Jointly constrained semidefinite bilinear programming with an application to Dobrushin curves

Abstract

We propose a branch-and-bound algorithm for minimizing a bilinear functional of the form \[ f(X,Y) = \mathrm{tr}((X\otimes Y)Q)+\mathrm{tr}(AX)+\mathrm{tr}(BY) , \] of pairs of Hermitian matrices $(X,Y)$ restricted by joint semidefinite programming constraints. The functional is parametrized by self-adjoint matrices $Q$, $A$ and $B$. This problem generalizes that of a bilinear program, where $X$ and $Y$ 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.

arXiv comment30 pages, 12 figures

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Cited by in corpus (5)

• Semidefinite programming hierarchies for constrained bilinear optimization
• Fidelity-Based Smooth Min-Relative Entropy: Properties and Applications
• Symmetric distinguishability as a quantum resource
• Upper Bounds on the Distillable Randomness of Bipartite Quantum States
• $k$-Positive Maps: New Characterizations and a Generation Method