Symmetry-Resolved Entanglement in Quantum Systems

• Symmetry-resolved entanglement is defined as the decomposition of quantum correlations into symmetry sectors characterized by conserved charges.
• The article explains analytical tools such as geometric replica methods, boundary CFT, and Toeplitz determinant techniques to compute sector-resolved entropies.
• Experimental protocols in cold atoms and quantum simulators validate these approaches, offering insights into topological phases and symmetry-protected phenomena.

Symmetry-resolved entanglement quantifies how quantum correlations in a many-body quantum system are distributed across the eigenstates of a subsystem's symmetry generator. In systems with internal symmetries—such as particle number, spin, or more general group-valued charges—the reduced density matrix of a subregion is block-diagonal in the associated quantum number basis, and the total entanglement entropy can be decomposed into sector contributions. This symmetry resolution provides a fine-grained view of entanglement, revealing universal and non-universal aspects tied to conservation laws, criticality, topological phases, and their corresponding field-theoretic and lattice descriptions. Theoretical developments encompass geometric replica methods, boundary conformal field theory, generalized correlation function and determinant approaches, group cohomology for SPT phases, and computational techniques including matrix product state formalism. This article surveys the mathematical definitions, analytical frameworks, key results, and experimental platforms related to symmetry-resolved entanglement in many-body and field-theoretic settings.

1. Mathematical Framework of Symmetry Resolution

Consider a global symmetry generator $Q$ in a quantum many-body system, with a subsystem $A$ described by a reduced density matrix $\rho_A$. The presence of the symmetry ensures that $[\rho_A, Q_A] = 0$, with $Q_A$ the restriction of $Q$ to $A$. $\rho_A$ therefore decomposes as:

$\rho_A = \bigoplus_{q} p(q) \, \rho_A(q),$

where $q$ labels the eigenvalues of $Q_A$, $p(q)$ is the probability of finding $q$ in $A$, and $\rho_A(q)$ is the normalized block in sector $q$. Symmetry-resolved Rényi and von Neumann entropies in each sector $q$ are defined as

$$S_n(q) = \frac{1}{1-n} \log \frac{\mathrm{Tr}[\rho_A(q)^n]}{p(q)^n}, \qquad S_1(q) = - \mathrm{Tr}[\rho_A(q)\log \rho_A(q)].$$

The total entropy separates as $S_n = \sum_q p(q) S_n(q) - \sum_q p(q)\log p(q)$, with the first term being the configurational (within-sector) entropy and the second the number (fluctuation) entropy.

To compute $S_n(q)$, one employs "charged moments" (or "twisted moments"):

$$Z_n(\alpha) = \mathrm{Tr}[\rho_A^n e^{i \alpha Q_A}],$$

and recovers the fixed-$q$ partition function by Fourier transform:

$$Z_n(q) = \int_{-\pi}^{\pi} \frac{d\alpha}{2\pi} e^{-i\alpha q} Z_n(\alpha).$$

In field theory, these structures are implemented by inserting fluxes (Aharonov-Bohm or more general) into replica path integrals, promoting twist fields to composite twist-flux operators.

2. Geometric and CFT Approaches to Symmetry Resolution

The geometric replica trick for symmetry resolution, originally developed for 1+1D conformal field theory (CFT), involves threading a symmetry flux (e.g., a U(1) phase) through the branch points of an $n$-sheeted Riemann surface. The composite twist fields $\mathcal{T}_\mathcal{V}$ behave as primaries with total scaling dimension

$$\Delta_n(\alpha) = \frac{c}{24}\left(n - \frac{1}{n}\right) + n[\Delta_\mathcal{V} + \bar{\Delta}_\mathcal{V}],$$

with $\Delta_\mathcal{V}$ (and antiholomorphic counterpart) encoding the flux dependence. For Luttinger liquid systems, $$\Delta_\mathcal{V} = \frac{K}{2}\left(\frac{\alpha}{2\pi}\right)^2$$.

The resulting charged moment scales as

$$Z_n(\alpha) \sim L^{ -(\frac{c}{6})(n - \frac{1}{n}) - \frac{2K}{n}\left( \frac{\alpha}{2\pi} \right)^2},$$

and a Fourier transform yields a Gaussian in $(q-\langle Q_A \rangle)$, implying the equipartition of entanglement at leading order.

In BCFT-based approaches, boundary conditions preserving the symmetry decompose the partition function into symmetry sector characters

$Z(q) = \sum_Q \chi_Q(q),$

allowing symmetry-resolved Rényi entropies $S_n(Q)$ to be read off directly to all orders in the UV cutoff, often revealing exact equipartition (e.g., $U(1)$ case) or its breakdown (e.g., nontrivial discrete symmetries with sector-dependent subleading corrections).

3. Applications in Lattice, Free, and Interacting Systems

Free-Fermionic Chains

In 1d free fermion chains (XX, Kitaev), the reduced density matrix for a subsystem is determined by the correlation matrix, and symmetry resolution leverages (generalized) Fisher-Hartwig techniques for Toeplitz determinants. The leading asymptotics reproduce CFT predictions, while subleading corrections introduce oscillatory and non-universal contributions, essential for fitting finite-size numerics and analyzing the approach to equipartition (Bonsignori et al., 2019, Fraenkel et al., 2019). For models with long-range couplings, the leading equipartition breaks at a different order (e.g., $O(1/\log L)$ rather than $O(1/\log^2 L)$) and is parameter dependent (Ares et al., 2022).

Higher Dimensions and Dimensional Reduction

In 2d, methods such as dimensional reduction decompose the system into chains (via Fourier transform in transverse directions) and sum one-dimensional results (Murciano et al., 2020). For free fermions, this introduces a multiplicative $\ell \log \ell$ scaling (logarithmic violation of the area law), while for free bosons only the zero-mode contributes additively. Equipartition at leading order persists, but subleading $q$-dependent corrections break it.

Non-Hermitian and Dissipative Systems

In non-Hermitian models (e.g., non-Hermitian SSH), the reduced density matrix may have complex or negative eigenvalues. Symmetry resolution is achieved by defining sectorwise positive-definite matrices using the absolute value $|\rho_A|$, allowing positive entropy in each sector even if the global entropy is negative or complex (Fossati et al., 2023). Out-of-equilibrium and dissipative free-fermion systems described by Lindblad equations can have symmetry-resolved dynamics analyzable via hydrodynamic (quasiparticle) descriptions. While the number entropy is dominated by dissipative terms, symmetry-resolved negativity captures the rise and decay of genuinely quantum entanglement (Murciano et al., 2023).

4. Symmetry-Protected Topological Phases and Group Cohomology

For SPT phases, symmetry-resolved entanglement provides a diagnostic of topological order beyond conventional invariants (Azses et al., 2020, Azses et al., 2022). In 1d systems protected by finite Abelian unitary symmetries, group cohomology techniques yield explicit path integral formulas on triangulated manifolds. The reduced density matrix decomposes into symmetry sectors with strictly identical spectra (equi-block decomposition) for maximally non-commutative cocycles: $Z_n(g) = 0$ for $g \neq e$. This leads to strictly degenerate symmetry-resolved entropies—a signature of SPT order. In higher dimensional SPTs, techniques based on matrix product state reductions extract 1d effective edge problems, with symmetry resolution naturally encoded in the MPO formalism (Azses et al., 2022).

5. Exact Results, Corrections, and Equipartition

Equipartition of entropy among symmetry sectors is a ubiquitous leading-order property in continuum CFTs, free lattice models, and certain topological phases. However, corrections—algebraic for discrete symmetries, logarithmic for U(1), and model-dependent in interacting or long-range systems—break strict equipartition at subleading orders (Estienne et al., 2020, Ares et al., 2022, Murciano et al., 2020). In BCFT, the ratio $Z_n(Q)/[Z_1(Q)]^n$ is often shown to be $Q$-independent for continuous symmetry with appropriate boundary conditions, ensuring equipartition to all orders in the UV cutoff (Giulio et al., 2022).

For symmetry-resolved entanglement in states with excitations, universal expressions for the excess entanglement are given in terms of subsystem fraction $r$, excitation number $k$, and charge $\epsilon$ (Capizzi et al., 2022, 2206.12223). For example, in free theories,

$$M_n^{(k^\epsilon)}(r;\alpha) = \sum_{j=0}^k \left[ \binom{k}{j} r^j (1-r)^{k-j} \right]^n e^{2\pi i \epsilon j \alpha}.$$

6. Experimental Proposals and Measurement

Experimental measurement of symmetry-resolved entanglement is feasible in quantum gas microscopes and cold atom experiments, where both replica-based protocols (via swap operations and Hong–Ou–Mandel interference) and direct measurements of conserved quantities in subsystems can reconstruct charge-resolved Rényi entropies (Goldstein et al., 2017). For $n=2$, protocols involve measurement of occupation numbers in spatial regions across two identical copies of a state, and extensions to higher $n$ are feasible in principle. Advances in quantum simulators and noisy intermediate-scale quantum devices are already enabling such direct access to block-decomposed entanglement structures.

7. Broader Implications and Outlook

Symmetry-resolved entanglement unifies quantum information, condensed matter theory, and high energy physics approaches to many-body entanglement. It reveals universal signatures at criticality, links entanglement spectra to underlying symmetry representations and topological invariants (group cohomology, quantum dimensions), and provides robust numerical and analytical tools for both integrable and non-integrable settings. Extensions unveiled rigorous results in higher dimensions (Huang et al., 12 Mar 2025), new diagnostics for topological phases with non-Abelian statistics (Arildsen et al., 7 Aug 2025), and universal corrections for out-of-equilibrium and dissipative scenarios.

The methodology, including composite twist fields, boundary CFT, Fisher–Hartwig and Toeplitz determinant asymptotics, and majorization for entanglement bounds, has refined our understanding of both total and sector-resolved quantum correlations. These advances have already found application in numerical tests, improved entropy bounds for SPT phases (Monkman et al., 2023), and predictions for bulk-boundary correspondences and entanglement spectra (Li–Haldane conjecture).

As experimental and computational capabilities advance, symmetry-resolved entanglement is poised to become a central diagnostic and quantitative tool in quantum matter, quantum computation, and beyond.