zbMATH — the first resource for mathematics

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 \(A,B\preceq C\).

81P15 Quantum measurement theory, state operations, state preparations
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
Full Text: EuDML
[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.