# Bells Inequalities, Computational Geometry and the Nature of Reality Theory and Experiment

Kim WinickProfessorUniversity of Michigan, Department of Electrical Engineering and Computer Science
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Consider a set of binary (0/1), random variables {X1,X2,…,Xn,Y1,Y2,…,Ym}, and let S1 = {1,2,… ,n}, S2 = {1,2,… ,m}, S12 = {(i, j): 1 Â‰ i Â‰ n, 1 Â‰ j Â‰ m}. Given sets Sx Âš† S1, Sy Âš† S2 and Sxy Âš† S12 and a set of single probability and correlation values C = {P(Xi = 1) = pxi for i Â^^ Sx, and P(Yj) = pyj for j Â^^ Sy, and P(Xi = 1,Yj = 1) = pxi,yj for (i, j) Â^^ Sxy}, can one determine the necessary and sufficient conditions on the values of pxi, pyj , pxi,yj such that there exists a joint probability mass function (pmf) for P(X1,X2,…,Xn,Y1,Y2,…,Ym) that when marginalized produces C? For example, when n = m = 1, a joint pmf P(X,Y) exists consistent with a specified px, py, px,y if and only if (1) px Â‰¥ px,y Â‰¥ 0 and (2) py Â‰¥ px,y Â‰¥ 0 and (3) px + py Â^’ px,y Â‰ 1. These later three conditions, which describe a polytope, are an example of a set of Bell's inequalities. Bell's inequalities have been widely studied and can be generated using probability theory and methods of computational geometry, which will be reviewed. Much of the mathematical theory of non-relativistic quantum mechanics was developed in the 1920's and 1930's. The theory has been able to predict experimental results, notably atomic spectra, to unprecedented precision, and thus few question its accuracy. The physical interpretation of the theory, however, has remained a source of controversy from its inception to present day, since the theory predicts outcomes that are consistent with experiment but inconsistent with our common perceptions of the natural world, e.g., violate Bell's inequalities. This contradiction is a profound assault on our notions concerning the natural world, i.e., objective realism (the properties of every object exist whether or not they are measured) and locality (action at a distance cannot propagate faster than the speed of light). A likely implication of these experiments is that the assumption of objective realism cannot be supported. Experiments, of varying degrees of sophistication, have been performed that produce violations of Bell's inequalities under a variety of assumptions (and hence potential "loopholes") such as fair sampling. Removal of these assumptions places stringent demands on the experimental apparatus. Thus finding Bell's inequalities and the corresponding experiments that reduce these demands, such as the requirement of high detector efficiency, is an area of active research. In this talk, we will review the historical and mathematical basis for Bell's inequalities, discuss our numerical search for Bell's inequalities that lower the detector efficiency required to obtain a Bell's inequality violation, and present our experimental results of a Bell's inequality violation using a pair of polarization entangled photons under the assumption of no detector enhancement (as described by Clauser and Horne (1974)), but without assuming fair sampling. The intended audience is not assumed to have prior knowledge of quantum mechanics or optics.
Kim Winick is a Professor in the Department of Electrical Engineering and Computer Science at the University of Michigan, Ann Arbor and a member of the Applied Physics Program. He received his B.S. degree in Electrical Engineering, summa cum laude, from the Pennsylvania State University in 1976, and his M.S. and Ph.D. degrees also in Electrical Engineering from the University of Michigan in 1977 and 1981, respectively. He attended the University of Michigan on a National Science Foundation Graduate Fellowship. Prior to his University appointment, he was a member of the technical staff at the Massachusetts Institute of Technology Lincoln Laboratory, where he worked on microwave and optical communication satellite systems. He has authored approximately seventy journal and conference publications in the areas of glass and crystal integrated optics, communications and information theory and has graduated eleven Ph.D. students. Professor Winick is a Fellow of the Optical Society of America, a Senior Member of Institute of Electrical and Electronics Engineers, and served as a Topical Editor of Optics Letters from 2004-2007. Professor Winick received the Teaching Excellence Award and the Outstanding Achievement Award from the Department of Electrical Engineering and Computer Science at the University of Michigan in 1997 and 2001, respectively. Hi research Interests include glass and crystal integrated optics, quantum optics, nanophotonics, communications, and information theory.