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Coexistence in interval effect algebras
Author:
Gejza Jenča
Journal:
Proc. Amer. Math. Soc. 139 (2011), 331-344
MSC (2010):
Primary 03G12; Secondary 06F20, 81P10
Posted:
July 29, 2010
MathSciNet review:
2729095
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Abstract: Motivated by the notion of coexistence of effect-valued observables, we give a characterization of coexistent subsets of interval effect algebras.
- 1.
M.
K. Bennett and D.
J. Foulis, Interval and scale effect algebras, Adv. in Appl.
Math. 19 (1997), no. 2, 200–215. MR 1459498
(98m:06024), http://dx.doi.org/10.1006/aama.1997.0535
- 2.
Ladislav
Beran, Orthomodular lattices, Mathematics and its Applications
(East European Series), D. Reidel Publishing Co., Dordrecht, 1985.
Algebraic approach. MR 784029
(86m:06015b)
- 3.
Paul
Busch, Pekka
J. Lahti, and Peter
Mittelstaedt, The quantum theory of measurement, 2nd ed.,
Lecture Notes in Physics. New Series m: Monographs, vol. 2,
Springer-Verlag, Berlin, 1996. MR 1419313
(98b:81019)
- 4.
P. Busch and H-J. Schmidt.
Coexistence of qubit effects. Quantum Information Processing, Quantum Information Processing 9:143-169, 2010.
- 5.
Paul
Busch, Marian
Grabowski, and Pekka
J. Lahti, Operational quantum physics, Lecture Notes in
Physics. New Series m: Monographs, vol. 31, Springer-Verlag, Berlin,
1995. MR
1356220 (96j:81022)
- 6.
C.
C. Chang, Algebraic analysis of many valued
logics, Trans. Amer. Math. Soc. 88 (1958), 467–490. MR 0094302
(20 #821), http://dx.doi.org/10.1090/S0002-9947-1958-0094302-9
- 7.
Ferdinand
Chovanec and František
Kôpka, Boolean D-posets, Tatra Mt. Math. Publ.
10 (1997), 183–197. Quantum structures
(Liptovský Ján, 1995). MR 1469294
(98c:03127)
- 8.
Anatolij
Dvurečenskij and Sylvia
Pulmannová, New trends in quantum structures,
Mathematics and its Applications, vol. 516, Kluwer Academic
Publishers, Dordrecht, 2000. MR 1861369
(2002h:81021)
- 9.
D.
J. Foulis and M.
K. Bennett, Effect algebras and unsharp quantum logics, Found.
Phys. 24 (1994), no. 10, 1331–1352. Special
issue dedicated to Constantin Piron on the occasion of his sixtieth
birthday. MR
1304942 (95k:06020), http://dx.doi.org/10.1007/BF02283036
- 10.
D.
J. Foulis and C.
H. Randall, Operational statistics. I. Basic concepts, J.
Mathematical Phys. 13 (1972), 1667–1675. MR 0416417
(54 #4491)
- 11.
Roberto
Giuntini and Heinz
Greuling, Toward a formal language for unsharp properties,
Found. Phys. 19 (1989), no. 7, 931–945. MR 1013913
(90j:81017), http://dx.doi.org/10.1007/BF01889307
- 12.
Stan
Gudder, Coexistence of quantum effects, Rep. Math. Phys.
63 (2009), no. 2, 289–303. MR 2519471
(2010i:81022), http://dx.doi.org/10.1016/S0034-4877(09)90005-2
- 13.
K.-E. Hellwig.
Coexistent effects in quantum mechanics. International Journal of Theoretical Physics, 2:147-155, 1969.
- 14.
Gejza
Jenča, Boolean algebras R-generated by MV-effect
algebras, Fuzzy Sets and Systems 145 (2004),
no. 2, 279–285. MR 2074002
(2005e:06018), http://dx.doi.org/10.1016/S0165-0114(03)00226-4
- 15.
G. Jenča.
Compatibility support mappings in effect algebras. Math. Slovaca (to appear). arXiv:math.RA/0910.2825.
- 16.
Gudrun
Kalmbach, Orthomodular lattices, London Mathematical Society
Monographs, vol. 18, Academic Press Inc. [Harcourt Brace Jovanovich
Publishers], London, 1983. MR 716496
(85f:06012)
- 17.
František
Kôpka, 𝐷-posets of fuzzy sets, Tatra Mt. Math.
Publ. 1 (1992), 83–87. Fuzzy sets (Liptovský
Mikuláš, 1992). MR 1230466
(94e:04008)
- 18.
František
Kôpka and Ferdinand
Chovanec, 𝐷-posets, Math. Slovaca 44
(1994), no. 1, 21–34. MR 1290269
(95i:03134)
- 19.
Karl
Kraus, States, effects, and operations, Lecture Notes in
Physics, vol. 190, Springer-Verlag, Berlin, 1983. Fundamental notions
of quantum theory; Lecture notes edited by A. Böhm, J. D. Dollard and
W. H. Wootters. MR 725167
(86j:81008)
- 20.
Pekka
Lahti and Sylvia
Pulmannová, Coexistent observables and effects in quantum
mechanics, Rep. Math. Phys. 39 (1997), no. 3,
339–351. MR 1477898
(98i:81013), http://dx.doi.org/10.1016/S0034-4877(97)89752-2
- 21.
Pekka
Lahti and Sylvia
Pulmannová, Coexistence vs. functional coexistence of
quantum observables, Rep. Math. Phys. 47 (2001),
no. 2, 199–212. MR 1836331
(2002d:81018), http://dx.doi.org/10.1016/S0034-4877(01)89037-6
- 22.
Pekka
Lahti, Sylvia
Pulmannova, and Kari
Ylinen, Coexistent observables and effects in a convexity
approach, J. Math. Phys. 39 (1998), no. 12,
6364–6371. MR 1656976
(99j:81020), http://dx.doi.org/10.1063/1.532643
- 23.
Günther
Ludwig, Foundations of quantum mechanics. I, Texts and
Monographs in Physics, Springer-Verlag, New York, 1983. Translated from the
German by Carl A. Hein. MR 690770
(85g:81014)
- 24.
Günther
Ludwig, Versuch einer axiomatischen Grundlegung der Quantenmechanik
und allgemeinerer physikalischer Theorien, Z. Physik
181 (1964), 233–260 (German). MR 0181244
(31 #5473)
- 25.
Daniele
Mundici, Interpretation of AF 𝐶*-algebras in
Łukasiewicz sentential calculus, J. Funct. Anal.
65 (1986), no. 1, 15–63. MR 819173
(87k:46146), http://dx.doi.org/10.1016/0022-1236(86)90015-7
- 26.
Sylvia
Pulmannová, Compatibility and decompositions of
effects, J. Math. Phys. 43 (2002), no. 5,
2817–2830. MR 1893702
(2003b:81012), http://dx.doi.org/10.1063/1.1462857
- 27.
Gian-Carlo
Rota, On the foundations of combinatorial theory. I. Theory of
Möbius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete
2 (1964), 340–368 (1964). MR 0174487
(30 #4688)
- 28.
Richard
P. Stanley, Enumerative combinatorics. Vol. I, The Wadsworth
& Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced
Books & Software, Monterey, CA, 1986. With a foreword by Gian-Carlo
Rota. MR
847717 (87j:05003)
- 29.
Peter
Stano, Daniel
Reitzner, and Teiko
Heinosaari, Coexistence of qubit effects, Phys. Rev. A (3)
78 (2008), no. 1, 012315, 9. MR 2491108
(2010d:81079), http://dx.doi.org/10.1103/PhysRevA.78.012315
- 1.
- M.K. Bennett and D.J. Foulis.
Interval and scale effect algebras. Advances in Applied Mathematics, 19:200-215, 1997. MR 1459498 (98m:06024)
- 2.
- L. Beran.
Orthomodular Lattices, Algebraic Approach. Kluwer, Dordrecht, 1985. MR 784029 (86m:06015b)
- 3.
- P. Busch, P. Lahti, and P. Mittelstaedt.
The Quantum Theory of Measurement. Springer-Verlag, 2nd edition, 1996. MR 1419313 (98b:81019)
- 4.
- P. Busch and H-J. Schmidt.
Coexistence of qubit effects. Quantum Information Processing, Quantum Information Processing 9:143-169, 2010.
- 5.
- P. Bush, M. Grabowski, and P. Lahti.
Operational Quantum Physics. Springer-Verlag, Berlin, 1995. MR 1356220 (96j:81022)
- 6.
- C.C. Chang.
Algebraic analysis of many-valued logics. Trans. Amer. Math. Soc., 88:467-490, 1959. MR 0094302 (20:821)
- 7.
- F. Chovanec and F. Kôpka.
Boolean -posets. Tatra Mt. Math. Publ, 10:183-197, 1997. MR 1469294 (98c:03127)
- 8.
- A. Dvurečenskij and S. Pulmannová.
New Trends in Quantum Structures. Kluwer, Dordrecht, and Ister Science, Bratislava, 2000. MR 1861369 (2002h:81021)
- 9.
- D.J. Foulis and M.K. Bennett.
Effect algebras and unsharp quantum logics. Found. Phys., 24:1325-1346, 1994. MR 1304942 (95k:06020)
- 10.
- D.J. Foulis and C.H. Randall.
Operational quantum statistics. I. Basic concepts. J. Math. Phys., 13:1667-1675, 1972. MR 0416417 (54:4491)
- 11.
- R. Giuntini and H. Greuling.
Toward a formal language for unsharp properties. Found. Phys., 19:931-945, 1989. MR 1013913 (90j:81017)
- 12.
- S. Gudder.
Coexistence of quantum effects. Reports on Mathematical Physics, 63(2):289-303, 2009. MR 2519471
- 13.
- K.-E. Hellwig.
Coexistent effects in quantum mechanics. International Journal of Theoretical Physics, 2:147-155, 1969.
- 14.
- G. Jenča.
Boolean algebras R-generated by MV-effect algebras. Fuzzy sets and systems, 145:279-285, 2004. MR 2074002 (2005e:06018)
- 15.
- G. Jenča.
Compatibility support mappings in effect algebras. Math. Slovaca (to appear). arXiv:math.RA/0910.2825.
- 16.
- G. Kalmbach.
Orthomodular Lattices. Academic Press, New York, 1983. MR 716496 (85f:06012)
- 17.
- F. Kôpka.
-posets of fuzzy sets. Tatra Mt. Math. Publ., 1:83-87, 1992. MR 1230466 (94e:04008)
- 18.
- F. Kôpka and F. Chovanec.
-posets. Math. Slovaca, 44:21-34, 1994. MR 1290269 (95i:03134)
- 19.
- K. Kraus.
States, Effects and Operations. Springer-Verlag, Berlin, 1983. MR 725167 (86j:81008)
- 20.
- P. Lahti and S. Pulmannová.
Coexistent observables and effects in quantum mechanics. Reports on Mathematical Physics, 39:339-351, 1997. MR 1477898 (98i:81013)
- 21.
- P. Lahti and S. Pulmannová.
Coexistence vs. functional coexistence of quantum observables. Reports on Mathematical Physics, 47:199-212, 2001. MR 1836331 (2002d:81018)
- 22.
- P. Lahti, S. Pulmannová, and K. Ylinen.
Coexistent observables and effects in convexity approach. Reports on Mathematical Physics, 39:6364-6371, 1998. MR 1656976 (99j:81020)
- 23.
- G. Ludwig.
Foundations of Quantum Mechanics. Springer-Verlag, Berlin, 1983. MR 690770 (85g:81014)
- 24.
- Günther Ludwig.
Versuch einer axiomatischen Grundlegung der Quantenmechanik und allgemeinerer physikalischer Theorien. Zeitschrift für Physik A Hadrons and Nuclei, 181(3):233-260, 1964. MR 0181244 (31:5473)
- 25.
- D. Mundici.
Interpretation of AF -algebras in Łukasziewicz sentential calculus. J. Functional Analysis, 65:15-53, 1986. MR 819173 (87k:46146)
- 26.
- S. Pulmannová.
Compatibility and decompositions of effects. Journal of Mathematical Physics, 43:2817-2830, 2002. MR 1893702 (2003b:81012)
- 27.
- Gian-Carlo Rota.
On the foundations of combinatorial theory. I. Theory of Möbius Functions. Probability Theory and Related Fields, 2:340-368, 1964. MR 0174487 (30:4688)
- 28.
- Richard P. Stanley.
Enumerative combinatorics, volume 1. Wadsworth and Brooks/Cole, Monterey, CA, 1986. MR 0847717 (87j:05003)
- 29.
- P. Stano, D. Reitzner, and T. Heinosaari.
Coexistence of qubit effects. Phys. Rev. A, 78:012315, 2008. MR 2491108 (2010d:81079)
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Additional Information
Gejza Jenča
Affiliation:
Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, Bratislava 813 68, Slovak Republic
Email:
gejza.jenca@stuba.sk
DOI:
http://dx.doi.org/10.1090/S0002-9939-2010-10554-3
PII:
S 0002-9939(2010)10554-3
Keywords:
Effect algebra,
coexistent observables
Received by editor(s):
September 26, 2009
Received by editor(s) in revised form:
March 19, 2010
Posted:
July 29, 2010
Additional Notes:
This research is supported by grant VEGA G-1/0080/10 of MŠ SR, Slovakia and by the Slovak Research and Development Agency under contract No. APVV-0071-06.
Communicated by:
Marius Junge
Article copyright:
© Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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