  The standard collisional model paradigm consists of a system that interacts
sequentially with identically prepared ancillas. After infinitely many
collisions, and under appropriate conditions, the system may converge to the
same state as the ancillas. This process, known as homogenization, is
independent of the ancilla initial state, being a property only of the
underlying dynamics. In this paper we extend this idea to locally identical,
but globally correlated, ancillas, and show that correlations break
homogenization. This is done numerically using a minimal qubit model, and
analytically using an exactly soluble Gaussian model. In both cases, we use
Hamiltonian graph states with cyclic graphs as the prototypical method for
building scalable many-body entangled ancillary states.
