Wigner negativity, random matrices and gravity

TL;DR

This paper probes how discrete Wigner negativity evolves under chaotic quantum dynamics in large-dimensional Hilbert spaces. Using random-matrix theory (GUE), replica tricks, and Haar integration, it shows that in a generic basis the Wigner negativity of a classical initial state grows to exponentially large values within O(1) time, while in the Krylov basis the growth is a sub-exponential power law for O(1) times and only becomes exponential at exponential times. The Krylov basis yields a tractable effective Hamiltonian (tri-diagonal) whose dynamics map to a q→0 limit of q-deformed JT gravity, establishing a concrete semi-classical description for chaotic quantum dynamics at sub-exponential times. The results support the Krylov basis as a natural, gravity-inspired framework for describing chaotic evolution and link basis-dependent non-classicality to gravitational dual descriptions of strongly coupled systems.

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.

Figures (4)

• Figure 1: A plot of $\min(H_{1/2}(q),H_{1/2}(p)$ vs log negativity for 30000 random pure states at $D=13$. The red dotted line indicates the lower bound in equation \ref{['inequality']}.
• Figure 2: The Wigner negativity in the computational basis for $D=101$: (left) the red curve shows the analytical formula in equation \ref{['avneg']} for GUE with $\overline{S(t)}$ given by equation \ref{['SFFGUE']}. The black points denote the numerically computed negativity for one specific choice of Hamiltonian drawn randomly from GUE. (right) The relative error $\frac{(\mathcal{N}_{\text{num.}}-\mathcal{N}_{\text{ana.}})}{\mathcal{N}_{\text{ana.}}}$ between the numerical and analytical formulas seems to be $O(1/D)$.
• Figure 3: The probability as a function of $k$ for $t=15$ and $D=1499$. Numerically, the wave function ( $|\langle k | e^{-itH}|0\rangle|^2$ ) is calculated for a single Hamiltonian drawn from GUE.
• Figure 4: $\log(\mathcal{N(\mathrm{t})})$ vs $\log(t)$ plot for different values of $D$. Here, the black dotted line represents the $\sqrt{t}$ curve.