On the Ryll-Nardzewski theorem for quantum stochastic processes

We prove a Ryll-Nardzewski Theorem for quantum stochastic processes, that shows that under natural assumptions which generalize the classical probability setting, the distributional symmetries of exchangeability and spreadability are the same. We further show that product states on twisted tensor products of C⁎-algebras provide a source of counterexamples to the Ryll-Nardzewski theorem, namely of quantum stochastic processes which are spreadable but not exchangeable. Furthermore, in this setting, we also analyze braidability of product states. We then prove an extended de Finetti Theorem for quantum stochastic processes whose distribution factorizes through twisted tensor products.