Uniqueness of purifications is equivalent to Haag duality
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908864057704448 |
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| author | van Luijk, Lauritz Stottmeister, Alexander Wilming, Henrik |
| author_facet | van Luijk, Lauritz Stottmeister, Alexander Wilming, Henrik |
| contents | 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 $\mathcal H$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_12911 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniqueness of purifications is equivalent to Haag duality van Luijk, Lauritz Stottmeister, Alexander Wilming, Henrik Quantum Physics Mathematical Physics Operator Algebras 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 $\mathcal H$. |
| title | Uniqueness of purifications is equivalent to Haag duality |
| topic | Quantum Physics Mathematical Physics Operator Algebras |
| url | https://arxiv.org/abs/2509.12911 |