| Title:
|
Hilbert-space-valued states on quantum logics (English) |
| Author:
|
Hamhalter, Jan |
| Author:
|
Pták, Pavel |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
37 |
| Issue:
|
1 |
| Year:
|
1992 |
| Pages:
|
51-61 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We analyze finitely additive orthogonal states whose values lie in a real Hilbert space. We call them $h$-states. We first consider the important case of $h$-states on a standard Hilbert logic $L(H)$ of projectors in $H$-we describe the $h$-states $s$: $L(H_1) \rightarrow H_2$, where $\text {dim } H_2 \leq$ \text {dim} H_1 < \infty$. In particular, we show that, up to a unitary mapping, every $h$-state $s$: $L(H)\rightarrow H(3\leq \text {dim } H < \infty)$ has to be concentrated on a one-dimensional projection. We also study the $h$-states $s$: $L(H_1)\rightarrow H_2$ for the case of $\text {dim } H_1 = \infty$. The results of the first part complement the papers [10] and [13]. In the second part we investigate $h$-states on general logics. Being motivated by the quantum axiomatics, the main question we ask here is as follow: Given a Hilbert space $H$ with $\text {dim } H < \infty$, what is the class of such logics $L$ that, for any Boolean subalgebra $B$ of $L$, every $h$-states $s$: $B \rightarrow H$ extends over $L$? We answer this question by finding a simple condition characterizing the class (Theorem 3.4]. It turns out that the class is considerably large-it contains e.g. all concrete logics-but, on the other hand, it does not contain all finite logics (we construct a counterexample in the appendix). (English) |
| Keyword:
|
Hilbert-space-valued state |
| Keyword:
|
$h$-state |
| Keyword:
|
finitely additive orthogonal vector states |
| Keyword:
|
normed measures |
| Keyword:
|
Hilbert logic |
| Keyword:
|
extension property |
| MSC:
|
03G12 |
| MSC:
|
06C15 |
| MSC:
|
28B05 |
| MSC:
|
46G10 |
| MSC:
|
46N50 |
| MSC:
|
81B10 |
| MSC:
|
81P10 |
| idZBL:
|
Zbl 0767.03034 |
| idMR:
|
MR1152157 |
| DOI:
|
10.21136/AM.1992.104491 |
| . |
| Date available:
|
2008-05-20T18:43:03Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/104491 |
| . |
| Reference:
|
[1] V. Alda: On 0-1 measure for projectors II.Aplikace matematiky 26 (1981), 57-58. Zbl 0459.28020, MR 0602402 |
| Reference:
|
[2] R. V. Kadison J. R. Ringrose: Fundamentals of the theory of operators algebras.Vol. I, Academic Press, Inc., 1986. MR 0859186 |
| Reference:
|
[3] A. Dvurečenskij S. Pulmannová: Random measures on a logic.Demonstratio Math XIV no. 2 (1981). MR 0632289 |
| Reference:
|
[4] A. Einstein B. Podolski N. Rosen: Can quantum-mechanical description of reality be considered complete?.Phys. Rev. 47 (1935), 777-780. 10.1103/PhysRev.47.777 |
| Reference:
|
[5] A. M. Gleason: Measures on the closed subspaces of Hilbert space.J. Math. Mech. 65 (1957), 885-893. MR 0096113 |
| Reference:
|
[6] R. J. Greechie: Orthomodular lattices admitting no states.Journ. Comb. Theory 10 (1971), 119-132. Zbl 0219.06007, MR 0274355, 10.1016/0097-3165(71)90015-X |
| Reference:
|
[7] S. Gudder: Stochastic Methods in Quantum Mechanics.North Holland, New York, 1979. Zbl 0439.46047, MR 0543489 |
| Reference:
|
[8] S. Gudder: Dispersion-free states and the existence of hidden variables.Proc. Amer. Math. Soc 19 (1968), 319-324. MR 0224339, 10.1090/S0002-9939-1968-0224339-X |
| Reference:
|
[9] A. Horn A. Tarski: Measures in Boolean algebras.Trans. Amer. Math. Soc. 64 (1948), 467-497. MR 0028922, 10.1090/S0002-9947-1948-0028922-8 |
| Reference:
|
[10] R. Jajte A. Paszkiewicz: Vector measures on the closed subspaces of a Hilbert space.Studia Math. T. LXIII (1978). MR 0632053 |
| Reference:
|
[11] G. Kalmbach: Measures on Hilbert Lattices.World Sci. Publ., Singapore, 1986. MR 0867884 |
| Reference:
|
[12] P. Kruszyňski: Vector measures on orthocomplemented lattices.Proceedings of the Koniklijke Akademie van Wetenschappen, Ser. A, 91 no. 4, December 19 (1988). MR 0976526 |
| Reference:
|
[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. |
| Reference:
|
[14] P. Pták: Exotic logics.Colloqium Mathematicum LIV, Fasc. 1 (1987), 1-7. MR 0928651 |
| Reference:
|
[15] P. Pták: Weak dispersion-free states and the hidden variables hypothesis.J. Math. Physics 24 (1983), 839-841. MR 0700618, 10.1063/1.525758 |
| Reference:
|
[16] P. Pták J. D. Wright: On the concreteness of quantum logics.Aplikace matematiky 30 č. 4 (1986), 274-285. MR 0795987 |
| Reference:
|
[17] F. Schultz: A characterization of state space of orthomodular lattices.Jour. Comb. Theory 17 (1974), 317-328. MR 0364042, 10.1016/0097-3165(74)90096-X |
| . |
