From Ehrenfest to Heisenberg scales: a hierarchy of mixing times for a phase-randomized quantum baker walk
Year: 2026
Type: preprint
Abstract: The quantum baker map is a standard model of quantum chaos, but rigorous analysis of how it spreads random perturbations remains difficult. Following the repeated ``baker plus refreshed diagonal perturbation'' architecture of Schack and Caves, we study a tractable strong-noise model based on the Balazs--Voros quantization: at each time step, its unitary propagator is followed by a diagonal unitary whose entries are independent and uniform on the unit circle. The resulting random propagator is a structured random walk on $\U(N)$, $N=2^k$. We quantify how quickly the baker dynamics spreads this basis-local randomness at three levels: ensemble means, two-copy statistics, and the full law of the accumulated propagator. The walk becomes an exact unitary $1$-design at time $k+1$, while its fixed-accuracy $2$-design mixing time is $Θ(\log N)$. Thus the induced state ensembles reproduce Haar means and two-copy statistics on the Ehrenfest scale. The full law mixes much more slowly. A phase-adapted path coupling gives normalized Wasserstein mixing in $O_\varepsilon(N)$ steps, whereas metric-entropy and Haar small-ball estimates give an $Ω_\varepsilon(N/\log N)$ lower bound and asymptotically maximal distance at every $t=o(N/\log N)$. The law remains singular with respect to Haar measure for every $t
Source: arXiv (Cornell University)
Author: Sepehri, Amir
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214770476