从 Ehrenfest 到 Heisenberg 尺度：相位随机化量子 baker 游走的混合时间层级
From Ehrenfest to Heisenberg scales: a hierarchy of mixing times for a phase-randomized quantum baker walk
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中文总结 AI 辅助
该研究分析相位随机化量子 baker 游走的混合时间层级，揭示态系综在 Ehrenfest 尺度随机化而完整酉群在近 Heisenberg 尺度探索。
中文摘要 AI 辅助
量子 baker 映射是量子混沌的标准模型，但对它如何传播随机扰动的严格分析仍然困难。遵循 Schack 和 Caves 的重复“baker 加刷新对角扰动”架构，我们研究了一个基于 Balazs--Voros 量子化的可处理的强噪声模型：在每个时间步，其酉传播子之后跟随一个对角酉矩阵，其元素独立且在单位圆上均匀分布。由此产生的随机传播子是在 $\U(N)$（$N=2^k$）上的结构化随机游走。我们在三个层面量化 baker 动力学传播这种基局域随机性的速度：系综平均、两拷贝统计以及累积传播子的完整分布。该游走在时间 $k+1$ 时成为一个精确的酉 $1$-设计，而其固定精度的 $2$-设计混合时间为 $\Theta(\log N)$。因此，诱导态系综在 Ehrenfest 尺度上再现 Haar 均值和两拷贝统计。完整分布的混合速度要慢得多。一个相位适应的路径耦合给出了在 $O_\varepsilon(N)$ 步内的归一化 Wasserstein 混合，而度量熵和 Haar 小球估计给出了 $\Omega_\varepsilon(N/\log N)$ 的下界，并且在每个 $t=o(N/\log N)$ 时达到渐近最大距离。对于每个 $t<N$，该分布相对于 Haar 测度仍然是奇异的。矩结果和 Wasserstein 上界由与 $B_N$ 关联的 unistochastic Markov 核驱动，这是一个仿射二进链，其非恒定 Fourier 模式在 $k$ 步后精确消失。由此产生的层级将 Ehrenfest 尺度的态系综随机化与近 Heisenberg 尺度的完整酉群探索区分开来。
英文摘要
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 $t<N$. 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.
