Bounding the Set of Classical Correlations of a Many-Body System.

We present a method to certify the presence of Bell correlations in experimentally observed statistics, and to obtain new Bell inequalities. Our approach is based on relaxing the conditions defining the set of correlations obeying a local hidden variable model, yielding a convergent hierarchy of semidefinite programs (SDP's). Because the size of these SDP's is independent of the number of parties involved, this technique allows us to characterize correlations in many-body systems. As an example, we illustrate our method with the experimental data presented in Science 352, 441 (2016)SCIEAS0036-807510.1126/science.aad8665.

[1]  N. Gisin,et al.  Looking for symmetric Bell inequalities , 2010, 1004.4146.

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

[3]  Carsten Lund,et al.  Nondeterministic exponential time has two-prover interactive protocols , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[4]  S. Pironio,et al.  All Clauser–Horne–Shimony–Holt polytopes , 2014 .

[5]  J. Aron.  About Hidden Variables , 1969 .

[6]  A. Acín,et al.  A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations , 2008, 0803.4290.

[7]  M. Lewenstein,et al.  Detecting nonlocality in many-body quantum states , 2014, Science.

[8]  Zizhu Wang,et al.  Entanglement and Nonlocality in Infinite 1D Systems. , 2016, Physical review letters.

[9]  John Bell.  Bell (1964): on the Einstein Podolsky Rosen Paradox [edit-1] , 2019 .

[10]  Nicolas Gisin,et al.  Classifying 50 years of Bell inequalities , 2014, 1404.1306.

[11]  N. J. Engelsen,et al.  Bell Correlations in Spin-Squeezed States of 500 000 Atoms. , 2017, Physical review letters.

[12]  Carsten Lund,et al.  Non-deterministic exponential time has two-prover interactive protocols , 2005, computational complexity.

[13]  Jean-Daniel Bancal,et al.  Bell correlations in a Bose-Einstein condensate , 2016, Science.

[14]  K. Svozil,et al.  Optimal tests of quantum nonlocality , 2000, quant-ph/0011060.

[15]  A. Fine.  Hidden Variables, Joint Probability, and the Bell Inequalities , 1982 .

[16]  A. Shimony,et al.  Proposed Experiment to Test Local Hidden Variable Theories. , 1969 .

[17]  J. Lasserre,et al.  Handbook on Semidefinite, Conic and Polynomial Optimization , 2012 .

[18]  Bruno Buchberger,et al.  Bruno Buchberger's PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal , 2006, J. Symb. Comput..

[19]  Rekha R. Thomas,et al.  Theta Bodies for Polynomial Ideals , 2008, SIAM J. Optim..

[20]  Rekha R. Thomas,et al.  Chapter 7: Spectrahedral Approximations of Convex Hulls of Algebraic Sets , 2012 .

[21]  Stefano Pironio,et al.  All CHSH polytopes , 2014, 1402.6914.

[22]  J. Lasserre.  Moments, Positive Polynomials And Their Applications , 2009 .

[23]  C. Śliwa.  Symmetries of the Bell correlation inequalities , 2003, quant-ph/0305190.

[24]  J. Ignacio Cirac,et al.  Energy as a Detector of Nonlocality of Many-Body Spin Systems , 2016, 1607.06090.

[25]  Tsuyoshi Ito,et al.  Deriving Tight Bell Inequalities for 2 Parties with Many 2-valued Observables from Facets of Cut Polytopes , 2004 .

[26]  Bernard Chazelle,et al.  An optimal convex hull algorithm in any fixed dimension , 1993, Discret. Comput. Geom..

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

[28]  Peter Wittek,et al.  Efficient Device-Independent Entanglement Detection for Multipartite Systems , 2016, 1612.08551.

[29]  M. Lewenstein,et al.  Translationally invariant multipartite Bell inequalities involving only two-body correlators , 2013, 1312.0265.

[30]  M. Lewenstein,et al.  Detecting non-locality in multipartite quantum systems with two-body correlation functions , 2013, 1306.6860.

[31]  C. Klempt,et al.  Nonlocality in many-body quantum systems detected with two-body correlators , 2015, 1505.06740.