Central Limit Theorem for Tensor Products of Free Variables

We establish a central limit theorem for tensor product random variables cₖ:=aₖ ⊗ aₖ, where (aₖ)k ∈ N is a free family of variables. We show that if the variables aₖ are centered, the limiting law is the semi-circle. Otherwise, the limiting law depends on the mean and variance of the variables aₖ and corresponds to a free interpolation between the semi-circle law and the classical convolution of two semi-circle laws.