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# Bernstein’s Socks, Polynomial-Time Provable Coherence and Entanglement

*Proceedings of the Eleventh International Symposium on Imprecise Probabilities: Theories and Applications*, PMLR 103:23-31, 2019.

#### Abstract

We recently introduced a bounded rationality approach for the theory of desirable gambles. It is based on the unique requirement that being nonnegative for a gamble has to be defined so that it can be provable in polynomial time. In this paper we continue to investigate properties of this class of models. In particular we verify that the space of Bernstein polynomials in which nonnegativity is specified by the Krivine-Vasilescu certificate is yet another instance of this theory. As a consequence, we show how it is possible to construct in it a thought experiment uncovering entanglement with classical (hence non quantum) coins.