How does the entanglement entropy of a many-body quantum system change after a single measurement?

TL;DR

This work investigates how a single measurement perturbs the entanglement entropy (EE) in a one-dimensional chain of non-interacting complex fermions, comparing quantum state diffusion (QSD), quantum jump (QJ), and projective measurement (PM) protocols. Using Gaussian-state formalism and trajectory sampling, EE for subsystem $A$ with $\ell = L/2$ is computed from the correlation matrix, with $S_\ell(t) = -\sum_i [\lambda_i \ln\lambda_i + (1-\lambda_i) \ln(1-\lambda_i)]$ where $\lambda_i$ are eigenvalues of $D^A$. The results reveal protocol- and boundary-dependent, non-Gaussian distributions for the EE changes: QSD yields Gaussian behavior at weak monitoring that morphs into exponential-like tails and a Zeno peak at strong monitoring, while QJ and PM produce asymmetric, zero-centered peaks with boundary sites dominating the fluctuations as monitoring strengthens. A non-Hermitian regeneration term $\delta S_{nH}$, governing EE growth between jumps, is site-independent and interacts with boundary effects to shape the full distribution. Overall, distribution-level analysis uncovers detailed entanglement dynamics under measurement, offering insights beyond averages and informing efforts to mitigate post-selection issues in monitoring-driven entanglement studies.

Abstract

For one-dimensional non-interacting complex fermions, we compute numerically the probability distribution of the change in the entanglement entropy (EE), after saturation, resulting from a single measurement of the occupation number by using different measurements protocols. In the thermodynamic limit, the system is in the area-law phase for any monitoring strength, however we can also study the distribution in the critical phase characterized by a logarithmic scaling of the EE with system size by considering a sufficiently weak monitoring strength. For the quantum jump and the projective measurement protocols, we observe clear deviations from Gaussianity characterized by broader and asymmetric tails, exponential for positive values of the change, and a peak at zero, likely a precursor of Zeno effect, that increases with the system size and the monitoring strength. Intriguingly, the distribution is spatially inhomogeneous. For sites around the boundary separating the subsystems defining the EE, the distribution is close to Gaussian with a broad support while for the rest of sites has asymmetric exponential tails and a much narrower support. As the monitoring strength increases, the full distribution is controlled by the boundary sites. For a quantum state diffusion protocol, where the change in EE is defined between two consecutive time steps, the distribution, Gaussian for weak monitoring, gradually develops symmetric exponential tails. For strong monitoring, the core turns from Gaussian to strongly peaked at zero suggesting the dominance of quantum Zeno effect.

Figures (16)

• Figure 1: The evolution of the entanglement entropy $S$ under different measurement protocols: \ref{['fig.QSD_L512_EE_vs_t']} QSD, \ref{['fig.QJ_L512_EE_vs_t']} QJ and \ref{['fig.PM_L512_EE_vs_t']} PM. The system size is $L=512$ and the subsystem size is $\ell = L/2$.
• Figure 2: The probability distribution of the global entanglement entropy under different measurement protocols: \ref{['fig.QSD_L512_Pro_S_vs_g_fit']} QSD, \ref{['fig.QJ_L512_Pro_EE_vs_g_fit']} QJ and \ref{['fig.PM_L512_Pro_EE_vs_g_fit']} PM. When the measurement strength is not very strong, the probability distributions agree well with a Gaussian distribution (red dashed line). The system size is $L=512$, and $S(\ell = L/2)$ is rescaled by its average $\overline{S(\ell = L/2)}$.
• Figure 3: The probability distribution of $\delta S_{qsd}$ for the QSD protocol Eq. (\ref{['eq:deltaqsd']}), employing Eq. (\ref{['eq:EE']}) after the saturation time of the average EE. The monitoring rates are $\gamma = 0.1, 0.5, 1.5$ and $3.0$ from \ref{['fig.QSD_g0p1_Pro_dS_vs_L_fit']} to \ref{['fig.QSD_g3p0_Pro_dS_vs_L_fit']}. The distribution agrees well with a Gaussian distribution in the weak monitoring limit. For intermediate monitoring strength, the distribution is well described by a simple function interpolating from Gaussian, for small $\delta S_{qsd}$, to exponential, for large $\delta S_{qsd}$. In the strong monitoring limit, the exponential tails are still observed. However, for small $\delta S_{qsd}$ , the distribution is strongly peaked around zero which indicates the dominance of quantum Zeno effect and provides an upper bound for the area law in the EE. We do not observe any size dependence in our results, see the text for an explanation of this feature.
• Figure 4: The probability distribution of the change in entanglement entropy $\Delta S_{qj}$ Eq. (\ref{['eq:Deltaqj']}), computed employing Eq. (\ref{['eq:EE']}), due to a quantum jump in the QJ protocol, for sufficiently late times so that EE has reached the saturation value. The $P(\Delta S_{qj})$ is rescaled by multiplying the system size $L$. The tail is independent of system size after rescaling, signaling the tail is dominated by the boundary region. The monitoring rates are $\gamma = 0.1, 0.5, 1.5$ and $3.0$ from \ref{['fig.QJ_g0p1_Pro_DSqj_vs_L']} to \ref{['fig.QJ_g3p0_Pro_DSqj_vs_L']}.
• Figure 5: $\Delta S_{qj}$ Eq. (\ref{['eq:Deltaqj']}) as a function of the measured sites $r = 1, \ldots, L$. The left plot is for the weak monitoring rate $\gamma=0.1$ and system size $L=1024$, while the right plot is for the strong monitoring rate $\gamma=3$ and the system size is $L=512$. Note the stark differences between boundary points, separating the two subsystems defining the EE, and the rest.