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A Possible Operational Motivation for the Orthocomplementation in Quantum Structures

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Abstract

In the foundations of quantum mechanics Gleason’s theorem dictates the uniqueness of the state transition probability via the inner product of the corresponding state vectors in Hilbert space, independent of which measurement context induces this transition. We argue that the state transition probability should not be regarded as a secondary concept which can be derived from the structure on the set of states and properties, but instead should be regarded as a primitive concept for which measurement context is crucial. Accordingly, we adopt an operational approach to quantum mechanics in which a physical entity is defined by the structure of its set of states, set of properties and the possible (measurement) contexts which can be applied to this entity. We put forward some elementary definitions to derive an operational theory from this State–COntext–Property (SCOP) formalism. We show that if the SCOP satisfies a Gleason-like condition, namely that the state transition probability is independent of which measurement context induces the change of state, then the lattice of properties is orthocomplemented, which is one of the ‘quantum axioms’ used in the Piron–Solèr representation theorem for quantum systems. In this sense we obtain a possible physical meaning for the orthocomplementation widely used in quantum structures.

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  1. Leo Apostel Center for Interdisciplinary Studies, Vrije Universiteit Brussel (VUB), Brussel, Belgium

    Bart D’Hooghe

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  1. Bart D’Hooghe
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Correspondence to Bart D’Hooghe.

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D’Hooghe, B. A Possible Operational Motivation for the Orthocomplementation in Quantum Structures. Found Phys 40, 1669–1680 (2010). https://doi.org/10.1007/s10701-010-9464-9

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  • DOI: https://doi.org/10.1007/s10701-010-9464-9

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