Related papers: Uniqueness of purifications is equivalent to Haag duality

Uniqueness of purifications is equivalent to Haag duality

URL: http://arxiv.org/abs/2509.12911v1
Date: Tue, 16 Sep 2025 10:05:17 GMT
Title: Uniqueness of purifications is equivalent to Haag duality
Authors: Lauritz van Luijk, Alexander Stottmeister, Henrik Wilming,
Abstract summary: We show that, if the two systems are modelled by commuting von algebra Neumanns $M_A$ and $M_B$ on a Hilbert space $mathcal H$, uniqueness of purifications is equivalent to Haag duality $M_A = M_B'$.
Score: 44.33169165028139
License: http://creativecommons.org/licenses/by/4.0/
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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