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arXiv · 2509.12911

Uniqueness of purifications is equivalent to Haag duality

Lauritz van Luijk·Alexander Stottmeister·Henrik Wilming
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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 $M_A = M_B'$. 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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Lauritz van Luijk, Alexander Stottmeister, Henrik Wilming. 2025-09-16. Uniqueness of purifications is equivalent to Haag duality. https://doi.org/10.1103/d7nm-gx37

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