Gleason’s theorem for composite systems. (English) Zbl 1534.81004
Summary: Gleason’s theorem [A. M. Gleason, J. Math. Mech. 6, 885–893 (1957; Zbl 0078.28803)] 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 \(\mathcal{P}(\mathcal{H})\) extend to positive linear functionals on the algebra of bounded operators \(\mathcal{B}(\mathcal{H})\). 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 \(\mathrm{I}_2\) factors). Here, we prove a generalisation of Gleason’s theorem [loc. cit.] 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 M. A. Neumark [C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 41, 359–361 (1943; Zbl 0061.25410)] and W. F. Stinespring [Proc. Am. Math. Soc. 6, 211–216 (1955; Zbl 0064.36703)] and second, we require consistency with respect to dynamical correspondences on the respective (local) algebras in the composition [E. M. Alfsen and F. W. Shultz, Commun. Math. Phys. 194, No. 1, 87–108 (1998; Zbl 0918.46060)]. 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.
{© 2023 The Author(s). Published by IOP Publishing Ltd}
{© 2023 The Author(s). Published by IOP Publishing Ltd}
MSC:
| 81P05 | General and philosophical questions in quantum theory |
| 62H20 | Measures of association (correlation, canonical correlation, etc.) |
| 46L10 | General theory of von Neumann algebras |
| 81P15 | Quantum measurement theory, state operations, state preparations |
| 51A05 | General theory of linear incidence geometry and projective geometries |
| 46C05 | Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) |
Keywords:
Gleason’s theorem; quantum foundations; quantum correlations; von Neumann algebras; Stinespring dilationReferences:
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