Topologically ordered matter carries exotic "anyonic" charges and stores information non-locally, shielded from local disturbance; the total topological charge of a region can be read from outside only by a measurement that encircles it. Quantum Darwinism, separately, says an observable is objective when the environment redundantly records it. This paper shows the two notions single out the same thing. For the toric code — a canonical solvable model — the paper proves, exactly and at arbitrary size, that a region's total topological charge is the objectively broadcast fact of its environment, while the local anyon content is not. Four results make this precise. The charge is recorded by a loop lying inside every fragment that encircles the region, so its redundancy grows with size. Each fragment holds a perfectly distinguishable copy, so the charge forms an exact broadcast structure. An individual anyon, by contrast, is invisible to any single fragment and cannot be copied. And the celebrated topological entanglement entropy is exactly the encoded, non-broadcast complement of the classical charge record — the region's entropy splits cleanly into the two. The same split gives an error-correction reading: the objective charge plays the role of the code's syndrome, and the hidden anyons its protected logical qubits. The abelian results are proved and verified by exact stabilizer computation at arbitrary size; the fully non-abelian broadcast is left open. In short: a topological phase publishes its total charge to the world while keeping its anyons private — and its celebrated hidden entanglement is exactly that private, non-broadcast complement.
Topologically ordered matter carries exotic "anyonic" charges and stores information non-locally, shielded from local disturbance; the total topological charge of a region can be read from outside only by a measurement that encircles it. Quantum Darwinism, separately, says an observable is objective when the environment redundantly records it. This paper shows the two notions single out the same thing. For the toric code — a canonical solvable model — the paper proves, exactly and at arbitrary size, that a region's total topological charge is the objectively broadcast fact of its environment, while the local anyon content is not. Four results make this precise. The charge is recorded by a loop lying inside every fragment that encircles the region, so its redundancy grows with size. Each fragment holds a perfectly distinguishable copy, so the charge forms an exact broadcast structure. An individual anyon, by contrast, is invisible to any single fragment and cannot be copied. And the celebrated topological entanglement entropy is exactly the encoded, non-broadcast complement of the classical charge record — the region's entropy splits cleanly into the two. The same split gives an error-correction reading: the objective charge plays the role of the code's syndrome, and the hidden anyons its protected logical qubits. The abelian results are proved and verified by exact stabilizer computation at arbitrary size; the fully non-abelian broadcast is left open. In short: a topological phase publishes its total charge to the world while keeping its anyons private — and its celebrated hidden entanglement is exactly that private, non-broadcast complement.
There exists a close relationship between topological quantum error-correcting codes and topological order in condensed matter systems. Indeed, a topological stabilizer code can be regarded as the ground state of a suitable Hamiltonian model, so that "wrong" syndromes correspond to excitations. These excitations are anyons, quasiparticles that carry a topological charge and exhibit exotic statistics. Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We call these defects twists. Twists give rise to new topological degrees of freedom in the ground state, useful as a quantum memory. Moreover, twists can be braided to perform topologically protected gates on these topological qubits. Thus, twists provide a new way to encode and compute with topological codes through code deformations. Because the properties of twists depend on the anyon model, codes with different anyon content give rise to different computational capabilities. E.g., in the well-known toric code a process where suitable twists are braided and fused has the same outcome as if they were Ising anyons. These are non-abelian anyons: braiding produces non-trivial gates on encoded qubits.
Decohering topological order (TO) is central to the many-body physics of open quantum matter and decoding transitions. We identify statistical mechanical models for decohering non-Abelian TOs, which have been crucial for understanding the error threshold of Abelian stabilizer codes. The decohered density matrix can be described by loop models, whose topological loop weight N is the quantum dimension of the decohering anyon-reducing to the Ising model if N=1. In particular, the Rényi-n moments correspond to n coupled O(N) loop models. Moreover, by diagonalizing the density matrix at maximal error rate, we connect the fidelity between two logically distinct ground states to random O(N) loop and spin models. We find a remarkable stability to quantum channels which proliferate non-Abelian anyons with large quantum dimension, with the possibility of critical phases for smaller dimensions. Intuitively, this stability is due to non-Abelian anyons not admitting finite-depth string operators. We confirm our framework with exact results for Kitaev quantum double models, and with numerical simulations for the non-Abelian phase of the Kitaev honeycomb model. Our work opens up the possibility of non-Abelian TO being robust against maximally proliferating certain anyons, which can inform error-correction studies of these topological memories.
We study the entanglement spectrum of topological systems hosting non-Abelian anyons. Akin to energy levels of a Hamiltonian, the entanglement spectrum is composed of symmetry multiplets. We find that the ratio between different eigenvalues within one multiplet is universal and is determined by the anyonic quantum dimensions. This result is a consequence of the conservation of the total topological charge. For systems with non-Abelian topological order, this generalizes known degeneracies of the entanglement spectrum, which are hallmarks of topological states. Experimental detection of these entanglement spectrum signatures may become possible in Majorana wires using multicopy schemes, allowing the measurement of quantum entanglement and its symmetry resolution.
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.
Non-Abelian anyon chains whose interactions favor a single fusion channel host quantum critical ground states described by minimal-model conformal field theories (CFTs) protected by a topological symmetry. We show that their measurement-only counterpart, where non-commuting fusion charge measurements act locally on the anyons of a chain of L sites, provides a fundamental example of a non-unitary CFT controlling a non-Abelian learning problem. Mapping the monitored dynamics to a random tensor network, we numerically find 1/L scaling Lyapunov gaps, logarithmic entanglement scaling with the chord length, and an effective central charge in the Shannon entropy of the measurement record. The conserved topological charge sharpens with dynamical exponent z=1 and is learnable by a decoder on an O(L) time scale. As the topological symmetry is a non-invertible symmetry of the SU(2)k anyon theories, we establish the monitored golden chain (k=3) as a non-Abelian generalization of the weakly (Kramers-Wannier) self-dual quantum critical point of the toric code (k=2), and the first of a family for k≥3.
Topological order (TO) provides a natural platform for storing and manipulating quantum information. However, its stability to noise has only been systematically understood for Abelian TOs. In this Letter, we exploit the nondeterministic fusion of non-Abelian anyons to inform active error-correction and design decoders where the fusion products, instead of flag qubits, herald the noise. This intrinsic heralding offers improved thresholds over those of Abelian counterparts. Furthermore, we use Bayesian inference to obtain a statistical mechanics model for fixed-point non-Abelian TOs with perfect measurements under any noise model, which yields the optimal threshold conditioned on measuring anyon syndromes. We numerically illustrate these results for D_{4}≅Z_{4}⋊Z_{2} TO. In particular, for non-Abelian charge noise and perfect syndrome measurement, we find a conditioned optimal threshold p_{c}=0.218(1), whereas an intrinsically heralded minimum-weight perfect matching (MWPM) decoder already gives p_{c}=0.208 42(2), outperforming standard honeycomb-lattice MWPM with p_{c}=0.158 60(1). Our Letter highlights how non-Abelian properties can enhance stability, rather than reduce it, and discusses potential generalizations for achieving fault tolerance.