Measurement-enhanced entanglement in a monitored superconducting chain

Abstract

A common view in monitored quantum dynamics is that local measurements suppress entanglement growth. We show that this intuition can fail in a one-dimensional spinful fermionic chain governed by a BCS Hamiltonian with pairing strength $Δ$ and subject to continuous, on-site, spin-resolved charge measurements at rate $γ$. Using free-fermion simulations and quasiparticle analysis, we show that pairing suppresses entanglement growth, while measurements suppress pairing. Their competition yields measurement-enhanced entanglement: for $Δ>0$, the steady-state entanglement $S_s$ increases with $γ$ over a finite interval $0<γ<γ_{\rm peak}$. This occurs because stronger measurements suppress pairing correlations, which would otherwise suppress entanglement growth. Using a nonlinear sigma-model calculation and free-fermion simulations, we provide evidence that for $Δ>0$ and small but finite $γ$, the steady-state entanglement scales as $S_s\sim \ln^2 L$. This implies that, in this setting, measurement-enhanced entanglement does not persist in the thermodynamic limit.

Figures (4)

• Figure 1: Monitored BCS dynamics. (a) We consider a 1D system of spinful fermions of $L$ sites, evolving under the BCS Hamiltonian [\ref{['eq:hamil']}], with hopping amplitude $J=1$. The BCS pairing is illustrated by the dashed circles, with amplitude $\Delta$. The system is subjected to continuous spin-resolved charge measurements of strength $\gamma$. (b) A three-way competition among entanglement growth (induced by fermion hopping), measurement, and pairing. (c) The schematic phase diagram for the scaling of the steady-state entanglement $\mathcal{S}_{s}$ as a function of the pairing amplitude $\Delta$ and measurement strength $\gamma$.
• Figure 2: Unmonitored evolution ($\gamma = 0$). (a) Time evolution of the entanglement ${\cal S}(t)$ for system size $L=500$ and various pairing strengths $\Delta$. The solid curves show the free-fermion simulations, while the dashed lines indicate the GGE predictions for the steady-state entanglement $\mathcal{S}_{s}$ (horizontal) and the entanglement-growth timescale $\tau_{\Delta}$ (vertical). (b) and (c) show the GGE-predicted entropy density $c_\Delta$ [cf. \ref{['Eq:cd']}] and the timescale $\tau_{\Delta}$ as functions of $\Delta$.
• Figure 3: Measurement-enhanced entanglement. System size $L=32$ here. (a) Evolution of the pairing-correlation amplitude $|\langle c_{j,\downarrow} c_{j+1,\uparrow} \rangle(t)|$ for unmonitored and monitored evolution ($\gamma=0,10$). (b) Evolution of the entanglement $\mathcal{S}(t)$ at pairing strength $\Delta = 2.0$, for various measurement strengths $\gamma = 0, 10, 70$. (c) The steady-state entanglement $\mathcal{S}_{s}$ as a function of $\gamma$, for various values of $\Delta$. The left inset shows a magnified view of the boxed region. Here, $\gamma_{\rm peak}$ denotes the measurement strength at which $\mathcal{S}_{s}$ reaches its maximum value. The right inset shows $\gamma_{\rm peak}$ as a function of $\Delta$ for various fixed $L$.
• Figure 4: Steady-state entanglement scaling. The steady-state entanglement $\mathcal{S}_{s}$ is shown as a function of $\ln^2 L$ at fixed pairing strength $\Delta = 2.0$ and for various measurement rates $\gamma$. The black dashed line denotes the unmonitored case ($\gamma = 0$), for which $\mathcal{S}_{s}$ exhibits volume-law scaling, while the colored dots correspond to the monitored cases. We fit the monitored data with the linear form $\mathcal{S}_{s} = \lambda_{\gamma,\Delta}\ln^2 L + c$, as indicated by the solid lines, and the inset shows the fitted slope $\lambda_{\gamma,\Delta}$.