Bipartite Matrix-Valued Tensor Product Correlations That are Not Finitely Representable

We consider the matrix-valued generalizations of bipartite tensor product quantum correlations and bipartite infinite-dimensional tensor product quantum correlations, respectively. These sets are denoted by Cq(n)(m,k) and Cqs(n)(m,k), respectively, where m is the number of inputs, k is the number of outputs, and n is the matrix size. We show that, for any m, k≥ 2 with (m, k) ≠ (2 , 2) , there is an n≤ 4 for which we have the separation Cq(n)(m,k)≠Cqs(n)(m,k).