We prove the conjecture by Diaconis and Eriksson (2006) that the Markov degree of the Birkhoff model is three. In fact, we prove the conjecture in a generalization of the Birkhoff model, where each voter is asked to rank a fixed number, say r, of can
didates among all candidates. We also give an exhaustive characterization of Markov bases for small r.
We propose a novel scheme to generate polarization entanglement from spatially-correlated photon pairs. We experimentally realized a scheme by means of a spatial correlation effect in a spontaneous parametric down-conversion and a modified Michelson
interferometer. The scheme we propose in this paper can be interpreted as a conversion process from spatial correlation to polarization entanglement.
