Information diagrams in the study of entanglement in symmetric multi-quDit systems and applications to quantum phase transitions in Lipkin-Meshkov-Glick D-level atom models

This paper employs information diagrams and generalized U(D) coherent states to analyze entanglement in symmetric multi-quDit systems, proposing the rank of reduced density matrices as a discrete order parameter to characterize quantum phase transitions in Lipkin-Meshkov-Glick models of D-level atoms.

The Gist

The Big Picture: Mapping the "Shape" of Entanglement

Imagine you have a huge room full of identical dancers (these are the quDits, or quantum particles). In a normal dance, everyone moves independently. But in a quantum dance, the dancers can become "entangled," meaning their movements are perfectly synchronized in a way that defies classical logic. If you look at just one dancer, you can't tell what they are doing without looking at the whole group.

The authors of this paper are trying to draw a map (called an "Information Diagram") to understand how "mixed up" or entangled these dancers are. They aren't just counting how many dancers are tangled; they are looking at the shape of that entanglement.

The Tools: The "Cat" and the "Map"

1. The Schrödinger's Cat (The DCAT)
Usually, quantum particles are in a "coherent" state, which is like a calm, predictable wave. But the authors are studying a special, chaotic state called a Schrödinger's Cat (or DCAT).

• The Analogy: Imagine a cat that is simultaneously sleeping and awake, or a coin that is spinning heads and tails at the same time. In this paper, they create a "super-cat" made of many atoms. This cat is a quantum superposition of two very different, macroscopic states. It's like a dance troupe where half the group is dancing a waltz and the other half is breakdancing, but they are doing it at the exact same time.

2. The Information Diagram (The Map)
To measure how tangled the dancers are, the authors use two different rulers:

• Linear Entropy: A simple ruler that measures how "messy" the state is.
• Von Neumann Entropy: A more complex, sophisticated ruler that measures the same thing but with more nuance.

They plot these two measurements against each other on a graph. This graph is the Information Diagram.

• The Shape of the Map: The paper shows that not every point on this graph is possible. The valid points form a specific shape (like a curved triangle). The edges of this shape are special; they represent the "extremal" or most extreme types of entanglement possible.
• The "Rank" (The Complexity Score): Inside this map, the authors track the Rank of the reduced density matrix. Think of the "Rank" as the number of different "colors" or "patterns" needed to describe the dance.
  • Rank 1: The dancers are all doing the exact same simple move (no entanglement).
  • Higher Rank: The dancers are doing a complex, multi-colored routine. The higher the rank, the more complex the entanglement.

The Experiment: The Lipkin-Meshkov-Glick (LMG) Model

The authors apply this map to a specific model of atoms called the Lipkin-Meshkov-Glick (LMG) model.

• The Setup: Imagine a group of 3-level atoms (like a three-way switch) that can interact with each other. You can turn up a "knob" (the interaction strength, $\lambda$) to make them interact more intensely.
• The Goal: They want to see what happens to the "dance" (the entanglement) as you turn up this knob. Specifically, they are looking for Quantum Phase Transitions (QPTs).
  • The Analogy: A phase transition is like water turning into ice. At a specific temperature, the water suddenly changes its fundamental nature. In this quantum dance, at a specific "knob setting," the way the atoms are entangled suddenly changes its fundamental nature.

The Discovery: The "Rank" as a Warning Sign

Here is the main finding of the paper, explained simply:

1. The Map Fills Up: When they plot the entanglement of these "Cat" states on their Information Diagram, the points fill up the bottom part of the map. This tells them that these specific quantum states have a very specific, constrained way of being entangled. They don't explore every possible type of entanglement; they stick to a specific "lane."

2. The Jump in Rank: As they turn up the interaction knob ($\lambda$), the Rank of the entanglement stays low for a while. Then, suddenly, it jumps.

   • At a low knob setting, the Rank is 1 (simple).
   • At a medium setting, it jumps to 2.
   • At a high setting, it jumps to 3 or 4.

3. The "Precursor" (The Canary in the Coal Mine): The authors discovered that these sudden jumps in Rank happen at the exact same moment the Quantum Phase Transition occurs.

   • The Metaphor: Usually, to detect a phase transition, you need to measure complex, continuous changes (like temperature or pressure). The authors found that you don't need a complex thermometer. You can just look at the Rank (the number of patterns). When the Rank suddenly changes from 2 to 3, you know instantly that a major shift in the system's nature has happened.

Why This Matters (According to the Paper)

• A New Tool: The authors propose using the Rank of the reduced density matrix as a "discrete order parameter." In plain English, this means it's a simple, whole-number switch that tells you exactly when the system is changing phases.
• Universality: They suggest that this "Rank jump" might be a universal way to detect these quantum changes in other similar systems, not just the one they studied.
• Simplicity: Instead of calculating complex, messy numbers to find a phase transition, you can just count the "colors" (Rank) in the quantum state. If the count changes, the phase has changed.

Summary

The paper is about drawing a map of quantum entanglement using "Schrödinger's Cat" states. They found that as you increase the interaction between atoms, the complexity of the entanglement (measured by "Rank") stays steady and then suddenly jumps. These jumps act like a perfect alarm bell, signaling exactly when the system undergoes a dramatic change in its quantum nature (a Phase Transition).

Technical Summary

Technical Summary: Information Diagrams in Symmetric Multi-quDit Systems and Lipkin-Meshkov-Glick Models

Problem Statement
The paper addresses the challenge of characterizing particle entanglement in symmetric multi-quDit systems, specifically those described by generalized "Schrödinger cat" states (parity-adapted coherent states). While entropy measures like von Neumann and linear entropy are standard tools for quantifying entanglement, the authors seek a more global, qualitative understanding of the relationship between these measures and the rank of reduced density matrices (RDMs). Furthermore, the paper aims to apply these information-theoretic tools to detect and characterize Quantum Phase Transitions (QPTs) in Lipkin-Meshkov-Glick (LMG) models of $D$-level identical atoms, proposing the rank of RDMs as a discrete order parameter.

Methodology
The authors employ a multi-step theoretical approach:

1. Information Diagrams: They utilize information diagrams, which plot pairs of entropy measures (specifically normalized linear entropy $L$ and von Neumann entropy $S$) against each other. The boundaries of the allowed region $\Delta$ in these diagrams are determined by "extremal" density matrices derived from convexity/concavity properties of entropy. These boundaries are defined by specific families of density matrices, $\rho_{\max}$ and $\rho_{\min}^{(k)}$, which depend on the dimension $d$ and a parameter $\epsilon$.
2. State Construction: The study focuses on $U(D)$-spin coherent states (DSCS) and their parity-adapted counterparts, known as $D$-component Schrödinger cat states (DCAT). These states are constructed as quantum superpositions of weakly-overlapping coherent wave packets, adapted to the $Z_2^{D-1}$ parity group. The authors specifically analyze the $D=2$ (qubits) and $D=3$ (qutrits) cases.
3. Reduced Density Matrices (RDMs): To quantify entanglement, the authors compute one- and two-quDit RDMs ($\rho_1$ and $\rho_2$) by tracing out the remaining $N-1$ or $N-2$ particles from the symmetric $N$-quDit DCAT states. They calculate the linear and von Neumann entropies for these RDMs in the thermodynamic limit ($N \to \infty$).
4. LMG Model Application: These tools are applied to a specific LMG Hamiltonian for $D=3$ atoms. The ground state is approximated using the variational method with the 3CAT state. The authors analyze the behavior of the stationary curve $(\alpha_0(\lambda), \beta_0(\lambda))$—which minimizes the energy surface—as a function of the interaction strength $\lambda$. They compare these analytical variational results with numerical solutions for finite $N$.

Key Contributions and Results

• Structure of Information Diagrams: The authors demonstrate that for one-qutrit RDMs derived from 3CAT states, the information diagram region $\Delta$ is completely filled. However, for two-qutrit RDMs, the region is only partially filled (specifically subregions $\Delta_2$ and $\Delta_3$), indicating that the maximum pairwise entanglement attainable in these states is lower than that of a maximally mixed RDM of the corresponding dimension.
• Entanglement and Rank: The study establishes a direct link between the position in the information diagram and the rank of the RDM. The boundaries of the diagram correspond to specific ranks (e.g., rank 1 for pure states, rank 2 for the curve $\rho_{\min}^{(1)}$). The authors show that the stationary curve of the 3CAT ground state typically lies on the lower boundary of the allowed region, suggesting these states minimize von Neumann entropy for a given linear entropy.
• High Coupling Limit: In the high coupling limit ($\lambda \to \infty$), the variational ground state parameters approach $(\alpha, \beta) = (1, 1)$. At this point, the one-qutrit RDM becomes maximally mixed (maximally entangled), while the two-qutrit RDM reaches a high but non-maximal entropy value, lying on the curve $\bar{\rho}_{\min}^{(3)}$.
• QPT Detection via Rank: The most significant application is the identification of the rank of the RDM as a discrete order parameter for QPTs. In the $D=3$ LMG model, the system exhibits three phases separated by two second-order QPTs at $\lambda = \epsilon/2$ and $\lambda = 3\epsilon/2$.
  • At the first transition ($\lambda = \epsilon/2$), the rank of the one- and two-quDit RDMs jumps from 1 to 2.
  • At the second transition ($\lambda = 3\epsilon/2$), the rank jumps from 2 to 3 (for 1-qutrit) and from 2 to 4 (for 2-qutrits).
  • Numerical simulations for finite $N$ confirm these jumps occur near the critical values, validating the rank as a robust precursor for QPTs.

Significance and Claims
The paper claims that information diagrams provide a powerful qualitative tool for visualizing the global structure of entanglement in symmetric multi-quDit systems. By mapping the RDMs of parity-adapted coherent states onto these diagrams, the authors reveal constraints on the achievable entanglement and rank.

Crucially, the authors propose that the rank of the reduced density matrix serves as a discrete, effective order parameter for detecting QPTs in LMG models. Unlike continuous measures (such as fidelity susceptibility), the rank changes abruptly at critical points, offering a clear signature of phase transitions. The authors suggest that while their analysis is restricted to even-parity states in the $D=3$ LMG model, the methodology is generalizable to other parity-adapted states and critical models. They conclude that the universality of extremal states at the boundaries of information diagrams likely underpins the behavior of variational ground states in these systems.