Why this matters. When information scrambles in a many‑body quantum system it spreads so thoroughly that no local measurement can recover it. Scrambling sets fundamental limits on quantum computers, underlies “fast thermalization” in cold‑atom experiments, and is thought to occur at the horizons of black holes. Measuring it in noisy, real‑world hardware is hard, because the textbook diagnostic—the out‑of‑time‑order correlator (OTOC)—assumes a perfect, noise‑free time‑reversal that experiments simply do not have.
Quantum scrambling is usually diagnosed with an out‑of‑time‑order correlator (OTOC). But real experiments rarely implement the perfect forward‑then‑backward time evolution that the textbook OTOC assumes. Two imperfections dominate:
- Imperfect time‑reversal – the Hamiltonian used for the backward evolution is not exactly the negative of the forward Hamiltonian.
- Decoherence – environmental noise (modeled here by a single‑qubit depolarizing channel at rate $\kappa$) acts throughout the protocol.
The central questions
Our work: arxiv 2505.00070 asks four big questions:
- How do these imperfections reshape the measured signal?
- Can we define an observable that “cancels” the most boring part of the noise and still carries scrambling physics?
- Does a true phase transition—similar to many‑body localization—emerge when the imperfections are strong?
- How should recent NMR measurements on nuclear‑spin ensembles be interpreted in this light?
The model in a nutshell
- All‑to‑all Brownian Hamiltonian—every pair of qubits couples with Gaussian noise that is resampled each instant, wiping out conservation laws.
- Mismatch parameter $r\in[0,1]$—correlates forward and backward noises: $r=1$ is perfect reversal, $r=0$ is totally uncorrelated.
- Depolarizing noise—each qubit suffers a simple depolarizing channel at rate $\kappa$.
- Solve first in the dilute limit $N\to\infty$ with $r,\kappa$ fixed, then add $1/N$ corrections for finite systems.
Meet the ROTOC—renormalized OTOC
The protocol (see above figure) is an abstraction of the experiment. It prepares a slightly perturbed infinite‑temperature state:
- Prepare the initial state, $\rho = \frac{I + \epsilon V}{2^N}$;
- Evolve forward, $\rho \to U \rho U^\dagger$;