Sequential product of quantum effects

Gheondea, Aurelian; Gudder, Stanley

doi:10.1090/S0002-9939-03-07063-1

Sequential product of quantum effects
HTML articles powered by AMS MathViewer

by Aurelian Gheondea and Stanley Gudder PDF

Proc. Amer. Math. Soc. 132 (2004), 503-512 Request permission

Abstract:

Unsharp quantum measurements can be modelled by means of the class $\mathcal {E}(\mathcal {H})$ of positive contractions on a Hilbert space $\mathcal {H}$, in brief, quantum effects. For $A,B\in \mathcal {E}(\mathcal {H})$ the operation of sequential product $A\circ B=A^{1/2}BA^{1/2}$ was proposed as a model for sequential quantum measurements. We continue these investigations on sequential product and answer positively the following question: the assumption $A\circ B\geq B$ implies $AB=BA=B$. Then we propose a geometric approach of quantum effects and their sequential product by means of contractively contained Hilbert spaces and operator ranges. This framework leads us naturally to consider lattice properties of quantum effects, sums and intersections, and to prove that the sequential product is left distributive with respect to the intersection.

References

W. N. Anderson Jr. and R. J. Duffin, Series and parallel addition of matrices, J. Math. Anal. Appl. 26 (1969), 576–594. MR 242573, DOI 10.1016/0022-247X(69)90200-5
Louis de Branges and James Rovnyak, Canonical models in quantum scattering theory, Perturbation Theory and its Applications in Quantum Mechanics (Proc. Adv. Sem. Math. Res. Center, U.S. Army, Theoret. Chem. Inst., Univ. of Wisconsin, Madison, Wis., 1965) Wiley, New York, 1966, pp. 295–392. MR 0244795
R. G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413–415. MR 203464, DOI 10.1090/S0002-9939-1966-0203464-1
P. A. Fillmore and J. P. Williams, On operator ranges, Advances in Math. 7 (1971), 254–281. MR 293441, DOI 10.1016/S0001-8708(71)80006-3
Stan Gudder and Gabriel Nagy, Sequential quantum measurements, J. Math. Phys. 42 (2001), no. 11, 5212–5222. MR 1861337, DOI 10.1063/1.1407837
Stan Gudder and Gabriel Nagy, Sequentially independent effects, Proc. Amer. Math. Soc. 130 (2002), no. 4, 1125–1130. MR 1873787, DOI 10.1090/S0002-9939-01-06194-9
È. L. Pekarev and Ju. L. Šmul′jan, Parallel addition and parallel subtraction of operators, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 2, 366–387, 470 (Russian). MR 0410429
Laurent Schwartz, Sous-espaces hilbertiens d’espaces vectoriels topologiques et noyaux associés (noyaux reproduisants), J. Analyse Math. 13 (1964), 115–256 (French). MR 179587, DOI 10.1007/BF02786620