The simplest proof of Lieb concavity theorem

Convex Entire Noncommutative Functions are Polynomials of Degree Two or Less

“…In another direction, variants of logarithms related to entropy and relative entropy are (even in several variables) matrix convex on positive matrices [Eff09,ENG11]. Applications and new proofs can be found in [Auj11,NEG13]. However, in keeping with our results here, none of these examples can extend to be real entire and thus each exhibits some kind of singular behavior.…”

Lieb's concavity theorem, matrix geometric means, and semidefinite optimization

“…where indicates the Löwner partial order on Hermitian matrices (i.e., A B ⇔ A − B positive semidefinite). This remarkable fact can be used to give a simple proof of Lieb's concavity theorem, see e.g., [NEG13]. The matrix geometric mean was recently shown in [Sag13] to have a semidefinite programming formulation.…”

Bounds of some relative operator entropies in a general form

“…concavity) of the perspective and generalized perspective functions are established. For some recent results on this subject we refer the readers to see [12,15,17]. Kubo and Ando [11] discussed the axiomatic theory for connections and established the existence of an affine order isomorphism between the class of connections and the class of positive operator monotone functions.…”