Relatively additive states on quantum logics

Pavel Pták; Hans Weber

Commentationes Mathematicae Universitatis Carolinae (2005)

  • Volume: 46, Issue: 2, page 327-338
  • ISSN: 0010-2628

Abstract

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In this paper we carry on the investigation of partially additive states on quantum logics (see [2], [5], [7], [8], [11], [12], [15], [18], etc.). We study a variant of weak states — the states which are additive with respect to a given Boolean subalgebra. In the first result we show that there are many quantum logics which do not possess any 2-additive central states (any logic possesses an abundance of 1-additive central state — see [12]). In the second result we construct a finite 3-homogeneous quantum logic which does not possess any two-valued 1-additive state with respect to a given Boolean subalgebra. This result strengthens Theorem 2 of [5] and presents a rather advanced example in the orthomodular combinatorics (see also [9], [13], [4], [6], [16], etc.). In the rest we show that Greechie logics allow for 2 -additive three-valued states, and in case of Greechie lattices we show that one can even construct many 2 -additive two-valued states. Some open questions are posed, too.

How to cite

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Pták, Pavel, and Weber, Hans. "Relatively additive states on quantum logics." Commentationes Mathematicae Universitatis Carolinae 46.2 (2005): 327-338. <http://eudml.org/doc/249531>.

@article{Pták2005,
abstract = {In this paper we carry on the investigation of partially additive states on quantum logics (see [2], [5], [7], [8], [11], [12], [15], [18], etc.). We study a variant of weak states — the states which are additive with respect to a given Boolean subalgebra. In the first result we show that there are many quantum logics which do not possess any 2-additive central states (any logic possesses an abundance of 1-additive central state — see [12]). In the second result we construct a finite 3-homogeneous quantum logic which does not possess any two-valued 1-additive state with respect to a given Boolean subalgebra. This result strengthens Theorem 2 of [5] and presents a rather advanced example in the orthomodular combinatorics (see also [9], [13], [4], [6], [16], etc.). In the rest we show that Greechie logics allow for $2$-additive three-valued states, and in case of Greechie lattices we show that one can even construct many $2$-additive two-valued states. Some open questions are posed, too.},
author = {Pták, Pavel, Weber, Hans},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {(weak) state on quantum logic; Greechie paste job; Boolean algebra; quantum logic; Boolean algebra; weak state},
language = {eng},
number = {2},
pages = {327-338},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Relatively additive states on quantum logics},
url = {http://eudml.org/doc/249531},
volume = {46},
year = {2005},
}

TY - JOUR
AU - Pták, Pavel
AU - Weber, Hans
TI - Relatively additive states on quantum logics
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2005
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 46
IS - 2
SP - 327
EP - 338
AB - In this paper we carry on the investigation of partially additive states on quantum logics (see [2], [5], [7], [8], [11], [12], [15], [18], etc.). We study a variant of weak states — the states which are additive with respect to a given Boolean subalgebra. In the first result we show that there are many quantum logics which do not possess any 2-additive central states (any logic possesses an abundance of 1-additive central state — see [12]). In the second result we construct a finite 3-homogeneous quantum logic which does not possess any two-valued 1-additive state with respect to a given Boolean subalgebra. This result strengthens Theorem 2 of [5] and presents a rather advanced example in the orthomodular combinatorics (see also [9], [13], [4], [6], [16], etc.). In the rest we show that Greechie logics allow for $2$-additive three-valued states, and in case of Greechie lattices we show that one can even construct many $2$-additive two-valued states. Some open questions are posed, too.
LA - eng
KW - (weak) state on quantum logic; Greechie paste job; Boolean algebra; quantum logic; Boolean algebra; weak state
UR - http://eudml.org/doc/249531
ER -

References

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  13. Pták P., Pulmannová S., Orthomodular Structures as Quantum Logics, Kluwer Academic Publishers, Dordrecht, 1991. MR1176314
  14. Sultanbekov F.F., Set logics and their representations, Internat. J. Theoret. Phys. 32 (1993), 11 2177-2186. (1993) Zbl0799.03081MR1254335
  15. Tkadlec J., Partially additive states on orthomodular posets, Colloquium Math. 62 (1991), 7-14. (1991) Zbl0784.03037MR1114613
  16. Trnková V., Automorphisms and symmetries of quantum logics, Internat. J. Theoret. Phys. 28 (1989), 1195-1214. (1989) MR1031603
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  18. Weber H., There are orthomodular lattices without non-trivial group valued states; a computer-based construction, J. Math. Anal. Appl. 183 (1994), 89-94. (1994) Zbl0797.06010MR1273434

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