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arXiv:2610.25329

Failure of disjoint additivity under magnetic Gauss constraints

Yi Liang

quant-phSubmitted Oct 18, 2026• Announced Oct 20, 2026• 0 citations
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Abstract

Under local magnetic Gauss constraints, vertex-disjoint regions can have commuting physical observable algebras that do not generate all observables on their union. We construct such counterexamples for every finite non-Abelian group on regular square tori, retaining the full magnetic physical space and original supports inside a fixed contractible disk. A peripheral projector selects inputs on which a joint transition preserves the constraints, while its loop factor alone fails the universal test over physical exterior completions. For a specified support family around a fixed eight-edge square loop, additivity fails if and only if the group is non-Abelian and each inner face is covered by at least one of its two prescribed peripheral edges. Leaving a face uncovered restores full additivity. For the same loop, additivity also holds with any region at vertex-set graph distance at least two. These results refute a recent group-double additivity proposition.

Comments
24 pages, 3 figures; includes Supplemental Material

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Under local magnetic Gauss constraints, vertex-disjoint regions can have commuting physical observable algebras that do not generate all observables on their union. We construct such counterexamples for every finite non-Abelian group on regular square tori, retaining the full magnetic physical space and original supports inside a fixed contractible disk. A peripheral projector selects inputs on which a joint transition preserves the constraints, while its loop factor alone fails the universal test over physical exterior completions. For a specified support family around a fixed eight-edge square loop, additivity fails if and only if the group is non-Abelian and each inner face is covered by at least one of its two prescribed peripheral edges. Leaving a face uncovered restores full additivity. For the same loop, additivity also holds with any region at vertex-set graph distance at least two. These results refute a recent group-double additivity proposition.
We study algebraic locality principles on a 2+1D closed lattice in the presence of a Gauss law for a non-invertible symmetry. Prior work in arXiv:2509.03589 showed that when enforcing the Gauss law of an invertible symmetry, the principle of "Haag duality" is preserved exactly, and "disjoint additivity" is preserved after appropriate treatment of discreteness artifacts. Here we show that for a large class of non-invertible on-site symmetries, Haag duality is preserved exactly only for sufficiently nice, "cuspless" regions. For cusped regions, we instead have a weak form of Haag duality that requires adding a collar. Our results apply to double models with a purely magnetic constraint, and to the more general framework of constraints induced by the on-site action of a Hopf algebra. In particular, we treat a class of extended string-net models explicitly. We also demonstrate disjoint additivity for double models based on a group, and a weakened form of disjoint additivity in the setting of a general Hopf algebra.
Earlier work on preferred decomposition for graph-local Hamiltonians established a layered picture rather than a single theorem: empirical structure selection under locality-sensitive selectors, identification of the correct quotient notion of uniqueness, local rigidity and local quantitative identifiability near exact realizations, stability under sufficiently small forbidden tails, and conditional global uniqueness once the exact zero set is known to be unique. The present paper shows that the unrestricted global step cannot be completed from graph locality alone. It gives an explicit connected counterexample family on star graphs: a commuting qubit Hamiltonian family preserved by an entangling star-CNOT circuit that exchanges leaf fields with edge Ising couplings while producing a genuinely inequivalent tensor-product decomposition. The same abstract Hamiltonian therefore has at least two exact admissible-support realizations, so exact zero-set identifiability fails on the physical quotient. The paper also shows why this does not contradict the earlier positive results. The counterexample lies entirely on a structurally degenerate locus excluded by the nondegeneracy hypotheses used in the local rigidity and exact reconstruction results: visible local directions are at most one-dimensional and the relevant complement algebras are abelian rather than full matrix algebras. The main foundational conclusion is sharp. Graph locality by itself does not determine a unique preferred decomposition even in the exact, static, connected setting. What remains open is a narrower problem: whether exact uniqueness holds on a suitably formulated nondegenerate locus that excludes commuting disentanglable obstruction classes of the type exhibited here.
The Page–Wootters construction and the three equivalent pictures built on it all require a bipartition of a kinematical space into a clock and a system, with commuting Hamiltonians and no interaction term. In the literature that bipartition is input, and its non-interaction clause is acknowledged as an idealisation. We observe that a reductive dual pair supplies it as output: two subgroups of a symplectic group that are each other’s full centraliser act on the oscillator representation with a multiplicity-free joint decomposition, so the two factors commute by theorem rather than by assumption, the tensor structure is canonical rather than chosen, and the sector label is one both factors must agree on. Of six requirements of the relational construction, four are met exactly, one is reduced to a condition on the pairing bijection, and one — the covariant POVM that makes a clock — is not supplied at all. Three consequences. The statement that the clock–system tensor product does not survive to the physical Hilbert space becomes: the constraint selects the diagonal of the joint decomposition. Whether commuting subalgebras form an exact dual pair is finite and checkable, and in a related arena it was checked and answered negatively — a gauge algebra of dimension 12 with a double centraliser of dimension 82 — so exactness must be measured, not assumed. And the oscillator representation is a representation of the double cover, whose covering sign no observable of the pair can see. Finally, an obstruction of the form “one universal clock cannot supply a per-mode cost” is shown to move from universal to uniform without declaring it; a clock universal as an operator but non-uniform in its weights is a third option the standard enumeration does not contain. We name it and do not claim it works. Every object here is an algebraic object on a symplectic arena.
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.
Local constraints govern the low-energy physics of frustrated matter, but familiar ice-type rules constrain flux-like quantities and are insensitive to handedness. Here we show that handedness itself can be imposed as an exact local quantum constraint without selecting an axis in spin space. We construct positive-semidefinite, SU(2)-invariant parent Hamiltonians whose complete zero-energy space on a tetrahedron has a prescribed chirality sign, rather than selecting a particular chiral wave function. For spin-1/2 the local term is a rank-one projector onto a chiral tetrahedral singlet, while for arbitrary spin it factorizes as B†BB^\dagger BB†B through a singlet-annihilation operator, with a completely characterized kernel given by the span of the globally rotated chiral color-ice states. For coherent states, the same zero-energy condition becomes an SSS-independent nonlinear constraint in which three spin directions determine the fourth through a Möbius transformation; compositions of these maps define constraint holonomies on extended lattices. Connecting the same local constraint in different ways produces qualitatively different collective regimes: corner-sharing lattices retain exponentially large quantum ground-state kernels, with rigorous lower bounds exceeding conventional ice benchmarks; edge-sharing lattices support subdimensional plane or line zero modes; while triangular constructions suppress nonuniform coherent deformations and contain the complete Anderson tower of tetrahedral magnetic order at exactly zero energy. Two inequivalent triangular coverings further show that harmonic zero-mode counting does not determine the size of the quantum kernel. These results establish a tractable setting in which local handedness, nonlinear constraint geometry, and quantum degeneracy can be disentangled and related directly to the connectivity of the constraint network.
The toric code can be constructed as a gauge theory of finite groups on oriented two dimensional lattices. Here we construct analogous models with the gauge fields belonging to groupoids, which are categories where every morphism has an inverse. We show that a consistent system can be constructed for an arbitrary groupoid and analyze the simplest example that can be seen as the analog of the Abelian Z2\mathbb{Z}_2Z2​ toric code. We find several exactly solvable models that have fracton-like features which include an extensive ground state degeneracy and excitations that are either immobile or have restricted mobility. Among the possibilities we study in detail the one where the ground state degeneracy scales as 2×2Nv2\times 2^{N_v}2×2Nv​, where NvN_vNv​ is the number of vertices in the lattice. The origin of this degeneracy can be traced to loop operators supported on both contractible and non-contractible loops. In particular, different non-contractible loops, along the same direction on a torus, result in different ground states. This is an exponential increase in the number of logical qubits that can be encoded in this code. Moreover the face excitations in this system are deconfined, free to move without an energy cost along certain directions of the lattice, whereas in certain other directions their movement incurs an energy cost. This places a restriction on the types of loop operators that contribute to the ground state degeneracy. The vertex excitations are immobile. The results are also extended to the groupoid analogs of Abelian ZN\mathbb{Z}_NZN​ toric codes.