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arXiv:2609.13550

An entropic characterization of Haag duality

Ruizhi Liu, Lauritz van Luijk

quant-phcond-mat.str-elmath-phmath.OASubmitted Sep 11, 2026• Announced Sep 15, 2026• 0 citations
v1 · Sep 11, 2026v2 · Oct 1, 2026
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Abstract

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 two-dimensional spin systems, let an annulus separate a finite region from the exterior. Haag duality then holds for a region $A$ if and only if the mutual information of the inner region and the exterior, conditioned on the part of $A$ 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.

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We prove Haag duality property of any translation invariant pure state on $\clb = \otimes_{\IZ}/!M_d(\IC), \;d \ge 2$, where $/!M_d(\IC)$ is the set of d×dd \times dd×d dimensional matrices over the field of complex numbers. We also prove a necessary and sufficient condition for a translation invariant factor state to be pure on $\clb$.
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.
Haag duality is a remarkable property in QFT stating that the commutant of the algebra of observables localized in some region of spacetime is exactly the algebra associated to the causally disconnected region. It is a strong condition on the local structure and has direct consequences on entanglement measures. It was first shown to hold for a free scalar field and causal diamonds by Araki in 1964 and later by many authors in different ways. In particular, Eckmann and Osterwalder (EO) used Tomita-Takesaki modular theory to give a direct proof. However, it is not straightforward to relate this proof to the works of Araki, since they rely on two forms of the canonical commutation relations (CCR), called Segal and Weyl formulations, while EO work as starting point assumes that duality holds in the so-called “first quantization” in the Weyl formulation. It is our purpose to first introduce the works of Araki in a more easy-to-read but still rigorous and self-contained fashion, and show how Haag duality is stated in the Segal and Weyl formulations and in both first and second quantizations (and their immediate combination). This permits to understand the setting of the EO proof of Haag duality. There is nothing essentially new in this manuscript, with the exception of what we consider a simplification of EO proof that uses the adjoint S⁎ of the Tomita operator S instead of introducing several auxiliary operators. We hope this note will be useful for those seeking to understand where Haag duality comes from in a free scalar QFT.
Abstract For pure states of multi-dimensional quantum lattice systems, which in a convenient computational basis have amplitude and phase structure of sufficiently rapid decorrelation, we construct high fidelity approximations of relatively low complexity. These are used for a conditional proof of area-law bounds for the states’ entanglement entropy. The condition is also shown to imply exponential decay of the state’s mutual information between disjoint regions, and hence exponential clustering of local observables. The applicability of the general results is demonstrated on the quantum Ising model in transverse field. Combined with available model-specific information on spin-spin correlations, we establish an area-law type bound on the entanglement in the model’s subcritical ground states, valid in all dimensions and up to the model’s quantum phase transition.