# Self-Concordance and Matrix Monotonicity with Applications to Quantum Entanglement Problems

Let $V$ be an Euclidean Jordan algebra and $\Omega$ be a cone of invertible squares in $V$. Suppose that $g:\mathbb{R}_{+} \to \mathbb{R}$ is a matrix monotone function on the positive semiaxis which naturally induces a function $\tilde{g}: \Omega \to V$. We show that $-\tilde{g}$ is compatible (in the sense of Nesterov-Nemirovski) with the standard self-concordant barrier $B(x) = -\ln\det(x)$ on $\Omega$. As a consequence, we show that for any $c \in \Omega$, the functions of the form $-\tr(c\tilde{g}(x)) + B(x)$ are self-concordant on $\Omega$. In particular, the function $x \mapsto -\tr(c\ln x)$ is a self-concordant barrier function on $\Omega$. Using these results, we apply a long-step path-following algorithm developed in [L.~Faybusovich and C.~Zhou Long-step path-following algorithm for solving symmetric programming problems with nonlinear objective functions. \emph{Comput Optim Appl}, 72(3):769-795, 2019] to a number of important optimization problems arising in quantum information theory. Results of numerical experiments and comparisons with existing methods are presented.

## Citation

preprint, University of Notre Dame, April, 2019