From Ehrenfest to Heisenberg scales: a hierarchy of mixing times for a phase-randomized quantum baker walk
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=2k. 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 Oε(N) steps, whereas metric-entropy and Haar small-ball estimates give an Ωε(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<N. The moment results and the Wasserstein upper bound are driven by the unistochastic Markov kernel associated with BN, an affine dyadic chain whose nonconstant Fourier modes vanish exactly after k steps. The resulting hierarchy separates Ehrenfest-scale randomization of state ensembles from near-Heisenberg-scale exploration of the full unitary group.
Related
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BibTeX
@misc{sepehri2026from,
title = {{From Ehrenfest to Heisenberg scales: a hierarchy of mixing times for a phase-randomized quantum baker walk}},
author = {Amir Sepehri},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32067},
eprint = {2609.32067},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32067},
}
RIS
TY - UNPB
TI - From Ehrenfest to Heisenberg scales: a hierarchy of mixing times for a phase-randomized quantum baker walk
AU - Amir Sepehri
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.32067
UR - https://arxiv.org/abs/2609.32067
AB - 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_ε(N) steps, whereas metric-entropy and Haar small-ball estimates give an Ω_ε(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<N. The moment results and the Wasserstein upper bound are driven by the unistochastic Markov kernel associated with B_N, an affine dyadic chain whose nonconstant Fourier modes vanish exactly after k steps. The resulting hierarchy separates Ehrenfest-scale randomization of state ensembles from near-Heisenberg-scale exploration of the full unitary group.
ER -