Jointly Constrained Semidefinite Bilinear Programming With an Application to Dobrushin Curves
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Abstract
We propose a branch-and-bound algorithm for minimizing a bilinear functional of the form <inline-formula> <tex-math notation="LaTeX">$f(X,Y) = \mathrm {tr}((X\otimes Y)Q)+ \mathrm {tr}(AX)+ \mathrm {tr}(BY) $ </tex-math></inline-formula>, of pairs of Hermitian matrices <inline-formula> <tex-math notation="LaTeX">$(X,Y)$ </tex-math></inline-formula> restricted by joint semidefinite programming constraints. The functional is parametrized by self-adjoint matrices <inline-formula> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$A$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$B$ </tex-math></inline-formula>. This problem generalizes that of a bilinear program, where <inline-formula> <tex-math notation="LaTeX">$X$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$Y$ </tex-math></inline-formula> belong to polyhedra. The algorithm converges to a global optimum and yields upper and lower bounds on its value in every step. Various problems in quantum information theory can be expressed in this form. As an example application, we compute Dobrushin curves of quantum channels, giving upper bounds on classical coding with energy constraints.