Late-time ensembles of quantum states in quantum chaotic systems

This paper demonstrates that in quantum chaotic systems with symmetries, late-time ensembles evolving from typical product states become statistically indistinguishable from Haar-random states at the level of finite moments, despite not ergodically exploring the entire Hilbert space, while also identifying specific atypical initial conditions that lead to distinguishable, non-universal behaviors.

The Gist

Imagine you have a giant, chaotic dance floor (the Hilbert Space) filled with millions of dancers (quantum states). In a perfectly chaotic system, if you let the music play long enough, the dancers should eventually mix so thoroughly that the crowd looks like a completely random, featureless blur. This is the idea of ergodicity: given enough time, the system explores every possible arrangement.

However, in the real world, there are rules. Maybe the dancers must keep their total number of red shirts constant (conservation of charge) or their total energy fixed. These rules act like invisible walls, preventing the dancers from exploring every single spot on the floor. They are trapped in a specific "zone" of the dance floor.

This paper asks a fascinating question: If the dancers are trapped in a zone by these rules, can we still tell the difference between their "trapped" dance and a truly random, unrestricted dance?

Here is the breakdown of their findings using simple analogies:

1. The Two Types of Starting Lines

The researchers looked at how the dance starts. They found two very different scenarios:

• The "Typical" Start (The Chaotic Mixer): Imagine starting with a group of dancers where everyone is wearing a random mix of red and blue shirts, but the average number of red shirts matches the rule. When the music starts, these dancers mix so wildly and thoroughly that, after a long time, the crowd looks indistinguishable from a truly random crowd. Even if you zoom in and look at the statistical details (like how often two specific dancers stand next to each other), you can't tell they were ever trapped.

  • The Takeaway: If you start with a "typical" state (like a simple product state where spins are just pointing up or down randomly), the system becomes so random that it effectively forgets the rules. It looks exactly like a "Haar-random" state (the gold standard of randomness).

• The "Atypical" Start (The Rigid Formation): Now, imagine starting with a very specific, rigid formation. For example, every dancer is wearing exactly the same number of red shirts, with zero variation. It's like a perfectly synchronized military drill. Even after the chaotic music plays for a long time, this group never becomes truly random. They retain a "fingerprint" of their rigid start.

  • The Takeaway: If you start with a state that has very low "variance" (it's too perfect, like a state with a fixed number of particles), the system remains "non-ergodic." You can easily tell the difference between this group and a random crowd by measuring simple things, like the entropy (disorder) of a small section of the floor.

2. The "Fine-Grained" vs. "Coarse-Grained" View

For a long time, scientists only looked at the "coarse-grained" view. This is like looking at the dance floor from a helicopter and saying, "It looks like a big, messy blur." At this level, both the "Typical" and "Atypical" groups look the same: they both have high disorder (Volume Law Entanglement).

But this paper looks at the "fine-grained" view. This is like getting a drone camera to hover right over the dancers and count exactly how many times specific patterns repeat.

• The Surprise: They found that for "Typical" starts, even the fine details look perfectly random. The rules (symmetries) didn't leave a trace.
• The Exception: For "Atypical" starts, the fine details still show the rigid structure of the beginning. The system didn't fully "scramble" the information.

3. The "Infinite Temperature" Paradox

Usually, we think of "infinite temperature" as the state of maximum chaos and randomness. The paper shows a twist: You can have a system at "infinite temperature" (high energy, middle of the spectrum) that is still not fully random if you start it in a very specific, low-variance way. It's like having a room full of people shouting (high energy) who are all shouting the exact same sentence in perfect unison (low variance). It's chaotic in energy, but not in information.

4. Why This Matters for Real Experiments

We are currently building quantum computers (using things like trapped ions or superconducting qubits) that are essentially these chaotic dance floors.

• Good News: If you prepare your quantum computer in a simple, easy-to-make state (like a product state), and let it run, it will naturally evolve into a state that is perfectly random. You don't need to do anything special to "scramble" the information; the chaos does it for you, even with conservation laws in place.
• Bad News (or interesting news): If you accidentally prepare a state that is too "perfect" (like a state with a fixed particle number), the system won't scramble properly. It will retain memory of its start, which could be a bug for randomization tasks, or a feature for storing information.

Summary Analogy

Imagine you are making a smoothie.

• The Rule: You must use exactly 10 strawberries and 10 blueberries (Conservation of Charge).
• Typical Start: You throw the fruit in the blender in a random pile. When the blender runs (chaos), the fruit mixes so perfectly that every spoonful looks exactly like a random mix of the whole. You can't tell the fruit was ever separate.
• Atypical Start: You stack the fruit in a perfect, alternating red-blue tower before turning on the blender. Even after the blender runs for a long time, the "tower" structure leaves a subtle statistical imprint. If you taste the smoothie carefully, you can tell it started as a tower, not a pile.

The paper's conclusion: Most of the time, nature acts like the "Typical Start." It mixes so well that the rules don't matter. But if you start with a "perfectly ordered" state, the rules stick, and the system never becomes truly random.

Technical Summary

1. Problem Statement

The paper addresses a fundamental gap in the understanding of quantum chaos and thermalization. While it is well-established that quantum chaotic systems evolve toward featureless states described by the Gibbs ensemble (or Eigenstate Thermalization Hypothesis) on average, the behavior of higher statistical moments remains unclear.

Specifically, the authors investigate:

• Symmetry Constraints: Physical systems often possess symmetries (e.g., charge, energy, magnetization) that constrain dynamics to a subspace of the Hilbert space (a "high-dimensional torus") rather than the full space.
• Ergodicity vs. Moments: Do these constraints leave measurable "fingerprints" in the late-time statistics of quantum states beyond the average?
• Initial Condition Dependence: Do low-entanglement initial states (e.g., product states), which are experimentally accessible, exhibit universal late-time behavior independent of their specific preparation, or do they retain memory of their initial symmetry properties?
• Timescales: Theoretical proofs of ergodicity often require exponentially long times. The paper asks what happens at "late times" accessible in current experiments.

2. Methodology

The authors employ a combination of analytical derivations using Random Matrix Theory (RMT) and numerical simulations on specific quantum models.

• Analytical Framework:

  • They define the late-time ensemble $\Phi_{T \to \infty}$ as states sampled at random times $t > T$ after a long evolution.
  • They compare this ensemble against the Haar-random ensemble (pure random states over the full Hilbert space) and constrained ensembles (random states restricted to specific symmetry sectors).
  • They analyze the moments of the symmetry operator $\hat{Q}$ (e.g., total magnetization or energy) in the initial state $|\Psi_0\rangle$. They define a state as typical if its distribution of symmetry outcomes matches that of a Haar-random state, and atypical if it deviates (specifically having lower variance).
  • They calculate the trace distance between the $k$-th moment density matrices of typical/atypical ensembles and the Haar ensemble.

• Numerical Models:

  • U(1) Random Quantum Circuits: 1D chains with brickwork unitary gates conserving total $Z$-magnetization.
  • Mixed Field Ising Model (MFIM): A chaotic Hamiltonian $\hat{H} = \sum \hat{Z}_j \hat{Z}_{j+1} + g \hat{X}_j + h \hat{Z}_j$ with broken translation and inversion symmetries.
  • Initial Conditions: They test "typical" product states (e.g., $|FMy\rangle$ with spins aligned in $y$, having large energy/magnetization variance) and "atypical" product states (e.g., $|AFMx\rangle$ or $|AFMz\rangle$ with alternating spins, having zero or reduced variance).
  • Observables: The primary metric is the von Neumann Entanglement Entropy (EE) of half-systems ($S_A$), analyzed via its distribution (mean and variance). They also verify results using second Rényi entropy and smaller subsystems.

3. Key Contributions and Results

A. Two Distinct Late-Time Regimes

The paper identifies two limiting regimes for late-time ensembles based on the variance of the conserved quantity in the initial state:

1. The Typical Regime (Haar-like):

   • Condition: Initial states where the distribution of the symmetry operator matches that of Haar-random states (e.g., product states with large fluctuations in charge/energy).
   • Result: Despite the dynamics being constrained by symmetries (not exploring the full Hilbert space), the late-time ensemble is indistinguishable from the Haar ensemble at the level of all finite statistical moments ($k$-moments) in the thermodynamic limit.
   • Implication: No local or non-local measurement of finite moments can distinguish these states from pure random states. The spatial locality of the initial state is effectively erased by chaotic dynamics.

2. The Atypical Regime (Constrained):

   • Condition: Initial states with negligible variance of the symmetry operator (e.g., fixed particle number states, energy eigenstates, or specific antiferromagnetic product states like $|AFMz\rangle$).
   • Result: These states evolve into non-universal ensembles that are distinguishable from the Haar ensemble. They behave like constrained random states (Haar states restricted to a specific symmetry sector).
   • Signature: These states exhibit $O(1)$ corrections to the Page entropy (the average entanglement entropy of random states) and larger statistical fluctuations compared to the Haar ensemble.

B. Quantitative Findings on Entanglement Entropy

• Typical States: The mean EE converges to the Page entropy ($\mu_H$) with exponentially small corrections. The variance is exponentially small.
• Atypical/Constrained States: The mean EE is lower than the Page entropy by a finite amount: $\Delta S \approx \frac{1}{2}[f + \log(1-f)]$ (where $f$ is the subsystem fraction). This indicates that eigenstates and atypical late-time states encode more structural information (due to spatial locality and conservation laws) than random states.
• Robustness: The "Typical" behavior is robust across different Hamiltonian parameters (as long as the system is chaotic), whereas "Atypical" behavior is highly sensitive to the specific initial condition's symmetry variance.

C. Distinction from Quantum Scars

The authors clarify that these atypical states are not "Quantum Many-Body Scars."

• Scars: Exhibit anomalously low entanglement (violating the volume law) due to fine-tuned phase space structures.
• Atypical States (this work): Still exhibit volume-law entanglement but with distinct $O(1)$ corrections and fluctuations. They are robust and ubiquitous, not requiring fine-tuning.

4. Significance and Implications

• Refining Quantum Chaos: The work suggests that the standard definition of quantum chaos (based on RMT level statistics and average EE) is insufficient. A complete definition must account for higher statistical moments and the role of initial conditions.
• Experimental Relevance: The results apply to states easily prepared in current quantum simulators (superconducting qubits, Rydberg atoms, trapped ions). It implies that while these devices cannot generate all possible states due to conservation laws, they can still generate states that are effectively random (Haar-like) if the initial state is chosen correctly (typical).
• Information Scrambling: It demonstrates that spatial locality, which imprints structure on eigenstates, is washed away by chaotic evolution for typical initial states, leading to maximal randomness. However, for atypical states, this locality persists in the late-time statistics.
• Thermalization Limits: The paper establishes that "infinite temperature" (midspectrum) dynamics can still be strongly non-ergodic in terms of higher moments if the initial state has low symmetry variance, challenging the notion that chaotic dynamics always leads to complete featurelessness.

Conclusion

The paper provides a rigorous framework for understanding how symmetries and initial conditions dictate the fine-grained structure of late-time quantum states. It establishes that while symmetries constrain the Hilbert space, typical product states effectively mix these constraints to produce Haar-random statistics, whereas atypical states retain a "memory" of their initial symmetry constraints, resulting in distinguishable, non-universal ensembles with reduced entanglement and larger fluctuations.