Observables on σ -MV algebras and σ -lattice effect algebras

Anna Jenčová; Sylvia Pulmannová; Elena Vinceková

Kybernetika (2011)

  • Volume: 47, Issue: 4, page 541-559
  • ISSN: 0023-5954

Abstract

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Effect algebras were introduced as abstract models of the set of quantum effects which represent sharp and unsharp properties of physical systems and play a basic role in the foundations of quantum mechanics. In the present paper, observables on lattice ordered σ -effect algebras and their “smearings” with respect to (weak) Markov kernels are studied. It is shown that the range of any observable is contained in a block, which is a σ -MV algebra, and every observable is defined by a smearing of a sharp observable, which is obtained from generalized Loomis-Sikorski theorem for σ -MV algebras. Generalized observables with the range in the set of sharp real observables are studied and it is shown that they contain all smearings of observables.

How to cite

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Jenčová, Anna, Pulmannová, Sylvia, and Vinceková, Elena. "Observables on $\sigma $-MV algebras and $\sigma $-lattice effect algebras." Kybernetika 47.4 (2011): 541-559. <http://eudml.org/doc/196559>.

@article{Jenčová2011,
abstract = {Effect algebras were introduced as abstract models of the set of quantum effects which represent sharp and unsharp properties of physical systems and play a basic role in the foundations of quantum mechanics. In the present paper, observables on lattice ordered $\sigma $-effect algebras and their “smearings” with respect to (weak) Markov kernels are studied. It is shown that the range of any observable is contained in a block, which is a $\sigma $-MV algebra, and every observable is defined by a smearing of a sharp observable, which is obtained from generalized Loomis-Sikorski theorem for $\sigma $-MV algebras. Generalized observables with the range in the set of sharp real observables are studied and it is shown that they contain all smearings of observables.},
author = {Jenčová, Anna, Pulmannová, Sylvia, Vinceková, Elena},
journal = {Kybernetika},
keywords = {lattice effect algebra; MV algebra; observable; state; Markov kernel; weak Markov kernel; smearing; generalized observable; state; observable; MV algebra; lattice effect algebra; Markov kernel; weak Markov kernel; smearing; generalized observable},
language = {eng},
number = {4},
pages = {541-559},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Observables on $\sigma $-MV algebras and $\sigma $-lattice effect algebras},
url = {http://eudml.org/doc/196559},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Jenčová, Anna
AU - Pulmannová, Sylvia
AU - Vinceková, Elena
TI - Observables on $\sigma $-MV algebras and $\sigma $-lattice effect algebras
JO - Kybernetika
PY - 2011
PB - Institute of Information Theory and Automation AS CR
VL - 47
IS - 4
SP - 541
EP - 559
AB - Effect algebras were introduced as abstract models of the set of quantum effects which represent sharp and unsharp properties of physical systems and play a basic role in the foundations of quantum mechanics. In the present paper, observables on lattice ordered $\sigma $-effect algebras and their “smearings” with respect to (weak) Markov kernels are studied. It is shown that the range of any observable is contained in a block, which is a $\sigma $-MV algebra, and every observable is defined by a smearing of a sharp observable, which is obtained from generalized Loomis-Sikorski theorem for $\sigma $-MV algebras. Generalized observables with the range in the set of sharp real observables are studied and it is shown that they contain all smearings of observables.
LA - eng
KW - lattice effect algebra; MV algebra; observable; state; Markov kernel; weak Markov kernel; smearing; generalized observable; state; observable; MV algebra; lattice effect algebra; Markov kernel; weak Markov kernel; smearing; generalized observable
UR - http://eudml.org/doc/196559
ER -

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