Gleason’s theorem for composite systems

Abstract

Gleason’s theorem (Gleason 1957 J. Math. Mech. 6 885) is an important result in the foundations of quantum mechanics, where it justifies the Born rule as a mathematical consequence of the quantum formalism. Formally, it presents a key insight into the projective geometry of Hilbert spaces, showing that finitely additive measures on the projection lattice extend to positive linear functionals on the algebra of bounded operators . Over many years, and by the effort of various authors, the theorem has been broadened in its scope from type I to arbitrary von Neumann algebras (without type factors). Here, we prove a generalisation of Gleason’s theorem to composite systems. To this end, we strengthen the original result in two ways: first, we extend its scope to dilations in the sense of Naimark (1943 Dokl. Akad. Sci. SSSR 41 359) and Stinespring (1955 Proc. Am. Math. Soc. 6 211) and second, we require consistency with respect to dynamical correspondences on the respective (local) algebras in the composition (Alfsen and Shultz 1998 Commun. Math. Phys. 194 87). We show that neither of these conditions changes the result in the single system case, yet both are necessary to obtain a generalisation to bipartite systems.