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Entanglement governs early-time growth of randomness in projected ensembles

This paper demonstrates that the initial growth of randomness in projected ensembles is governed by bipartite entanglement, showing that short-time frame potentials are determined by subsystem purity and establishing a direct link between local randomness and global spectral properties.

Original authors: Thuwaragesh Jayachandran, Wai-Keong Mok

Published 2026-08-25

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CC BY 4.0

Original authors: Thuwaragesh Jayachandran, Wai-Keong Mok

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quiet, isolated world of a quantum system, particles evolve according to strict, reversible rules, yet when we look at just a small piece of that system, it often appears to have settled into a calm, random equilibrium. This phenomenon, known as thermalization, explains how the chaotic motion of individual atoms gives rise to the predictable laws of heat and temperature. For decades, physicists have understood that if you ignore the rest of the system and focus only on a small part, that part eventually looks like a standard thermal state. However, a deeper layer of this story has only recently come into focus. Scientists realized that the simple average view misses a richer structure: if you were to measure the rest of the system in a specific way, the small part would be left in a collection of different pure states, each with its own probability. This collection, called a projected ensemble, reveals how randomness is distributed across the system's possible configurations. The big question has been how this randomness grows. Does it appear instantly, or does it build up slowly? And what physical mechanism drives the very first moments of this process?

A team of researchers at the California Institute of Technology has now traced the origin of this randomness back to its source, showing that the initial burst of disorder is governed entirely by the entanglement between the part being watched and the part being measured. Entanglement is a uniquely quantum connection where two systems become so linked that the state of one cannot be described without the other. The researchers studied what happens in the first few moments after a system starts evolving from a state where its parts are completely separate and unentangled. They found that the speed at which randomness appears in the small subsystem is set by a specific timescale determined by how strongly the two parts interact. Crucially, they discovered that during these early moments, the specific way the outside world is measured does not matter at all. The randomness grows purely because the two parts of the system are becoming entangled, regardless of the tools used to observe the outside.

To reach this conclusion, the team analyzed the mathematical structure of these projected ensembles using a tool that measures how "random" a collection of quantum states is. They focused on the very beginning of the evolution, starting from a moment when the system was perfectly ordered and unentangled. By calculating how the randomness changed over tiny fractions of a second, they proved that the growth is dictated by the purity of the small subsystem, which is a direct measure of how much entanglement has formed with the rest of the system. Their calculations showed that up to a certain level of precision, the details of the measurement—such as which specific direction or basis the outside particles are measured in—have no effect. The influence of the measurement choice only begins to appear much later, after the initial entanglement has already established the pace of randomness. This finding clarifies that the mechanism for the onset of quantum randomness is universal and rooted in the fundamental interaction between the system's parts, rather than in the specific details of how an observer looks at it.

The researchers also explored what happens when the system's energy levels are chosen randomly, a scenario that mimics the chaotic behavior found in many complex quantum materials. In these cases, they uncovered a surprising link between the local randomness of the small subsystem and the global properties of the entire system's energy spectrum. They found that the rate at which randomness grows in the small part is directly controlled by the statistical fluctuations of the energy levels of the whole system. This connection reveals a hierarchy in how quantum chaos manifests: the global chaos of the entire system sets the stage, but the local randomness emerges through the specific channel of entanglement. The study suggests that while the long-term behavior of these systems depends heavily on how they are measured, the very first steps toward disorder are a pure consequence of the system's internal dynamics.

The team tested these ideas using numerical simulations of a one-dimensional chain of interacting particles, a model known to exhibit chaotic behavior. They started with the particles in a simple, unentangled state and watched how the randomness evolved as the system interacted with itself. The simulations confirmed their theoretical predictions: the growth of randomness matched the rate of entanglement generation perfectly in the early stages. They also checked what happened when the initial state of the outside part did not cover all possible measurement outcomes evenly. In those cases, the agreement between the theory and the simulation broke down quickly, confirming that the initial simplicity of the process relies on the system having a broad, uniform starting point. These results provide a clear picture of how quantum systems transition from order to disorder, identifying entanglement as the primary engine that drives the emergence of randomness in the earliest moments of evolution.

This work shifts the focus from the final, settled state of a quantum system to the dynamic process of how it gets there. By isolating the early-time behavior, the researchers have shown that the complex interplay between measurement and dynamics simplifies into a single, dominant factor: the generation of entanglement. This insight helps explain why different quantum systems, despite having different internal structures or being measured in different ways, often exhibit similar patterns of thermalization. It suggests that the universal behavior seen in quantum chaos is not a coincidence of late-time statistics but is rooted in the fundamental way quantum information spreads through a system. The study does not claim to solve the entire mystery of quantum thermalization, but it firmly establishes the rules for the opening act, showing that before the measurement details can shape the outcome, the system must first build the quantum connections that make randomness possible.

Technical Summary: Entanglement governs early-time growth of randomness in projected ensembles

Problem Statement
While the late-time behavior of projected ensembles—statistical distributions of pure states on a subsystem obtained via projective measurements on its complement—is well-understood through the lens of "deep thermalization" and the emergence of Scrooge or Haar ensembles, the mechanism governing the initial growth of quantum randomness remains unclear. Previous research has largely focused on equilibrium states or the dynamical approach to $k$ -designs at late times. This paper addresses the short-time regime, asking what physical quantity dictates the onset of randomness in a projected ensemble generated from an initially unentangled state.

Methodology
The authors analyze a bipartite quantum system $AB$ initialized in a product state $|\Psi_0\rangle = |\phi_A\rangle \otimes |\phi_B\rangle$ . The system evolves under a time-independent Hamiltonian $H$ for a short time $t$ . Subsystem $B$ (the bath) is measured in a fixed orthonormal basis $\{|z\rangle\}$ , generating a projected ensemble $\mathcal{E} = \{p(z); |\psi_z\rangle\}$ on subsystem $A$ .

To quantify the randomness of $\mathcal{E}$ , the authors employ the frame potential $F^{(k)}(t)$ , defined as the purity of the $k$ -th moment operator of the ensemble. A lower frame potential indicates higher randomness, with the Haar ensemble representing the minimum. The analysis proceeds in two stages:

Arbitrary Hamiltonians: Deriving a general expansion of $F^{(k)}(t)$ for small $t$ under the assumption that the initial bath state $|\phi_B\rangle$ has full support in the measurement basis.
Unitarily Invariant Ensembles: Specializing to ensembles of Hamiltonians invariant under unitary conjugation (e.g., the Gaussian Unitary Ensemble, GUE) to relate the projected ensemble's frame potential to global spectral properties.

Key Contributions and Results

1. Entanglement as the Primary Driver (Result 1 & 2)
For arbitrary Hamiltonians, the authors derive that the early-time growth of the frame potential is determined entirely by the bipartite entanglement between subsystems $A$ and $B$ .

Cubic Order Independence: Up to cubic order in time ( $O(t^3)$ ), the $k$ -th frame potential $F^{(k)}(t)$ is given by:
$F^{(k)}(t) = (F^{(1)}(t))^k + O(t^4)$
where $F^{(1)}(t)$ is the subsystem purity (related to the second Rényi entanglement entropy).
Basis Independence: Because entanglement is basis-independent, the specific choice of measurement basis on the bath does not influence the randomness growth up to $O(t^3)$ . Measurement basis effects only appear at quartic order ( $O(t^4)$ ) and higher.
Timescale: The initial growth rate is governed by the entanglement timescale $T_{\text{ent}}$ , defined by the interaction Hamiltonian's ability to drive the system out of its initial product state. Specifically, $F^{(k)}(t) \approx 1 - 2k(t/T_{\text{ent}})^2$ . This establishes that the timescale for the onset of local randomness is identical to the timescale for entanglement generation.

2. Connection to Spectral Properties (Result 3 & 4)
For unitarily invariant Hamiltonian ensembles (like the GUE), the authors connect local randomness to global spectral statistics.

Spectral Variance: The initial growth of randomness depends solely on the spectral variance of the Hamiltonian ensemble.
The "2k → k" Correspondence: The ensemble-averaged $k$ -th frame potential of the projected ensemble is directly related to the $4k$ -point spectral form factor $R_{4k}(t)$ of the global unitary dynamics. In the limit of large subsystem dimensions ( $D_A, D_B \gg 1$ ):
$\mathbb{E}_{H} [F^{(k)}(t)] \approx \frac{R_{4k}(t)}{D^{4k}}$
Equivalently, this relates the projected ensemble's $k$ -th moment to the $2k$ -th frame potential of the global unitary ensemble. This reveals a hierarchy where the randomness of the global unitary dynamics at order $2k$ controls the initial growth of randomness in the projected ensemble at order $k$ .

3. Numerical Validation
The theoretical predictions were tested using:

Mixed-Field 1D Ising Model: Numerical simulations confirmed that $F^{(k)}(t) \approx (F^{(1)}(t))^k$ holds for short times ( $Jt \lesssim 0.2$ ) in chaotic regimes.
Gaussian Unitary Ensemble (GUE): Simulations verified the relationship between the projected frame potential and the spectral form factor, showing that the approximation improves as subsystem dimensions increase.
Full-Support Assumption: The authors demonstrated that if the initial bath state lacks full support in the measurement basis (e.g., concentrated in a single basis state), the short-time validity window shrinks, consistent with the theoretical requirement that $p_{\text{min}}$ (minimum initial probability) must not be exponentially small.

Significance and Claims
The paper claims to identify entanglement as the universal mechanism governing the onset of quantum randomness in projected ensembles, distinct from the late-time regime where measurement basis selection is critical for forming $k$ -designs.

Hierarchy of Randomness: The work establishes a hierarchy where entanglement dictates the leading-order behavior at short times, while measurement-basis effects are subleading.
Local-Global Link: It provides a direct quantitative link between local randomness (projected ensemble) and global spectral properties (spectral form factors) for unitarily invariant ensembles, complementing existing theorems that link approximate unitary $2k$ -designs to state $k$ -designs.
Limitations and Scope: The authors modestly note that their results apply to the early-time regime and infinite-temperature settings. They do not claim that the initial growth diagnoses quantum chaos (as it depends only on spectral variance), but rather that it sets the stage for when finer signatures of deep thermalization and chaotic dynamics emerge. The extension of the "2k → k" correspondence to generic chaotic Hamiltonians remains an open question.

Technical Summary: Entanglement governs early-time growth of randomness in projected ensembles

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