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An order for quantum observables. (English) Zbl 1141.81008
Logical order on the set $$S(H)$$ of self-adjoint operators acting on a Hilbert space $$H$$ is introduced and investigated. This order, motivated by generalized effect algebra, is defined as follows: $$A\preceq B$$ if there is a $$C\in S(H)$$ such that $$A$$ and $$C$$ have orthogonal ranges and $$A+C=B$$. This order can be characterized in terms of the spectral measures as follows: $$A\leq B$$ if and only if for the corresponding spectral measures $$P^A(\Delta )\subset P^B(\Delta )$$ whenever $$\Delta$$ is a Borel set not containing 0. Among others it is shown that for each $$A\in S(H)$$ with the range projection $$P_A$$ the set $$\{B\in S(H)\:, B\preceq A\}$$ endowed with the logical order is isomorphic to the $$\sigma$$-orthomodular projection lattice $$L_A=\{ P\: P\leq P_A, PA=AP\}.$$ As a consequence, $$\bigl (S(H), \preceq \bigr )$$ is a near lattice in the sense that $$A\land B$$, $$A\lor B$$ exist if and only if there is $$C\in S(H)$$ such that $$A,B\preceq C$$.

##### MSC:
 81P15 Quantum measurement theory, state operations, state preparations 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
##### Keywords:
effects; logical order of operators
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##### References:
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