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 MA and MB on a Hilbert space H, then uniqueness of purifications is equivalent to Haag duality MA=MB′. In particular, the uniqueness of purifications can fail in systems with infinitely many degrees of freedom—even when MA and MB 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.