Many-Body Chirality in Topological Stabilizer States

• Many-body chirality is a quantum-information property where a state’s mirror image cannot be recovered by finite-depth local operations, establishing intrinsic chiral topological order.
• It is characterized by mirror-invariance tests on anyon braiding and spin statistics, which reveal chiral features undetectable by conventional measures such as chiral central charge.
• The framework underpins chiral color codes, where four-partite entanglement uncovers robust anomalies and guides the engineering of time-reversal and parity breaking in stabilizer models.

Many-body chirality in topological stabilizer states is a quantum-information theoretic property that manifests as an intrinsic distinction between a quantum state and its mirror image (complex conjugate), when no finite-depth local operation can relate one to the other. This property provides a state-based, basis-independent characterization of chiral topological order and exposes forms of chirality that are invisible to conventional diagnostics such as chiral central charge, edge anomalies, or modular commutators. In topological stabilizer codes, notably including chiral color codes and $\mathbb{Z}_d^{(k)}$ fixed-point models, many-body chirality is realized by an obstruction rooted in the anyon data of the underlying phase, and can be rigorously formulated in terms of local operation (LO) and local unitary (LU) channels on the many-body wavefunction. This framework reveals genuine multipartite effects that are strictly four-partite, undetectable via standard tripartite entanglement measures, and tied to deep algebraic and group-cohomological invariants of the code.

1. Foundational Definition: Many-Body Chirality as Local Operation Obstruction

The central notion is that of an LO-chiral state. Let $\rho$ be a many-body state (pure or mixed) on a lattice with local Hilbert spaces; denote $\rho^*$ as its entrywise complex conjugate in the computational basis. The following holds:

• LU-chirality: $\rho$ is LU-chiral if there exists no finite-depth local unitary $U$ (acting on physical sites) such that $U\,\rho^*\,U^\dagger = \rho$.
• LO-chirality: $\rho$ is LO-chiral if no finite-depth local quantum channel $U$ (possibly using constant-density ancillas) implements $$\text{Tr}_{\text{anc}}[\,U(\rho\otimes |0^n\rangle\langle 0^n|)U^\dagger] = \rho^*$$. Equivalently, for pure $|\psi\rangle$, LO-chirality is the non-existence of finite-depth $\rho$0 such that $\rho$1.

This definition makes many-body chirality a property of the global entanglement structure, rather than of any specific Hamiltonian or edge observable (Ellison et al., 18 Jun 2026, Vardhan et al., 13 Mar 2025).

2. Stabilizer States, Mirror-Invariance, and the Chirality Theorem

The information-theoretic content of chirality is particularly robust in stabilizer codes realizing abelian anyon theories of the form $\rho$2. These theories are generated by anyons of order $\rho$3 with braiding and spin

$\rho$4

and mirror conjugation corresponds to $\rho$5.

The decisive result is:

• Chirality-Mirror Equivalence ((Ellison et al., 18 Jun 2026) Theorem 3.5): A stabilizer state $\rho$6 realizing $\rho$7 anyons is LO-chiral if and only if the anyon theory is not mirror-invariant:

  $\rho$8

  where the isomorphism requires a permutation $\rho$9 of anyon types with $\rho^*$0 and $\rho^*$1 $\rho^*$2.

The S-matrix criterion gives a computable check: $\rho^*$3 is mirror-invariant iff $\rho^*$4 with $\rho^*$5 such that $\rho^*$6 for all entries.

This criterion exposes chirality even in models where all standard diagnostics vanish, e.g., for $\rho^*$7 theories with $\rho^*$8 (Ellison et al., 18 Jun 2026).

3. Chiral Color Codes: Construction and Anomalous Order

Chiral color codes (Lee et al., 22 Sep 2025) provide explicit stabilizer Hamiltonians that realize many-body chirality via a group-cohomological twist. The construction is as follows:

• Lattice and Stabilizers: Defined on a 4-valent, 4-colorable 3D lattice with a bipartition $\rho^*$9. Qudits of dimension $\rho$0 are placed on vertices. Face stabilizer generators $\rho$1 depend on a chiral parameter $\rho$2 and take mixed Pauli forms: $\rho$3, etc.
• Chiral Parameter $\rho$4: When $\rho$5, the code reduces to an ordinary (non-chiral) color code. For $\rho$6, the code Hamiltonian admits no finite-depth local circuit mapping it to a time-reversal invariant code, and the associated 4-cocycle classifies the underlying anomaly in $\rho$7.

On the boundary, the code realizes a twisted abelian anyon theory $\rho$8 with nontrivial fusion, modular S- and T-matrices: $\rho$9 These are non-real for $U$0, ensuring explicitly chiral surface order.

4. Multipartite Nature: Four-Partite but not Three-Partite Chirality

Chirality in this sense is intrinsically a four-region phenomenon. The key results are:

• For $U$1 stabilizer states, any reduction to three contiguous regions admits a decomposition pairing each generator in $U$2 to a corresponding term in $U$3, and no tripartite obstruction to local conjugation exists.
• In contrast, a reduction to four regions $U$4 yields an obstruction exactly when $U$5, as the factorization of edge-channel operators and their T-junction phases cannot be consistently matched under conjugation unless a suitable $U$6 exists [(Ellison et al., 18 Jun 2026), Theorem 5.4].

This distinguishes many-body chirality sharply from resource-theoretic and entropic properties detected by tripartite entanglement or modular commutators.

5. Imaginarity: A Weaker Notion than Chirality

Beyond chirality lies the concept of LU-imaginarity: a state $U$7 is LU-imaginary if no finite-depth local unitary $U$8 makes $U$9 real (i.e., $U\,\rho^*\,U^\dagger = \rho$0 in the computational basis). Theorem 6.1 in (Ellison et al., 18 Jun 2026) establishes:

• For all $U\,\rho^*\,U^\dagger = \rho$1 with $U\,\rho^*\,U^\dagger = \rho$2, any stabilizer realization of $U\,\rho^*\,U^\dagger = \rho$3 is LU-imaginary, whether or not it is LO-chiral.
• Thus, certain states (e.g., $U\,\rho^*\,U^\dagger = \rho$4 models) exhibit imaginarity without chirality, highlighting distinct forms of non-classical complex structure in many-body states.

Imaginarity may be quantified by the minimal trace distance to the set of real density matrices, minimized over finite-depth local unitaries.

6. Quantitative Measures: Chiral Log-Distance and Modular Commutators

A faithful, basis-independent metric for bipartite chirality is the chiral log-distance: $U\,\rho^*\,U^\dagger = \rho$5 where $U\,\rho^*\,U^\dagger = \rho$6 denotes the Uhlmann fidelity. This measure vanishes if and only if $U\,\rho^*\,U^\dagger = \rho$7 is non-chiral and provides lower bounds on various stabilizer "magic" monotones (e.g., stabilizer nullity, stabilizer fidelity), but is generically non-computable due to the non-convex local unitary maximization (Vardhan et al., 13 Mar 2025).

Alternatively, computable additive witnesses such as nested modular commutators $U\,\rho^*\,U^\dagger = \rho$8 are available: $U\,\rho^*\,U^\dagger = \rho$9 where $\rho$0 is the modular Hamiltonian. These functionals are invariant under local unitaries, additive under tensor products, odd under $\rho$1, and vanish for separable states.

Qubit stabilizer states, including the ground spaces of all topological stabilizer codes without group-cohomology twist, satisfy $\rho$2 and are non-chiral for any partition and any code type (e.g., toric, color, surface codes) (Vardhan et al., 13 Mar 2025).

7. Implications and Limitations of Standard Diagnostics

Conventional markers such as vanishing chiral central charge ($\rho$3), modular commutator invariants, or boundary anomaly do not reliably diagnose many-body chirality. Specifically, models with $\rho$4 and vanishing modular commutator can be LO-chiral if their anyon data are not mirror-invariant. Conversely, ground states of certain chiral theories may be neither edge-gappable nor real-representable by any finite-depth basis change (Ellison et al., 18 Jun 2026).

A plausible implication is that many-body chirality as an LO-obstruction encompasses forms of time-reversal and parity breaking invisible to traditional measures, and that it can be sharply characterized by the full topological data (fusion, braiding, spin) of the system via the mirror-invariance criterion.

Table: Summary of Chirality Diagnostics and Their Applicability

Diagnostic	Detects Chirality?	Counterexamples
Chiral central charge $\rho$5	Not sufficient/necessary	$\rho$6 with $\rho$7 is LO-chiral
Modular commutator	Not sufficient/necessary	All stabilizer models: $\rho$8, but may be LO-chiral
Mirror-invariance test	Necessary and sufficient for stabilizer LO-chirality	—

8. Outlook and Open Directions

The state-based framework for many-body chirality established by these results provides a platform for further study of chiral topological order outside the scope of usual edge-based or entanglement-based measures. Open directions include extension to non-abelian topological orders, continuum field theory analogs, and the investigation of chirality in interacting and symmetry-enriched phases. The group-cohomological twist mechanism, as realized in chiral color codes, represents a paradigm for engineering models with precise, information-theoretic chiral invariants and robust four-partite multipartite obstructions (Lee et al., 22 Sep 2025).