Quantum Measures on Finite Effect Algebras with the Riesz Decomposition Properties
Tóm tắt
One kind of generalized measures called quantum measures on finite effect algebras, which fulfil the grade-2 additive sum rule, is considered. One basis of vector space of quantum measures on a finite effect algebra with the Riesz decomposition property (RDP for short) is given. It is proved that any diagonally positive symmetric signed measure
$$\lambda $$
on the tensor product
$$E\otimes E$$
can determine a quantum measure
$$\mu $$
on a finite effect algebra
$$E$$
with the RDP such that
$$\mu (x)=\lambda (x\otimes x)$$
for any
$$x\in E$$
. Furthermore, some conditions for a grade-2 additive measure
$$\mu $$
on a finite effect algebra
$$E$$
are provided to guarantee that there exists a unique diagonally positive symmetric signed measure
$$\lambda $$
on
$$E\otimes E$$
such that
$$\mu (x)=\lambda (x\otimes x)$$
for any
$$x\in E$$
. At last, it is showed that any grade-
$$t$$
quantum measure on a finite effect algebra
$$E$$
with the RDP is essentially established by the values on a subset of
$$E$$
.
Tài liệu tham khảo
Bauer, H.: Measure and Integration Theory. Walter de Gruyter, Berlin (2001)
Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. Second Ser. 37, 823–834 (1936)
Combarro, E.F., Miranda, P.: On the structure of the \(k\)-additive fuzzy measures. Fuzzy Sets Syst. 161, 2314–2327 (2010)
Dvurečenskij, A.: Tensor product of difference posets and effect algebras. Int. J. Theor. Phys. 34, 1337–1348 (1995)
Dvurečenskij, A.: Tensor product of difference posets. Trans. Am. Math. Soc. 347, 1043–1057 (1995)
Dvurečenskij, A., Pulmannová, S.: D-test spaces and difference posets. Rep. Math. Phys. 34, 151–170 (1994)
Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer Academic Publishers, Dordrecht, and Ister Science, Bratislava (2000)
Dvurečenskij, A., Chovanec, F., Rybáriková, E.: D-hommorphisms and atomic \(\sigma \)-complete Boolean D-posets. Soft Comput. 4, 9–18 (2000)
Foulis, D., Bennett, M.K.: Effect algebras and unsharp quantum logics. Found. Phys. 24, 1325–1346 (1994)
Gell-Mann, M., Hartle, J.B.: Classical equations for quantum systems. Phys. Rev. D 47, 3345–3382 (1993)
Grabisch, M.: k-Order additive discrete fuzzy measures and their representation. Fuzzy Sets Syst. 92, 167–189 (1997)
Griffiths, R.B.: Consistent histories and the interpretation of quantum mechanics. J. Stat. Phys. 36, 219–272 (1984)
Griffiths, R.B.: Consistent Quantum Theory. Cambridge University Press, Cambridge (2002)
Gudder, S.: A histories approach to quantum mechanics. J. Math. Phys. 39, 5772–5788 (1998)
Gudder, S.: Morphisms, tensor products and \(\sigma \)-effect algebras. Rep. Math. Phys. 42, 321–346 (1998)
Gudder, S.: Quantum measure and integration theory. J. Math. Phys. 50, 123509 (2009)
Gudder, S.: Finite quantum measure spaces. Am. Math. Mon. 117, 512–527 (2010)
Gudder, S.: Quantum measure theory. Math. Slovaca 60, 681–700 (2010)
Gudder, S.: An anhomomorphic logic for quantum mechanics. J. Phys. A 43, 095302 (2010)
Gudder, S.: Quantum integrals and anhomomorphic logics. J. Math. Phys. 51, 112101 (2010)
Gudder, S.: Quantum measures and the coevent interpretation. Rep. Math. Phys. 67, 681–700 (2011)
Isham, C.: Quantum logic and the histories approach to quantum theory. J. Math. Phys. 35, 2157–2185 (1994)
Jenča, G., Pulmannová, S.: Orthocomplete effect algebras. Proc. Am. Math. Soc. 131, 2663–2671 (2003)
Kalmbach, G.: Orthomodular Lattices. Academic Press, London (1983)
Kôpka, F., Chovanec, F.: D-posets. Math. Slovaca 44, 21–34 (1994)
Mesiar, R.: k-Order additive measures. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 6, 561–568 (1999)
Mesiar, R.: Generlizations of k-order additive discrete measures. Fuzzy Sets Syst. 102, 423–428 (1999)
Pulmannová, S.: Difference posets and the histories approach to quantum theories. Int. J. Theor. Phys. 34, 189–210 (1995)
Salgado, R.: Some identities for the quantum measure and its generalizations. Mod. Phys. Lett. A 17, 711–728 (2002)
Sorkin, R.: Quantum mechanics as quantum measure theory. Mod. Phys. Lett. A 9, 3119–3127 (1994)
Sorkin, R.: Quantum mechanics without the wave function. J. Phys. A 40, 3207–3231 (2007)
Surya, S., Wallden, P.: Quantum covers in quantum measure theory. Found. Phys. 40, 585–606 (2010)
Xie, Y., Yang, A., Ren, F.: Super quantum measures on finite spaces. Found. Phys. 43, 1039–1065 (2013)