Random unitary matrices associated to a graph

We analyze composed quantum systems consisting of $k$ subsystems, each described by states in the $n$-dimensional Hilbert space. Interaction between subsystems can be represented by a graph, with vertices corresponding to individual subsystems and edges denoting a generic interaction, modeled by random unitary matrices of order $n^2$. The global evolution operator is represented by a unitary matrix of size $N=n^k$. We investigate statistical properties of such matrices and show that they display spectral properties characteristic to Haar random unitary matrices provided the corresponding graph is connected. Thus basing on random unitary matrices of a small size $n^2$ one can construct a fair approximation of large random unitary matrices of size $n^{k}$. Graph--structured random unitary matrices investigated here allow one to define the corresponding structured ensembles of random pure states.