Self-concordance and matrix monotonicity with applications to quantum entanglement problems

Leonid Faybusovich and Cunlu Zhou

Applied Mathematics and Computation, 2020, vol. 375, issue C

Abstract: Let V be a Euclidean Jordan algebra and Ω be a cone of invertible squares in V. Suppose that g:R+→R is a matrix monotone function on the positive semiaxis which naturally induces a function g˜:Ω→V. We show that −g˜ is compatible (in the sense of Nesterov–Nemirovski) with the standard self-concordant barrier B(x)=−lndet(x) on Ω. As a consequence, we show that for any c ∈ Ω, the functions of the form −tr(cg˜(x))+B(x) are self-concordant on Ω. In particular, the function x↦−tr(clnx) is a self-concordant barrier function on Ω. 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. 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.

Keywords: Self-concordant functions; Matrix monotone functions; Quantum entanglement; Quantum relative entropy; Interior-point method; Long-step path-following (search for similar items in EconPapers)
Date: 2020
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:375:y:2020:i:c:s0096300320300400

DOI: 10.1016/j.amc.2020.125071

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