Wigner negativity, random matrices and gravity

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.

💡 Research Summary

The paper investigates how the negativity of the discrete Wigner function—a quasi‑probability distribution defined on a finite‑dimensional phase space—grows under chaotic quantum dynamics, and how this growth depends on the choice of computational basis. The authors focus on two contrasting bases: a generic (random) orthonormal basis and the Krylov basis constructed from successive applications of the Hamiltonian to the initial state.

First, the authors review the discrete Wigner function for odd‑prime Hilbert‑space dimensions, its basic properties (realness, normalization, marginalization to position or momentum probabilities), and the discrete Moyal equation governing its time evolution. They emphasize that negativity (the sum of absolute values minus one) quantifies the “magic” resource needed for universal quantum computation and directly bounds the classical simulation cost via the Gottesman‑Knill theorem.

The central technical contribution is the analytic computation of the ensemble‑averaged Wigner negativity for a chaotic Hamiltonian drawn from the Gaussian Unitary Ensemble (GUE). Because negativity involves an absolute value, the authors employ the replica trick (taking an $n\to0$ limit of moments) and a diagrammatic expansion to handle disorder averaging. They find that the average negativity $\langle N(t)\rangle$ is expressed in terms of the spectral form factor $K(t)=\langle|\mathrm{Tr},e^{-iHt}|^{2}\rangle$, which is well‑studied in random‑matrix theory. For times of order one (i.e., independent of the Hilbert‑space dimension $D$), the negativity already scales as $D^{\alpha}$ with a positive exponent $\alpha$, meaning that it becomes exponentially large in $\log D$. Consequently, in a generic basis the Wigner function quickly loses any classical‑like interpretation, and the associated quantum state becomes intractable for classical simulation.

In contrast, the Krylov basis—obtained by orthogonalizing the set ${|\psi_{0}\rangle, H|\psi_{0}\rangle, H^{2}|\psi_{0}\rangle,\dots}$—provides a dramatically different picture. In the large‑$D$ limit and for $O(1)$ times, the dynamics projected onto the Krylov subspace is governed by an effective Hamiltonian that is tridiagonal and exactly solvable. Remarkably, this effective theory coincides with the $q\to0$ limit of $q$‑deformed Jackiw‑Teitelboim (JT) gravity, which is known to describe the double‑scaled SYK model. Using this effective description, the authors derive a rigorous bound on the growth of negativity in the Krylov basis: $N_{\text{Krylov}}(t)\lesssim \sqrt{t}$ (or more precisely $O(\sqrt{t})$). Thus, for any fixed $O(1)$ evolution time the negativity remains polynomially bounded and does not explode with $D$. This demonstrates that the Krylov basis furnishes a semi‑classical set of variables that capture chaotic quantum dynamics efficiently at sub‑exponential times.

Numerical simulations confirm that, despite the early‑time power‑law behavior, the Krylov negativity eventually saturates to an exponentially large value at times that scale exponentially with $D$ (i.e., $t\sim e^{cD}$). This signals the breakdown of the effective JT‑gravity description and marks a transition to a regime where the state again becomes highly non‑classical. The authors acknowledge that an analytic understanding of this late‑time saturation remains an open problem.

The discussion highlights several broader implications. First, the stark contrast between generic and Krylov bases suggests that the choice of basis can dramatically affect the perceived “classicality” of quantum dynamics, with the Krylov basis being optimal for representing chaotic evolution in a way that postpones the onset of large Wigner negativity. Second, the identification of the effective Krylov Hamiltonian with $q\to0$ JT gravity provides a concrete bridge between random‑matrix descriptions of quantum chaos and low‑dimensional gravitational theories, hinting that aspects of quantum gravity may emerge from the structure of Hilbert‑space bases rather than from spacetime geometry alone. Finally, the work motivates further studies on specific physical models (e.g., SYK, spin chains) to test whether the Krylov‑basis advantage persists beyond the GUE ensemble, and to develop theoretical tools capable of capturing the exponential‑time regime where negativity becomes large.

In summary, the paper establishes that while generic bases lead to rapid, exponential growth of Wigner negativity—rendering classical simulation impossible even at short times—the Krylov basis dramatically suppresses this growth, allowing a semi‑classical effective description (identifiable with $q\to0$ JT gravity) to remain valid up to sub‑exponential times. This result deepens our understanding of the interplay between quantum chaos, computational complexity, and emergent gravitational dynamics.