map, and the mixture weights become outcome probabilities through the basin assignment. The derivation is geometric rather than decision-theoretic, does not assume the probability interpretation of the density matrix, requires no ensemble averaging, and applies to single systems. Perturbations to the contraction map produce bounded corrections to the Born probabilities, providing a falsifiable prediction: any measured deviation from Born statistics constrains the perturbation strength. This work connects quantum measurement theory to the geometry of projective Hilbert space and contraction dynamics, offering a new perspective on the origin of quantum probabilities that complements existing approaches by Gleason, Zurek, and Deutsch-Wallace.

•Mar 15, 2026•0 citations

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

Born Statistics from Decoherence and Contraction Dynamics on Projective Hilbert Space

Elythian

We derive the Born rule as a consequence of two physical processes: environment-induced decoherence, which selects the pointer-basis decomposition of the density matrix, and a contraction map on projective Hilbert space, which assigns each pointer state to a definite outcome basin. The contraction map partitions projective space into basins of attraction whose boundaries are determined by the measurement projectors. We prove that the decoherence-selected decomposition of the reduced density matrix, combined with the basin assignment, reproduces the Born probabilities p_i = ⟨ψ|Π_i|ψ⟩ exactly for projective measurements in any finite dimension, including dimension 2 where Gleason's theorem does not apply. For generalized measurements (POVMs), the result extends via Naimark dilation. The derivation resolves the "probability problem" of decoherence-based approaches: standard decoherence produces the correct diagonal density matrix but leaves open why diagonal entries should be interpreted as outcome probabilities. Our answer is geometric — each diagonal component corresponds to a pointer state lying deep inside a definite basin of the contraction map, and the mixture weights become outcome probabilities through the basin assignment. The derivation is geometric rather than decision-theoretic, does not assume the probability interpretation of the density matrix, requires no ensemble averaging, and applies to single systems. Perturbations to the contraction map produce bounded corrections to the Born probabilities, providing a falsifiable prediction: any measured deviation from Born statistics constrains the perturbation strength. This work connects quantum measurement theory to the geometry of projective Hilbert space and contraction dynamics, offering a new perspective on the origin of quantum probabilities that complements existing approaches by Gleason, Zurek, and Deutsch-Wallace.

•Jan 1, 2026•0 citations
Abstract We derive Born’s rule and the density-operator formalism for quantum systems with Hilbert spaces of dimension two or larger. Our extension of Gleason’s theorem only relies upon the consistent assignment of probabilities to the outcomes of projective measurements and their classical mixtures. This assumption is significantly weaker than those required for existing Gleason-type theorems valid in dimension two.
We extend Gleason's theorem to the two-dimensional Hilbert space of a qubit by invoking the standard axiom that describes composite quantum systems. The tensor-product structure allows us to derive density matrices and Born's rule for d=2d=2d=2 from a simple requirement: the probabilities assigned to measurement outcomes must not depend on whether a system is considered on its own or as a subsystem of a larger one. In line with Gleason's original theorem, our approach assigns probabilities only to projection-valued measures, while other known extensions rely on considering more general classes of measurements. This extension of Gleason's theorem to two-dimensional systems is shown to remain valid for some foil theories of quantum theory.
It has recently been claimed by De Zela that Gleason's theorem, for probability measures on the lattice of projection operators, can be extended to qubits by adding assumptions related to continuity and the existence of 'eigenstates'. This amounts to a claim of the derivation of Born's rule for Hermitian qubit observables. I point out a simple counterexample, and the flaw in De Zela's derivation (these are equally applicable to the repetition of the derivation given in a recent Reply). I also briefly discuss a valid extension to qubits given by Busch.
We derive the Born rule as a consequence of two physical processes: environment-induced decoherence, which selects the pointer-basis decomposition of the density matrix, and a contraction map on projective Hilbert space, which assigns each pointer state to a definite outcome basin. The contraction map partitions projective space into basins of attraction whose boundaries are determined by the measurement projectors. We prove that the decoherence-selected decomposition of the reduced density matrix, combined with the basin assignment, reproduces the Born probabilities p_i = ⟨ψ|Π_i|ψ⟩ exactly for projective measurements in any finite dimension, including dimension 2 where Gleason's theorem does not apply. For generalized measurements (POVMs), the result extends via Naimark dilation. The derivation resolves the "probability problem" of decoherence-based approaches: standard decoherence produces the correct diagonal density matrix but leaves open why diagonal entries should be interpreted as outcome probabilities. Our answer is geometric — each diagonal component corresponds to a pointer state lying deep inside a definite basin of the contraction map, and the mixture weights become outcome probabilities through the basin assignment. The derivation is geometric rather than decision-theoretic, does not assume the probability interpretation of the density matrix, requires no ensemble averaging, and applies to single systems. Perturbations to the contraction map produce bounded corrections to the Born probabilities, providing a falsifiable prediction: any measured deviation from Born statistics constrains the perturbation strength. This work connects quantum measurement theory to the geometry of projective Hilbert space and contraction dynamics, offering a new perspective on the origin of quantum probabilities that complements existing approaches by Gleason, Zurek, and Deutsch-Wallace.
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.
We derive the Born rule as a consequence of two physical processes: environment-induced decoherence, which selects the pointer-basis decomposition of the density matrix, and a contraction map on projective Hilbert space, which assigns each pointer state to a definite outcome basin. The contraction map partitions projective space into basins of attraction whose boundaries are determined by the measurement projectors. We prove that the decoherence-selected decomposition of the reduced density matrix, combined with the basin assignment, reproduces the Born probabilities p_i = ⟨ψ|Π_i|ψ⟩ exactly for projective measurements in any finite dimension, including dimension 2 where Gleason's theorem does not apply. For generalized measurements (POVMs), the result extends via Naimark dilation. The derivation resolves the "probability problem" of decoherence-based approaches: standard decoherence produces the correct diagonal density matrix but leaves open why diagonal entries should be interpreted as outcome probabilities. Our answer is geometric — each diagonal component corresponds to a pointer state lying deep inside a definite basin of the contraction map, and the mixture weights become outcome probabilities through the basin assignment. The derivation is geometric rather than decision-theoretic, does not assume the probability interpretation of the density matrix, requires no ensemble averaging, and applies to single systems. Perturbations to the contraction map produce bounded corrections to the Born probabilities, providing a falsifiable prediction: any measured deviation from Born statistics constrains the perturbation strength. This work connects quantum measurement theory to the geometry of projective Hilbert space and contraction dynamics, offering a new perspective on the origin of quantum probabilities that complements existing approaches by Gleason, Zurek, and Deutsch-Wallace.