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Journal of Physics A Mathematical and Theoretical

Automorphisms of Hilbert space effect algebras

Peter Šemrl

Published Apr 17, 2015• 7 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8113/48/19/195301

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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 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, 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.