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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)
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References:
[1] BUSCH P.-GRABOWSKI M.-LAHTI P. J.: Operational Quantum Physics. Springer-Verlag, Berlin, 1995. · Zbl 0863.60106
[2] DAVIES E. B.: Quantum Theory of Open Systems. Academic Press, New York, 1976. · Zbl 0388.46044
[3] DVUREČENSKIJ A.-PULMANNOVÁ S.: New Trends in Quantum Structures. Kluwer Acad. Publ., Dordrecht, 2000. · Zbl 0987.81005
[4] FOULIS D. J.-BENNETT M. K.: Effect algebras and unsharp quantum logics. Found. Phys. 24 (1994), 1325-1346. · Zbl 1213.06004
[5] GIUNTINI R.-GREULING H.: Toward a formal language for unsharp properties. Found. Phys. 19 (1989), 931-945.
[6] GUDDER S.: A structure for quantum measurements. Rep. Math. Phys. 55 (2005), 249-267. · Zbl 1098.81014
[7] HOLEVO A. S.: Probabilistic and Statistical Aspects of Quantum Theory. North-Holland, Amsterdam, 1982. · Zbl 0497.46053
[8] KADISON R.: Order properties of bounded self-adjoint operators. Proc. Amer. Math. Soc. 34 (1951), 505-510. · Zbl 0043.11501
[9] KRAUS K.: States, Effects and Operations. Springer-Verlag, Berlin, 1983. · Zbl 0545.46049
[10] LUDWIG G.: Foundations of Quantum Mechanics, Vols. I, II. Springer, Berlin, 1983/1985.
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