发表机构
Tata Institute of Fundamental Research（塔塔基础研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究在JT引力中，通过Wigner函数负性度量Hartle-Hawking态的复杂性，发现其早期为正且冻结，晚期饱和至上限，并指出长度基适合描述混沌量子动力学。
AI 中文摘要
Wigner函数的负性给出了在经典计算机上模拟量子态复杂性的一个具有操作意义的度量。我们在Jackiw-Teitelboim引力中，于长度基下研究了Hartle-Hawking态在时间演化下该负性的增长。在盘级近似下，且在半经典参数$\beta$的首阶，Hartle-Hawking态是一个静止于Liouville壁转折点处的最小不确定度高斯波包，中心位于$x=2\log(2\beta/\pi)$，且$\langle p^2\rangle=\pi^2/8\beta$。其Wigner函数为正，因此对于所有次指数时间，其负性为$1+o(1)$；由于时间反演对称性，它关于$t$是偶函数，并且一旦反射波包与壁解耦，它就完全冻结，因为自由演化是Clifford剪切。在晚期时间，负性饱和接近其上限，为$\sqrt{2/\pi}\sqrt{d_{\rm eff}(\beta)}$，其中$d_{\rm eff}(\beta)=Z(\beta)^2/Z(2\beta)$，该值需要离散谱。与谱形状因子不同，负性没有斜坡：双边界虫洞贡献相对于盘项被抑制了$e^{-2S_0}$，而盘项（不同于谱形状因子的盘项）不会衰减。精确存活振幅$Z(\beta+it)/Z(\beta)$给出扩展复杂性随$\sigma_E^2t^2$增长，而种子归一化负性在盘级近似下是精确存活概率的倒数。我们将此视为证据，表明长度基非常适合用于在次指数时间内、大$e^{S_0}$情况下混沌量子动力学的对偶半经典有效描述。
英文摘要
The Wigner function, in general, takes on negative values, and the amount of negativity in the Wigner function gives an operationally meaningful measure of the complexity of simulating the quantum state on a classical computer. In this paper, we study the growth of Wigner negativity of the Hartle-Hawking state of Jackiw-Teitelboim gravity under time evolution. We work in the gravitational length basis, at genus zero, and at high temperature, $β\ll 1$. Our main analytic result is simple: to leading order in $β$ the state is a Gaussian wavepacket of minimum uncertainty, sitting at rest at the turning point of the Liouville wall. Its Wigner function is therefore positive, and the negativity is $1+δ(β,t)$, where the correction $δ$ is smaller than any power of $β$. We show that $δ$ is an even function of time, so there is no linear growth at $t=0$, and that once the packet has reflected off the wall its leading-order evolution is a Clifford shear, which cannot change the negativity. We also numerically study the time evolution of the Wigner function and its negativity. We observe that $δ$ stays below $10^{-11}$ at $t=0$ and through the reflection, then rises slowly over a few tens of $β/π$, and then approaches a late-time value consistent with $δ_\infty(β) \simeq 0.08\, e^{-4.4/β}$ for $0.25 \le β\le 1$: the lower the temperature, the larger the late-time negativity. The exact survival amplitude $Z(β+it)/Z(β)$ fixes the spread complexity to order $t^4$, and it also fixes the seed-normalised Wigner diagnostic of our earlier work (arXiv:2607.04065, arXiv:2607.17346). We take this as evidence that the length basis is ideally suited for a dual, semi-classical description of the dynamics at genus zero. Beyond genus zero the length basis is overcomplete, and we make no statement about finite $e^{S_0}$.
Comments19 pages, 1 figure, 1 table
