Abstract
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
I
2
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.
Funder
Foundational Questions Institute
Silicon Valley Community Foundation
EPSRC
Subject
General Physics and Astronomy,Mathematical Physics,Modeling and Simulation,Statistics and Probability,Statistical and Nonlinear Physics
Reference53 articles.
1. Quantum theory from five reasonable axioms;Hardy,2001
2. Informational derivation of quantum theory;Chiribella;Phys. Rev. A,2011
3. Information and the reconstruction of quantum physics;Jaeger;Ann. Phys.,2019
4. Measures on the closed subspaces of a Hilbert space;Gleason;J. Math. Mech.,1957
5. Measures on projections and physical states;Christensen;Commun. Math. Phys.,1982
Cited by
1 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献
1. Variations on the Choi–Jamiołkowski isomorphism;Journal of Physics A: Mathematical and Theoretical;2024-06-17