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Beyond Entanglement Entropy: Spectral Entropy of Reduced States in Random Quantum Circuits

This paper proposes spectral entropy as a refined diagnostic tool that captures the detailed eigenvalue structure of reduced density matrices in random quantum circuits, revealing distinct growth dynamics, late-time complexification, and unique responses to noise that standard entanglement entropy overlooks.

Original authors: Jim Dorsey

Published 2026-10-13

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Jim Dorsey

Original paper licensed under CC BY 4.0 (https://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 quantum world, the rules of everyday life dissolve. Particles can exist in multiple states at once, and two objects can become so deeply linked that measuring one instantly reveals the state of the other, no matter how far apart they are. This deep connection is called entanglement, and for decades, physicists have used a single number to measure how strong it is. This number, known as entanglement entropy, acts like a thermometer for quantum complexity. It tells scientists how much information is shared between two parts of a system. When this number grows, it means the system is scrambling information, mixing it up so thoroughly that it becomes impossible to trace back to where it started. This scrambling is the engine behind everything from how black holes might work to why quantum computers could one day solve problems that are impossible for today's machines. However, this single number has a blind spot. It captures the total amount of uncertainty but ignores the detailed shape of the mix. It is like knowing the average temperature of a room without knowing if the heat is coming from a single hot spot or a gentle, uniform warmth.

A new study by Jim Dorsey at Miami University of Science and Technology suggests that looking deeper into the mix reveals a hidden layer of complexity that standard measurements miss. The researcher focused on the specific pattern of numbers that describe a quantum system's state, a pattern known as the spectrum. While the standard thermometer measures the total heat, this new approach, called spectral entropy, measures how that heat is distributed. By running massive computer simulations of random quantum circuits—digital models where quantum gates are applied in a random sequence—the team watched how these systems evolved over time. They found that while the standard measure of entanglement stops growing once the system reaches a certain level of chaos, the new spectral measure keeps changing. It continues to grow long after the system should have settled, revealing a slow, ongoing process of restructuring that was previously invisible.

The study began by simulating two types of quantum circuits to see how they scrambled information. One type used gates chosen from a vast, continuous range of possibilities, representing a truly random and chaotic system. The other used a more restricted set of gates, which are easier to calculate but still capable of creating complex states. The researchers started with a simple, unentangled state and applied layers of random operations, watching how the quantum information spread. They tracked the standard entanglement entropy and the new spectral entropy side by side. In the beginning, both numbers rose quickly together, indicating that the system was rapidly becoming entangled. But as the simulation progressed, a clear difference emerged. The standard entanglement entropy grew until it hit a ceiling, a point where the system appeared to have reached its maximum possible disorder. At this stage, the standard view suggested the system had finished its work of scrambling.

However, the spectral entropy told a different story. Even after the standard measure had stopped growing, the spectral measure continued to rise. In the simulations, this late-time growth amounted to an increase of about eight percent beyond the point where the standard measure had saturated. This finding implies that the system was not actually finished. The internal structure of the quantum state was still evolving, becoming more uniform and less structured in a way that the single-number thermometer could not detect. To understand why this happens, the researchers looked at the spacing between the numbers in the spectrum. They found that as time went on, these numbers rearranged themselves to follow a pattern typical of chaotic systems, a transition that took longer than the initial scrambling of information. This suggests that the journey to true quantum chaos has two distinct phases: a fast phase where information spreads, and a slower phase where the internal structure of that information refines itself.

The study also tested how these systems reacted when they were disturbed by measurements, which act like a force that tries to pull the entangled parts apart. When the researchers introduced measurements into the simulation, they observed a sharp transition. At low rates of measurement, the system remained highly entangled. At high rates, the entanglement collapsed. The standard measure showed this as a sudden, sharp drop. The spectral entropy, however, showed a smoother, more gradual decline. Even in the phase where the system appeared to be disentangled, the spectral measure remained higher than the standard measure. This indicates that even when a system looks simple, there is still a subtle, residual complexity in the way its parts are connected, a nuance that is lost when looking only at the total amount of entanglement.

To ensure these results were not just an artifact of the computer simulation, the team compared the random circuits with the more restricted ones. In the restricted circuits, the two measures of entropy were identical and did not show the late-time growth. This confirmed that the extra complexity seen in the random circuits was a genuine feature of that specific type of chaos, not a mathematical error. The researchers also checked how the results changed as they increased the size of the simulated system. They found that the time it took for the spectral entropy to settle grew in direct proportion to the size of the system, just as the standard entropy did, but it took longer. This scaling suggests that the phenomenon is a fundamental property of large quantum systems, not an artifact of the small size of the simulation.

The implications of this work are significant for how scientists understand the complexity of the universe. It suggests that the story of quantum scrambling is not finished when the standard measures say it is. There is a deeper, slower process of structural refinement that continues long after the initial chaos has set in. This hidden layer of complexity might be crucial for understanding how quantum systems thermalize, or how they might store information in ways that are robust against noise. While the study was conducted through numerical simulation and not on physical hardware, the findings offer a new lens through which to view quantum dynamics. It proposes that to truly understand the complexity of a quantum state, one must look not just at how much information is shared, but at the intricate shape of the distribution of that information. This new perspective turns a single number into a rich landscape, revealing that even in a system that appears to have reached its limit, there is still a quiet, ongoing evolution taking place.

Technical Summary: Beyond Entanglement Entropy: Spectral Entropy of Reduced States in Random Quantum Circuits

Problem Statement
Entanglement entropy has long served as the primary diagnostic for complexity and information scrambling in quantum many-body systems, particularly within the context of random quantum circuits. However, as a single scalar value derived from the von Neumann entropy ( $S_{vN} = -\text{Tr}(\rho_A \log \rho_A)$ ), it collapses the full eigenvalue spectrum of the reduced density matrix ( $\rho_A$ ) into a single number. This coarse-graining discards structural information regarding the shape of the eigenvalue distribution, level repulsion signatures, and the specific transition from integrable to chaotic behavior. While the entanglement spectrum is known to contain richer information than the entropy alone, a specific diagnostic that quantifies the "spread" of this spectrum independently of the total entanglement magnitude has been underexplored. This work addresses the question of whether a spectral measure can reveal late-time complexification and spectral restructuring that standard entanglement entropy fails to resolve.

Methodology
The study employs numerical simulations of random quantum circuits to compute and compare spectral entropy against standard entanglement measures.

Circuit Models: Two ensembles are analyzed: Haar-random circuits (gates drawn from the Haar measure on $U(4)$ ) and Clifford circuits (gates from the Clifford group). Both utilize a one-dimensional brickwork architecture with open boundary conditions.
Definitions: The spectral entropy ( $S_{spec}$ ) is defined as the Shannon entropy of the eigenvalue distribution of the reduced density matrix: $S_{spec} = -\sum_i \lambda_i \log \lambda_i$ , where $\{\lambda_i\}$ are the eigenvalues of $\rho_A$ . This is formally identical to the von Neumann entropy but is interpreted here as a measure of the distribution's shape rather than just the entanglement magnitude.
Simulation Protocol: The author simulates system sizes ( $L$ ) ranging from 8 to 16 qubits. For each circuit depth ( $t$ ), the reduced density matrix for a contiguous subsystem of size $|A| = L/2$ is computed. Eigenvalues are obtained via numerical diagonalization (for Haar circuits) or stabilizer formalism (for Clifford circuits).
Measurement Protocols: To study disentanglement, projective measurements are introduced at a rate $p$ after each unitary layer, allowing for the investigation of measurement-induced phase transitions.
Statistical Analysis: Ensemble averages are performed over 100–500 circuits. The study also analyzes level-spacing statistics (adjacent eigenvalue ratios) to characterize the transition toward random-matrix behavior.

Key Contributions and Results
The paper presents five primary findings derived from the numerical data:

Distinct Growth Timescales: In Haar-random circuits, spectral entropy and von Neumann entropy grow similarly at early times ( $t < 3$ ). However, as the circuit depth increases, a divergence emerges. While von Neumann entropy saturates at the Page value (dictated by subsystem size), spectral entropy continues to grow. The ratio $S_{spec}/S_{vN}$ increases from close to 1 at early times to approximately 1.15 by $t = 8$ , which occurs prior to the von Neumann saturation depth.
Late-Time Complexification: Spectral entropy exhibits slow growth even after von Neumann entropy has saturated. By $t = 16$ (well beyond the saturation depth), the spectral entropy has increased by approximately 8% relative to its value at the von Neumann saturation depth. This indicates that the entanglement spectrum continues to restructure and become more uniform long after the total entanglement magnitude has maxed out.
Finite-Size Scaling: The saturation depth for spectral entropy ( $t_{spec}$ ) scales linearly with system size $L$ , but with a larger slope than the von Neumann saturation depth ( $t_{vN}$ ). The difference $t_{spec} - t_{vN}$ grows from ~1.5 layers at $L=8$ to ~4 layers at $L=16$ , suggesting the late-time growth is a genuine thermodynamic feature rather than a finite-size artifact.
Dependence on Gate Set (Haar vs. Clifford): A critical distinction is found between circuit ensembles. In Clifford circuits, the eigenvalues of stabilizer states are restricted to integer powers of $1/2$ , resulting in a uniform distribution over non-zero eigenvalues. Consequently, for Clifford circuits, $S_{spec}$ coincides exactly with $S_{vN}$ (up to a logarithmic factor), and no late-time divergence occurs. This highlights that the spectral restructuring observed in Haar circuits is a feature of the continuous, generic random dynamics, not present in the discrete Clifford dynamics.
Response to Measurements: In the presence of projective measurements, spectral entropy exhibits a smoother crossover at the measurement-induced phase transition compared to the sharp transition seen in von Neumann entropy. In the area-law phase ( $p > p_c$ ), spectral entropy decays more slowly than von Neumann entropy, indicating that the reduced density matrix retains residual spectral structure (a spread over a few eigenvalues) even when the system is effectively disentangled.
Spectral Statistics: The level-spacing statistics of the entanglement spectrum evolve from a structured form (low adjacent ratios) at early times to a Wigner-Dyson distribution (approaching Gaussian Unitary Ensemble values) at late times. This transition occurs on the same timescale as the late-time growth of spectral entropy, linking the metric directly to the onset of random-matrix statistics in the entanglement spectrum.

Significance and Claims
The paper positions spectral entropy not as a replacement for entanglement entropy, but as a complementary diagnostic tool. Its significance lies in its ability to resolve details of the complexification process that are invisible to standard scalar measures:

It reveals a regime of late-time complexification where the eigenvalue distribution continues to evolve toward uniformity even after the system has reached the Page curve saturation.
It provides a finer spectral lens on quantum scrambling, distinguishing between the total magnitude of correlations (captured by $S_{vN}$ ) and the structural organization of those correlations (captured by $S_{spec}$ ).
It offers a more nuanced view of measurement-induced phase transitions, suggesting that the transition in the entanglement spectrum is a broader crossover than the sharp transition in entanglement entropy.

The author concludes that while entanglement entropy characterizes the amount of quantum correlations, spectral entropy characterizes the structure of those correlations, offering a more refined understanding of thermalization and complexity growth in random quantum circuits. The paper acknowledges limitations regarding system size (restricted to $L \le 16$ due to exact diagonalization) and the current lack of efficient experimental protocols for estimating full spectral entropy, noting that future work is needed to extend these findings to larger systems and practical quantum devices.

Technical Summary: Beyond Entanglement Entropy: Spectral Entropy of Reduced States in Random Quantum Circuits

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