2016/09/16 by Stephen Brierley, Miguel Navascues, Miguel Navascués +5 · 3 citations
Computer Science · Mathematics · Physics and Astronomy · #Advanced Optimization Algorithms Research #FOS: Mathematics #FOS: Physical sciences #Optimization and Control (math.OC) #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Physics (quant-ph) #math.OC #quant-ph
paper · pdf · doi:10.48550/arxiv.1609.05011
Added reference to the related paper arXiv:1609.06269
openalex publication_date 2016/09/16 · arxiv created 2017/01/05 · arxiv updated 2017/01/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present a scheme, based on Gilbert's algorithm for quadratic minimization [SIAM J. Contrl., vol. 4, pp. 61-80, 1966], to prove separation between a point and an arbitrary convex set S⊂ℝn via calls to an oracle able to perform linear optimizations over S. Compared to other methods, our scheme has almost negligible memory requirements and the number of calls to the optimization oracle does not depend on the dimensionality n of the underlying space. We study the speed of convergence of the scheme under different promises on the shape of the set S and/or the location of the point, validating the accuracy of our theoretical bounds with numerical examples. Finally, we present some applications of the scheme in quantum information theory. There we find that our algorithm out-performs existing linear programming methods for certain large scale problems, allowing us to certify nonlocality in bipartite scenarios with upto 42 measurement settings. We apply the algorithm to upper bound the visibility of two-qubit Werner states, hence improving known lower bounds on Grothendieck's constant KG(3). Similarly, we compute new upper bounds on the visibility of GHZ states and on the steerability limit of Werner states for a fixed number of measurement settings.