Gleason's Theorem for a Qubit as Part of a Composite System
Vincenzo Fiorentino, Stefan Weigert
TL;DR
This work addresses extending Gleason's theorem to a qubit by enforcing consistency of measurement probabilities under subsystem–composite system relations rather than enlarging the measurement class. By embedding subsystem measurements into a bipartite system with a tensor-product Hilbert space and enforcing composition compatibility, the authors define marginal frame functions and show that, for $d\ge 3$, all frame functions are marginal and correspond to density operators via $f(P)=\mathrm{Tr}(P\rho)$, while for $d=2$ marginal frame functions arise from partial traces of joint states and are in one-to-one correspondence with $2\times2$ density matrices. The central result is that every marginal frame function on a qubit is representable as $f(P)=\mathrm{Tr}(P\rho)$ with $\rho\in\mathcal{S}(\mathcal{H}_2)$, thereby recovering density matrices and Born's rule for the qubit from standard quantum-postulate of composite systems. The approach avoids appealing to POVMs and highlights the crucial role of system composition in the reconstruction of quantum theory, reinforcing the idea that quantum structure is tightly linked to multi-partite tensor-product composition.
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.