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Partial order of quantum effects. (English) Zbl 0829.46060
The authors examine the order structure of the set of effects and some of its subsets. They show: The set of effects is not a lattice with respect to its natural order. Projection operators do have the greatest lower bounds (and the least upper bounds) in that set, but there are also other (incomparable) effects which share this property. However, the coexistence, the commutativity, and the regularity of a pair of effects are not sufficient for the existence of their infima and suprema. The structure of the range of an observable (as a normalized POV measure) can vary from that of a commutative Boolean to a noncommutative non-Boolean subset of effects.

MSC:
46N50 Applications of functional analysis in quantum physics
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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References:
[1] DOI: 10.2307/2372173 · Zbl 0042.35001
[2] DOI: 10.2307/2372173 · Zbl 0042.35001
[3] DOI: 10.1007/BF00672820 · Zbl 0806.03040
[4] DOI: 10.1090/S0002-9947-1965-0206736-3
[5] DOI: 10.1063/1.529811 · Zbl 0769.60101
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