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Probe Signatures of Quantum Chaos with Quantum Sensing

This paper establishes a relationship between the spectral form factor and quantum Fisher information via quantum sensing protocols, demonstrating that the self-averaging nature of the latter provides a less noisy experimental signature of quantum chaos compared to direct spectral measurements.

Original authors: Isaías Vallejo-Fabila, Jing Yang

Published 2026-10-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Isaías Vallejo-Fabila, Jing Yang

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 microscopic world of quantum physics, systems are often described by their energy levels, the specific amounts of energy a collection of particles can hold. For decades, physicists have been fascinated by the difference between systems that are orderly and predictable, and those that are chaotic and unpredictable. A key tool for spotting this chaos is a mathematical fingerprint called the spectral form factor. This fingerprint reveals how the energy levels of a system are arranged relative to one another. If the levels are spaced out in a random, repelling pattern, the system is chaotic; if they follow a simple, predictable rhythm, it is orderly. However, measuring this fingerprint directly in a real laboratory is notoriously difficult. The signal is incredibly noisy, and in large systems, the measurement fluctuates wildly from one experiment to the next, making it nearly impossible to see the underlying pattern without averaging thousands of trials.

This difficulty has left a gap between the theoretical understanding of quantum chaos and the ability to observe it in real materials. Researchers need a way to probe this chaos that is robust enough to survive the noise of the real world. A team of physicists at Zhejiang University and Nordita has now proposed a solution that bridges this gap by turning the problem of chaos detection into a problem of precision measurement. They have discovered a fundamental link between the spectral form factor and a quantity known as quantum Fisher information. While the former is a measure of chaos that is notoriously hard to pin down, the latter is a standard tool in quantum sensing used to determine how precisely a system can detect a tiny, unknown signal. The researchers found that by using a quantum sensor to measure how well a system can detect a known signal, they can indirectly read out the chaotic fingerprint of that system, but with a crucial advantage: the measurement is stable and reliable.

The researchers began by setting up a theoretical experiment involving a quantum system, which could be a collection of atoms or spins, and a "sensing" signal. They imagined two ways to interact with the system. In one method, called the continuous protocol, the system and the sensing signal are applied at the same time. In the other, called the quench protocol, the system evolves on its own for a while, and then the sensing signal is applied. By analyzing the mathematics of these interactions, they derived a clear relationship: the quantum Fisher information, which quantifies the system's sensitivity to the signal, is directly connected to the time-integrated spectral form factor. This means that instead of trying to measure the chaotic fingerprint directly, scientists can measure the system's sensitivity to a known signal and mathematically extract the fingerprint from that data.

A major breakthrough in their work is the discovery that this new method is "self-averaging." In the world of quantum chaos, the direct measurement of the spectral form factor is not self-averaging; this means that if you run the experiment once, the result might be very different from the next run, and you need an enormous number of repetitions to find the true average. In contrast, the quantum Fisher information they proposed to measure is self-averaging. Even in a single run of the experiment, the result is typically very close to the average. This property makes the measurement far less noisy and much more feasible for real-world experiments. The researchers proved mathematically that this stability holds true for both short and long periods of time, and their computer simulations suggest it remains stable even at intermediate times.

To test their ideas, the team ran detailed numerical simulations on three different types of quantum systems. They looked at random matrices, which are the theoretical gold standard for chaotic systems, as well as two specific models of interacting spins: one with disorder and one that is clean but chaotic. In every case, the simulations confirmed that the quantum Fisher information tracked the chaotic behavior of the system exactly as their theory predicted. They also checked the stability of the measurement by looking at how much the results fluctuated when they changed the specific details of the system. They found that for the chaotic systems, the fluctuations in their proposed measurement were negligible, whereas the direct measurement of the chaotic fingerprint remained highly unstable. This confirms that their method effectively filters out the noise that usually plagues these experiments.

The implications of this work extend beyond just a new way to measure chaos. The researchers showed that the stability of their method relies on the statistical independence of the system's energy levels and its quantum states. When this independence holds, the measurement is robust. They also noted that while the direct measurement of the chaotic fingerprint often shows a "correlation hole"—a specific dip in the data that is a hallmark of chaos—this feature can be lost when the data is averaged over time. However, because their method allows them to reconstruct the full fingerprint from the stable measurements, they can still recover this crucial feature. This suggests that their approach does not just provide a noisy average, but a complete and accurate picture of the system's chaotic nature.

The paper concludes by highlighting the practical path forward. Since quantum Fisher information is already being measured in various experimental platforms, such as those using cold atoms or trapped ions, the researchers believe their proposal offers a realistic and immediate way to detect quantum chaos. By coupling a known signal to an unknown many-body system and measuring the resulting sensitivity, scientists can finally probe the chaotic signatures of complex quantum matter with high precision. This work reveals a deep and unexpected connection between the noise inherent in quantum sensing and the fundamental signatures of chaos, turning a measurement challenge into a powerful diagnostic tool.

Problem Statement
The spectral form factor (SFF), defined as the Fourier transform of the two-point spectral correlation, serves as a fundamental diagnostic for quantum chaos in many-body systems. However, its experimental detection remains challenging because the SFF is not self-averaging; its fluctuations are large, particularly at late times, making direct measurement in single realizations of quantum systems noisy and difficult. Conversely, Quantum Fisher Information (QFI) is a central quantity in quantum metrology, quantifying the precision limit for estimating unknown parameters and witnessing multipartite entanglement. While QFI has been linked to dynamical susceptibility and entanglement, a direct, experimentally accessible relationship between QFI and the SFF (and thus quantum chaos signatures) has not been established. The paper addresses the problem of how to probe the chaotic nature of a many-body Hamiltonian—specifically distinguishing between integrable (Poisson statistics) and chaotic (Wigner-Dyson statistics) regimes—using quantum sensing protocols, while overcoming the noise issues associated with direct SFF measurement.

Methodology
The authors propose a quantum sensing framework where an unknown many-body system Hamiltonian $H$ is coupled to a known sensing Hamiltonian $\Pi_\theta = \sum_\alpha \theta_\alpha P_\alpha$ for a duration $T$ . Two protocols are analyzed:

Continuous Protocol: The system and sensing Hamiltonians are applied simultaneously ( $H_\theta = H + \Pi_\theta$ ).
Quench Protocol: The system evolves under $H$ for time $T$ , followed by evolution under $\Pi_\theta$ for time $T$ .

The study focuses on the Geometric Tensor (GT), whose real part is the QFI. The authors perform analytical and numerical investigations involving:

Ensemble Averaging: Calculating the average GT over Haar-random initial states and specific Hamiltonian ensembles (Isospectral ensembles, Gaussian Unitary Ensemble (GUE), and realistic disordered Hamiltonians like the Full-Random-Field XXX and Mixed-Field Ising models).
Twirling Operations: Utilizing "isospectral twirling" and "randomized coupling" (twirling the coupling operators $P_\alpha$ with Haar-random unitaries) to map realistic Hamiltonian ensembles to isospectral ensembles where eigenstates are Haar-random.
Statistical Analysis: Investigating the typicality and self-averaging properties of the QFI and the time-integrated SFF by analyzing their relative variance (RV) over Hamiltonian ensembles.
Numerical Simulation: Simulating spin-1/2 systems with open boundary conditions to validate analytical predictions and compare deterministic versus randomized coupling protocols.

Key Contributions and Results

QFI-SFF Relation: The paper establishes a fundamental analytical relationship between the average QFI and the SFF. For the continuous protocol with Haar-random initial states and isospectral Hamiltonian ensembles, the average QFI is shown to be proportional to the time-integrated SFF, $I(T) = \int_0^T (T-t)K(t)dt$ , where $K(t)$ is the SFF. Specifically, $\langle\langle Q^{ctn}_{\alpha\beta}(T) \rangle\rangle \propto [2N^2 I(T) - T^2]$ .
Self-Averaging of QFI: A critical finding is that while the SFF itself is not self-averaging (its relative variance remains $\sim 1$ at late times), the time-integrated SFF $I(T)$ , and consequently the average QFI, exhibits self-averaging properties. The authors prove that when eigenvalues and eigenstates are statistically independent, the relative variance of the time-integrated SFF vanishes as the system size $N \to \infty$ at late times ( $T \gg N$ ). Numerical results suggest this self-averaging holds even at intermediate timescales for both random matrix ensembles (GUE) and realistic chaotic Hamiltonians.
Protocol Comparison:
- The Quench Protocol yields an average QFI that is independent of the system Hamiltonian, rendering it incapable of distinguishing between integrable and chaotic dynamics.
- The Continuous Protocol successfully encodes chaos signatures into the QFI.
- For realistic Hamiltonians where eigenstates are not strictly Haar-random, the authors propose a randomized coupling protocol (twirling the coupling operators). This technique effectively restores the link between the average QFI and the SFF, making the relationship robust for generic chaotic systems.
Experimental Feasibility: The paper proposes that by performing "QFI tomography" (measuring QFI at different timescales using known weak signals), one can indirectly extract the SFF via the second time derivative of the averaged QFI. Because QFI is self-averaging, this indirect measurement is significantly less noisy than direct SFF measurement.

Significance
The paper claims to reveal a fundamental connection between quantum noise (quantified by QFI) and the signatures of quantum chaos (quantified by SFF). By establishing that QFI is self-averaging while SFF is not, the work provides a viable experimental pathway to probe quantum chaos in many-body systems using quantum sensing techniques. This approach circumvents the difficulty of measuring non-self-averaging quantities directly. The results suggest that quantum chaos can be diagnosed by engineering many-body interactions to boost sensing precision, and conversely, that quantum sensing protocols can be used to detect the spectral statistics of unknown Hamiltonians. The authors note that since QFI measurement has been realized in various experimental platforms, their proposal offers a feasible method to measure SFF signatures, potentially aiding in the investigation of phenomena like many-body localization and chaos-to-integrability transitions.

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