Complexity study of the Hartle-Hawking state in JT gravity

Abstract

The negativity of the Wigner function gives an operationally meaningful measure of the complexity of simulating a quantum state on a classical computer. We study its growth for the Hartle-Hawking state under time evolution in Jackiw-Teitelboim gravity, in the length basis. At the disk level, and to leading order in the semiclassical parameter $β$, the Hartle-Hawking state is a minimum-uncertainty Gaussian wavepacket at rest at the turning point of the Liouville wall, centred at $x=2\log(2β/π)$ with $\langle p^2\rangle=π^2/8β$. Its Wigner function is positive, so its negativity is $1+o(1)$ for all sub-exponential times; it is even in $t$ by time-reversal symmetry, and exactly frozen once the reflected packet decouples from the wall, because free evolution is a Clifford shear. At late times the negativity saturates close to its upper bound, at $\sqrt{2/π}\sqrt{d_{\rm eff}(β)}$ with $d_{\rm eff}(β)=Z(β)^2/Z(2β)$, a value that requires the discrete spectrum. Unlike the spectral form factor, the negativity displays no ramp: the two-boundary wormhole contribution is suppressed by $e^{-2S_0}$ relative to a disk term that, unlike that of the spectral form factor, does not decay. The exact survival amplitude $Z(β+it)/Z(β)$ gives the spread complexity growing as $σ_E^2t^2$, and the seed-normalised negativity is, at the disk level, the inverse of the exact survival probability. We take this as evidence that the length basis is ideally suited for a dual, semi-classical effective description of chaotic quantum dynamics for large $e^{S_0}$ at sub-exponential times.