Measuring the scrambling of quantum information

TL;DR

The paper lays out a general protocol for measuring out-of-time-order correlators (OTOCs) to diagnose scrambling of quantum information, using forward time evolution and interferometric or distinguishability schemes to access F(t) or |F(t)|^2. It provides a concrete cavity-QED realization of a nonlocal kicked-top model, analyzes dissipative effects via quantum trajectories, and derives cooperativity and detuning requirements to observe scrambling and chaos-like dynamics. By connecting OTOCs to semiclassical chaos and black-hole information dynamics, the work furnishes experimentally feasible pathways to study scrambling and Lyapunov growth in controllable quantum systems and to compare with holographic bounds. Overall, the study offers detailed protocols and practical design criteria for probing scrambling in nonlocal spin models and, more broadly, for exploring fundamental limits on information spreading in quantum many-body dynamics.

Abstract

We provide a protocol to measure out-of-time-order correlation functions. These correlation functions are of theoretical interest for diagnosing the scrambling of quantum information in black holes and strongly interacting quantum systems generally. Measuring them requires an echo-type sequence in which the sign of a many-body Hamiltonian is reversed. We detail an implementation employing cold atoms and cavity quantum electrodynamics to realize the chaotic kicked top model, and we analyze effects of dissipation to verify its feasibility with current technology. Finally, we propose in broad strokes a number of other experimental platforms where similar out-of-time-order correlation functions can be measured.

Figures (6)

• Figure 1: (a) Protocols for measuring $F(t)$. (i) Given a control qubit $\mathcal{C}$, the interferometric protocol can measure $F$ for the system $\mathcal{S}$ by applying different sequences of operators in the two interferometer arms. (ii) Without a control qubit, the distinguishability protocol can access $|F|^2$. (b) $F(t)$ can be measured with either local or global operators $V, W$, as shown for a spin chain in the upper branch of the interferometer, with control qubit ${\cal C}$ in green (left of chain).
• Figure 2: Scheme for measuring out-of-time-order correlators. (a) Atomic ensemble ${\cal S}$ and control qubit ${\cal C}$ in an optical cavity are driven by control fields $\Omega_{\uparrow}, \Omega_{\downarrow}, \Omega_{{\cal C}}$. (b) Control fields $\Omega_{\uparrow,\downarrow}$ and cavity coupling $g$ mediate pairwise interactions in the ensemble ${\cal S}$ via 4-photon Raman transitions.
• Figure 3: Interferometric protocol for unitary $S_x^2$ dynamics at $N = 50$. For an initial coherent state $\left| \hat{y} \right\rangle$ and rotation angle $\phi = \pi / 4$, $\text{Re}[F]$ (green) exhibits decay at short times (a,b), a quiescent period (c), and subsequent oscillations (d). Inset: states of the two interferometer arms at various times, illustrated by Wigner quasiprobability distributions.
• Figure 4: Interferometric protocol for the kicked top. (a) Unitary time-ordered correlators $\vert G(t) \vert$ (thin yellow) and out-of-time-order correlators $\vert F(t) \vert$ (thick blue) for atom numbers $N = 50,100,200,300,400,500$ (light to dark), $k = 3$, $\phi = 1/\sqrt{N}$, and initial state $e^{- i S_y \pi / 4} e^{- i S_z \pi / 4} \left| \hat{x} \right\rangle$. (b) Unitary (solid) and dissipative (dashed) evolution of $\vert G(t) \vert$ (thin yellow) and $\vert F(t) \vert$ (thick blue) for $N = 100$. Dissipative evolution is calculated at $\eta = 100$ and $\delta = 10 \kappa$ from 200 quantum trajectories; error bars are statistical. Horizontal axes show kick number (black) and mean number of photons lost by decay processes in measuring $F$ (blue) and $G$ (yellow) SM.
• Figure 5: Scheme for measuring out-of-time-order correlators. (a) Atomic ensemble ${\cal S}$ and control qubit ${\cal C}$ in an optical cavity are driven by control fields $\Omega_{\uparrow}, \Omega_{\downarrow}, \Omega_{{\cal C}}$. (b) Control fields $\Omega_{\uparrow,\downarrow}$ and cavity coupling $g$ mediate pairwise interactions in the ensemble ${\cal S}$ via 4-photon Raman transitions.