Coexistency on Hilbert Space Effect Algebras and a Characterisation of Its Symmetry Transformations
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Communications in
Mathematical Physics
Coexistency on Hilbert Space Effect Algebras and a Characterisation of Its Symmetry Transformations György Pál Gehér1 , Peter Šemrl2,3 1 Department of Mathematics and Statistics, University of Reading, Whiteknights, P.O. Box 220, Reading
RG6 6AX, UK E-mail: [email protected]; [email protected]
2 Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia 3 Institute of Mathematics, Physics, and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia.
E-mail: [email protected] Received: 21 September 2019 / Accepted: 19 August 2020 Published online: 30 September 2020 – © The Author(s) 2020
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. 1. Introduction 1.1. On the classical mathematical formulation of quantum mechanics. Throughout this paper H will denote a complex, not necessarily separable, Hilbert space with dimension at least 2. In the classical mathematical formulation of quantum mechanics such a space is used to describe experiments at the atomic scale. For instance, the famous Stern– Gerlach experiment (which was one of the firsts showing the reality of the quantum spin) can be described using the two-dimensional Hilbert space C2 . In the classical formulation of quantum mechanics, the space of all rank-one projections P1 (H ) plays an important role, as its elements represent so-called quantum pure-states (in particular in the Stern–Gerlach experiment they represent the quantum spin). The so-called transition probability between two pure states P, Q ∈ P1 (H ) is the number tr P Q, where tr denotes György Pál Gehér was supported by the Leverhulme Trust Early Career Fellowship, ECF-2018-125. He was also partly supported by the Hungarian National Research, Development and Innovation Office-NKFIH (K115383) Peter Šemrl was supported by Grants N1-0061, J1-8133, and P1-0288 from ARRS, Slovenia
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the trace. For the physical meaning of this quantity we refer the interested reader to e.g. [33]. A very important cornerstone of the mathematical foundations of quantum mechanics is Wigner’s theorem, which states the following. Wigner’s Theorem. Given a bijective map φ : P1 (H )
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