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

TL;DR

This paper extends Gleason's theorem to qubits within composite systems, deriving density matrices and Born's rule from the tensor-product structure and measurement independence.

Contribution

It provides a new derivation of Gleason's theorem for qubits using the composite system framework, focusing on measurement independence.

Findings

Derived density matrices for qubits from composite system assumptions

Established Born's rule for qubits based on measurement independence

Reinforced Gleason's theorem applicability to composite quantum systems

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.