Convex separation from convex optimization for large-scale problems

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 $S\subset\mathbb{R}^{n}$ 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 $K_G(3)$. 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.

[1]  Stefan Wolf,et al.  Can non-local correlations be discriminated in polynomial time? , 2016, ArXiv.

[2]  Teiko Heinosaari,et al.  Adaptive strategy for joint measurements , 2016, 1604.08724.

[3]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[4]  Jacek Gondzio,et al.  Solving large-scale optimization problems related to Bell's Theorem , 2012, J. Comput. Appl. Math..

[5]  Werner,et al.  Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. , 1989, Physical review. A, General physics.

[6]  H. Crowder,et al.  Solving Large-Scale Symmetric Travelling Salesman Problems to Optimality , 1980 .

[7]  S. Wehner,et al.  Bell Nonlocality , 2013, 1303.2849.

[8]  E. Gilbert.  An Iterative Procedure for Computing the Minimum of a Quadratic Form on a Convex Set , 1966 .

[9]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[10]  V. Scarani,et al.  Testing the dimension of Hilbert spaces. , 2008, Physical review letters.

[11]  I. Pitowsky.  Quantum Probability ― Quantum Logic , 1989 .

[12]  R. Cleve,et al.  Nonlocality and communication complexity , 2009, 0907.3584.

[13]  J. Bell.  On the Einstein-Podolsky-Rosen paradox , 1964 .

[14]  William H. Cunningham,et al.  Testing membership in matroid polyhedra , 1984, J. Comb. Theory, Ser. B.

[15]  T. V'ertesi,et al.  More efficient Bell inequalities for Werner states , 2008, 0806.0096.

[16]  Lawrence M. Ioannou,et al.  Convex Separation from Optimization via Heuristics , 2006, ArXiv.

[17]  Xianqing Li-Jost,et al.  Towards Grothendieck constants and LHV models in quantum mechanics , 2015, 1501.05507.

[18]  Jacek Gondzio,et al.  Interior point methods 25 years later , 2012, Eur. J. Oper. Res..

[19]  Valerio Scarani.  The device-independent outlook on quantum physics (lecture notes on the power of Bell's theorem) , 2013 .

[20]  N. Gisin,et al.  Grothendieck's constant and local models for noisy entangled quantum states , 2006, quant-ph/0606138.

[21]  D Cavalcanti,et al.  Quantum steering: a review with focus on semidefinite programming , 2016, Reports on progress in physics. Physical Society.

[22]  Adrien Feix,et al.  Characterizing finite-dimensional quantum behavior , 2015, 1507.07521.

[23]  D. J. Saunders,et al.  Experimental EPR-steering using Bell-local states , 2009, 0909.0805.