First appearance of quasiprobability negativity in quantum many-body dynamics

TL;DR

This work addresses when quasiprobability negativity first appears in real-time quantum dynamics of many-body systems. It introduces the first-time negativity $t_{\rm FTN}$ of the Margenau-Hill quasiprobability $q^{mn}_{\gamma\delta}(t)$ as an operational marker for the onset of contextual quantum interference under sequential local measurements, and analyzes it in a 1D Ising chain with transverse and longitudinal fields. In the integrable regime, negativity occurs only for $\sigma_z$ probes with two dynamical regimes around the critical point $h_z = J$, displaying weak system-size dependence and a separation-dependent light-cone structure; finite temperature broadens this feature and can suppress negativity. Breaking integrability via a longitudinal field $h_x$ enables negativity for both $\sigma_z$ and $\sigma_x$, with an overarching $t_{\rm FTN} \sim 1/h_z$ tail at large fields, while KDQ-based quantum speed limits provide a geometric bound that does not always match the actual onset. Overall, $t_{\rm FTN}$ is a practical probe of real-time coherence and contextuality in many-body dynamics, compatible with current platforms implementing sequential measurements and extensible to higher dimensions.

Abstract

Quasiprobability distributions capture aspects of quantum dynamics that have no classical counterpart, yet the dynamical emergence of their negativity in many-body systems remains largely unexplored. We introduce the \emph{first-time negativity} (FTN) of the Margenau-Hill quasiprobability as a dynamical indicator of when local measurement sequences in an interacting quantum system begin to exhibit genuinely nonclassical behavior. Using the Ising chain, we show that FTN discriminates clearly between interaction-dominated and field-dominated regimes, is systematically reshaped by temperature, and responds sensitively to the breaking of integrability. When measurements are performed on different sites, FTN reveals a characteristic spatio-temporal structure that reflects the finite-time spreading of operator incompatibility across the lattice. We further compare the numerical onset of negativity with a recently proposed quantum speed limit (QSL) for quasiprobabilities, which provides a geometric benchmark for the observed dynamics. Our results identify FTN as a practical and experimentally accessible probe of real-time quantum coherence and contextuality, directly suited to current platforms capable of sequential weak and strong measurements.

Figures (7)

• Figure 1: First-time negativity $t_{\rm FTN}$ (in units of $1/J$) for the integrable transverse-field Ising model ($h_x=0$). (a) Zero temperature: $t_{\rm FTN}$ as a function of transverse field $h_z$ for various system sizes (see legend), shown on a log-log scale with $J=1$. The black dotted line indicates the quantum speed-limit bound $t_{\rm QSL}$ for $N=2$. (b) Zero temperature: $t_{\rm FTN}$ vs. $h_z$ in the range $h_z\in[10^{-2},1]$, shown on a log-linear scale. (c) Zero temperature: $t_{\rm FTN}$ vs. $h_z$ in the range $h_z\in[1,10]$, shown on a log--log scale. (d) Finite temperature: $\log(t_{\rm FTN})$ as a function of $h_z$ and inverse temperature $\beta=1/k_B T$ (with $k_B=1$) for $N=8$ and $J=1$.
• Figure 2: FTN $t_{\rm FTN}$ (in units of $1/J$) as a function of the transverse field $h_z$ on a log-log scale in the integrable transverse-field Ising model. The quasiprobability is constructed from local observables $V=\sigma_z^m$ and $W=\sigma_z^n$, with $m=1$ fixed and $n$ varied from $1$ to $N/2$ (so that $d = |n-m|$). Parameters: $J=1$, $N=8$, maximum simulation time $10^{5}$, and time step $10^{-3}$.
• Figure 3: First-time negativity $t_{\rm FTN}$ (in units of $1/J$) in the nonintegrable Ising chain with a longitudinal field $h_x$. (a,c) $t_{\rm FTN}$ as a function of the transverse field $h_z$ for several fixed values of $h_x$ (see legends). (b,d) $t_{\rm FTN}$ as a function of both $h_z$ and $h_x$. In panels (a,b) the local observable is $V=W=\sigma_z^1$, while in panels (c,d) it is $V=W=\sigma_x^1$. In all cases $N=8$ and $J=1$.
• Figure 4: First-time negativity and quantum speed-limit time for boundary and bulk observables in the integrable transverse-field Ising model. (a) First-time negativity $t_{\rm FTN}$ as a function of the transverse field $h_z$ for different single-site positions with $m=n$ (boundary and bulk, see legend). (b) Corresponding quantum speed-limit time $T_{\rm QSL}$ for the onset of possible nonpositivity of the KD quasiprobability, plotted under the same conditions. All data are at zero temperature, with $N=8$, $J=1$, and $h_x=0$.
• Figure 5: Exact and asymptotic behavior of the first-time negativity in the transverse-field Ising model for $N=2$ and $J=1$. The blue line shows the exact $t_{\rm FTN}$ as a function of the transverse field $h_z$. Orange squares indicate the weak-field approximation $t_{\rm FTN} = \frac{\pi}{2J} - \sqrt{2h_z/J^3}$ (valid for $h_z \ll J$), and yellow circles indicate the strong-field approximation $t_{\rm FTN} = \frac{\pi h_z}{J^2 + 8h_z^2}$ (valid for $h_z \gg J$).