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Sequential product of quantum effects

Authors: Aurelian Gheondea and Stanley Gudder
Journal: Proc. Amer. Math. Soc. 132 (2004), 503-512
MSC (2000): Primary 47B65, 81P15, 47N50, 46C07
Published electronically: July 2, 2003
MathSciNet review: 2022376
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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.

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  • 1. W. N. Anderson, Jr. and R. J. Duffin: Series and parallel addition of matrices, J. Math. Anal. Appl., 26 (1969), 576-594. MR 39:3904
  • 2. L. de Branges and J. Rovnyak: Canonical models in quantum scattering theory, in 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) pp. 295-392, Wiley, New York, 1966. MR 39:6109
  • 3. R. G. Douglas: On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc., 17 (1966), 413-415. MR 34:3315
  • 4. P. A. Fillmore and J. P. Williams: On operator ranges, Advances in Math., 7 (1971), 254-281. MR 45:2518
  • 5. S. Gudder and G. Nagy: Sequential quantum measurements, J. Math. Phys., 42 (2001), 5212-5222. MR 2002h:81032
  • 6. S. Gudder and G. Nagy: Sequentially independent effects, Proc. Amer. Math. Soc., 130 (2002), 1125-1130. MR 2002i:81014
  • 7. E. L. Pekarev and Yu. L. Shmulyan: Parallel addition and parallel subtraction of operators [Russian], Izv. Akad. Nauk SSSR, Ser. Mat., 40 (1976), 366-387. MR 53:14178
  • 8. L. Schwartz: Sous-espaces Hilbertiens d'espaces vectoriel topologiques et noyaux associés (noyaux reproduisants), J. Analyse Math., 13 (1964), 115-256. MR 31:3835

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Additional Information

Aurelian Gheondea
Affiliation: Institutul de Matematică al Academiei Române, C.P. 1-764, 014700 Bucureşti, România
Address at time of publication: Department of Mathematics, Bilkent University, 06533 Ankara, Turkey

Stanley Gudder
Affiliation: Department of Mathematics, University of Denver, Denver, Colorado 80208

Received by editor(s): August 29, 2002
Received by editor(s) in revised form: October 17, 2002
Published electronically: July 2, 2003
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2003 American Mathematical Society

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