Measurement-induced Lévy flights of quantum information

TL;DR

This work shows that free fermions in 1D subjected to frustrated local density measurements exhibit measurement-induced Lévy flights, yielding fractal entanglement growth $S(\ell) \propto \ell^{1/3}$ at a critical misalignment $\theta=\pi$. Using a replicated Keldysh nonlinear sigma model, the authors derive a diffusive spatial kernel with a long-range temporal coupling $\mathcal{B}(\omega) \sim |\omega|^{-1/2}$ at $\theta=\pi$, which drives Lévy-type spreading of quantum information. Away from the critical point, the system crosses over from superdiffusive to diffusive and eventually localized behavior as $\theta$ deviates from $\pi$, with a finite localization length that diverges at the critical point. Numerical simulations of Gaussian states corroborate the analytic predictions, showing robust $S \propto L^{1/3}$ scaling in the presence of weak measurements and quantifying finite-size and symmetry-class–driven quantum corrections. Overall, the study demonstrates how intricate fractal-scaling entanglement can arise from local Hamiltonians and measurements, and it highlights a new universality class for measurement-induced quantum dynamics.

Abstract

We explore a model of free fermions in one dimension, subject to frustrated (non-commuting) local measurements across adjacent sites, which resolves the fermions into non-orthogonal orbitals, misaligned from the underlying lattice. For maximal misalignment, superdiffusive behavior emerges from the vanishing of the measurement-induced quasiparticle decay rate at one point in the Brillouin zone, which generates fractal-scaling entanglement entropy $S \propto \ell^{1/3}$ for a subsystem of length $\ell$. We derive an effective non-linear sigma model with long-range couplings responsible for Lévy flights in entanglement propagation, which we confirm with large-scale numerical simulations. When the misalignment is reduced, the entanglement exhibits, with increasing $\ell$, consecutive regimes of superdiffusive, $S\propto \ell^{1/3}$, diffusive, $S\propto \ln \ell$, and localized, $S = \rm{const}$, behavior. Our findings show how intricate fractal-scaling entanglement can be produced for local Hamiltonians and measurements.

Figures (8)

• Figure 1: (a) Model: free-fermion chain of size $L$ continuously monitored via frustrated measurements with a misalignment $\theta$. (b) "Phase diagram" of this system showing superdiffusion at $\theta=\pi$, with characteristic regimes of the entanglement scaling indicated by different colors. (c) Half-chain entanglement entropy $S$ as a function of $L$ for different values of the measurement strength $\gamma$ at fixed $\theta = \pi$. The dashed line shows the extensive ("ballistic") behavior $S\sim L$, while the dotted line shows the fractal scaling $S\sim L^{1/3}$ corresponding to superdiffusion in 1+1-dimensional space-time.
• Figure 2: (a) Collapse of $S / \ell_0$, where $S$ is half-chain entanglement entropy, as a function of $L / \ell_0$ for $\theta = \pi$ and measurement rates from $\gamma=0$ to $\gamma=4$. The inset shows results for $\gamma=\infty$, which are of the form $S = s(L) L^{1/3}$, where $s(L)$ exhibits a slow crossover from a finite value at $L \sim 30 \ell_0$ to a slightly smaller finite value at $L \to \infty$ due to quantum corrections, see SM SuppMat. (b) $d \ln S / d \ln L$ as a function of the system size. Legend in (a) applies in (b). The dashed lines show the ballistic behavior $S\sim L$, while the dotted lines show superdiffusion $S\sim L^{1/3}$ in the thermodynamic limit.
• Figure 3: (a) Correlation function $C(q)$ as a function of rescaled momentum $\tilde{q} \ell_0$, where $\tilde{q} = 2 \sin(q/2)$ and $\ell_0 = \sqrt{(2 J / \gamma)^2 +1/4}$, for $\gamma=4$ and different values of $\theta/\pi$. The system size used is $L = 320$. The dashed line shows the superdiffusive behavior $C(q)/q \sim q^{-1/3}$. The inset shows $C(q) / q^{2/3}$, which saturates at $q\to 0$ for $\theta=\pi$. (b) Particle number covariance $G_{AB}$ as a function of system size $L$ for different values of $\theta/\pi$. Dashed lines are fits to $\sim\exp(-L/4\ell_\text{loc})$ for $L\ge 128$. The extracted localization length is shown in the inset, along with a power-law fit $\ell_\text{loc} \sim |1-\theta/\pi|^\nu$, with exponent $\nu\approx 2.33(3)$ consistent with the analytical asymptotics \ref{['eq:localization_length']} at numerically accessible scales.
• Figure S1: Numerical results for the ratio $S/\mathcal{C}^{(2)}$ for a subsystem of length $L/2$ at $\theta = \pi$, for various monitoring strength $\lambda$ and system size $L$. The dashed line shows the value of $\pi^2/3$.
• Figure S2: Illustration of the crossover behavior of $S(L)$ given by Eq. \ref{['eq:SLSpeculation']} with the following choice of parameters: $s_0 = 3$, $\delta s^{\text{(AIII)}} = 1$, $\delta s^{\text{(BDI)}} = 2$. For the sake of comparison, it is assumed that both classes are realized at the same finite value of $\gamma\sim 1$, such that the Gaussian results for these classes coincide (dashed line), while the quantum corrections differ by the factor of 2, yielding different prefactors at $L\to \infty$ (dotted and dash-dotted lines for classes AIII and BDI, respectively). Strictly speaking, in our model, class BDI is only realized at $\gamma=\infty$, for which case $s_0$ is reduced and the role of quantum corrections is further enhanced. Note that the crossover spans several orders of magnitude, and the apparent power-law exponent appears smaller in the intermediate regime. At the same time, for not too large values of $\gamma$, the crossover for the AIII class takes place between two rather close values of the prefactor, as the blue curve in the plot shows. (For small $\gamma$, the relative effect of the quantum correction is still smaller.)