Wigner negativity, random matrices and gravity

TL;DR

This paper investigates how Wigner negativity, a measure of quantum complexity, evolves under chaotic dynamics, showing that the Krylov basis minimizes early growth and relates to semi-classical gravity models.

Contribution

It demonstrates that the Krylov basis suppresses early Wigner negativity growth in chaotic systems, linking quantum chaos to semi-classical gravity via random matrix theory.

Findings

Wigner negativity grows exponentially for generic bases within O(1) time.

In the Krylov basis, negativity grows at most polynomially, becoming exponential only at exponential times.

The effective description relates to the q→0 limit of q-deformed JT gravity.

Abstract

Given a choice of an ordered, orthonormal basis for a $D$-dimensional Hilbert space, one can define a discrete version of the Wigner function -- a quasi-probability distribution which represents any quantum state as a real, normalized function on a discrete phase space. 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. Further, Wigner negativity also gives a lower bound on an entropic measure of spread complexity. In this paper, we study the growth of Wigner negativity for a generic initial state under time evolution with chaotic Hamiltonians. In arXiv:2402.13694, a perturbative argument was given to show that the Krylov basis minimizes the early time growth of Wigner negativity in the large-$D$ limit. Using tools from random matrix theory, here we show that for a generic choice of basis, the Wigner negativity for a classical initial state becomes exponentially large in an $O(1)$ amount of time evolution. On the other hand, we show that in the Krylov basis the negativity grows at most as a power law, and becomes exponentially large only at exponential times. We take this as evidence that the Krylov basis is ideally suited for a dual, semi-classical effective description of chaotic quantum dynamics for large-$D$ at sub-exponential times. For the Gaussian unitary ensemble, this effective description is the $q\to 0$ limit of $q$-deformed JT gravity.