Gleason-Type Theorem for Projective Measurements, Including Qubits: The Born Rule Beyond Quantum Physics

Abstract

Born’s quantum probability rule is traditionally included among the quantum postulates as being given by the squared amplitude projection of a measured state over a prepared state, or else as a trace formula for density operators. Both Gleason’s theorem and Busch’s theorem derive the quantum probability rule starting from very general assumptions about probability measures. Remarkably, Gleason’s theorem holds only under the physically unsound restriction that the dimension of the underlying Hilbert space $\mathcal {H}$ must be larger than two. Busch’s theorem lifted this restriction, thereby including qubits in its domain of validity. However, while Gleason assumed that observables are given by complete sets of orthogonal projectors, Busch made the mathematically stronger assumption that observables are given by positive operator-valued measures. The theorem we present here applies, similarly to the quantum postulate, without restricting the dimension of $\mathcal {H}$ and for observables given by complete sets of orthogonal projectors. We also show that the Born rule applies beyond the quantum domain, thereby exhibiting the common root shared by some quantum and classical phenomena.