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Uniqueness of bounded observables. (English) Zbl 0840.03048
The uniqueness condition states that if the expectations of two bounded observables on a quantum logic are equal in every state, then the observables are equal. A quarter of a century ago, S. Gudder [“Uniqueness and existence properties of bounded observables”, Pac. J. Math. 19, 81-93, Corr. ibid. 588-590 (1966; Zbl 0149.23603)] noticed that there were several cases when expectations do separate the bounded observables, and posed the question whether an arbitrary reasonable quantum logic satisfies the condition. Later, some negative results in this connection appeared. The main result of the paper under review gives a negative answer to Gudder’s question for \(\sigma\)-orthomodular lattices with a strongly order-determining set of states. The paper also includes a short overview of the history of the problem and a summary of a recently developed construction technique needed for building up the needed counter-example.
MSC:
03G12 Quantum logic
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
06C15 Complemented lattices, orthocomplemented lattices and posets
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References:
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