Local-Gauge-Symmetric Decoherence

• Local-gauge-symmetric decoherence is a quantum process that preserves logical qubits by ensuring decoherence channels commute with local gauge transformations.
• It employs polar decompositions and gauge-link integrations to construct channels that act nontrivially only on gauge-junk while protecting essential quantum information.
• The approach allows robust information storage even in critical mixed phases, highlighting resource-theoretic limits and practical stabilization of logical subsystems.

Local-gauge-symmetric decoherence refers to quantum processes on composite systems that strictly respect local gauge symmetry constraints. In such settings, decoherence channels are constructed so that all their action—at the level of both Kraus operators and statistical mixing—commutes with local gauge transformations, ensuring that physically meaningful subsystems, such as logical qubits in subsystem codes, remain unaffected while the rest of the system ("gauge-junk") can undergo nontrivial mixed-state dynamics. This principle governs quantum information preservation, phase structure, and the algebraic and resource-theoretic properties of quantum operations in lattice gauge theories and quantum codes.

1. Structural Framework of Local-Gauge-Symmetric Decoherence

Decoherence respecting a local gauge principle is realized by quantum channels $\mathcal{E}$ whose Kraus operators $K_\alpha\in\mathcal{G}$, the gauge group, and that commute with all local gauge constraints. In the context of the $\mathbb{Z}_2$ gauge–Higgs model (GHM), the full Hilbert space is composed of matter fields at vertices, dual-matter at plaquettes, and gauge fields on links. The physical state space $\mathcal{H}_{\rm phys}$ is defined as the stabilizer-invariant subspace

$$G_{v}\,\lvert\psi\rangle = \lvert\psi\rangle, \qquad B_{p}\,\lvert\psi\rangle = \lvert\psi\rangle,$$

where each generator $G_{v}=X_{v}\prod_{\ell\in v}\sigma^x_\ell$ and $B_p=Z_p\prod_{\ell\in p}\sigma^z_\ell$ implements local gauge invariance and magnetic flux conservation, respectively. The gauge symmetry group $\mathcal{G}$ contains all local symmetry transformations, and its center $\mathcal{Z}(\mathcal{G})$ consists of the stabilizer subgroup.

A local-gauge-symmetric decoherence channel thus has the form

$$\mathcal{E}^g(\rho) = \prod_{(p,p')}\big[(1-p_x)\rho + p_x (X_p \sigma^x_{p,p'} X_{p'}) \rho (X_p \sigma^x_{p,p'} X_{p'})\big],$$

where all Kraus operators commute with the gauge constraints, ensuring trivial action on the logical subsystem $\mathcal{L}$ and nontrivial action on the gauge-junk $\bar{\mathcal{L}}$.

2. Polar Decompositions, Process Modes, and Gauged Channels

Process modes formalism (Cirstoiu et al., 2017) decomposes arbitrary symmetric channels with respect to a local gauge group $G^{\times n}$. Any channel $\mathcal{E}$ is expressed as a sum over process modes $\Phi_k^\lambda$ labeled by irreps $\lambda$: $$\mathcal{E} = \sum_{\lambda,k} \alpha_{\lambda,k}\Phi^\lambda_k,$$ with $\lambda=0$ specifying the symmetric part, while nonzero $\lambda$ indicate symmetry-breaking components. This structure underlies the polar decomposition: $$\mathcal{E} = \sum_{\lambda,k} a_\lambda Y_{\lambda,k}(\mathbf{x}) \Phi_k^\lambda,$$ where $Y_{\lambda,k}$ are generalised spherical harmonics and $a_\lambda$ quantify symmetry-breaking resources. Locally gauge-invariant—or "gauged"—channels are built by inserting link systems (gauge-links), process gauge couplings, and enforcing local symmetry via a $G$-twirl,

$$\widetilde\Phi_{x,l,y} := \int_{G\times G} \left(\mathcal{U}_{g_x}\otimes\mathcal{U}_{(g_x,g_y)}\otimes\mathcal{U}_{g_y}\right) [\Phi'_{x,l,y}] dg_x\,dg_y,$$

thereby generating channels invariant under all local gauge actions. Gauge-links transform covariantly under the gauge group at their endpoints.

3. Decoherence, "Gauging Out," and Logical Subsystem Preservation

The core mechanism of logical-state preservation under local-gauge-symmetric decoherence is the restriction of all Kraus operators to the gauge group. In the $\mathbb{Z}_2$ gauge–Higgs subsystem code, this leads to a factorization

$$C^{(P,S_Z)} = \mathcal{L} \otimes \bar{\mathcal{L}}^{(P,S_Z)},$$

where $\mathcal{L}$ corresponds to the encoded logical qubit, and $\bar{\mathcal{L}}$ ("gauge-junk") houses all decohered degrees of freedom. Even maximal-gauge symmetric decoherence (e.g., measurement in the $X$-basis for $J=0$, $p_x=1/2$) does not affect logical operators $L_x$, $L_z$; the mixed-state stabilizer group is constructed from the center of the enlarged gauge group inclusive of those projectors.

When mapped to the toric code via a local Clifford $U=U_v U_p$, logical operators are preserved under the same conditions, as the channel acts only on the gauge-junk. This mapping facilitates transfer of results on decoherence and phase structure between the gauge-Higgs model and toric code.

Sufficient and necessary preservation criterion: $$[K_\alpha, G_v] = 0,\; [K_\alpha, B_p] = 0 \Longrightarrow \mathcal{E}(\overline{X}) = \overline{X},\; \mathcal{E}(\overline{Z}) = \overline{Z}.$$ This determines when the logical subsystem remains noiseless under decoherence.

4. Mixed-State Phase Structure and Criticality under Gauge-Symmetric Noise

Upon subjecting the gauge-Higgs code to local-gauge-symmetric decoherence, the system exhibits a rich phase diagram. Through a mapping to the classical random-bond Ising model (RBIM) on the Nishimori line, the following transitions are found:

• At $J=0$, entropy and purity transitions occur at $p_c \approx 0.1094$ and $p\approx 0.178$, respectively.
• For $p_x=0$, one recovers the standard Fradkin–Shenker pure-state transition at $J=J_c$.
• The $(J,p_x)$ plane is partitioned into four regions: deconfined/topological, critical mixed, gauged-out mixed topological order, and weakly mixed.

Notably, in the "critical mixed" phase, the gauge-junk becomes critical (diverging correlation length and large operator variance), but the logical qubit remains noiseless. A plausible implication is that the presence of a preserved logical sector across all mixed phases is a robust non-local signature of gauge symmetry in decoherence processes.

5. Resource-Theoretic Interpretation and Irreversibility

Gauge-covariant decoherence is naturally situated in the resource theory of asymmetry. Free states satisfy a local twirl condition $\rho = \mathcal{G}[\rho]$ (with $\mathcal{G}$ the local gauge twirl), enforcing a generalized Gauss law. Free operations—exactly locally gauge-invariant channels—commute with the gauge twirl: $$\Lambda\circ\mathcal{G} = \mathcal{G}\circ\Lambda = \Lambda.$$ When decoherence is not gauge-invariant, resource monotones such as the relative entropy of asymmetry or frameness strictly decrease under gauge-covariant operations: $A(\widetilde{\mathcal{D}}(\rho)) \leq A(\rho).$ For abelian symmetries (e.g., U(1)), resource broadcasting is possible; for non-abelian groups, noncommutativity imposes irreducible limits, reflected in the impossibility of perfect resource re-use.

6. Stability of Logical Qubits Under Environmental and Dynamical Perturbations

While logical qubits in locally gauge-symmetric decoherence channels remain noiseless at the level of the channel, their practical robustness is determined by the coupling between $\mathcal{L}$ and $\bar{\mathcal{L}}$. For dynamical perturbations modeled as

$$U_{\Delta t} = \exp[-i\Delta t (\hat{L}_0 \otimes \hat{O}_G)],$$

with $\hat{O}_G$ anticommutes with the decoherence operator, the leading-order decay of logical fidelity is set by the variance of $\hat{O}_G$ in the gauge-junk sector,

$F \approx -2 (\Delta t)^2 \mathrm{Var}(\hat{O}_G).$

In the critical mixed phase, where $\mathrm{Var}(\hat{O}_G)$ diverges, logical-qubit dephasing is significantly enhanced. This suggests that the logical sector's practical stability is non-uniform across the mixed-state phase diagram and is acutely sensitive to criticality and correlations in the gauge-junk.

7. Explicit Examples and Special Cases

Specific constructions of locally gauge-symmetric decoherence channels are illustrated for both abelian (U(1)) and non-abelian (SU(2)) cases (Cirstoiu et al., 2017):

• U(1) dephasing channels combine phase-encoding gauge-links and dephasing maps, implementing local symmetry via integrals over classical references.
• SU(2) depolarizing channels leverage spin-system gauge-links and intertwiner operators; local gauge invariance is enforced by full averaging over local group rotations.

In summary, local-gauge-symmetric decoherence provides a model-independent algebraic and resource-theoretic mechanism for preserving non-local quantum information in lattice gauge theories and subsystem codes—even when the rest of the system becomes highly mixed or critical. Its practical limits of stability are governed by both the structure of gauge-covariant channels and the mixed-state properties of gauge qubits in the presence of environmental coupling (Cirstoiu et al., 2017, Kuno et al., 11 Nov 2025).