Curator's Take
AI Commentary
This article introduces entangling power as a quantitative probe of quantum ergodicity and mixing, establishing a clear hierarchy that separates long‑time averaged randomness from true convergence to Haar‑typical behavior. By showing that even simple two‑qubit gates fail to be ergodic while many‑body kicked Ising chains can be ergodic (and, when nonintegrable, genuinely mixing), the work provides a fresh diagnostic for scrambling and thermalization that complements recent random‑circuit and out‑of‑time‑order‑correlator studies. The proposed Lyapunov‑like exponent gives hardware designers a practical figure of merit to gauge how quickly a processor’s dynamics approach ideal random unitary evolution, informing both benchmarking and error‑mitigation strategies.
— Mark Eatherly
Summary
We study quantum dynamics through the lens of entanglement generation and characterize the underlying unitary evolution by the distinct signatures it imprints on the time-dependent entangling power. For a unitary operator, we characterize ergodicity by the equality between its long-time-averaged entangling power and the Haar-averaged linear entropy. We define mixing more stringently as the convergence of the time-dependent entangling power itself to the Haar value at long times. Within this framework, we establish the ergodic hierarchy of dynamical behavior, showing in particular that mixing implies ergodicity, whereas ergodicity does not necessarily imply mixing. As an application, we find that two-qubit unitary gates are neither ergodic nor mixing: their long-time-averaged entangling power can take only four discrete values, none of which coincides with the Haar average. We then investigate many-body dynamics using the kicked Ising chain and find that the long-time-averaged entangling power converges to the Haar value in both integrable and nonintegrable cases, indicating ergodicity. Remarkably, however, the nonintegrable chain exhibits mixing, whereas the integrable chain, despite being ergodic, is demonstrably nonmixing. We also introduce a Lyapunov-like exponent to characterize the rate at which the time-dependent entangling power approaches its saturation value. We find that this exponent increases systematically with the degree of integrability breaking in the many-body system. Our results establish entanglement generation as a useful framework for characterizing dynamical systems and reveal qualitatively different signatures of integrability beyond conventional diagnostics.