Universal bound on the cardinality of local hidden variables in networks

We present an algebraic description of the sets of local correlations in arbitrary networks, when the parties have finite inputs and outputs. We consider networks generalizing the usual Bell scenarios by the presence of multiple uncorrelated sources. We prove a finite upper bound on the cardinality of the value sets of the local hidden variables. Consequently, we find that the sets of local correlations are connected, closed and semialgebraic, and bounded by tight polynomial Bell-like inequalities.

[1]  Miguel Navascués,et al.  The inflation technique solves completely the classical inference problem , 2017 .

[2]  S. Basu,et al.  Algorithms in real algebraic geometry , 2003 .

[3]  Imre Bárány,et al.  Notes About the Carathéodory Number , 2012, Discrete & Computational Geometry.

[4]  Peter J Seiler,et al.  SOSTOOLS: Sum of squares optimization toolbox for MATLAB , 2002 .

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

[6]  Nicolas Gisin,et al.  Nonlinear Bell Inequalities Tailored for Quantum Networks. , 2015, Physical review letters.

[7]  T. Fritz,et al.  Entropic approach to local realism and noncontextuality , 2012, 1201.3340.

[8]  Armin Tavakoli,et al.  Quantum correlations in connected multipartite Bell experiments , 2015, 1509.08491.

[9]  Nihat Ay,et al.  Information-theoretic inference of common ancestors , 2010, Entropy.

[10]  Zeilinger,et al.  Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits , 2000, Physical review letters.

[11]  N. Gisin,et al.  A relevant two qubit Bell inequality inequivalent to the CHSH inequality , 2003, quant-ph/0306129.

[12]  Nicholas I. M. Gould,et al.  SIAM Journal on Optimization , 2012 .

[13]  Robert W. Spekkens,et al.  Causal Inference via Algebraic Geometry: Feasibility Tests for Functional Causal Structures with Two Binary Observed Variables , 2015, 1506.03880.

[14]  Rafael Chaves,et al.  Polynomial Bell Inequalities. , 2015, Physical review letters.

[15]  N. Gisin,et al.  Bilocal versus nonbilocal correlations in entanglement-swapping experiments , 2011, 1112.4502.

[16]  Hendrik B. Geyer,et al.  Journal of Physics A - Mathematical and General, Special Issue. SI Aug 11 2006 ?? Preface , 2006 .

[17]  Matthew F Pusey,et al.  Theory-independent limits on correlations from generalized Bayesian networks , 2014, 1405.2572.

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

[19]  Robert W. Spekkens,et al.  The Inflation Technique for Causal Inference with Latent Variables , 2016, Journal of Causal Inference.

[20]  J. S. BELLt.  Einstein-Podolsky-Rosen Paradox , 2018 .

[21]  William Slofstra,et al.  THE SET OF QUANTUM CORRELATIONS IS NOT CLOSED , 2017, Forum of Mathematics, Pi.

[22]  Itamar Pitowsky,et al.  Correlation polytopes: Their geometry and complexity , 1991, Math. Program..

[23]  J. Rotman.  An Introduction to Algebraic Topology , 1957 .

[24]  Armin Tavakoli,et al.  Nonlocal correlations in the star-network configuration , 2014, 1409.5702.

[25]  D. Gross,et al.  Causal structures from entropic information: geometry and novel scenarios , 2013, 1310.0284.

[26]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[27]  Elie Wolfe,et al.  Identifying nonconvexity in the sets of limited-dimension quantum correlations , 2015, 1506.01119.

[28]  A. Tavakoli.  Bell-type inequalities for arbitrary noncyclic networks , 2015, 1510.05977.

[29]  A. Falcon.  Physics I.1 , 2018 .

[30]  Pablo A. Parrilo,et al.  Semidefinite programming relaxations for semialgebraic problems , 2003, Math. Program..

[31]  Kiel T. Williams,et al.  Extreme quantum entanglement in a superposition of macroscopically distinct states. , 1990, Physical review letters.

[32]  J. Lofberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004, 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508).

[33]  T. Fritz.  Beyond Bell's theorem: correlation scenarios , 2012, 1206.5115.

[34]  William Slofstra,et al.  Tsirelson’s problem and an embedding theorem for groups arising from non-local games , 2016, Journal of the American Mathematical Society.

[35]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[36]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[37]  N. Gisin,et al.  Characterizing the nonlocal correlations created via entanglement swapping. , 2010, Physical review letters.