Sequential products on effect algebras. (English) Zbl 1023.81001

A measurement composed of two measurements represented by operation \(A,B\) is considered. The authors introduce axiomatically a more general notion of sequential product on an effect algebra and study its properties. The two elements \(A,B\) are called sequentially independent if \(A\circ B=B\circ A\). The relation between this property and compatibility is clarified. The sequential center is the set of elements which are sequentially independent to all other elements; it coincides with the set of sharp central elements. The case when the sequential center is isomorphic to a fuzzy set system is characterized.
The existence of a sequential product appears to be a strong restriction. Boolean algebras are the only finite effect algebras admitting a sequential product. If a map preserves the sequential product then it completely preserves the effect algebra structure of the sharp elements. The existence of horizontal sums is characterized and some examples of horizontal sums and tensor products are given.


81P15 Quantum measurement theory, state operations, state preparations
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
03E72 Theory of fuzzy sets, etc.
06C15 Complemented lattices, orthocomplemented lattices and posets
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[1] Bennett, M. K.; Foulis, D. J., Adv. Appl. Math., 91, 200-215 (1977)
[2] Busch, P.; Lahti, P. J.; Middlestaedt, P., (The Quantum Theory of Measurements (1991), Springer: Springer Berlin)
[3] Busch, P.; Grabowski, M.; Lahti, P. J., (Operation Quantum Physics (1995), Springer: Springer Berlin) · Zbl 0863.60106
[4] Davies, E. B., (Quantum Theory of Open Systems (1976), Academic Press: Academic Press New York) · Zbl 0388.46044
[5] Dvurečenskij, A., Trans. Amer. Math. Soc., 147, 1043-1057 (1995) · Zbl 0859.03031
[6] Dvurečenskij, A.; Pulmannová, S., (New Trends in Quantum Structures (2000), Kluwer: Kluwer Dordrecht) · Zbl 0987.81005
[7] Foulis, D. J.; Bennett, M. K., Found. Phys., 24, 1325-1346 (1994)
[8] Giuntini, R.; Greuling, H., Found. Phys., 19, 931-945 (1989)
[9] Greechie, R. J.; Foulis, D. J.; Pulmannová, S., The center of an effect algebra, Order, 12, 91-106 (1995) · Zbl 0846.03031
[10] Gudder, S., J. Math. Phys., 39, 5772-5788 (1998) · Zbl 0935.81005
[11] Gudder, S., Rep. Math. Phys., 42, 321-346 (1998) · Zbl 1008.81002
[12] S. Gudder and G. Nagy: Sequentially independent effects, Proc. Amer. Math. Soc.; S. Gudder and G. Nagy: Sequentially independent effects, Proc. Amer. Math. Soc. · Zbl 1016.47020
[13] S. Gudder and G. Nagy: Sequentially quantum measurements, J. Math. Phys.; S. Gudder and G. Nagy: Sequentially quantum measurements, J. Math. Phys. · Zbl 1018.81005
[14] Kôpka, F., D-Posets and fuzzy sets, Tatra Mountains Publ., 1, 83-87 (1992) · Zbl 0797.04011
[15] Kôpka, F.; Chovanec, F., Math. Slovaca, 44, 21-34 (1994) · Zbl 0789.03048
[16] Kraus, K., (States, Effects, and Operations (1983), Springer: Springer Berlin) · Zbl 0545.46049
[17] Ludwig, G., (Foundations of Quantum Mechanics (1983), Springer: Springer Berlin) · Zbl 0509.46057
[18] Molnar, L., Letts. Math. Phys., 51, 37-45 (2000) · Zbl 1072.81535
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