We prove that the set of quantum correlations for a bipartite system of 5 inputs and 2 outputs is not closed. Our proof relies on computing the correlation functions of a graph, which is a concept that we introduce.2010 Mathematics Subject Classification. Primary 46L05; Secondary 47L90.

We introduce and study the entanglement breaking rank of an entanglement breaking channel. We show that the entanglement breaking rank of the channel Z:Md→Md defined by Z(X)=1d+1(X+Tr(X)Id) is d2 if and only if there exists a symmetric informationally complete positive operator-valued measure in dimension d.

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A. We consider correlations, , , arising from measuring a maximally entangled state using measurements with two outcomes each, constructed from projections that add up to . We show that the correlations , robustly self-test the underlying states and measurements. To achieve this, we lift the group-theoretic Gowers-Hatami based approach for proving robust self-tests to a more natural algebraic framework. A key step is to obtain an analogue of the Gowers-Hatami theorem allowing to perturb an "approximate" representation of the relevant algebra to an exact one.For = 4, the correlations , self-test the maximally entangled state of every odd dimension as well as 2-outcome projective measurements of arbitrarily high rank. The only other family of constant-sized self-tests for strategies of unbounded dimension is due to Fu (QIP 2020) who presents such self-tests for an infinite family of maximally entangled states with even local dimension. Therefore, we are the first to exhibit a constant-sized self-test for measurements of unbounded dimension as well as all maximally entangled states with odd local dimension.

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