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Hilbert-space-valued states on quantum logics. (English) Zbl 0767.03034

The authors study finitely additive orthogonal vector states (=normed measures) with values in a real Hilbert space (so called \(h\)-states). It is shown that there is no \(h\)-state on a Hilbert logic \(L(M)\) of projections in \(H\), \(\dim H \geq 3\), to a Hilbert space of a lesser dimension. Further, a Hilbert space \(H\), \(\dim H \geq 3\), has a finite dimension iff every \(h\)-state on \(L(H)\) to \(H\) is, up to a unitary mapping, concentrated on a one-dimensional projection.
Finally, a characterization of logics with extension property with respect to \(h\)-states to a finite dimensional Hilbert space is given.
Reviewer: J.Tkadlec (Praha)

MSC:

03G12 Quantum logic
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
46G10 Vector-valued measures and integration
46N50 Applications of functional analysis in quantum physics
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References:

[1] V. Alda: On 0-1 measure for projectors II. Aplikace matematiky 26 (1981), 57-58. · Zbl 0459.28020
[2] R. V. Kadison J. R. Ringrose: Fundamentals of the theory of operators algebras. Vol. I, Academic Press, Inc., 1986. · Zbl 0601.46054
[3] A. Dvurečenskij S. Pulmannová: Random measures on a logic. Demonstratio Math XIV no. 2 (1981).
[4] A. Einstein B. Podolski N. Rosen: Can quantum-mechanical description of reality be considered complete?. Phys. Rev. 47 (1935), 777-780. · Zbl 0012.04201 · doi:10.1103/PhysRev.47.777
[5] A. M. Gleason: Measures on the closed subspaces of Hilbert space. J. Math. Mech. 65 (1957), 885-893. · Zbl 0078.28803
[6] R. J. Greechie: Orthomodular lattices admitting no states. Journ. Comb. Theory 10 (1971), 119-132. · Zbl 0219.06007 · doi:10.1016/0097-3165(71)90015-X
[7] S. Gudder: Stochastic Methods in Quantum Mechanics. North Holland, New York, 1979. · Zbl 0439.46047
[8] S. Gudder: Dispersion-free states and the existence of hidden variables. Proc. Amer. Math. Soc 19 (1968), 319-324. · Zbl 0162.01202 · doi:10.2307/2035519
[9] A. Horn A. Tarski: Measures in Boolean algebras. Trans. Amer. Math. Soc. 64 (1948), 467-497. · Zbl 0035.03001 · doi:10.2307/1990396
[10] R. Jajte A. Paszkiewicz: Vector measures on the closed subspaces of a Hilbert space. Studia Math. T. LXIII (1978). · Zbl 0407.46058
[11] G. Kalmbach: Measures on Hilbert Lattices. World Sci. Publ., Singapore, 1986. · Zbl 0656.06012
[12] P. Kruszyňski: Vector measures on orthocomplemented lattices. Proceedings of the Koniklijke Akademie van Wetenschappen, Ser. A, 91 no. 4, December 19 (1988). · Zbl 0819.46036
[13] R. Mayet: Classes equationelles de trellis orthomodulaires et espaces de Hilbert. These pour obtenir Docteur d’Etat es Sciences, Université Claude Bernard-Lyon (France), 1987.
[14] P. Pták: Exotic logics. Colloqium Mathematicum LIV, Fasc. 1 (1987), 1-7. · Zbl 0639.03063
[15] P. Pták: Weak dispersion-free states and the hidden variables hypothesis. J. Math. Physics 24 (1983), 839-841. · Zbl 0508.60006 · doi:10.1063/1.525758
[16] P. Pták J. D. Wright: On the concreteness of quantum logics. Aplikace matematiky 30 č. 4 (1986), 274-285. · Zbl 0586.03050
[17] F. Schultz: A characterization of state space of orthomodular lattices. Jour. Comb. Theory 17 (1974), 317-328. · Zbl 0317.06007 · doi:10.1016/0097-3165(74)90096-X
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