Semidefinite programming hierarchies for constrained bilinear optimization

Abstract We give asymptotically converging semidefinite programming hierarchies of outer bounds on bilinear programs the form $${\mathrm {Tr}}\big [H(D\otimes E)\big ]$$ Tr [ H ( D ? E ) ] , maximized with respect to constraints D and E . Applied problem approximate error correction in quantum information theory, this gives efficiently computable success probability codes any dimension. The first level our corresponds a previously studied relaxation (Leung Matthews IEEE Trans Inf Theory 61(8):4486, 2015) positive partial transpose can be added sufficient criterion for exact convergence at given hierarchy. To quantify worst case speed sum-of-squares hierarchies, we derive novel de Finetti theorems that allow imposing linear approximating state. In particular, finite channels, quantifying closeness convex hull product channels as well local operations classical forward communication assisted channels. As special constitutes version Fuchs-Schack-Scudo’s asymptotic theorem Finally, proof methods answer question Brandão Harrow (Proceedings forty-fourth annual ACM symposium theory computing, STOC’12, p 307, 2012) by improving approximation factor no symmetry from $$O(d^{k/2})$$ O d k / 2 {poly}}(d,k)$$ poly , where d denotes dimension k number copies.