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arXiv:1204.6460

Observables on Quantum Structures

Anatolij Dvurečenskij, Mária Kuková

Published Apr 29, 2012• 1 citation
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Abstract

An observable on a quantum structure is any $σ$-homomorphism of quantum structures from the Borel $σ$-algebra into the quantum structure. We show that our partial information on an observable known only for all intervals of the form $(-\infty,t)$ is sufficient to determine uniquely the whole observable defined on quantum structures like $σ$-MV-algebras, $σ$-effect algebras, Boolean $σ$-algebras, monotone $σ$-complete effect algebras with the Riesz Decomposition Property, the effect algebra of effect operators of a Hilbert space, and a system of functions, and an effect-tribe.

DOI
10.48550/arxiv.1204.6460

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An observable on a quantum structure is any σσσ-homomorphism of quantum structures from the Borel σσσ-algebra into the quantum structure. We show that our partial information on an observable known only for all intervals of the form (−∞,t)(-\infty,t)(−∞,t) is sufficient to determine uniquely the whole observable defined on quantum structures like σσσ-MV-algebras, σσσ-effect algebras, Boolean σσσ-algebras, monotone σσσ-complete effect algebras with the Riesz Decomposition Property, the effect algebra of effect operators of a Hilbert space, and a system of functions, and an effect-tribe.
An observable effects a schematization of the Quantum event structure by correlating Boolean algebras picked by measurements with the Borel algebra of the real line. In a well-defined sense Boolean observables play the role of coordinatizing objects in the Quantum world, by picking Boolean figures and subsequently opening Boolean windows for the perception of the latter, interpreted as local measurement charts. A mathematical scheme for the implementation of this thesis is being proposed based on Category theoretical methods. The scheme leads to a manifold representation of Quantum structure in terms of topos-theoretical Boolean reference frames. The coordinatizing objects give rise to structure preserving maps with the modeling objects as their domains, effecting finally an isomorphism between quantum event algebra objects and Boolean localization systems for the masurement of observables.
Given a quantum state in the finite-dimensional Hilbert space Cn, the range of possible values of a quantum observable is usually identified with the discrete spectrum of eigenvalues of a corresponding Hermitian matrix. Here any such observable is identified with (i) an 'ortho-measurable' function defined on the Boolean 'ortho-algebra' generated by the eigenspaces that form an orthogonal decomposition of Cn, and (ii) a 'numerically identified' orthogonal decomposition of Cn. The latter means that each subspace of the orthogonal decomposition can be uniquely identified by its own attached real number, just as each eigenspace of a Hermitian matrix can be uniquely identified by the corresponding eigenvalue. Furthermore, any density matrix on Cn is identified with a Bayesian prior 'ortho-probability' measure defined on the linear subspaces that make up the Boolean ortho-algebra induced by its eigenspaces. Then any pure quantum state is identified with a degenerate density matrix, and any mixed state with a probability measure on a set of orthogonal pure states. Finally, given any quantum observable, the relevant Bayesian posterior probabilities of measured outcomes can be found by the usual trace formula that extends Born's rule. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.
Our basic structure is a finite-dimensional complex Hilbert space HHH. We point out that the set of effects on HHH form a convex effect algebra. Although the set of operators on HHH also form a convex effect algebra, they have a more detailed structure. W introduce sequential products of effect and operations. Although these have already been studied, we introduce the new concept of sequential products of effects with operations and operations with effects. We then consider various special types of operations. After developing properties of these concepts, the results are generalized to include observables and instruments. In particular, sequential products of observables with instruments and instruments with observables are developed. Finally, we consider conditioning and coexistence of observables and instruments.