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Maps on states preserving generalized entropy of convex combinations. (English) Zbl 1370.15028
Summary: Let $$S(H)$$ be the set of all linear positive-semidefinite self-adjoint Trace-one operators (states) on $$H$$ where $$H$$ is an at least two-dimensional finite-dimensional real or complex Hilbert space or at least three-dimensional left quaternionic Hilbert space of dimension $$n$$. Given a strictly convex function $$f : [0, 1] \mapsto \mathbb{R}$$, for any $$\rho \in S(H)$$ we define $$F(\rho) = \sum_i f(\lambda_i)$$, where $$\lambda_1, \lambda_2, \ldots, \lambda_n$$ are the eigenvalues of $$\rho$$ counted with multiplicities. In this note, we completely describe maps $$\phi : S(H) \rightarrow S(H)$$ having the property $$F(t \rho +(1 - t) \sigma) = F(t \phi(\rho) +(1 - t) \phi(\sigma))$$ for all $$t \in [0, 1]$$ and every $$\rho, \sigma \in S(H)$$. It turns out that $$\phi(\rho) = U \rho U^\ast$$, $$\rho \in S(H)$$, where $$U$$ is a real-linear isometry of $$H$$. Note that there is no surjectivity assumption and that our result in particular improves the description of maps preserving the von Neumann entropy of convex combinations of states in the complex Hilbert space. It can as well be applied to preserving Schatten or some other strictly convex norms of convex combinations of states.

##### MSC:
 15A86 Linear preserver problems 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
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##### References:
 [1] Bhatia, R., Matrix analysis, (1997), Springer-Verlag New York [2] Djoković, D.Ž.; Smith, B. H., Quaternionic matrices: unitary similarity, simultaneous triangularization and some trace identities, Linear Algebra Appl., 428, 890-910, (2008) · Zbl 1135.15012 [3] Drnovšek, R., The von Neumann entropy and unitary equivalence of quantum states, Linear Multilinear Algebra, 61, 10, 1391-1393, (2013) · Zbl 1321.15013 [4] Fošner, A.; Kuzma, B.; Kuzma, T.; Sze, N.-S., Maps preserving matrix pairs with zero Jordan product, Linear Multilinear Algebra, 59, 1, 507-529, (2011) · Zbl 1222.15031 [5] Gaál, M., Maps preserving a new version of quantum f-divergence, Banach J. Math. Anal., (2017) · Zbl 06841252 [6] Gehér, Gy. P., An elementary proof for the non-bijective version of Wigner’s theorem, Phys. Lett. A, 378, 30, 2054-2057, (2014) · Zbl 1331.46066 [7] He, K.; Yuan, Q.; Hou, J., Entropy-preserving maps on quantum states, Linear Algebra Appl., 467, 243-253, (2015) · Zbl 1320.47039 [8] Karamata, J., Sur une inégalité relative aux fonctions convexes, Publ. Math. Univ. Belgrade, 1, 145-148, (1932), (in French) · JFM 58.0211.01 [9] Li, Y.; Busch, P., Von Neumann entropy and majorization, J. Math. Anal. Appl., 408, 1, 384-393, (2013) · Zbl 1308.81050 [10] Liping, H., Extremum principles of eigenvalues for self-conjugate quaternion matrix, J. Math. Res. Exposition, 17, 1, 101-104, (1997) · Zbl 0912.15013 [11] Molnár, L., Orthogonality preserving transformations on indefinite inner product spaces: generalization of uhlhorn’s version of Wigner’s theorem, J. Funct. Anal., 194, 2, 248-262, (2002) · Zbl 1010.46023 [12] Molnár, L.; Nagy, G.; Szokol, P., Maps on density operators preserving quantum f-divergences, Quantum Inf. Process., 12, 2309-2323, (2013) · Zbl 1270.81021 [13] Neshveyev, S.; Størmer, E., Dynamical entropy in operator algebras, (2006), Springer · Zbl 1109.46002 [14] Nielsen, M.; Chuang, I., Quantum computation and quantum information, (2010), Cambridge University Press [15] Rockafellar, R. T., Convex analysis, (1970), Princeton UP · Zbl 0202.14303 [16] Rodman, L., Topics in quaternion linear algebra, (2014), Princeton UP · Zbl 1304.15004 [17] Rossignoli, R.; Canosa, N.; Ciliberti, L., Generalized entropic measures of quantum correlations, (2011) · Zbl 1279.81018 [18] Schumacher, B., Quantum coding, Phys. Rev. A, 51, 4, 2738, (1995) [19] Šemrl, P., Generalized symmetry transformations on quaternionic indefinite inner product spaces: an extension of quaternionic version of Wigner’s theorem, Comm. Math. Phys., 242, 3, 579-584, (2003) · Zbl 1053.46012 [20] Turnšek, A., A variant of Wigner’s functional equation, Aequationes Math., 89, 949-956, (2015) · Zbl 1321.39025 [21] Xi, B.-J., A general form of the minimum-maximum theorem for eigenvalues of self-conjugate quaternion matrices, Nei Monggol Daxue Xuebao Ziran Kexue, 22, 4, 455-458, (1991) · Zbl 1332.15087
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