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arXiv · 2511.15607

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

Vincenzo Fiorentino·Stefan Weigert
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Abstract

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.

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Vincenzo Fiorentino, Stefan Weigert. 2025-11-19. Gleason's Theorem for a Qubit as Part of a Composite System. https://doi.org/10.1103/8dyw-6ktq

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