Jump to ContentJump to Main Navigation
Show Summary Details
In This Section

Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

6 Issues per year


IMPACT FACTOR 2016: 0.346
5-year IMPACT FACTOR: 0.412

CiteScore 2016: 0.42

SCImago Journal Rank (SJR) 2015: 0.485
Source Normalized Impact per Paper (SNIP) 2015: 0.886

Mathematical Citation Quotient (MCQ) 2015: 0.27

Online
ISSN
1337-2211
See all formats and pricing
In This Section
Volume 67, Issue 2 (Apr 2017)

Issues

Linear algebraic proof of Wigner theorem and its consequences

Jáchym Barvínek
  • Department of MathematicsFaculty of Electrical Engineering Czech Technical University in Prague Technická 2 166 27 Prague 6 Czech Republic
/ Jan Hamhalter
  • Department of MathematicsFaculty of Electrical Engineering Czech Technical University in Prague Technická 2 166 27 Prague 6 Czech Republic
  • Email:
Published Online: 2017-04-28 | DOI: https://doi.org/10.1515/ms-2016-0273

Abstract

We present new proof of non-bijective Wigner theorem on symmetries of quantum systems using only basic linear algebra. It is based on showing that any non-zero Jordan ∗-homomorphism between matrix algebras preserving rank-one projections is implemented by either a unitary or an anitiunitary map. As a new application we extend hitherto known results on preservers of quantum relative entropy to infinite quantum systems.

Keywords: Jordan homomorphisms; Wigner theorem; relative quantum entropy

MSC 2010: Primary 15A86; Secondary 81P45

References

  • [1]

    Barvínek, J.: Quantum Entropy and its Preservation, Bachelor thesis, CVUT, 2013 (In Czech).

  • [2]

    Cassinelli, G.—De Vito, E.—Levrero, A.—Lahti, P.: Symmetry groups in quantum mechanics and the theorem of Wigner on the symmetry transformations, Rev. Math. Phys. 8 (1997), 921–941.

  • [3]

    Emch, G.: Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Dover Publishing, 1972.

  • [4]

    Gehér, P.: An elementary proof for the non-bijective version of Wigner’s theorem, Phys. Lett. A 387 (2014), 2054–2057. [Web of Science]

  • [5]

    Hamhalter, J.: Quantum Measure Theory, Kluwer Publishers, Dordrecht, Boston, London, 2003.

  • [6]

    Hanche-Olsen, H.—Størmer, E.: Jordan Operator Algebras, Pitman, 1984.

  • [7]

    Herstein, I. N.: Jordan homomorphisms, Trans. Amer. Math. Soc. 81 (1956), 331–341.

  • [8]

    Hou, J.: Rank-preserving linear maps on B(K), Sci. China Ser. A 32 (1989), 929–940.

  • [9]

    Jacobson, N.—Rickart, C.: Jordan homomorphisms of rings, Trans. Amer. Math. Soc. 69 (1950), 479–502.

  • [10]

    Kadison, R. V.: Isometries of operator algebras, Ann. of Math. 54 (1951), 325–338.

  • [11]

    Kadison, R. V.—Ringrose, J. R.: Fundamentals of the Theory of Operator Algebras American Mathematical Society, Vol. I, II, III, IV, 1994.

  • [12]

    Molnar, L.: Wigner’s unitary-antiunitary theorem via Herstein theorem on Jordan homomorphisms, J. Nat. Geom. 10 (1996), 137–148.

  • [13]

    Molnar, L.: Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces. Springer, 2007.

  • [14]

    Molnar, L.: Maps on states preserving the relative entropy. J. Math. Phys. 49 (2008), 0321114. [Web of Science]

  • [15]

    Molnar, L.—Szokol, P.: Maps on states preserving the relative entropy II., Linear Algebra Appl. 432 (2010), 3343–3350.

  • [16]

    Nielsen, M. A.—Chuang, I. J.: Quantum Computation and Quantum Information, Cambridge University Press, 2001.

  • [17]

    Palmer, T. W.: Banch algebras and the general theory of *-algebras, I, II, University Press, Cambridge, 1994.

  • [18]

    Ohya, M.—Petz, D.: Quantum Entropy and Its Use. Texts and Monographs in Physics, Springer Verlag, 1993.

  • [19]

    Petz, D.: Quantum Information Theory and Quantum Statistics, Springer Verlag, Berlin, Heidelberg, 2008.

  • [20]

    Størmer, E.: On the Jordan structure of C*-algebras, Trans. Amer. Math. Soc. 120 (1965), 438–447.

  • [21]

    Størmer, E.: Positive Linear Maps of Operator Algebras, Springer, 2013.

  • [22]

    Simon, R.—Mukunda, N.—Chaturvedi, S.—Srinavasan, V.: Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics, Phys. Lett. A 372 (2008), 6847–6852. [Web of Science]

  • [23]

    Simon, R.—Mukunda, N.—Chaturvedi, S.—Srinavasan, V.—Hamhalter, J.: Comment on: Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics [Phys. Lett. A 372 (2008), 6847], Phys. Lett. A 378 (2014), 2332–2335.

  • [24]

    Weinberger, S.: The Quantum Theory of Fields, Cambridge, USA, 1995.

  • [25]

    Wigner, E. P.: Group Theory and Its Applications to the Quantum Theory of Atomic Spectra, Academic Press Inc., New York, 1959.

About the article


Received: 2014-04-14

Accepted: 2015-05-15

Published Online: 2017-04-28

Published in Print: 2017-04-25



Citation Information: Mathematica Slovaca, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2016-0273. Export Citation

Comments (0)

Please log in or register to comment.
Log in