标题： 复杂度增长和Krylov-Wigner函数 显示英文标题

标题： Complexity Growth and the Krylov-Wigner function

摘要： 对于一个$D$维希尔伯特空间中的任何状态，以及选择一个基，可以定义一个离散的维格纳函数——一种准概率分布，它在离散相空间上表示该状态。 维格纳函数通常可以取负值，而维格纳函数中的负性在操作上具有作为量子计算资源的意义。 在本说明中，我们研究了在混沌哈密顿量下的时间演化中，通用初始状态的维格纳负性的增长。 我们引入了克雷洛夫-维格纳函数，即相对于克雷洛夫基（带有适当相位）定义的维格纳函数，并表明这种基的选择在大$D$极限下最小化了早期时间的维格纳负性增长。 我们认为这是证据，表明克雷洛夫基（带有适当相位）非常适合用于大$D$时混沌量子动力学的对偶、半经典描述。 我们还数值研究了初始纯态在随机矩阵理论中克雷洛夫-维格纳函数及其负性的时间演化。 我们观察到负性大致经历了三个阶段：在时间$O(\sqrt{D})$内逐渐上升，然后出现陡峭的上升阶段，最后接近其上限$\sqrt{D}$。 显示英文摘要

摘要： For any state in a $D$-dimensional Hilbert space with a choice of basis, one can define a discrete version of the Wigner function -- a quasi-probability distribution which represents the state on a discrete phase space. The Wigner function can, in general, take on negative values, and the amount of negativity in the Wigner function has an operational meaning as a resource for quantum computation. In this note, we study the growth of Wigner negativity for a generic initial state under time evolution with chaotic Hamiltonians. We introduce the Krylov-Wigner function, i.e., the Wigner function defined with respect to the Krylov basis (with appropriate phases), and show that this choice of basis minimizes the early time growth of Wigner negativity in the large $D$ limit. We take this as evidence that the Krylov basis (with appropriate phases) is ideally suited for a dual, semi-classical description of chaotic quantum dynamics at large $D$. We also numerically study the time evolution of the Krylov-Wigner function and its negativity in random matrix theory for an initial pure state. We observe that the negativity broadly shows three phases: it rises gradually for a time of $O(\sqrt{D})$, then hits a sharp ramp and finally saturates close to its upper bound of $\sqrt{D}$.