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Maps on Quantum States in $C^{\ast }$ -algebras Preserving von Neumann Entropy or Schatten $p$ -norm of Convex Combinations

  • Marcell Gaál (a1)
Abstract

Very recently, Karder and Petek completely described maps on density matrices (positive semidefinite matrices with unit trace) preserving certain entropy-like convex functionals of any convex combination. As a result, maps could be characterized that preserve von Neumann entropy or Schatten $p$ -norm of any convex combination of quantum states (whose mathematical representatives are the density matrices). In this note we consider these latter two problems on the set of invertible density operators, in a much more general setting, on the set of positive invertible elements with unit trace in a $C^{\ast }$ -algebra.

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This work was partially supported by the Hungarian National Research, Development and Innovation Office – NKFIH Reg.No. K115383.

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[1] Beneduci, R. and Molnár, L., On the standard K-loop structure of positive invertible elements in a C -algebra . J. Math. Anal. Appl. 420(2014), 551562. https://doi.org/10.1016/j.jmaa.2014.05.009.
[2] Dixmier, J., Von Neumann algebras. North-Holland Mathematical Library, 27, North-Holland Publishing Company, Amsterdam-New York, 1981.
[3] Fack, T. and Kosaki, H., Generalized s-numbers of 𝜏-measurable operators . Pacific J. Math. 123(1986), 269300. https://doi.org/10.2140/pjm.1986.123.269.
[4] Farenick, D., Jaques, S., and Rahaman, M., The fidelity of density operators in an operator-algebraic framework . J. Math. Phys. 57(2016), 102202. https://doi.org/10.1063/1.4965876.
[5] Herstein, I. N., Jordan homomorphisms . Trans. Amer. Math. Soc. 81(1956), 331341. https://doi.org/10.1090/S0002-9947-1956-0076751-6.
[6] Jenčová, A., Geodesic distances on density matrices . J. Math. Phys. 45(2004), 17871794. https://doi.org/10.1063/1.1689000.
[7] Kadison, R. V. and Ringrose, J. R., Fundamentals of the theory of operator algebras . Vol II., Pure and Applied Mathematics, 100, Academic Press, Orlando, FL, 1986.
[8] Karder, M. and Petek, T., Maps on quaternion states preserving generalized entropy of convex combination . Linear Algebra Appl. 532(2017), 8698. https://doi.org/10.1016/j.laa.2017.06.003.
[9] Molnár, L., Maps on the positive definite cone of a C*-algebra preserving certain quasi-entropies . J. Math. Anal. Appl. 447(2017), 206221. https://doi.org/10.1016/j.jmaa.2016.09.067.
[10] Molnár, L. and Nagy, G., Transformations on density operators that leave the Holevo bound invariant . Int. J. Theor. Phys. 53(2014), 32733278. https://doi.org/10.1007/s10773-013-1638-8.
[11] Palmer, T. W., Banach algebras and the general theory of ∗-algebras . Vol. I. Encyclopedia of Mathematics and its Applications, 49, Cambridge University Press, Cambridge, 1994. https://doi.org/10.1017/CBO9781107325777.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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