Symmetry-Resolved Multipartite Entanglement

• The paradigm decomposes global entanglement into symmetry-resolved contributions by analyzing intra-sector purities and inter-sector coherences.
• It establishes a clear methodology using Haar ensembles and finite-size scaling to reveal equipartition effects in multipartite quantum systems.
• Experimental protocols, including swap tests and ancilla-assisted circuits, enable measurement of sector-specific and off-diagonal contributions in complex quantum phases.

The symmetry-resolved multipartite entanglement paradigm is a framework for decomposing measures of genuine multipartite entanglement into contributions associated with specific symmetry sectors—typically “charge” sectors defined by a locally acting symmetry on each particle—and their quantum coherences. This approach generalizes the established notions of symmetry-resolved bipartite entanglement in many-body systems, extending the analysis to measures that quantify genuine multipartite correlations, such as the global entanglement Q, and elaborates their algebraic and physical decomposition under local symmetries. The paradigm elucidates how multipartite quantum correlations are distributed across symmetry (charge) subspaces and their associated quantum coherences, with direct implications for theoretical characterizations and experimental protocols in quantum many-body physics and quantum simulation.

1. Multipartite Global Entanglement and Symmetry Decomposition

The global entanglement measure Q, originally introduced by Meyer and Wallach, quantifies genuine multipartite entanglement for pure states of n distinguishable particles. For a system where each site has a Hilbert space of dimension $d$, Q is the averaged single-site linear entropy: $$Q = \frac{d}{d-1} \langle 1 - \operatorname{Tr}(\rho_k^2) \rangle_k$$ where %%%%1%%%% is the reduced density matrix of site $k$, and the average runs over all sites.

When the system features a local symmetry $G$ (acting via a local unitary representation $R$), the local Hilbert space decomposes as $\mathcal{H} = \bigoplus_\alpha \mathcal{H}_\alpha$ into orthogonal charge sectors, each labeled by an index $\alpha$ (charge sector). The projector $P_\alpha$ selects the sector $\mathcal{H}_\alpha$. The reduced density matrix admits the decomposition: $$\rho_k = \sum_\alpha P_\alpha \rho_k P_\alpha + \sum_{\alpha < \beta} \left(P_\alpha \rho_k P_\beta + P_\beta \rho_k P_\alpha\right)$$ Separating Q into symmetry-resolved contributions involves:

• Sector-wise terms $Q_\alpha$, dependent on the purity inside each charge sector.
• Interference (coherence) terms $Q_{\alpha\beta}$, quantifying quantum correlations between different charge sectors.

The explicit decomposition is

$$Q = \sum_\alpha Q_\alpha + \sum_{\alpha < \beta} Q_{\alpha\beta}$$

with

$$\begin{aligned} Q_\alpha &= \frac{2d}{d-1} \langle f_2^{(d)}(\rho_{k,\alpha}) \rangle_k \ Q_{\alpha\beta} &= \frac{2d}{d-1} \langle p_{k,\alpha} p_{k,\beta} + f_2^{(d)}(\rho_{k,\alpha\beta}) \rangle_k \end{aligned}$$

where

• $$f_2^{(d)}(A) = \frac{(\operatorname{Tr}A)^2 - \operatorname{Tr}(A^2)}{2}$$
• $\rho_{k,\alpha} = P_\alpha \rho_k P_\alpha$ (sector $\alpha$ block)
• $p_{k,\alpha} = \operatorname{Tr}(P_\alpha \rho_k)$ (probability in sector $\alpha$)
• $$\rho_{k,\alpha\beta} = P_\alpha \rho_k P_\beta + P_\beta \rho_k P_\alpha$$ (coherence between sectors $\alpha,\beta$)

If the symmetry is Abelian such that each $\mathcal{H}_\alpha$ is one-dimensional, then $Q_\alpha=0$ and the entire global entanglement comes from $Q_{\alpha\beta}$, i.e., from inter-sector coherences.

2. Equipartition, Haar Ensembles, and Scaling

For Haar-random pure states, the average sector contributions and interference terms take the form: $$Q_\alpha = \frac{d_\alpha(d_\alpha - 1)}{d(d-1)}F(n,d) \qquad Q_{\alpha\beta} = \frac{2d_\alpha d_\beta}{d(d-1)}F(n,d)$$ where $d_\alpha$ is the dimension of sector $\mathcal{H}_\alpha$, and $F(n,d)$ is a finite-size correction. The leading term of each $Q_{\alpha\beta}$ corresponds to the probability that two basis states (chosen randomly) fall in sectors $\alpha$ and $\beta$, respectively. This “equipartition” indicates that the average contributions per available pair of sectors are uniform, subject to scaling by the sector sizes and underlying Hilbert-space dimension. Finite-size corrections follow a universal (power-law) scaling with $n$ and $d$.

Such equipartition results are in agreement with the analysis of symmetry-resolved entanglement in Haar ensembles in bipartite and multipartite settings for large $(n,d)$ (Jain, 16 Sep 2025).

3. Physical Interpretation: Sector-Wise Versus Inter-Sector Coherence

The decomposition highlights that in multipartite settings, particularly for Abelian symmetries, all nonlocal global entanglement may be encoded exclusively in the off-diagonal (coherence) terms between different symmetry sectors. In contrast, bipartite symmetry-resolved entanglement entropies generally admit nontrivial intra-sector contributions. This underscores the nontrivial role of global phase coherence across charge sectors in multipartite entanglement.

This feature is particularly significant for phases where the pattern of coherence conditioned on charge sector is an order parameter, such as in symmetry-protected topological (SPT) order or in quantum many-body states built from SPT resource states (Azses et al., 2020, Goldstein et al., 2017).

4. Connections to Symmetry-Resolved Entanglement in Many-Body Systems

The paradigm builds on deep analogies with symmetry-resolved bipartite entanglement, where the reduced density matrix of a subsystem block-diagonalizes into charge (or quantum number) sectors, and the entanglement entropy is resolved into contributions from each sector: $$S_n(q) = \frac{1}{1-n} \log \frac{\mathcal{Z}_n(q)}{\mathcal{Z}_1(q)^n}$$ with $\mathcal{Z}_n(q)$ obtained by threading a fictitious Aharonov-Bohm flux and Fourier transforming over the phase (Goldstein et al., 2017, Bonsignori et al., 2019, Fraenkel et al., 2019, Gaur, 12 Feb 2024). In free-fermion and CFT frameworks, symmetry resolution reveals subtle distinctions between universal leading scaling (often equipartitioned among sectors) and oscillatory, model-dependent subleading corrections (breaking equipartition). In higher multipartite settings, the structure is further enriched by the presence of inter-sector coherences which dominate in many scenarios.

Notably, for generic symmetry-protected topological phases classified by group cohomology, explicit calculations show equi-block spectral degeneracies in symmetry-resolved entanglement spectra, indicating uniformity of contributions per sector (Azses et al., 2020).

5. Experimental Accessibility and Quantum Circuit Protocols

The framework provides methods for experimentally extracting the symmetry-resolved contributions to multipartite entanglement:

• Sector-wise contributions $Q_\alpha$ can be extracted by post-selecting on symmetry (charge) measurements on each local system. Once projected to a sector, linear entropy can be measured using two copies of the state (purity estimation via swap tests [Brennen, et al.]).
• Inter-sector coherences $Q_{\alpha\beta}$ require quantum circuits to measure the off-diagonal elements preserved before a charge measurement. The construction involves entangling ancilla qubits with two physical copies of the $k$th particle and applying controlled-unitary operations:

$U_{k,\alpha\beta} = P_\alpha \otimes P_\beta$

Hadamard transformations and ancilla measurements then yield statistics such as $$\langle Z \rangle = \operatorname{Tr}(\rho_k P_\alpha \rho_k P_\beta)$$, which are algebraically related to $Q_{\alpha\beta}$.

These protocols are realistic for present-day quantum simulators leveraging standard swap-test and ancillary qubit methodologies.

6. Implications for Complex Quantum Phases

The paradigm has direct relevance for the paper of SPT phases, measurement-based quantum computation, and topologically ordered phases. The characterization of multipartite entanglement via the distribution and structure of inter-sector coherence enables refined diagnostics for exotic phases where order is not simply captured by conventional local observables, but rather by the existence and robustness of symmetrically protected global coherence across symmetry sectors (Azses et al., 2020).

In randomized or Haar-random ensembles, the equipartition results and scaling laws provide theoretical benchmarks for expected multipartite entanglement structure in typical states.

7. Summary Table: Symmetry-Resolved Decomposition of Q for Pure States

Contribution	Formula (per site k)	Physical Interpretation
Sector intra-block	$$Q_\alpha = \frac{2d}{d-1} \langle f_2^{(d)}(\rho_{k,\alpha}) \rangle$$	Purity loss within charge sector α
Inter-sector coherence	$$Q_{\alpha\beta} = \frac{2d}{d-1} \langle p_{k,\alpha} p_{k,\beta} + f_2^{(d)}(\rho_{k,\alpha\beta}) \rangle$$	Coherence between sectors α, β

If $\mathcal{H}_\alpha$ is 1-dimensional (e.g., for qubits under Abelian symmetries), $Q_\alpha = 0$ and only coherences encode multipartite entanglement.

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The symmetry-resolved multipartite entanglement paradigm thus provides a powerful algebraic and operational decomposition of multipartite quantum correlations, revealing the sector-specific and coherence-based structure underpinning global quantum entanglement. The approach enables new insights into the classification of quantum phases, experimental access to multipartite entanglement signatures, and a detailed understanding of the interplay between symmetry and multipartite correlations in complex quantum systems (Jain, 16 Sep 2025).