Symmetry-resolved genuine multi-entropy: Haar random and graph states

TL;DR

This work analyzes symmetry-resolved genuine multi-entropies (GM) for states with a global U(1) charge, focusing on Haar-random states and random graph states. It derives analytic expressions for symmetry-resolved GM in the thermodynamic limit for Haar states and confirms them numerically for finite systems, showing that symmetry resolution largely preserves the multi-partite entanglement profile of fully Haar states. For graph states, higher-partite entanglement exhibits code-like structure and deviates from Haar behavior, though projection onto fixed charge sectors makes the curves resemble Haar results in shape albeit with sector-dependent weights. The results illuminate how conservation laws and state structure influence genuinely multipartite entanglement and have potential implications for black-hole evaporation models and holography, where higher-partite entanglement plays a crucial role.

Abstract

We study the symmetry-resolved genuine multi-entropy, a measure that captures genuine multi-partite entanglement, in Haar random states and random graph states in the presence of a conserved quantity. For Haar random states, we derive explicit formulae for the genuine multi-entropy under a global $U(1)$ symmetry in the thermodynamic limit, and find that its dependence on subsystem sizes closely resembles that of fully Haar random states without a conserved charge. We also perform numerical analyses, focusing on spin systems for both Haar random and graph states. For random graph states, our numerical analyses reveal distinctive features of their multi-partite entanglement structure and we contrast them with the Haar random case.

Figures (8)

• Figure 1: The contraction pattern for the reduced density matrix $\rho_{AB}=\tr_C(\ket{\psi}\bra{\psi})$ used to construct the tripartite Rényi multi-entropy $S^{\IfInteger{3}{(3)}{\mathtt{(3)}}}_2$. Figure taken from Ref. Iizuka:2025ioc.
• Figure 2: A plot of the tripartite multi-entropy curves \ref{['MES3']} for different global charge densities $n_Q$.
• Figure 3: A plot of the tripartite genuine multi-entropy \ref{['GME3']} curves for different global charge densities $n_Q$. The quadripartite genuine multi-entropy \ref{['GME4']} also follows a similar curve.
• Figure 4: Numerical evaluation for the symmetry-resolved $\mathtt{k}=3$ and $\mathtt{k}=4$ (genuine) $n=2$ multi-entropy curves on $N=12$ qubits with global charge $Q=2,4,6$. The top row shows multi-entropies and the bottom row shows genuine multi-entropies. For each $(N_R,Q)$ point we sample $100$ random vectors from the uniform Haar ensemble within the sector Hilbert space. The darkness of the data points represents the number of samples (darker = more). The large variance and the smoothed out transitions ( i.e., $N_R=N/\mathtt{k}$ for both $S^{\IfInteger{k}{(k)}{\mathtt{(k)}}}$ and ${\rm GM}^{\IfInteger{k}{(k)}{\mathtt{(k)}}}$, as well as $N_R=2$ for ${\rm GM}^{\IfInteger{k}{(k)}{\mathtt{(k)}}}$) are expected from finite dimensional corrections to the analytical results in the thermodynamic limit.
• Figure 5: Numerical evaluations of the second Rényi entropy (purity) of a random graph state on $N=12$ qubits. The detailed setup is the same as in Fig. \ref{['fig:graph_S34']}.