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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Sequentially independent effects


Authors: Stan Gudder and Gabriel Nagy
Journal: Proc. Amer. Math. Soc. 130 (2002), 1125-1130
MSC (2000): Primary 47B15, 47B65; Secondary 81P10, 81P15
Published electronically: October 1, 2001
MathSciNet review: 1873787
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Abstract: A quantum effect is a yes-no measurement that may be unsharp. An effect is represented by an operator $E$ on a Hilbert space that satisfies $0\le E\le I$. We define effects $E_1,E_2,\ldots ,E_n$ to be sequentially independent if the result of any sequential measurement of $E_1,E_2,\ldots,E_n$ does not depend on the order in which they are measured. We show that two effects are sequentially independent if and only if they are compatible. That is, their corresponding operators commute. We also show that three effects are sequentially independent if and only if all permutations of the product of their corresponding operators coincide. It is noted that this last condition does not imply that the three effects are mutually compatible.


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Additional Information

Stan Gudder
Affiliation: Department of Mathematics, University of Denver, Denver, Colorado 80208
Email: sgudder@cs.du.edu

Gabriel Nagy
Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506
Email: nagy@math.ksu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06194-9
PII: S 0002-9939(01)06194-9
Keywords: Sequential independence, measurements, effects, positive operators, quantum mechanics.
Received by editor(s): September 19, 2000
Received by editor(s) in revised form: October 27, 2000
Published electronically: October 1, 2001
Additional Notes: The second author was partially supported by NSF grant DMS 9706858
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2001 American Mathematical Society