Generalized distillability conjecture and generalizations of Cauchy-Bunyakovsky-Schwarz inequality and Lagrange identity

Let rho_k, k=1,2,...,m, be the critical Werner state in a bipartite d_k by d_k quantum system, i.e., the one that separates the 1-distillable Werner states from those that are 1-indistillable. We propose a new conjecture (GDC) asserting that the tensor product of rho_k is 1-indistillable. This is much stronger than the familiar conjecture saying that a single critical Werner state is indistillable. We prove that GDC is true for arbitrary m provided that d_k is bigger than 2 for at most one index k. We reformulate GDC as an intriguing inequality for four arbitrary complex hypermatrices of type d_1 x ... x d_m. This hypermatrix inequality is just the special case n=2 of a more general conjecture (CBS conjecture) for 2n arbitrary complex hypermatrices of the same type. Surprisingly, the case n=1 turns out to be quite interesting as it provides hypermatrix generalization of the classical Lagrange identity. We also formulate the integral version of the CBS conjecture and derive the integral version of the hypermatrix Lagrange identity.