Quantum Rényi relative entropies on density spaces of C⁎-algebras: Their symmetries and their essential difference

Abstract

We extend the definitions of different types of quantum Rényi relative entropy from the finite dimensional setting of density matrices to density spaces of C * -algebras. We show that those quantities (which trivially coincide in the classical commutative case) are essentially different on non-commutative algebras in the sense that none of them can be transformed to another one by any surjective transformation between density spaces. Besides, we determine the symmetry groups of density spaces corresponding to each of those quantum Rényi relative entropies and find that they are identical. Similar results concerning the Umegaki and the Belavkin-Staszewksi relative entropies are also presented.2010 Mathematics Subject Classification. Primary: 46L40, 47B49, 81P45. We next collect some useful properties of Jordan *-isomorphisms that we will use in what follows. First, any Jordan *-isomorphism J :holds for every nonnegative integer n, see 6.3.2 Lemma in [16]. In particular, J is unital meaning that J sends the identity to the identity. Since J is clearly positive (in fact, it preserves the order between self-adjoint elements in both directions), it is bounded. Indeed, more is true: J is an isometry with respect to the C * -norm. By Proposition 1.3 in [22], J preserves invertibility, namely we have J (X −1 ) = J (X ) −1 for every invertible element X ∈ A . It follows that J preserves the spectrum and, using continuous function calculus, from (18) we deduce that J ( f (X )) = f (J (X )) holds for any self-adjoint element X ∈ A s and continuous real function f on the spectrum of X .

Transformations preserving the norm of means between positive cones of general and commutative $C^*$-algebras

“…In particular, it follows from Kadison's extension [17] of the classical Banach-Stone theorem that the positive surjective linear isometries between C * -algebras are exactly the Jordan * -isomorphisms. We also have the following special case of Theorem 9 in [12] (also see Theorem 16 in [23]).…”

“…If, on the positive definite cone A −1 + of a unital C * -algebra A, we replace the arithmetic mean by the geometric mean and the C * -norm by the trace norm (corresponding to a faithful tracial positive linear functional), then we have got a quantity which is a variant of quantum Rényi relative entropy. Hence, the maps which preserve that quantity are kinds of quantum symmetries, and the description of them is presented in [23].…”

Maps on positive cones in operator algebras preserving power means

“…This means that ψ is a dilation (or, in other words, homothety) between the positive definite cones A ++ and B ++ . We proved in Theorem 18 in [16] that the existence of a non-isometric dilation between the positive definite cones of C *algebras implies that the underlying algebras are necessarily commutative. This completes the proof of the statement.…”

On Some Properties of the Norm of the Spectral Geometric Mean

“…In [8], the authors studied the maps preserving the spectral geometric mean and many interesting results are obtained. There are many interesting and important results related to the norms of means (see [3] [9] [11] [13] and references therein).…”