Uniqueness of purifications is equivalent to Haag duality

Abstract

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 modelled by commuting von Neumann algebras $M_A$ and $M_B$ on a Hilbert space $\mathcal{H}$, then uniqueness of purifications is equivalent to Haag duality $M_A=M_B^{\prime }$. 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 $B(\mathcal{H})$ and hence allow for local tomography of all density matrices on $\mathcal{H}$.

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