Symmetry-Resolved Genuine Multi-Entropy

• Symmetry-resolved genuine multi-entropy is a measure that isolates truly multipartite entanglement by dissecting contributions from distinct charge sectors under superselection constraints.
• The formalism utilizes Rényi entropies, cyclic permutations, and Fourier transformations to extract k-partite information in systems with conserved quantities like a global U(1) charge.
• Analytical and numerical studies in Haar-random and graph states demonstrate its efficacy in distinguishing operational entanglement, with implications for quantum simulation and resource theory.

Symmetry-resolved genuine multi-entropy quantifies the amount of genuinely multipartite entanglement or correlation present in a quantum system when resolved with respect to additive or locally conserved symmetries, such as a global U(1) charge. This diagnostic refines conventional multi-entropy or multi-partite information by dissecting contributions that stem specifically from distinct charge sectors or symmetry sectors, thereby clarifying the underlying structure and operational capabilities of multipartite quantum states under superselection or symmetry constraints.

1. Formalism and Definitions

Let a quantum system be divided into $k$ distinct subsystems $R_1,\ldots,R_k$ with Hilbert space $\mathcal{H} = \bigotimes_{i=1}^k \mathcal{H}_{R_i}$. For a pure state $|\psi\rangle$, the $k$-partite $n$-Rényi multi-entropy is defined as

$$S^{(n)}_{[k]} = \frac{1}{1-n}\ln\frac{Z^{(n)}_{[k]}}{\Bigl(Z_{[k]}^{(1)}\Bigr)^n}$$

where $Z^{(n)}_{[k]}$ is a symmetric contraction

$$Z_{[k]}^{(n)} := \langle\psi|^{\otimes n^{k-1}}\,\Sigma_1\,\Sigma_2\,\cdots\,\Sigma_k\,|\psi\rangle^{\otimes n^{k-1}}$$

with $\Sigma_i$ the cyclic permutation (twist) of the $i$th subsystem's indices among the $n^{k-1}$ folded copies. For $k=2$, $S_{[2]}^{(n)}$ recovers the standard bipartite Rényi entropy.

The genuine multipartite multi-entropy, denoted here as $\operatorname{GM}[k]^{(n)}$ (Editor's term), extracts the truly $k$-partite component by constructing the uniquely defined linear combination of the multi-entropies for all nontrivial subsystems, which is nonzero if and only if the $k$-body entanglement is present and vanishes if the state can be constructed from lower-partite entanglement alone. For $k=3$,

$$\operatorname{GM}[3]^{(n)}(A:B:C) = S_{[3]}^{(n)}(A:B:C) - \frac12\Bigl[S_{[2]}^{(n)}(A) + S_{[2]}^{(n)}(B) + S_{[2]}^{(n)}(C)\Bigr]$$

and for $k=4$ (with $a=1/3$),

$$\operatorname{GM}[4]^{(n)}(A:B:C:D) = S_{[4]}^{(n)} -\frac13\sum_{\text{3-1 splits}} S_{[3]}^{(n)} + \frac13\sum_{\text{2-2 splits}} S_{[2]}^{(n)}$$

In the symmetry-resolved context, e.g. in the presence of a global U(1) charge, each subsystem's reduced density matrix block-diagonalizes in the local charge basis: $$\rho_{R_i} = \bigoplus_{q_i} p_{q_i} \,\rho_{R_i}(q_i)$$ with $p_{q_i}=\mathrm{Tr}[\Pi_{q_i}\rho_{R_i}]$. The symmetry-resolved multi-entropy $S_{[k]}^{(n)}(q_1,\dots,q_k)$ refers to the contraction performed within the $(q_1,\dots, q_k)$ block, and the symmetry-resolved genuine entropy is constructed via the same linear combinations, applied to these block entropies.

2. Symmetry Resolution: Charge Sectors and Fourier Methods

The symmetry resolution is operationalized by decomposing the global Hilbert space (or local reduced density matrices) into subspaces labeled by symmetry eigenvalues, such as total charge for U(1) symmetry. In field-theoretic systems---notably free compact bosons---multi-charged moments are constructed as generating functions: $$Z_n(\{\alpha_j\}) = \mathrm{Tr}\Big[\rho_A^n \exp\Big(i\sum_{j=1}^n\alpha_j \hat Q_{A_j}\Big)\Big]$$ which, in conformal field theory, are computed as partition functions or correlation functions of vertex operators on replica Riemann surfaces of genus $g=n-1$ (Gaur et al., 2023).

The charged moments are then projected to fixed-charge sectors via Fourier transform: $$\mathcal{Z}_n(\{q_j\}) = \prod_{j=1}^n \frac{1}{2\pi}\int_{-\pi}^{\pi} d\alpha_j\, e^{-i\alpha_j q_j}\, Z_n(\{\alpha_j\})$$ and the symmetry-resolved $n$-Rényi entropy in sector $\{q_j\}$ is

$$S_n(\{q_j\}) = \frac{1}{1-n} \ln\left[ \frac{\mathcal{Z}_n(\{q_j\})}{\left(\mathcal{Z}_1(\{q_j\})\right)^n} \right]$$

3. Genuine Multi-Partite Rényi Entropy and Equipartition

The symmetry-resolved genuine n-partite Rényi multi-entropy is defined as

$$I_n^{(g)}(q_1, ..., q_n) = \sum_{j=1}^n S_n(q_j) - S_n(q_1,\dots, q_n)$$

with $S_n(q_j)$ the single-interval (marginal) entropy in charge sector $q_j$, and $S_n(q_1,...,q_n)$ the joint entropy of all intervals (Gaur et al., 2023). This construction isolates the information shared genuinely by all $n$ subsystems, suppressing contributions from correlations restricted to fewer parts.

To leading order in the large-interval (large-length) limit---the "equipartition regime"---one finds

$$I_n^{(g)}(q_1,\dots, q_n) \approx I_n^{(g,0)} - \tfrac12 \ln\frac{\prod_j \ln\ell_j}{\ln\left(\prod_i\ell_i\prod_{i<j}y_{ij}\right)} - \frac{\pi^2 n}{2K(1-n)} \Bigl[\bm q^T Y_n^{-1} \bm q - n\, \bm q^T Y_1^{-1} \bm q \Bigr]$$

where $I_n^{(g,0)}$ is the conventional (unresolved) genuine $n$-partite entropy, $\ell_j$ are interval sizes, $y_{ij}$ are cross-ratios, $Y_n$ is a covariance matrix of interval logs, and $K$ the Luttinger parameter. The $O(1)$ and $q$-dependent corrections account for charge fluctuations and break exact equipartition.

At the self-dual radius ($K=1$), all symmetry-resolved expressions for the compact boson map directly to those of the massless Dirac fermion, indicating universality of symmetry-resolved multipartite entropy in this regime.

4. Analytical and Numerical Results for Haar Random and Graph States

In finite-size spin systems, especially spin chains or qubit arrays with conserved charge, symmetry-resolved genuine multi-entropy has been computed analytically and numerically for both Haar-random pure states and random graph (stabilizer) states (Iizuka et al., 2 Nov 2025).

For Haar-random states in a fixed total charge $Q$ sector and for subsystems of size $N_{R_i}$, the typical dimension of the fixed-charge block is

$$d_i(q_i) \approx \exp\left[N_{R_i} s\left(q_i/N_{R_i}\right)\right], \qquad s(x) = -x\ln x - (1-x)\ln(1-x)$$

and, in the thermodynamic limit,

$$S_{[k]}^{(n)}(N;Q;N_R) \approx \begin{cases} S_Q (k-1)N_R & N_R < N/k \ S_Q (N-N_R) & N_R > N/k \end{cases}$$

with $S_Q = s(n_Q)$, $n_Q=Q/N$. The genuine curves $\operatorname{GM}[k]$ are piecewise-linear, with kinks and slopes determined as in the unconstrained Page curve but rescaled by $S_Q < \ln 2$.

For random graph states (Erdős–Rényi graphs), bipartite Rényi entropy is set by the $\mathbb{Z}_2$-rank of the adjacency submatrix. Genuine tripartite $\operatorname{GM}[3]^{(2)}$ is identically zero for graph states, while at $n=3$, $\operatorname{GM}[3]^{(3)}$ detects the number of extractable GHZ states via adjacency ranks. Quadripartite $\operatorname{GM}[4]^{(2)}$ satisfies a linear relation with tripartite information $I_3$ specific to stabilizer graphs. After projecting onto charge sectors, the structure of symmetry-resolved genuine multi-entropy in graph states interpolates towards Haar-random typicality, but discretization persists due to underlying code structure.

5. Relation to Mutual Information and Connections with Quantum Information Theory

The symmetry-resolved genuine multi-entropy generalizes the symmetry-resolved mutual information to multipartite contexts. In the bipartite case, mutual information can be symmetry-resolved by computing joint probability distributions for local charges and integrating over constraints with Fourier techniques. In the multipartite framework, the definition and Fourier transformation machinery naturally extend to $n$-body joint distributions and multiparameter integrals.

This enables precise quantitative diagnostics of entanglement and correlation structure in physical systems where symmetry is not only present but constrains dynamics or measurement, such as cold atomic gases with particle-number conservation, or systems exhibiting superselection sectors.

Experimentally, for small systems, symmetry-resolved multi-entropy can be accessed using two-copy SWAP tests on subsystems. One first projects each copy to desired charge sectors, and then performs entanglement measurements on the post-selected ensemble (Jain, 16 Sep 2025). For more complex or larger systems, randomized measurement protocols and classical shadows techniques enable reconstruction of quadratic functionals such as the symmetry-resolved linear entropy.

6. Comparative Behavior and Universality in Physical Models

In models such as the free compact boson CFT, symmetry-resolved multi-entropy exhibits leading-order equipartition of entropic contributions among charge sectors, paralleling the "Page curve" in random state entanglement. Numerical checks against spin chain tight-binding models confirm analytic predictions (Gaur et al., 2023).

In stabilizer graph states, the structure of multipartite entanglement is extremely non-generic, with higher-partite genuine measures vanishing in certain Rényi indices. Projecting onto symmetry sectors "regularizes" this structure, smoothing entropies towards those observed in Haar-random ensembles, though some lattice-scale discretization remains evident.

A plausible implication is that symmetry-resolved genuine multi-entropy is a robust tool for distinguishing truly multipartite entanglement in generic versus code-like many-body states, and for quantifying physically operational entanglement under superselection constraints. This bears immediate relevance for resource theory of entanglement, black hole information modeling, and the study of many-body localization phenomena.

7. Extensions, Generalizations, and Experimental Outlook

The symmetry-resolved genuine multi-entropy formalism applies to arbitrary locally-acting abelian symmetries beyond U(1): general abelian group charges, and, with suitable modifications, to non-abelian cases via Cartan decompositions or irrep block analysis (Jain, 16 Sep 2025). It admits analytic computation in Haar ensembles and symmetry-protected models, and generalizes to arbitrary Rényi indices, system dimensions, and subsystem topologies.

Experimental realization can leverage SWAP-based measurement protocols, charge-resolved projective measurements, and randomized measurement schemes for quadratic observables. The formalism is anticipated to find applications in quantum simulation, condensed matter, and quantum information, particularly in tracking the fate of multipartite entanglement in presence of symmetries, and in enriching the operational meaning of entropy measures under physical constraints.