Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-14T01:30:26.372Z Has data issue: false hasContentIssue false
  • English
  • Français

TRANSFORMATIONS ON DENSITY OPERATORS PRESERVING GENERALISED ENTROPY OF A CONVEX COMBINATION

Part of: Basic linear algebra

Published online by Cambridge University Press:  03 May 2018

MARCELL GAÁL  and
GERGŐ NAGY
MARCELL GAÁL*
Affiliation:
Functional Analysis Research Group, University of Szeged, H-6720 Szeged, Aradi vértanúk tere 1, Hungary email [email protected]
GERGŐ NAGY
Affiliation:
Institute of Mathematics, University of Debrecen, H-4002 Debrecen, PO Box 400, Hungary email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We aim to characterise those transformations on the set of density operators (which are the mathematical representatives of the states in quantum information theory) that preserve a so-called generalised entropy of one fixed convex combination of operators. The characterisation strengthens a recent result of Karder and Petek where the preservation of the same quantity was assumed for all convex combinations.

Keywords

entropydensity operatorsquantum statespreserversconvex combination

MSC classification

Primary: 15A86: Linear preserver problems
Secondary: 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 98 , Issue 1 , August 2018 , pp. 102 - 108
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

This work was supported by the National Research, Development and Innovation Office NKFIH Reg. No. K-115383.

References

Barvínek, J. and Hamhalter, J., ‘Linear algebraic proof of Wigner theorem and its consequences’, Math. Slovaca 67 (2017), 371–386.Google Scholar
Chen, H. Y., Gehér, Gy. P., Liu, C. N., Molnár, L., Virosztek, D. and Wong, N. C., ‘Maps on positive definite operators preserving the quantum 𝜒𝛼 2 -divergence’, Lett. Math. Phys. 107 (2017), 2267–2290.CrossRefGoogle Scholar
Drnovšek, R., ‘The von Neumann entropy and unitary equivalence of quantum states’, Linear Multilinear Algebra 61 (2013), 1391–1393.Google Scholar
Gaál, M. and Molnár, L., ‘Transformations on density operators and on positive definite operators preserving the quantum Rényi divergence’, Period. Math. Hung. 74 (2017), 88–107.Google Scholar
Gaál, M. and Nagy, G., ‘Preserver problems related to quasi-arithmetic means of invertible positive operators’, Integr. Equ. Oper. Theory 90 (2018), to appear.Google Scholar
Gehér, Gy. P., ‘An elementary proof for the non-bijective version of Wigner’s theorem’, Phys. Lett. A 378 (2014), 2054–2057.Google Scholar
Győry, M., ‘A new proof of Wigner’s theorem’, Rep. Math. Phys. 54 (2004), 159–167.Google Scholar
Karder, M. and Petek, T., ‘Maps on states preserving generalized entropy of convex combinations’, Linear Algebra Appl. 532 (2017), 86–98.Google Scholar
Li, Y. and Busch, P., ‘Von Neumann entropy and majorization’, J. Math. Anal. Appl. 408 (2013), 384–393.Google Scholar
Molnár, L., Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces, Lecture Notes in Mathematics, 1895 (Springer, Berlin, 2007).Google Scholar
Molnár, L. and Nagy, G., ‘Isometries and relative entropy preserving maps on density operators’, Linear Multilinear Algebra 60 (2012), 93–108.Google Scholar
Molnár, L. and Nagy, G., ‘Transformations on density operators that leave the Holevo bound invariant’, Int. J. Theor. Phys. 53 (2014), 3273–3278.Google Scholar
Molnár, L. and Szokol, P., ‘Transformations preserving norms of means of positive operators and nonnegative functions’, Integral Equations Operator Theory 83 (2015), 271–290.Google Scholar
Nagy, G., ‘Preservers for the p-norm of linear combinations of positive operators’, Abstr. Appl. Anal. 2014 (2014), 434121.Google Scholar
Neshveyev, S. and Størmer, E., Dynamical Entropy in Operator Algebras (Springer, Berlin, 2006).Google Scholar
Nielsen, M. and Chuang, I., Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2010).Google Scholar
Virosztek, D., ‘Maps on quantum states preserving Bregman and Jensen divergences’, Lett. Math. Phys. 106 (2016), 1217–1234.Google Scholar