Maps on positive definite operators preserving the quantum \(\chi _\alpha ^2\)-divergence
- 154 Downloads
We describe the structure of all bijective maps on the cone of positive definite operators acting on a finite and at least two-dimensional complex Hilbert space which preserve the quantum \(\chi _\alpha ^2\)-divergence for some \(\alpha \in [0,1]\). We prove that any such transformation is necessarily implemented by either a unitary or an antiunitary operator. Similar results concerning maps on the cone of positive semidefinite operators as well as on the set of all density operators are also derived.
KeywordsPositive definite operators Quantum \(\chi _\alpha ^2\)-divergence Preservers
Mathematics Subject ClassificationPrimary 46N50 47B49
Gy. P. Gehér and L. Molnár were supported by the “Lendület” Program (LP2012-46/2012) of the Hungarian Academy of Sciences and by the National Research, Development and Innovation Office - NKFIH, Grant No. K115383. D. Virosztek was supported by the “Lendület” Program (LP2012-46/2012) of the Hungarian Academy of Sciences, by the National Research, Development and Innovation Office - NKFIH, Grant No. K104206, and by the “For the Young Talents of the Nation” scholarship program (NTP-EFÖ-P-15-0481) of the Hungarian State. The project was also supported by the joint venture of Taiwan and Hungary MOST-HAS, Grant No. 104-2911-1-110-508.
- 7.Molnár, L.: Selected Preserver Problems on Algebraic Structures of Linear Operators and On Function Spaces, Lecture Notes in Mathematics, vol. 1895, 236 p, Springer, Berlin, Heidelberg (2007)Google Scholar
- 14.Petz, D., Ghinea, C.: Introduction to quantum Fisher information. In: Rebolledo, R., Orszag, M. (eds.) Quantum Probability and Related Topics, QP-PQ: Quantum Probability and White Noise Analysis, vol. 27, pp. 261–281. World Scientific Publishing, Hackensack (2011)Google Scholar