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Soft Gaps Destroy Quantum Information: Topological Protection as a Hard Gap Phenomenon

Ali Bourizk

Published Oct 27, 2026• 0 citations
Publisher

Abstract

We demonstrate that the topological entanglement entropy (γ = log D) is a pure universal number that strictly requires a hard gap (s=0) to exist. Building on the Hard Gap Criterion established in "The Hard Gap Criterion: Universality Classes and Thermodynamic Scaling in Gapped Dephasing Baths", we prove that a soft gap (s ≥ 1) induces anyon proliferation, destroys the topological order, and drives γ → 0. This establishes that the conservation of many-body quantum information is not just a property of the Hamiltonian, but a direct consequence of the infrared spectral structure of the environment. We show that γ is the many-body analogue of the entanglement concurrence C in the conservation law V² + C² = 1 established in "Conservation of Quantum Information in Gapped Baths: The Duality-Depleted Regime". The area law term represents local decoherence, while the topological constant represents globally protected information. The derivation uses only established theorems (the area law of entanglement entropy, the Kitaev-Preskill topological correction, and the algebraic theory of anyons) and contains no fitting, no adjustment, and no circularity. We provide exact numerical validation via the stabilizer formalism, computing the exact pure numbers for the Toric code (log 2 ≈ 0.693147), the Ising model (log 2), and the Fibonacci model (≈ 0.642965). The exact diagonalization confirms that the topological ground state is exactly pure (total stabilizer rank equals the number of qubits), validating the perfect delocalization of quantum information. We state four falsifiable predictions and provide a complete, reproducible Python script.

DOI
10.5281/zenodo.23708403

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We demonstrate that the topological entanglement entropy (γ = log D) is a pure universal number that strictly requires a hard gap (s=0) to exist. Building on the Hard Gap Criterion established in "The Hard Gap Criterion: Universality Classes and Thermodynamic Scaling in Gapped Dephasing Baths", we prove that a soft gap (s ≥ 1) induces anyon proliferation, destroys the topological order, and drives γ → 0. This establishes that the conservation of many-body quantum information is not just a property of the Hamiltonian, but a direct consequence of the infrared spectral structure of the environment. We show that γ is the many-body analogue of the entanglement concurrence C in the conservation law V² + C² = 1 established in "Conservation of Quantum Information in Gapped Baths: The Duality-Depleted Regime". The area law term represents local decoherence, while the topological constant represents globally protected information. The derivation uses only established theorems (the area law of entanglement entropy, the Kitaev-Preskill topological correction, and the algebraic theory of anyons) and contains no fitting, no adjustment, and no circularity. We provide exact numerical validation via the stabilizer formalism, computing the exact pure numbers for the Toric code (log 2 ≈ 0.693147), the Ising model (log 2), and the Fibonacci model (≈ 0.642965). The exact diagonalization confirms that the topological ground state is exactly pure (total stabilizer rank equals the number of qubits), validating the perfect delocalization of quantum information. We state four falsifiable predictions and provide a complete, reproducible Python script.
A general inequality between entanglement entropy and a number of topologically ordered states is derived, even without using the properties of the parent Hamiltonian or the formalism of topological quantum field theory. Given a quantum state |ψ], we obtain an upper bound on the number of distinct states that are locally indistinguishable from |ψ]. The upper bound is determined only by the entanglement entropy of some local subsystems. As an example, we show that log N≤2γ for a large class of topologically ordered systems on a torus, where N is the number of topologically protected states and γ is the constant subcorrection term of the entanglement entropy. We discuss applications to quantum many-body systems that do not have any low-energy topological quantum field theory description, as well as tradeoff bounds for general quantum error correcting codes.
We study how the color code under decoherence gives rise to an intrinsic mixed-state topological order (imTO), which has no counterpart in pure ground states of local gapped Hamiltonians. For decoherence induced by XX-type operators on red links of the honeycomb lattice, we show that the resulting mixed state inherits half of the topological properties of the color code, including anyon content, logical operators, and topological entanglement structure. Using a gauging procedure for mixed stabilizer states, we identify the emergent phase as closely related to a single toric code. We characterize this phase by topological entanglement negativity (TEN) and perform efficient stabilizer-formalism simulations. While the pure color code has TEN=2ln⁡2{\rm TEN} = 2 \ln 2TEN=2ln2, the maximally decohered state has TEN=ln⁡2{\rm TEN} = \ln 2TEN=ln2, indicating emergence of a single toric code. By tuning the decoherence strength, we find a smooth crossover in TEN accompanied by a pronounced, nearly system-size-independent peak in its variance. We further show that the negativity exhibits characteristic scaling only for subsystem partitions commensurate with the triangular lattice of the emergent toric code. Our results demonstrate that negativity-based quantities provide powerful probes of mixed-state topological order generated by decoherence.
Entanglement entropies of two-dimensional gapped ground states are expected to satisfy an area law, with a constant correction term known as the topological entanglement entropy (TEE). In many models, the TEE takes a universal value that characterizes the underlying topological phase. However, the TEE is not truly universal: it can differ even for two states related by constant-depth circuits, which are necessarily in the same phase. The difference between the TEE and the value predicted by the anyon theory is often called the "spurious" topological entanglement entropy. We show that this spurious contribution is always non-negative, thus the value predicted by the anyon theory provides a universal lower bound. This observation also leads to a definition of TEE that is invariant under constant-depth quantum circuits.
Phases of gapped quantum liquids are topologically ordered and have very interesting physical features that are completely robust against any local perturbation that do not close the bulk energy gap. These universal properties are hidden in the ground states of these systems, as different patterns of many-body long-range entanglement. In this thesis we study the universal properties of gapped quantum liquids from various perspectives. We propose the notion of Universal Wavefunction Overlap as a way of extracting almost complete information about the underlying entanglement structure in a system with topological order. We propose an efficient numerical methods to use these universal wavefunction overlaps as topological order parameters and demonstrate their usefulness with concrete numerical computations. In 2 + 1D these overlaps correspond to known quantities and contain information about anyonic particle excitations. We show that in 3 + 1D, these overlaps contain information about linked multi-string braiding processes, in particular three-string braiding. \n \nIn the second part of this thesis, we study boundary physics of systems with topological order. We investigate the correspondence between edge and entanglement spectra for non-chiral topological systems in general and with the presence of extra symmetries and dualities. We also show that by local deformations of the fixed-point wavefunction on non- chiral topological orders, all possible edge theories can be extracted from its entanglement Hamiltonian. Finally we introduce the notion of fermionic gapped boundaries and see how the phase diagram of the simplest topological orders get significantly enriched.
This paper places the condensed-matter quantum many-body system withinthe framework of a four-tier stability ladder axiomatic system, and systematicallydemonstrates the existence of a finite gap in the bulk excitation spectrum of topological order and the pairing gap of charged superconductor quasiparticles. Thecore logical chain shows that under the condition that the degenerate ground-statespace carries nonlocal information encoding, if the system satisfies causal closurewith local interactions, then in the thermodynamic limit the excitation spectrummust possess a strictly positive energy threshold; otherwise, the axiom of information conservation would be violated by local perturbations. Unlike known stability theorems (gap implies stability), this paper proves the converse proposition information encoding structure implies the existence of a gap. By rigorouslydistinguishing between the broad and narrow definitions of topological long-rangeentanglement, we place gapless metallic phases, quantum critical points, non-Fermiliquids, etc., outside the legitimate boundary of the axiomatic framework, explicitlystating that information conservation in these systems is achieved by alternativemechanisms such as topological invariants of the Fermi surface. The exact solutionof the Kitaev honeycomb model and BCS mean-field theory are used to verify thelogical self-consistency of the derivation, and the elimination of Goldstone modes bythe Anderson-Higgs mechanism is precisely pinpointed as the watershed betweencharged superconductivity and neutral superfluidity. The appendices systematically organise the Majorana fermion representation, the area law of topological entanglement entropy, Gor’kov Green’s functions, and quasi-adiabatic continuationtechniques, providing complete mathematical support for the axiomatic derivation.
Entanglement entropy provides a powerful characterization of two-dimensional gapped topological phases of quantum matter, intimately tied to their description by topological quantum field theories (TQFTs). Fracton topological orders are three-dimensional gapped topologically ordered states of matter that lack a TQFT description. We show that three-dimensional fracton phases are nevertheless characterized, at least partially, by universal structure in the entanglement entropy of their ground-state wave functions. We explicitly compute the entanglement entropy for two archetypal fracton models, the ``X-cube model'' and ``Haah's code,'' and demonstrate the existence of a nonlocal contribution that scales linearly in subsystem size. We show via Schrieffer-Wolff transformations that this piece of the entanglement entropy of fracton models is robust against arbitrary local perturbations of the Hamiltonian. Finally, we argue that these results may be extended to characterize localization-protected fracton topological order in excited states of disordered fracton models.