⚛️ high-energy theory

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

This paper investigates symmetry-resolved genuine multi-entropy in Haar random and random graph states with conserved quantities, deriving explicit thermodynamic limit formulae for the former and using numerical analyses to reveal distinctive multi-partite entanglement features in the latter compared to the Haar random case.

Original authors: Norihiro Iizuka, Simon Lin

Published 2026-01-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Norihiro Iizuka, Simon Lin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant, complex party where everyone is holding hands in a giant, invisible web of connection. In the world of quantum physics, this "holding hands" is called entanglement. Usually, scientists measure how connected two groups of people are by looking at just one pair of groups (like Group A and Group B). This is like measuring how much two friends are talking to each other.

But what if the party has a secret rule? What if everyone is also sorted by a specific trait, like wearing a red shirt or a blue shirt? And what if we want to know how connected the groups are only among the people wearing red shirts?

This paper is about a new way of measuring those connections, specifically looking at groups of three or four people (not just pairs) while respecting that "red shirt" rule. The authors call this "Symmetry-Resolved Genuine Multi-Entropy." That's a mouthful, so let's break it down with some everyday analogies.

1. The Problem: The "Coarse" Ruler

Imagine you have two different groups of three people:

Group 1: Three people standing in a circle, all holding hands with each other.
Group 2: Three people where two are holding hands, and the third is just watching.

If you only look at the "pair" connections (who is holding hands with whom), you might think these groups are similar. But if you look at the whole group, they are totally different. The first group has a special "team spirit" that the second one lacks.

The paper says that old measuring tools (called entanglement entropy) are like a ruler that only measures pairs. They miss the special "team spirit" of groups larger than two. The authors want to build a new tool that can see this "team spirit" (which they call Genuine Multi-Entropy).

2. The Two Types of Parties They Studied

The authors tested their new tool on two very different types of "parties" (quantum states):

A. The "Chaotic" Party (Haar Random States)

Imagine a party where everyone is dancing wildly and randomly, with no choreography. This represents a Haar random state. It's the most chaotic, unpredictable system you can have.

The Finding: When the authors applied their "red shirt" rule (symmetry resolution) to this chaotic party, they found something surprising. Even though they forced everyone to sort by color, the "team spirit" of the groups looked almost exactly the same as if they hadn't sorted them at all.
The Metaphor: It's like sorting a chaotic dance floor by shoe size. Even after sorting, the dancers are still dancing just as wildly and randomly as before. The "shape" of the connection didn't change; it just got scaled down slightly.

B. The "Structured" Party (Graph States)

Now, imagine a party where everyone is following a strict instruction manual based on a map (a graph). This represents Graph States, which are used in quantum computers.

The Finding: These parties are very different. Before sorting, they looked somewhat like the chaotic party. But when the authors applied the "red shirt" rule, the results were weird and distinct.
The Metaphor: Imagine a marching band. If you sort them by hat color, the pattern of their formation changes in a very specific, rigid way. Unlike the chaotic dancers, the "team spirit" here didn't look like a random mess. It had a unique, structured signature that was different from the chaotic party, even after sorting.

3. The "Team Spirit" Detector (Genuine Multi-Entropy)

The authors had to be careful. Sometimes, a group of three people might look connected just because two of them are holding hands, and the third is holding hands with one of them. That's not a true "three-way" connection.

They created a formula to subtract out those simple "pair" connections to find the Genuine connection—the kind that only exists when all three (or four) are involved together.

For the Chaotic Party: The "Genuine Team Spirit" appeared exactly when you expected it to, following a predictable curve.
For the Structured Party: The "Genuine Team Spirit" was often zero or followed a very strange, stepped pattern (like a staircase instead of a smooth hill). This tells us that structured quantum systems (like those in computers) don't have the same kind of deep, multi-person "team spirit" as chaotic systems do.

4. Why Does This Matter? (The Black Hole Connection)

The authors mention that this isn't just about abstract math; it relates to Black Holes.

Think of a black hole as a giant, chaotic party that is slowly evaporating (losing particles).
Real black holes have rules (like conservation of energy or electric charge).
The paper suggests that if you only look at simple "pair" connections, you might think the black hole behaves like a chaotic mess. But if you look at the "Genuine Team Spirit" of larger groups, you might see that the black hole's behavior is actually more complex or different than we thought.
The Takeaway: If we want to understand how black holes work, we can't just use the old "pair" rulers. We need these new "multi-person" tools, especially when we respect the rules (symmetries) that nature follows.

Summary

Old Way: Measuring how connected two groups are.
New Way: Measuring how connected three or four groups are together, while respecting a specific rule (like a conserved charge).
Result 1: In totally chaotic systems, the rules don't change the "shape" of the connection much.
Result 2: In structured systems (like quantum computers), the rules change the connection pattern significantly, revealing a unique, rigid structure that chaotic systems don't have.
Goal: To better understand the deep, hidden connections in complex systems like black holes.

Technical Summary: Symmetry-Resolved Genuine Multi-Entropy in Haar Random and Graph States

Problem Statement
Entanglement entropy is a standard diagnostic for bipartite entanglement but fails to capture the richer structure of genuine multi-partite entanglement. While measures like multi-entropy and genuine multi-entropy (GM) have been developed to address this, their behavior under symmetry resolution—specifically in the presence of a conserved global charge (e.g., $U(1)$ charge)—remains largely unexplored. Most existing literature on symmetry-resolved entanglement focuses on bipartite systems. Furthermore, there is a need to understand how conservation laws modulate genuine multi-partite entanglement in the context of black hole physics, where total energy and charge are intrinsic constraints. This paper investigates the symmetry-resolved genuine multi-entropy (GM) for two distinct ensembles: Haar random states and random graph states.

Methodology
The authors employ a combination of analytical derivations in the thermodynamic limit and numerical simulations on finite spin chains ( $N=12$ ).

Theoretical Framework:
- Symmetry Resolution: The Hilbert space is decomposed into sectors labeled by total charge $Q$ . For a system divided into $k$ subsystems, the Hilbert space further factorizes into subsectors labeled by local charges $(q_1, \dots, q_k)$ . The reduced density matrix becomes block-diagonal.
- Haar Random States: The authors derive analytical formulae for the averaged Rényi multi-entropy and genuine multi-entropy in the thermodynamic limit ( $N, Q, q \gg 1$ ). They utilize the factorization of the Haar measure into products of measures over subsectors and apply the Page curve logic, noting that the measure is sharply peaked around uniform charge distribution.
- Graph States: These are stabilizer states defined by a graph $G$ and generated by applying controlled-Z gates to Hadamard-initialized qubits. The authors utilize the property that the entanglement entropy of stabilizer states is determined by the $\mathbb{Z}_2$ rank of the adjacency matrix sub-blocks. They derive explicit formulae for non-symmetry-resolved GM in terms of these ranks and perform numerical projections onto specific charge sectors.
Numerical Analysis:
- Simulations were performed on $N=12$ qubits.
- For Haar random states, random vectors were sampled from the uniform Haar ensemble within specific charge sectors ( $Q=2, 4, 6$ ).
- For graph states, random graphs were generated using the Erdős–Rényi model ( $p=0.5$ ). The states were projected onto charged sectors using projectors $\Pi_q$ to study symmetry-resolved properties.
- The study focused on Rényi index $n=2$ (and $n=3$ for specific graph state properties) for tripartite ( $k=3$ ) and quadripartite ( $k=4$ ) partitions.

Key Contributions and Results

Haar Random States:
- Analytical Formulae: The authors derive explicit expressions for symmetry-resolved multi-entropy $S^{(k)}_n$ and genuine multi-entropy $GM^{(k)}_n$ in the thermodynamic limit.
- Scaling Behavior: They find that the symmetry-resolved curves closely resemble those of fully Haar random states without conserved charges. The primary effect of the conserved charge is a global scaling factor, $S_Q = -n_Q \log n_Q - (1-n_Q) \log(1-n_Q)$ , where $n_Q = Q/N$ .
- Shape Preservation: The functional dependence on subsystem size (the "Page curve" shape) remains unchanged, merely scaled by $S_Q$ . For example, the tripartite genuine multi-entropy $GM^{(3)}$ exhibits a piecewise linear behavior with transitions at specific ratios of subsystem size to total size, scaled by $S_Q$ .
Graph States:
- Bipartite Similarity: In the bipartite case ( $k=2$ ), random graph states exhibit Page-curve-like behavior similar to Haar random states, as the adjacency matrices are typically full rank.
- Multi-partite Distinctiveness: Unlike Haar random states, graph states show distinct features in higher-partite entanglement.
  - Tripartite Case ( $k=3$ ): For $n=2$ , the genuine tripartite multi-entropy $GM^{(3)}_2$ is identically zero for all graph states. This is attributed to the fact that stabilizer states can be decomposed into GHZ states and Bell pairs, and $GM^{(3)}_2$ vanishes for GHZ states. Non-zero values appear only for $n=3$ .
  - Quadripartite Case ( $k=4$ ): The authors find a specific constraint for graph states: $4GM^{(4)}_2 + I_{3, n=2} = 0$ , where $I_3$ is the tripartite information. This indicates a highly constrained entanglement structure distinct from generic Haar random states.
- Symmetry Resolution Effects: When graph states are projected onto specific charge sectors, the discrete nature of their entanglement (integer multiples of $\log 2$ ) "smears out," making them appear more "Haar-typical." However, qualitative differences remain, such as the relative weighting between different charge sectors, which differs from the Haar random case.

Significance and Claims
The paper claims to initiate the systematic study of symmetry-resolved multi-partite entanglement. Its primary significance lies in:

Robustness of Typicality: For Haar random states, the imposition of a global $U(1)$ symmetry does not qualitatively alter the multi-partite entanglement structure; it only rescales the magnitude. This suggests that typicality arguments in black hole physics may remain robust even with a few conserved charges ("hairs").
Limitations of Code States: The study highlights that stabilizer/graph states, often used as toy models for holographic error correction, possess a fundamentally different genuine multi-partite structure compared to Haar random states. Specifically, their multi-partite entanglement is highly constrained (e.g., vanishing $GM^{(3)}_2$ ) and does not naturally reproduce the generic behavior expected in chaotic systems or black hole evaporation.
Diagnostic Utility: The authors emphasize that genuine multi-entropy provides a necessary diagnostic beyond symmetry-resolved entanglement entropy to distinguish between generic chaotic dynamics and structured code-like states, which is crucial for modeling black hole evaporation dynamics accurately.

The paper concludes that while symmetry-resolved entanglement entropy captures the bipartite spectrum, it is insufficient to determine the genuine higher-partite structure, necessitating the use of measures like GM to fully characterize the entanglement dynamics in systems with conservation laws.