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arXiv:2509.12911 · Physical Review Letters

Uniqueness of Purifications Is Equivalent to Haag Duality

Lauritz van Luijk, Alexander Stottmeister, Henrik Wilming

Published Jan 5, 2026• 0 citations
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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 modeled by commuting von Neumann algebras M_{A} and M_{B} on a Hilbert space 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(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.

DOI
10.1103/d7nm-gx37

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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 M_{A} and M_{B} on a Hilbert space 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(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.
In systems with infinitely many degrees of freedom, fundamental results from quantum information theory can fail. An important example is the uniqueness of purifications: Even when two subsystems, described by commuting von Neumann algebras AAA and BBB, are tomographically complete, purifications of a state on AAA need not be related by unitaries in BBB. It was recently shown that this occurs precisely when Haag duality fails, i.e., when the commutant B′B'B′ is strictly larger than AAA. This raises the question of which fundamental entanglement properties survive in such a setting. We show that, for a pure global state, the ability to steer any ensemble decomposition of the marginal state on AAA by measurements on BBB is equivalent to the existence of a state-preserving conditional expectation from B′B'B′ onto AAA. This establishes a direct connection between quantum steering and subfactor theory. The key observation is that steering is equivalent to the existence of extensions of ensemble decompositions from AAA to B′B'B′. Working with general Jordan algebras, we prove that unital positive maps have state-preserving left inverses if and only if ensemble decompositions can be lifted. For the inclusion A↪B′A\hookrightarrow B'A↪B′, a left inverse is precisely a conditional expectation, yielding the characterization above.
We show that a superselction sector describing a single nonabelian anyon violates Haag duality, which is equivalent to the fundamental quantum-information theoretic principle of uniqueness of purifications. We provide a simple physics argument as well as a rigorous proof using sector theory, and instantiate our result concretely in Levin-Wen models. We also show that the associated ground state does not allow for quantum steering despite being a pure state. Finally, we show that this ground state violates approximate Haag duality, disproving the conjecture that all gapped ground states satisfy this condition. Our result implies that there are gapped phases of matter where Haag duality fails at every point in the phase.
Motivated by Haag duality in quantum spin systems, we show that Haag duality for a pair of commuting factors generated by increasing sequences of finite-dimensional matrix algebras is equivalent to the asymptotic vanishing of a suitable conditional mutual information. For spin systems, let an annulus separate a finite region from the exterior. Haag duality then holds for a region AAA if and only if the mutual information of the inner region and the exterior, conditioned on the part of AAA inside the annulus, vanishes as the outer radius becomes large. As a corollary, assuming a strict area law with subleading corrections, Haag duality holds for single cones, and it holds for unions of two or more disjoint cones precisely when the topological entanglement entropy vanishes. For translation-invariant pure states on spin chains, half-chain Haag duality is equivalent to vanishing entropy density.
We prove Haag duality for cone-like regions in the ground state representation corresponding to the translational invariant ground state of Kitaev’s quantum double model for finite abelian groups. This property says that if an observable commutes with all observables localized outside the cone region, it actually is an element of the von Neumann algebra generated by the local observables inside the cone. This strengthens locality, which says that observables localized in disjoint regions commute. As an application, we consider the superselection structure of the quantum double model for abelian groups on an infinite lattice in the spirit of the Doplicher–Haag–Roberts program in algebraic quantum field theory. We find that, as is the case for the toric code model on an infinite lattice, the superselection structure is given by the category of irreducible representations of the quantum double.
In our previous article [arXiv:2307.12552], we introduced local topological order (LTO) axioms for abstract quantum spin systems which allow one to access topological order via a boundary algebra construction. Using the LTO axioms, we produced a canonical pure state on the quasi-local algebra, which gives a net of von Neumann algebras associated to a poset of cones in Rn\mathbb{R}^nRn. In this article, motivated by [arXiv:2509.23734], we introduce an axiom for LTOs which ensures Haag duality for cone-like regions using Tomita-Takesaki theory. We prove this axiom is satisfied for all known topologically ordered commuting projector models. We thus get an independent proof of Haag duality for the Levin-Wen string net models originally proved in [arXiv:2509.23734]. We also give a reflection positivity axiom for LTOs, connecting to the recent article [arXiv:2510.20662]. We again prove this axiom is satisfied for all known topologically ordered commuting projector models about some Z/2\mathbb{Z}/2Z/2-reflection symmetry.
We prove that Haag duality holds for cones in the toric code model. That is, for a cone Λ, the algebra RΛ{\mathcal{R}_{\Lambda}}RΛ​ of observables localized in Λ and the algebra RΛc{\mathcal{R}_{\Lambda^c}}RΛc​ of observables localized in the complement Λ c generate each other’s commutant as von Neumann algebras. Moreover, we show that the distal split property holds: if Λ1⊂Λ2{\Lambda_1 \subset \Lambda_2}Λ1​⊂Λ2​ are two cones whose boundaries are well separated, there is a Type I factor N{\mathcal{N}}N such that RΛ1⊂N⊂RΛ2{\mathcal{R}_{\Lambda_1} \subset \mathcal{N} \subset \mathcal{R}_{\Lambda_2}}RΛ1​​⊂N⊂RΛ2​​ . We demonstrate this by explicitly constructing N{\mathcal{N}}N .