Information phases of partial projected ensembles generated from random quantum states and scrambling dynamics

This paper introduces partial projected ensembles to characterize quantum information distribution in tripartite systems, revealing distinct information phases defined by the scaling of Holevo information that uncover a measurement-invisible quantum-correlated phase and provide a finer probe of scrambling dynamics than conventional entanglement measures.

The Gist

Imagine you have a giant, incredibly complex puzzle made of billions of tiny pieces (these are quantum bits, or qubits). In the quantum world, these pieces are all tangled together in a giant knot of information. Usually, when scientists try to understand this knot, they look at just one small section of it and ignore the rest. They ask, "How messy is this small section?" This is like looking at a single slice of a cake and guessing the flavor of the whole thing based only on that slice.

This paper introduces a much smarter way to look at the puzzle. Instead of just ignoring the rest of the cake, imagine you have a friend (let's call them Alice) who looks at the rest of the puzzle and takes a photo of it. Then, she sends you a description of what she saw. Based on her description, you can reconstruct what the small section you are holding must look like.

The paper explores what happens when you do this, but with a twist: sometimes Alice sends you a partial description. She might say, "I saw a red piece here, but I forgot what was next to it."

Here is the breakdown of their discovery using simple analogies:

1. The "Partial Projected Ensemble" (The Partial Clue)

Think of the whole quantum system as a massive library.

• The Old Way: You look at one book (a subsystem) and ignore the rest. You only see the average of all the books around it.
• The New Way (Projected Ensemble): You ask a librarian (the "bath" or environment) to look at all the other books and tell you which specific book is next to yours. This gives you a whole collection of possibilities for your book, depending on what the librarian saw.
• The Twist (Partial): Sometimes the librarian is forgetful. She tells you about some books but loses the notes for others. You end up with a "mixed" collection of possibilities. This is the Partial Projected Ensemble.

2. The "Holevo Information" (The Detective's Score)

The authors needed a way to measure how much real information you gain from the librarian's clues. They used a tool called Holevo Information.

Imagine you are trying to guess a secret code.

• If the librarian's clues are useless, your guess doesn't change no matter what she says. Your "score" is zero.
• If the librarian's clues are perfect, you know the exact code. Your "score" is high.
• The Holevo Information is the score that tells you: "How much does my understanding of my book change based on what the librarian tells me?"

3. The Two "Information Phases" (The Big Discovery)

The most exciting part of the paper is that they found the system behaves in two completely different ways depending on the size of the pieces involved. They call these Information Phases.

Phase A: The "Invisible" Phase (Measurement-Invisible)

• The Scenario: Imagine the "bath" (the part the librarian looks at) is huge, and the part you are holding is small.
• The Result: Even though the librarian is looking at a massive amount of data, her clues tell you nothing new about your book. Your book looks exactly the same no matter what she says.
• The Analogy: It's like trying to guess a specific grain of sand on a beach by looking at the ocean. The ocean is so big and chaotic that it "scrambles" the information so thoroughly that the grain of sand looks completely random and independent of the ocean's waves.
• Why it matters: This is a "Measurement-Invisible" phase. The quantum information is there, it's just so deeply scrambled that measuring the environment doesn't help you see it. It's a new kind of quantum correlation that doesn't exist in simple two-part systems.

Phase B: The "Visible" Phase (Measurement-Visible)

• The Scenario: Now, imagine the part you are holding is bigger than the part the librarian is looking at.
• The Result: Suddenly, the librarian's clues become incredibly powerful. Every time she describes a different part of the ocean, your book changes drastically. You can learn a lot!
• The Analogy: Now you are holding a large chunk of the beach, and the librarian is looking at a small puddle. What happens in the puddle tells you a lot about the shape of your chunk of sand. The information is "visible."

4. The "Phase Transition" (The Tipping Point)

The paper shows that there is a sharp line between these two phases. It's not a slow fade; it's like flipping a switch.

• If the "bath" is just a tiny bit bigger than your piece, you are in the Invisible phase.
• If your piece gets just a tiny bit bigger than the bath, you instantly jump to the Visible phase.

This is a "Phase Transition," similar to how water suddenly turns to ice at 0°C. Here, the "temperature" is the relative size of the quantum pieces.

5. Real-World Chaos (The Dynamic Circuits)

The authors didn't just look at static, random puzzles. They also watched these systems evolve over time, like watching a chaotic dance of particles (using "quantum circuits").

• They found that even in a chaotic, messy dance, these two phases appear naturally.
• Time Scales: In a system where everything talks to everything (all-to-all), the "Invisible" phase happens almost instantly. But in a system where particles only talk to their neighbors (like a 1D line), it takes longer for the information to scramble enough to become "invisible."

Why Should You Care?

This paper changes how we think about quantum information.

1. It's not just about Entanglement: Traditional physics says, "If two things are entangled, they are connected." This paper says, "Not always! Sometimes they are entangled, but the connection is so scrambled that measuring one tells you nothing about the other."
2. Deep Thermalization: It helps explain how complex quantum systems "forget" their past and settle into a random state, but with a hidden layer of structure that we can now detect.
3. Future Tech: Understanding these "invisible" phases is crucial for building quantum computers. If you want to protect information (quantum memory), you might want to hide it in this "Invisible" phase where it's scrambled but still there, safe from prying eyes (measurements).

In a nutshell: The authors discovered that in the quantum world, there is a "Goldilocks zone" of size. If your piece of the puzzle is too small compared to the rest, the rest of the universe becomes a "black box" that hides all its secrets from you, even though they are technically connected. This is a new, fundamental way that quantum information hides in plain sight.

Technical Summary

1. Problem Statement

The paper addresses the limitations of conventional entanglement measures (such as von Neumann entropy, Rényi entropy, and logarithmic negativity) in characterizing the information structure of quantum many-body systems.

• The Gap: Traditional measures rely on the Reduced Density Matrix (RDM), which is obtained by tracing out the environment. This process discards all information about the specific measurement outcomes of the environment, effectively "coarse-graining" the state.
• The Context: Recent developments in "Deep Thermalisation" and "Projected Ensembles" (PE) suggest that retaining information about measurement outcomes on a subsystem (the "bath") reveals a richer statistical structure than the RDM alone.
• The Specific Challenge: While PEs of pure states are well-studied, the behavior of Partial Projected Ensembles (PPEs)—where measurement outcomes on part of the bath are discarded, resulting in an ensemble of mixed states—remains poorly understood. The authors ask: Can we define an information-theoretic framework for PPEs that reveals distinct "information phases" and transitions that are invisible to standard entanglement measures?

2. Methodology

The authors employ a combination of rigorous analytical derivations and numerical simulations to study tripartite quantum systems ($N$ qubits) divided into three extensive subsystems:

• $R$: The subsystem of interest (receiving the conditional states).
• $S$: The measured subsystem (outcomes $o_S$ are retained).
• $E$: The traced-out "bath" (outcomes are discarded).

Key Tools and Frameworks:

1. Partial Projected Ensemble (PPE): Defined as the ensemble of mixed states $\{\rho_R(o_S)\}$ on $R$, conditioned on measurement outcomes $o_S$ on $S$, with $E$ traced out.
2. Holevo Information ($\chi$): The authors identify the Holevo information as the central metric. It quantifies the mutual information between the classical measurement outcomes $S$ and the quantum states on $R$:
   $$\chi(\mathcal{E}_{PPE}) = S_{vN}(\rho_R) - \sum_{o_S} p(o_S) S_{vN}(\rho_R(o_S))$$
   This measures the distinguishability of the ensemble members from the average state.
3. Universal Ensembles:
   • They utilize the Generalized Hilbert-Schmidt Ensemble (gHSe) as the universal limit for PPEs generated from Haar-random states when no conservation laws exist.
   • They prove that the moments of the PPE converge to the gHSe under specific conditions on subsystem sizes.
4. Analytical & Numerical Approach:
   • Analytical: They derive the spectral distribution of the conditional states $\rho_R(o_S)$ in the thermodynamic limit ($N \to \infty$) using concentration inequalities derived from the convergence of the second moment to the gHSe.
   • Numerical: They simulate chaotic quantum circuits (both all-to-all and 1+1D brickwork geometries) to observe the dynamical emergence of these phases and analyze the timescales involved.

3. Key Contributions

1. Definition of Information Phases: The paper establishes that the scaling of the Holevo information with system size $N$ defines distinct "information phases," separated by sharp, non-analytic transitions.
2. Discovery of the Measurement-Invisible Phase (MIQC): The authors rigorously prove the existence of a phase where the Holevo information vanishes exponentially with $N$, despite the system possessing extensive entanglement. This phase has no analogue in bipartite systems.
3. Refinement of Entanglement Phase Diagrams: By contrasting the Holevo information phase diagram with the logarithmic negativity phase diagram, the authors show that the Holevo information reveals "fine structure" (sub-phases) that standard entanglement measures miss.
4. Dynamical Realization: They demonstrate that these static information phases emerge dynamically in chaotic quantum circuits, providing a physical mechanism for their observation.

4. Key Results

A. The Information Phase Diagram (Haar-Random States)

The behavior of the Holevo information $\chi$ depends on the relative sizes of the subsystems, parameterized by $\gamma = |R|/N$ and $p = |S|/N$. Two distinct phases emerge:

1. Measurement-Invisible Quantum-Correlated (MIQC) Phase:

   • Condition: Occurs when $R$ is small relative to the bath $E$ (specifically $p \leq 1 - 2\gamma$).
   • Scaling: $\chi \sim e^{-cN}$ (Exponential decay).
   • Physical Meaning: The measurement outcomes on $S$ provide no information about the state of $R$. The conditional states $\rho_R(o_S)$ are all identical to the average state $\rho_R$.
   • Significance: This proves the existence of a phase where quantum correlations exist (finite entanglement/negativity) but are "invisible" to measurements on $S$ because the effect is mediated and screened by the traced-out subsystem $E$.

2. Measurement-Visible Quantum-Correlated (MVQC) Phase:

   • Condition: Occurs when $R$ is large relative to $E$ (specifically $p > 1 - 2\gamma$).
   • Scaling: $\chi \sim N$ (Volume law / Linear growth).
   • Physical Meaning: The measurement outcomes on $S$ significantly alter the state of $R$. The ensemble has high distinguishability.

3. The Transition:

   • The transition occurs at the critical line $p_c = 1 - 2\gamma$.
   • This transition is sharp and non-analytic, distinct from the entanglement phase transitions defined by logarithmic negativity (which occur at different boundaries, e.g., $p+\gamma=0.5$).

B. Comparison with Entanglement Measures

• Logarithmic Negativity ($N_{RS}$): Describes the entanglement between $R$ and $S$. It identifies phases like "Decoupled," "Maximally Entangled," and "Entanglement Saturation."
• Holevo Information ($\chi$): Cuts across these entanglement phases.
  • Crucially, the MIQC phase exists within the "Entanglement Saturation" region of the entanglement diagram.
  • This reveals that extensive entanglement does not guarantee that measurements on one part of the system affect another; the third subsystem ($E$) can mediate and screen these effects.

C. Dynamical Emergence in Chaotic Circuits

The authors simulate 2-local random unitary circuits to show these phases emerge dynamically:

• All-to-All Circuit: The information phases emerge rapidly. The timescale $t^*$ for the MIQC phase scales logarithmically with $N$ ($t^* \sim \ln N$), while the MVQC phase emerges almost instantly ($t^* \sim N^0$).
• 1+1D Brickwork Circuit: Due to spatial locality and light-cone constraints, the emergence of both phases is slower, with timescales scaling linearly with system size ($t^* \sim N$).
• Conclusion: The static information phase diagram is robustly realized in the long-time limit of chaotic dynamics.

5. Significance

• Beyond Entanglement: The work demonstrates that entanglement is not the sole descriptor of quantum information structure. The "measurement-invisible" phase highlights a form of scrambling where information is delocalized such that local measurements yield no predictive power, even in the presence of strong entanglement.
• Deep Thermalisation: It extends the concept of deep thermalisation to mixed-state ensembles (PPEs), showing that even with partial information loss, universal statistical properties (gHSe) emerge, but the information accessibility (Holevo info) depends critically on subsystem geometry.
• Quantum Error & Scrambling: The findings have implications for quantum error correction and scrambling dynamics. The MIQC phase suggests a regime where errors or measurements on a subsystem are effectively "shielded" by the environment, a phenomenon relevant for understanding the stability of quantum information in noisy, many-body environments.
• Theoretical Rigor: The paper provides the first rigorous proof of the existence of the MIQC phase in tripartite Haar-random states, moving beyond heuristic arguments to analytical proofs based on moment convergence and spectral concentration.

In summary, this paper introduces a new lens (Holevo information of PPEs) to view quantum many-body systems, revealing a rich phase structure of information accessibility that is fundamentally distinct from, and more detailed than, traditional entanglement phase diagrams.