Uniqueness of Purifications Is Equivalent to Haag Duality

The uniqueness of purifications of quantum states on a system A up to local unitary transformations on a purifying system B is central to quantum information theory. We show that, if the two systems are modeled by commuting von Neumann algebras M_{A} and M_{B} on a Hilbert space H, then uniqueness of purifications is equivalent to Haag duality M_{A}=M_{B}^{'}. In particular, the uniqueness of purifications can fail in systems with infinitely many degrees of freedom-even when M_{A} and M_{B} are commuting factors that jointly generate B(H) and hence allow for local tomography of all density matrices on H. We present a simple argument showing that the uniqueness of purifications, and, hence, Haag duality, fail in ground state sectors of topologically ordered models at renormalization group fixed points on infinite two-dimensional lattices, partitioned into the union of two spatially separated cones and its complement.