<p>We prove a multiplicative ergodic theorem for bistochastic completely positive (bcp) linear cocycles acting on finite-dimensional matrix algebras, giving an invariant splitting described explicitly in terms of the multiplicative domains of the underlying bcp maps. As an application of our theorem, we classify when compositions of random bcp maps are asymptotically entanglement breaking, and use this classification to show that occasionally positive partial transpose bcp maps are asymptotically entanglement breaking. We conclude by demonstrating a certain class of bcp linear cocycles are almost surely entanglement breaking in finite time.</p>