Matrix monotonicity and self-concordance: how to handle quantum entropy in optimization problems Leonid Faybusovich ∗ Takashi Tsuchiya † August 2014 Abstract Let g be a continuously differentiable function whose derivative is matrix monotone on positive semi-axis. Such a function induces a function φ(x)= tr(g(x)) on the cone of squares of an arbitrary Euclidean Jordan algebra. We show that φ(x) − ln det(x) is a self-concordant function on the interior of the cone. We also show that − ln(t−φ(x))−ln det(x) is  5 3 (r +1)-self-concordant barrier on the epigraph of φ, where r is the rank of the Jordan algebra. The case φ(x)= tr(x ln x) is discussed in detail. Key words: quantum entropy, matrix monotonicity, self-concordance 1 Introduction A substantial amount of optimization problems arising in quantum information theory,quantum statistical physics ,information geometry ([9]) and other areas (for the impressive list of various applications see [2])requires dealing with the so-called quantum or von Neumann entropy which is the function of the form Tr(XLnX ), where X is positive semi-definite complex Hermitian matrix. In present paper we propose a self-concordant barrier for this function which, in principle, allows one to include the above mentioned problems into general interior point polynomial framework. We develop the corresponding theory within the formalism of Euclidean Jordan algebras which indicates possible generalizations of this concept. Moreover, proposed approach allows us to consider relatively broad class of functions, namely, primitives of matrix monotone functions on positive semi-axis. It turns out that the standard lndet barrier is compatible (in the sense of the theory of self-concordant functions) with functions from this class. The plan of the paper is as follows. In section 2 we briefly describe some Jordan-algebraic concepts. In section 3 we introduce a class functions on the cone of squares of an Euclidean Jordan algebra associated with matrix monotone functions on positive semi-axis. We then show that the standard lndet self-concordant barrier is compatible with every * University of Notre Dame, Department of Mathematics (Email: leonid.faybusovich.1@nd.edu) This author is supported in part by Simmons Foundation Grant 275013. † National Graduate Institute for Policy Studies (Email: tsuchiya@grips.ac.jp). This author is supported in part with Grant-in-Aid for Scientific Research (B), 24310112 from the Japan Society for the Promotion of Sciences. 1