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  • Open Access

Gleason's theorem for a qubit as part of a composite system

Vincenzo Fiorentino* and Stefan Weigert†

  • *Contact author: vincenzo.fiorentino@york.ac.uk
  • †Contact author: stefan.weigert@york.ac.uk
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Phys. Rev. A 113, 062221 – Published 23 June, 2026

DOI: https://doi.org/10.1103/8dyw-6ktq

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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Outline

Information

Phys. Rev. A 113, 062221– Published 23 June, 2026
  • Received 16 December 2025
  • Accepted 28 May 2026
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