Coherence-induced deep thermalization transition in random permutation quantum dynamics

This paper reports a novel phase transition in random permutation quantum dynamics where the projected ensemble shifts between a maximally entropic "deep thermalization" phase and a minimally entropic "classical" phase, a phenomenon driven by input coherence and measurement basis that remains invisible to the subsystem's density matrix despite universal infinite-temperature thermalization.

The Gist

Imagine you have a giant, chaotic dance floor filled with thousands of dancers. This is your quantum system. The paper describes a specific type of dance called "Random Permutation Dynamics."

Here's the twist: In this dance, the dancers never actually change their steps or create new moves (superpositions). They just swap places with each other randomly. If Dancer A was at spot 1 and Dancer B was at spot 2, they might swap, but they never become a "ghostly mix" of being in both spots at once. It's like shuffling a deck of cards: the cards move, but they remain distinct cards.

The researchers asked a fascinating question: If we watch just a small group of dancers (a subsystem) while the rest of the floor (the environment) is shuffled and then measured, what does that small group look like?

The Two Worlds of the Dance

The paper discovers that this system can exist in two completely different "universes" or phases, depending on how much coherence (a fancy word for quantum "waviness" or superposition) is injected into the system at the start.

1. The "Chaos Phase" (Deep Thermalization)

• The Setup: You start with a deck of cards that is already mixed up in a "superposition" (like a card that is half-ace and half-king until you look at it).
• The Result: When you shuffle the deck and look at just two cards, they look like they were pulled from a perfectly random, infinite deck. Every possible combination is equally likely.
• The Metaphor: Imagine looking at a single pixel on a TV screen that is displaying a high-definition, static-filled image. The pixel looks like pure, random noise. It has lost all memory of where it came from. This is called Deep Thermalization. The system has "forgotten" its past so thoroughly that it looks like pure randomness.

2. The "Classical Phase" (No Deep Thermalization)

• The Setup: You start with a deck of cards that is perfectly ordered (all Aces, then all Kings, etc.) and you shuffle them.
• The Result: Even after shuffling, if you look at just two cards, they don't look like random noise. They look like specific, distinct cards. You can tell exactly which card is which. The system hasn't "forgotten" its structure; it just rearranged the pieces.
• The Metaphor: Imagine shuffling a deck of red and blue cards. If you look at a small handful, you can still clearly see "Red, Blue, Red." The pattern is preserved. This is the "Classical Bit-String" phase. It's orderly, not chaotic.

The Big Surprise: The Invisible Switch

Here is the most mind-bending part of the paper:

Usually, when a system changes from "ordered" to "chaotic," you can see it by looking at the average behavior (like the temperature of the gas). But in this quantum dance, the average behavior never changes.

• The Density Matrix (The Average): No matter which phase you are in, if you just ask "What is the average state of these two dancers?", the answer is always the same: "They look like hot, random noise."
• The Hidden Switch: The difference only shows up if you look deeper, at the higher moments (the specific patterns and correlations between the dancers).
  • In the Chaos Phase, the specific patterns are complex and random.
  • In the Classical Phase, the specific patterns are simple and rigid.

It's like two different songs that sound exactly the same if you only listen to the volume (the average), but if you listen to the lyrics (the higher details), one is a jazz improvisation and the other is a marching band.

The Trigger: Coherence

What makes the system switch from the "Classical" march to the "Chaos" jazz?

Coherence.

Think of coherence as the "quantum fuel" or "superposition energy."

• If you inject too little coherence (start with a very ordered state), the system stays in the Classical Phase. The shuffling isn't enough to break the order.
• If you inject enough coherence (start with a "wavy" state), the system snaps into the Chaos Phase. The shuffling spreads that quantum "waviness" everywhere, creating true randomness.

The paper shows there is a sharp tipping point. It's like a light switch. You can slowly turn the "coherence knob," and nothing happens until you hit a specific number. Click! Suddenly, the system flips from being orderly to being deeply random.

Why Does This Matter?

1. New Kind of Physics: We usually think of "phase transitions" (like ice melting to water) as things you can see with a thermometer. This paper shows a new kind of transition that is invisible to thermometers but visible to quantum detectives looking at the fine print.
2. Quantum Information: It tells us how information is stored and scrambled. In the "Classical" phase, information is easy to find (it's just rearranged). In the "Chaos" phase, information is hidden in complex correlations, making it hard to retrieve.
3. Universality: The researchers found this happens in many different types of quantum systems, as long as the system shuffles things around without creating new "wavy" states on its own.

Summary Analogy

Imagine a library where books are constantly being swapped between shelves by a robot (the random permutation).

• The Classical Phase: The books are all red. When the robot swaps them, the shelves still look like they are full of red books. If you pull one out, it's just a red book. The library looks boring and predictable.
• The Chaos Phase: The books are a mix of colors, but they are also "glowing" (coherent). When the robot swaps them, the glowing spreads. If you pull a book out, it looks like a blur of every color in the universe. The library looks like a magical, random explosion of light.

The paper proves that you can switch the library from "boring red" to "magical blur" just by changing how many "glowing" books you start with, even though the robot's shuffling rules never change. And the most surprising part? If you just count the average number of books on a shelf, it looks the same in both cases! You have to look at the colors to see the difference.

Technical Summary

1. Problem Statement

The paper addresses a fundamental question in non-equilibrium quantum statistical mechanics: Does the "Projected Ensemble" (PE) of a local subsystem always thermalize to a universal, maximally entropic distribution?

• Context: Recent work established that for generic chaotic quantum systems, the PE (the collection of post-measurement wavefunctions of a subsystem $A$ obtained by measuring its complement $B$) converges to a "deep thermalized" state. This state is typically the Haar ensemble (uniform distribution over the Hilbert space), representing a stronger form of equilibration than the standard thermalization of the reduced density matrix (RDM) to a Gibbs state.
• The Gap: It was unknown whether this "deep thermalization" is a universal law or if there exist dynamical regimes where the PE fails to reach the Haar ensemble, even if the RDM appears thermal. Specifically, the authors investigate systems where the dynamics preserve quantum coherence (superposition) rather than scrambling it into complex entanglement.

2. Methodology

The authors employ a combination of rigorous analytical proofs, information-theoretic resource theory, and numerical simulations to study Random Permutation Dynamics (RPD).

• Model System: They study $N$-qubit systems evolving under a global random permutation unitary $U_\pi$, which shuffles computational basis states $|z\rangle \to |\pi(z)\rangle$ without creating new superpositions. This models the late-time behavior of circuits composed of local permutation gates.
• Two Microscopic Models:
  1. Tilted-Basis Model: Initial states and measurement bases are uniform product states defined by Bloch angles $(\theta_0, \phi_0)$ and $(\theta_m, \phi_m)$. The angle $\theta_m$ tunes the measurement basis from the $z$-axis to the $x$-axis.
  2. Mixed-Basis Model: A discrete analog where the initial state and measurement basis are mixtures of computational basis states ($|0\rangle$) and superposition states ($|Y_+\rangle$). Parameters $\alpha_0$ and $\alpha_m$ control the fraction of superposition in the input and measurement, respectively.
• Analytical Tools:
  • Weingarten Calculus: Used to perform averages over the symmetric group $S_{2^N}$ to compute moments of the PE.
  • Maximum Entropy Principle (MEP): Applied to predict the limiting form of the PE based on equivalence classes of measurement outcomes.
  • Resource Theory of Coherence: The transition is framed in terms of the relative entropy of coherence ($C_r$), quantifying the amount of superposition relative to the computational basis.
• Numerical Approach: Simulations of the PE for various system sizes ($N=12$ to $24$) to compute trace distances between the PE and target ensembles (Haar vs. Classical Bit-String) and to perform finite-size scaling analysis.

3. Key Contributions

The paper identifies a sharp phase transition in the projected ensemble that is invisible to standard local observables.

1. Discovery of Deep Ergodicity Breaking: The authors demonstrate that while the reduced density matrix $\rho_A$ always thermalizes to the infinite-temperature state (maximally mixed) regardless of parameters, the Projected Ensemble undergoes a phase transition between two distinct universality classes:
   • Deep Thermalized Phase: The PE converges to the Haar ensemble ($E_{Haar}$), where states are uniformly distributed over the entire Hilbert space (maximal entropy).
   • Deep Non-Ergodic Phase: The PE converges to the Classical Bit-String Ensemble ($E_{Cl}$), where states are uniformly distributed only over the computational basis states (minimal entropy).
2. Coherence as the Order Parameter: The transition is driven by the total coherence injected into the system by the initial state and the measurement basis.
   • Low coherence $\to$ Classical Bit-String Ensemble.
   • High coherence $\to$ Haar Ensemble.
3. Analytical Determination of Phase Boundaries:
   • For the Mixed-Basis Model, they derive an exact analytical condition for the transition: $\alpha_0 + \alpha_m = 1$.
   • For the Tilted-Basis Model, they predict the critical angle $\theta_m^*$ by mapping the coherence of the mixed-basis model to the tilted-basis parameters, yielding $\theta_m^* \approx 0.181\pi$ (close to the numerically observed $0.193\pi$).
4. Universality: They argue that this phenomenon is generic for any "coherence-preserving" scrambling dynamics (where the dynamics do not generate new superpositions but scramble existing ones). They verify this by extending the analysis to include diagonal phase gates, showing the transition persists.

4. Key Results

• Theorem 1 (Regular Thermalization): Proved that for almost all random permutation states, the local RDM $\rho_A$ converges to the maximally mixed state $I/d_A$, regardless of the measurement basis. This confirms that standard thermalization is robust.
• Theorem 2 (Emergence of Classical Ensemble): Proved that for $z$-basis measurements (zero coherence injection), the projected state collapses to a single computational basis state $|z_A\rangle$ with unit probability in the thermodynamic limit. This defines the $E_{Cl}$ phase.
• Numerical Phase Diagram:
  • In the Tilted-Basis Model, varying the measurement angle $\theta_m$ reveals a critical point $\theta_m^* \approx 0.193\pi$. Below this angle, the PE is $E_{Cl}$; above it, the PE is $E_{Haar}$.
  • In the Mixed-Basis Model, the transition occurs sharply at $\alpha_0 + \alpha_m = 1$.
• Finite-Size Scaling:
  • The Tilted-Basis Model exhibits a continuous transition with a critical exponent $\nu \approx 1.3$.
  • The Mixed-Basis Model exhibits a first-order-like transition with $\nu \approx 1.0$.
• Invisibility to Local Observables: The transition is strictly a "higher-moment" phenomenon. The first moment (RDM) shows no signature of the transition, distinguishing it from standard symmetry-breaking or localization transitions.

5. Significance and Implications

• New Form of Ergodicity Breaking: This work introduces a novel class of ergodicity breaking that does not prevent the system from reaching infinite temperature (thermalization) but prevents the system from reaching the "deep" thermal state (Haar randomness). It challenges the universality of the Maximum Entropy Principle in the context of projected ensembles.
• Resource-Driven Transitions: It establishes coherence as a thermodynamic resource that can drive phase transitions in quantum dynamics, analogous to how energy or charge conservation drives symmetry-breaking transitions.
• Experimental Relevance: Since RPD can be modeled by local random permutation gates (brickwork circuits), this transition is experimentally accessible in current quantum simulators (e.g., neutral atoms or superconducting qubits) where measurement-induced transitions are already being studied.
• Generalization: The framework suggests that other quantum resources (e.g., imaginarity, magic, or non-Gaussianity) could similarly drive transitions between "resource-rich" (deeply thermalized) and "resource-poor" (ergodicity-breaking) phases in the projected ensemble.

In summary, the paper reveals that the "deep" structure of quantum thermalization is fragile and depends critically on the interplay between initial state coherence and measurement basis, leading to a sharp phase transition that is undetectable by standard local thermodynamic observables.