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arXiv:1911.09490

Continuous coexistency preservers on effect algebras

Michiya Mori, Peter Šemrl

Published Nov 21, 2019• 0 citations
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Abstract

Let $H$ be a finite-dimensional Hilbert space, $\dim H \ge 2$. We prove that every continuous coexistency preserving map on the effect algebra $E(H)$ is either a standard automorphism of $E(H)$, or a standard automorphism of $E(H)$ composed with the orthocomplementation. We present examples showing the optimality of the result.

DOI
10.48550/arxiv.1911.09490

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Let HHH be a finite-dimensional Hilbert space, dim⁡H≥2\dim H \ge 2dimH≥2. We prove that every continuous coexistency preserving map on the effect algebra E(H)E(H)E(H) is either a standard automorphism of E(H)E(H)E(H), or a standard automorphism of E(H)E(H)E(H) composed with the orthocomplementation. We present examples showing the optimality of the result.
Let H be a Hilbert space and E ( H ) the effect algebra on H . A bijective map is called an ortho-order automorphism of E ( H ) if for every we have and . The classical theorem of Ludwig states that every such ϕ is of the form , , for some unitary or antiunitary operator U . It is also known that each bijective map on E ( H ) preserving order and coexistency in both directions is of the same form. Can we improve these two theorems by relaxing the bijectivity assumption and/or replacing the above preserving properties by the weaker assumptions of preserving above relations in one direction only and still get the same conclusion? For both characterizations of automorphisms of effect algebras we will prove the optimal versions and give counterexamples showing the optimality of the obtained results.
Let H be a Hilbert space and E(H) the effect algebra on H, that is, E(H) is the set of all self-adjoint operators A: H→H satisfying 0⩽A⩽I. The effect algebra can be equipped with several operations and relations that are important in mathematical foundations of quantum mechanics. Automorphisms with respect to these operations or relations are called symmetries. We present a new method that can be used to describe the general form of such maps. The main idea is to reduce this kind of problem to the study of adjacency-preserving maps. The efficiency of this approach is illustrated by reproving some known results as well as by obtaining some new theorems.
Abstract Spectral automorphisms have been introduced in [IVANOV, A.—CARAGHEORGHEOPOL, D.: Spectral automorphisms in quantum logics, Internat. J. Theoret. Phys. 49 (2010), 3146–3152]_in an attempt to construct, in the abstract framework of orthomodular lattices, an analogue of the spectral theory in Hilbert spaces. We generalize spectral automorphisms to the framework of effect algebras with compression bases and study their properties. Characterizations of spectral automorphisms as well as necessary conditions for an automorphism to be spectral are given. An example of a spectral automorphism on the standard effect algebra of a finite-dimensional Hilbert space is discussed and the consequences of spectrality of an automorphism for the unitary Hilbert space operator that generates it are shown. The last section is devoted to spectral families of automorphisms and their properties, culminating with the formulation and proof of a Stone type theorem (in the sense of Stone’s theorem on strongly continuous one-parameter unitary groups — see, e.g. [REED, M.#x2014;SIMON, B.: Methods of Modern Mathematical Physics, Vol. I, Acad. Press, New York, 1975]) for a group of spectral automorphisms.
Abstract The Hilbert space effect algebra is a fundamental mathematical structure which is used to describe unsharp quantum measurements in Ludwig’s formulation of quantum mechanics. Each effect represents a quantum (fuzzy) event. The relation of coexistence plays an important role in this theory, as it expresses when two quantum events can be measured together by applying a suitable apparatus. This paper’s first goal is to answer a very natural question about this relation, namely, when two effects are coexistent with exactly the same effects? The other main aim is to describe all automorphisms of the effect algebra with respect to the relation of coexistence. In particular, we will see that they can differ quite a lot from usual standard automorphisms, which appear for instance in Ludwig’s theorem. As a byproduct of our methods we also strengthen a theorem of Molnár.