Home | About Journal | Web Links | E-mail Alerts | RSS RSS Icon | Browse
Previous Article Next Article

Algebraic complementarity in quantum theory

Source: J. Math. Phys. 51, 015215 (2010); doi:10.1063/1.3276681

Published 29 January 2010

KEYWORDS and PACS
Keywords
PACS
  • 03.65.Fd
    Algebraic methods in quantum mechanics
  • 03.65.Ta
    Foundations of quantum mechanics; measurement theory
  • 02.10.Ud
    Linear algebra
  • 03.67.-a
    Quantum information
  • YEAR: 2010
RELATED DATABASES

To view database links for this article,
you need to log in.
To view database links for this article,
you need to log in.
PUBLICATION DATA
Publisher:
AIP is a member of CrossRef AIP
Dénes Petz
Alfréd Rényi Institute of Mathematics, P.O. Box 127, H-1364 Budapest, Hungary and Department for Mathematical Analysis, BUTE, P.O. Box 91, H-1521 Budapest, Hungary
This paper is an overview of the concept of complementarity, the relation to state estimation, to Connes–Størmer conditional (or relative) entropy, and to uncertainty relation. Complementary Abelian and noncommutative subalgebras are analyzed. All the known results about complementary decompositions are described and several open questions are included. The paper contains only few proofs, typically references are given. ©2010 American Institute of Physics
History: Received 30 October 2009; accepted 1 December 2009; published 29 January 2010
Permalink: http://link.aip.org/link/?JMAPAQ/51/015215/1

REFERENCES (41)

For access to fully linked references, you need to log in. For access to fully linked references, you need to Log in.
  1. Accardi, L., “Some trends and problems in quantum probability,” Lect. Notes Math. 1055, 1 (1984).
  2. Bagan, E., Ballester, M. A., Gill, R. D., Monras, A., and Munoz-Tapia, R., “Optimal full estimation of qubit mixed states,” Phys. Rev. A 73, 032301 (2006).
  3. Baier, T. and Petz, D., Rep. Math. Phys. (in press).
  4. Bandyopadhyay, S., Boykin, P. O., Roychowdhury, V., and Vatan, F., “A new proof for the existence of mutually unbiased bases,” Algorithmica 34, 512 (2002).
  5. Brubeta, D., “Optimal eavesdropping in quantum cryptography with six states,” Phys. Rev. Lett. 81, 3018 (1998).
  6. Busch, P. and Lahti, P. J., “The complementarity of quantum observables: theory and experiment,” Riv. Nuovo Cim. 18, 1 (1995).
  7. Cassinelli, G. and Varadarajan, V. S., “On Accardi's notion of complementary observables,” Infinite Dimen. Anal., Quantum Probab., Relat. Top. 5, 135 (2002).
  8. Choda, M., “Relative entropy for maximal Abelian subalgebras of matrices and the entropy of antistochastic matrices,” Int. J. Math. 19, 767 (2008).
  9. Connes, A. and Størmer, E., “Entropy of II1 von Neumann algebras,” Acta Math. 134, 289 (1975).
  10. D'Ariano, G. M., Paris, M. G. A., and Sacchi, M. F., “Quantum tomography,” Adv. Imaging Electron Phys. 128, 205 (2003).
  11. Fischer, D. G. and Freyberger, M., “Estimating mixed quantum states,” Phys. Lett. A 273, 293 (2000).
  12. Kraus, K., “Complementary observables and uncertainty relations,” Phys. Rev. D 35, 3070 (1987).
  13. Krishna, M. and Parthasarathy, K. R., “An entropic uncertainty principle for quantum measurements,” Sankhya: Indian J. Statistics 64, 842 (2002).
  14. Maassen, H. and Uffink, I., “Generalized entropic uncertainty relations,” Phys. Rev. Lett. 60, 1103 (1988).
  15. Nathanson, M. and Ruskai, M. B., “Pauli diagonal channels constant on axes,” J. Phys. A: Math. Theor. 40, 8171 (2007).
  16. von Neumann, J., Mathematische Grundlagen der Quantenmechanik (Springer, Berlin, 1981).
  17. Neshveyev, S. and Størmer, E., Dynamical Entropy in Operator Algebras (Springer-Verlag, Berlin, 2006).
  18. Ohno, H., “Quasi-orthogonal subalgebras of matrix algebras,” Linear Algebr. Appl. 429, 2146 (2008).
  19. Ohno, H., Petz, D., and Szántó, A., “Quasi-orthogonal subalgebras of 4×4 matrices,” Linear Algebr. Appl. 425, 109 (2007).
  20. Ohno, H. and Petz, D., “Generalizations of Pauli channels,” Acta Math. Hung. 124, 165 (2009).
  21. Ohya, M. and Petz, D., Quantum Entropy and Its Use (Springer-Verlag, Heidelberg, 1993)
    22. Quantum Entropy and Its Use, 2nd ed. (Springer-Verlag, Heidelberg, 2004).
  22. Oppenheim, J., Horodecki, K., Horodecki, M., Horodecki, P., and Horodecki, R., “A new type of complementarity between quantum and classical information,” Phys. Rev. A 68, 022307 (2003).
  23. Pauli, W., General Principles of Quantum Mechanics (Springer, Berlin, 1980).
  24. Petz, D., “Complementarity in quantum systems,” Rep. Math. Phys. 59, 209 (2007).
  25. Petz, D., “Complementary subalgebras. Problems to solve,” Annales Univ. Sci. Budapest. Sect. Math.51, 117 (2008).
  26. Petz, D., Hangos, K. M., Szántó, A., and Szöllősi, F., “State tomography for two qubits using reduced densities,” J. Phys. A 39, 10901 (2006).
  27. Petz, D., Hangos, K. M., and Magyar, A., “Point estimation of states of finite quantum systems,” J. Phys. A 40, 7955 (2007).
  28. Petz, D. and Kahn, J., “Complementary reductions for two qubits,” J. Math. Phys. 48, 012107 (2007).
  29. Petz, D., Szántó, A., and Weiner, M., “Complementarity and the algebraic structure of 4-level quantum systems,” J. Infin. Dim. Analysis Quantum Prob. 12, 99 (2009).
  30. Petz, D. and Ruppert, L. (unpublished).
  31. Pittenger, A. O. and Rubin, M. H., “Mutually unbiased bases, generalized spin matrices and separability,” Linear Algebr. Appl. 390, 255 (2004).
  32. Popa, S., “Orthogonal pairs of *-subalgebras in finite von Neumann algebras,” J. Oper. Theory 9, 253 (1983).
  33. Rastegin, A. E., “Statement of uncertainty principle for quantum measurements in terms of the Rényi entropies,” e-print arXiv:0807.2691.
  34. Rédei, M., Quantum Logic in Algebraic Approach, Fundamental Theories of Physics (Kluwer Academic, Dordrecht, 1998), Vol. 91.
  35. Schwinger, J., “Unitary operator bases,” Proc. Natl. Acad. Sci. U.S.A. 46, 570 (1960).
  36. Tadej, W. and Zyczkowski, K., “A concise guide to complex Hadamard matrices,” Open Syst. Inf. Dyn. 13, 133 (2006).
  37. Watatani, Y. and Wierzbicki, J., “Commuting squares and relative entropy for two subfactors,” J. Funct. Anal. 133, 329 (1995).
  38. Wehner, S. and Winter, A., “Entropic uncertainty relations–A survey,” e-print arXiv:0907.3704.
  39. Weiner, M., “A gap for the maximum number of mutually unbiased bases,” e-print arXiv:0902.0635.
  40. Wootters, W. K. and Fields, B. D., “Optimal state determination by mutually unbiased measurements,” Ann. Phys. 191, 363 (1989).
  41. Weyl, H., Theory of Groups and Quantum Mechanics (Methuen, London, 1931).

CITING ARTICLES
For access to citing articles, you need to log in.
For access to citing articles, you need to Log in.
ADVERTISEMENT