On the properties of spectral effect algebras

Anna Jenčová,Martin Plávala

doi:10.22331/q-2019-06-03-148

Anna Jenčová, Martin Plávala

Open Access

https://doi.org/10.22331/q-2019-06-03-148

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Abstract

The aim of this paper is to show that there can be either only one or uncountably many contexts in any spectral effect algebra, answering a question posed in [S. Gudder, Convex and Sequential Effect Algebras, (2018), arXiv:1802.01265]. We also provide some results on the structure of spectral effect algebras and their state spaces and investigate the direct products and direct convex sums of spectral effect algebras. In the case of spectral effect algebras with sharply determining state space, stronger properties can be proved: the spectral decompositions are essentially unique, the algebra is sharply dominating and the set of its sharp elements is an orthomodular lattice. The article also contains a list of open questions that might provide interesting future research directions.

Highlights

Spectrality plays an important role in quantum theory
On one hand spectral theorem for operators over Hilbert spaces is a well known result found in many textbooks, on the other hand various forms of spectrality are used in operational derivations of quantum theory
In this article we have proved that there can be either only one or uncountably many contexts contained in a spectral effect algebra as well as few other results concerning spectral effect algebras

Summary

Introduction

Spectrality plays an important role in quantum theory. On one hand spectral theorem for operators over Hilbert spaces is a well known result found in many textbooks, on the other hand various forms of spectrality are used in operational derivations of quantum theory. In the special case of quantum theory over a finite-dimensional complex Hilbert space H, selfadjoint operators such that 0 ≤ A ≤ 1 are called effects, here A ≥ 0 means that A is positive semi-definite and 1 is the identity operator. It is well-known that the effects are used in description of quantum measurements [10]. Necessary and sufficient conditions were obtained for the algebra to be affinely isomorphic to the set of effects over a complex Hilbert space In this case, the algebra must have infinitely many contexts which corresponded to the rank projective measurements.

Properties of convex effect algebras

Contexts and spectral effect algebras

Number of contexts in a spectral effect algebra

Compositions of spectral effect algebras

Sharply determining state spaces

Conclusions and open questions