Diagnosing chaos with projected ensembles of process tensors

This paper introduces the projected process ensemble as a unified framework for diagnosing quantum chaos in many-body systems, demonstrating that its higher moments reveal characteristic entanglement structures that more sharply distinguish chaotic from integrable dynamics than previously studied chaos quantifiers.

The Gist

Imagine you are watching a complex magic show. The magician (the quantum system) performs a series of tricks. Sometimes, the tricks are predictable and follow a strict, repeating pattern (like a clockwork toy). Other times, the tricks seem completely random, chaotic, and impossible to predict (like a tornado).

For a long time, scientists have tried to figure out a simple way to tell the difference between a "clockwork" system and a "tornado" system. They have used various tools to measure the "chaos," but many of these tools have a flaw: they sometimes get fooled. A very regular, predictable system can look chaotic to these tools, making it hard to tell them apart.

This paper introduces a new, sharper way to diagnose chaos in quantum systems. Here is how they did it, explained through simple analogies:

1. The "Butterfly" Recording Device

First, the authors use a concept called a Process Tensor. Think of this as a super-advanced video camera that doesn't just record the final picture, but records every possible version of the show simultaneously.

• The Setup: Imagine the magician performs a trick, and you have to choose how to watch it (e.g., from the left, from the right, or with a filter).
• The Recording: The Process Tensor creates a giant "library" of all possible outcomes. For every choice you make (every intervention), there is a corresponding "output state" (the result of the trick).
• The "Butterfly" Space: The authors call the space where all these choices live the "Butterfly Space." It's like a control room where every possible sequence of button presses is recorded.

2. The Old Tools: Why They Got Fooled

The paper looks at two previous tools used to measure chaos:

• Quantum Dynamical Entropy (QDE): This measures how much the system "forgets" its past. If you poke a chaotic system, it scrambles information quickly. If you poke a regular system, it might also scramble information if you poke it enough times. The problem is that some boring, regular systems (like free-floating particles) can look just as chaotic as the real tornadoes when using this tool.
• Spatiotemporal Entanglement (STE): This tool looks at how the "scrambling" spreads through space and time. It's better than the first tool, but it still struggles to tell the difference between a "regular but complex" system and a truly "chaotic" one when the system gets very large.

3. The New Solution: The "Projected Process Ensemble" (PPE)

To fix this, the authors invented a new method called the Projected Process Ensemble (PPE).

The Analogy: The "Classroom Test"
Imagine you are a teacher trying to figure out if a class of students is truly chaotic (randomly shouting answers) or just following a hidden script (reciting a poem).

• The Old Way (QDE/STE): You ask the class one question and look at the average noise level. Sometimes, a class reciting a poem loudly can sound just as noisy as a chaotic class.
• The New Way (PPE): Instead of just asking one question, you ask the class a specific sequence of questions (interventions).
  • You record the answer for every single possible sequence of questions you could ask.
  • Now, you don't just look at the average noise. You look at the distribution of the answers.
  • The Key Insight:
    • Chaotic Systems: No matter which sequence of questions you ask, the answers are all wildly different and look like they were pulled from a completely random hat. The "spread" (variance) of these answers is tiny because they are all equally random.
    • Regular Systems: The answers depend heavily on which questions you asked. Some sequences give similar answers, others give very different ones. The "spread" is huge.

4. What They Found

The authors ran massive computer simulations (like running the magic show millions of times on a supercomputer) using different types of "magicians" (quantum models):

• The Tornado (Chaotic): These systems showed a very specific signature. When you looked at the spread of their answers, it was incredibly small and consistent, matching what you would expect from pure randomness.
• The Clockwork (Integrable/Regular): These systems showed a much wider spread. Their answers were not uniformly random; they depended on the specific path taken.
• The Frozen (Many-Body Localized): These systems barely moved at all, showing very little chaos.

The "Measurement" Twist:
The paper also tested what happens if you "peek" at the system (measure it) during the process.

• If you use deterministic interventions (like pressing a button that always does the same thing), the chaotic systems look perfectly random.
• If you use non-deterministic interventions (like a coin flip that might collapse the state), the "chaos" gets a bit dampened. It's like the act of watching the magic trick too closely makes the trick less wild. However, even with this dampening, the chaotic systems still looked distinct from the regular ones.

Summary

The paper argues that to truly diagnose chaos in a quantum system, you shouldn't just look at the "average" behavior. Instead, you should look at the entire family of possible outcomes generated by different sequences of actions.

• Chaotic systems are like a perfect random number generator: no matter how you try to trick them, they always produce a perfectly uniform, random spread of results.
• Regular systems are like a complex machine: they produce results that vary depending on exactly how you press the buttons.

By analyzing the "variance" (the spread) of these results, the authors found a way to clearly distinguish between true chaos and systems that just look chaotic, solving a problem that previous tools couldn't handle.

Technical Summary

Technical Summary: Diagnosing Chaos with Projected Ensembles of Process Tensors

Problem Statement
Quantum chaos lacks a unique, universally agreed-upon definition, and existing quantifiers often yield inconsistent results. While classical chaos is characterized by the growth of dynamical entropy (Kolmogorov-Sinai entropy) distinguishing random, chaotic, and non-chaotic dynamics, the quantum analogue—the Alicki-Fannes Quantum Dynamical Entropy (QDE)—fails to unambiguously distinguish chaotic from non-chaotic dynamics in many-body settings. Specifically, the QDE exhibits maximal growth rates not only in chaotic systems (e.g., Sachdev-Ye-Kitaev models) but also in regular dynamics such as free fermions and Lindblad-Bernoulli shifts. Similarly, Spatiotemporal Entanglement (STE), which attempts to capture the complexity of output states, struggles to distinguish chaotic from interacting-integrable dynamics in the thermodynamic limit. The authors identify a need for a framework that captures both the unpredictability of observed dynamics and the complexity of conditional quantum states to resolve these deficiencies.

Methodology
The authors utilize the process tensor formalism, which provides a general representation of a quantum system evolving under repeated interventions. They construct a pure process tensor $|\Upsilon\rangle$ by applying a sequence of local interventions (Kraus operators) to a system initially in a pure state, entangling the "butterfly space" (temporal interventions) with the "remainder space" (spatial output states).

To diagnose chaos, the paper introduces and analyzes three hierarchical probes:

1. Quantum Dynamical Entropy (QDE): Defined as the Rényi-2 entanglement entropy across the bipartition of the butterfly space ($B$) and the remainder space ($R$). This measures the sensitivity of output states to different intervention sequences.
2. Spatiotemporal Entanglement (STE): An extension of QDE that considers entanglement across bipartitions mixing spatial and temporal degrees of freedom ($BR_1 : R_2$), aiming to capture information scrambling.
3. Projected Process Ensemble (PPE): A novel contribution where the authors define an ensemble of pure output states conditioned on a fixed basis of local interventions. They analyze the statistical moments of the bipartite entanglement entropy within this ensemble. Specifically, they compute the mean (first moment) and variance (related to the fourth moment) of the entanglement distribution.

The study employs exact diagonalization on several paradigmatic 1D spin-chain and fermionic models:

• Chaotic: XXZ model with next-to-nearest neighbor interactions; Interacting Aubry-André (IAA) model with weak disorder.
• Integrable: XXZ model (Bethe ansatz integrable); Free fermion model.
• Many-Body Localized (MBL): IAA model with strong disorder.

The authors exploit $U(1)$ symmetry (particle number conservation) to restrict the process tensor to a single symmetry sector, enabling simulations of larger system sizes ($L \approx 14$). They compare results against the Haar ensemble (random unitary dynamics) as a benchmark for maximal randomness. Both deterministic (unitary) and non-deterministic (projective measurement) interventions are considered.

Key Contributions and Results

• Limitations of QDE and STE: Numerical simulations confirm that QDE grows linearly and saturates near the maximal value for both chaotic and non-chaotic (integrable, free fermion) systems in finite sizes, failing to distinguish them in the thermodynamic limit. STE improves upon this by distinguishing MBL from chaotic dynamics but still fails to clearly separate chaotic from interacting-integrable dynamics at large system sizes.
• Projected Process Ensemble (PPE) as a Superior Probe: The authors demonstrate that higher-order moments of the PPE provide a fine-grained diagnostic for chaos.
  • Mean Entanglement: For chaotic systems, the mean bipartite entanglement of the PPE approaches the Haar-random value (Page curve behavior).
  • Variance (Coefficient of Variation): The most significant finding is the scaling of the variance. For chaotic systems, the variance of the entanglement entropy vanishes exponentially with system size, indicating that any sequence of local unitary interventions generates maximal entanglement. In contrast, non-chaotic systems (integrable, free fermions, MBL) exhibit significantly larger variances that do not decay exponentially.
• Intervention Dependence: The distinction is sharpest for deterministic interventions (unitary operations). When using non-deterministic interventions (projective measurements), the disentangling effect of measurements reduces the mean entanglement and increases the variance for all models, making the distinction between chaotic and non-chaotic dynamics more difficult, though still observable.
• Temporal Correlations: The study links the linear growth of QDE to the suppression of non-Markovian temporal correlations. Chaotic systems exhibit rapid decay of mutual information between temporal regions, approaching Markovian behavior, whereas non-chaotic systems retain long-lived non-Markovian correlations.

Significance and Claims
The paper claims to provide a unified framework for analyzing the complexity of unitary and monitored many-body dynamics. By introducing the Projected Process Ensemble, the authors offer a method that overcomes the deficiencies of QDE and STE, particularly the inability to distinguish chaotic from interacting-integrable dynamics.

The authors assert that the PPE moments, specifically the exponential decay of the variance with system size, serve as a robust "fingerprint" of chaos in quantum stochastic processes. This approach elucidates how chaos manifests in spatiotemporal correlations, suggesting that chaotic dynamics generate output states that are indistinguishable from Haar-random ensembles in terms of their entanglement structure, provided the initial state is low-entangled and the interventions are deterministic.

The work is modest regarding experimental realization, noting that constructing the full process tensor requires post-selection and scales exponentially with the number of interventions. However, the authors suggest that the exponential scaling of the variance difference between chaotic and non-chaotic systems could make these signatures observable even with a relatively modest number of interventions ($n_B \sim 10$) in future experimental setups, potentially informing the study of measurement-induced phase transitions.