Partial projected ensembles and spatiotemporal structure of information scrambling

TL;DR

The paper introduces the partial projected ensemble (PPE) to probe the spatiotemporal structure of information scrambling in out-of-equilibrium quantum many-body systems. It develops a tripartite framework on $R,E,S$ and connects PPE fluctuations and bit-string probabilities (PoPs) to the causal lightcones governing scrambling, showing how these quantities reveal both linear and logarithmic lightcones in ergodic and MBL regimes, respectively. Through Floquet brickwork circuits, the non-integrable kicked Ising chain, its self-dual point, and the $oldsymbol{ extell{}}$-bit model, the authors establish universal late-time behaviour (gHS, Erlang, Beta) and quantify exponential suppression of correlations with the discarded region $E$. The results provide a practical diagnostic toolkit for deep thermalisation and scrambling, with clear experimental-accessible signatures via PoPs and bit-string measurements, and open avenues for studying higher dimensions, conservation laws, and noise effects.

Abstract

Thermalisation and information scrambling in out-of-equilibrium quantum many-body systems are deeply intertwined: local subsystems dynamically approach thermal density matrices while their entropies track information spreading. Projected ensembles--ensembles of pure states conditioned on measurement outcomes of complementary subsystems--provide higher-order probes of thermalisation, converging at late times to universal maximum-entropy ensembles. In this work, we introduce the partial projected ensemble (PPE) as a framework to study how the spatiotemporal structure of scrambling is imprinted on projected ensembles. The PPE consists of an ensemble of mixed states induced on a subsystem by measurements on a spatially separated part of its complement, tracing out the remainder, naturally capturing scenarios involving discarded outcomes or noise-induced losses. We show that statistical fluctuations of the PPE faithfully track the causal lightcone of information spreading, revealing how scrambling dynamics are encoded in ensemble structure. In addition, we demonstrate that the probabilities of bit-string probabilities (PoPs) associated with the PPE exhibit distinct dynamical regimes and provide an experimentally accessible probe of scrambling. Both PPE fluctuations and PoPs display exponential sensitivity to the size of the discarded region, reflecting exponential degradation of quantum correlations under erasure. We substantiate these findings using the non-integrable kicked Ising chain, combining numerics in the ergodic regime with exact results at its self-dual point. We extend our analysis to a many-body localised (MBL) regime numerically, along with analytic results for the $\ell$-bit model. The linear and logarithmic lightcones characteristic of ergodic and MBL regimes emerge naturally from PPE dynamics, establishing it as a powerful tool for probing scrambling and deep thermalisation.

Figures (16)

• Figure 1: Schematic representation of the partial projected ensemble. The system is tripartitioned into $R$, $E$, and $S$. The degrees of freedom in $E$ are traced out whereas those in $S$ are projectively measured in a 1-local basis. Conditioned on the outcome $o_S$, the state of $R$ is denoted as $\rho_R(o_S)$ (see Eq. \ref{['eq:rhoR-os']}) and appears with probability $p({o_S})$ given in Eq. \ref{['eq:p-os']}.
• Figure 2: Schematic plot showing the lightcones of two operators, initially supported solely on $R$ and $S$ respectively. There exists a threshold timescale proportional to the separation between $R$ and $S$, $t_\ast\propto L_E$, such that the lightcones do not overlap for $t<t_\ast$ and do so for $t>t_\ast$.
• Figure 3: Numerical results for $\Delta(t,L_E)$ for the kicked Ising chain in the ergodic regime. The upper left panel shows $\Delta(t,L_E)$ as a heatmap in the $(t,L_E)$ plane where the linear lightcone is evident. The upper right panel shows $\Delta(t,L_E)$ as a function of $t$ for different $L_E$ where the onset happens at $t=\lfloor \frac{L_E}{2}\rfloor+1$ implying $t_\ast(L_E) = \lfloor \frac{L_E}{2}\rfloor$ making the linear lightcone quantitative. The lower left panel shows that $\braket{\Delta_\infty(L_E)}$ decays as $2^{-L_E}$ (indicated by the black dashed line). Using the onset timescale and the saturation value, the data for different $L_E$ can be collapsed onto each other as shown in the lower right panel. For the data $L_R=1$ and $L_S=15$, and the parameters are mentioned in Eq. \ref{['eq:params']}.
• Figure 4: The PoPs over the PPE (Eq. \ref{['eq:pop-ppe']}) for the ergodic kicked Ising chain. Different panels correspond to different times and $L_E=6$ which implies $t_\ast(L_E)=3$. The data consistently shows the PoPs to be Dirac-delta functions at unity for $t\leq t_\ast$ whereas for $t>t_\ast$, they deviate from it. At longer times, the PoPs approach the universal Erlang distribution characterised by the Hilbert-space dimension of $L_E$ indicated by the black dashed line. For these plots, $L_R=3$ and $L_S=15$ and other parameters of the model are same as in Eq. \ref{['eq:params']}.
• Figure 5: Comparison between ${\rm PoP_{b{\text{-}}str}}(\tilde{p},\rho_{RS})$ and the Mellin convolution of ${\rm PoP_{b{\text{-}}str}}(\tilde{p},\rho_{R})$ and ${\rm PoP_{b{\text{-}}str}}(\tilde{p},\rho_{S})$, the latter denoted by $M(\tilde{p},\rho_R,\rho_S)$. The left two panels show the PoPs at $t<t_\ast(L_E)$ (top) and $t>t_\ast(L_E)$ (bottom). There is excellent agreement between the two distributions in the former case and a clear distinction in the latter case. The right panel shows the KLd between the two distributions as a function of time $t$. For these plots, $L_E=6$, and as expected the KLd is vanishingly small for $t\leq t_\ast(L_E=6)=3$ whereas it picks up a finite value for $t>t_\ast(L_E)$. Note that these results for a single time-evolved state of the kicked Ising chain with parameters in Eq. \ref{['eq:params']}. Other parameters are $L_R=3$ and $L_S=15$. As an aside note that ${\rm PoP_{b{\text{-}}str}}(\tilde{p},\rho_{RS})$ at late times is well described the Erlang distribution (black dashed line) characterised by the Hilbert-space dimension of $E$. For the data we have $L_R=1$.