Monitored Fluctuating Hydrodynamics

TL;DR

The paper develops a general hydrodynamic, replicated MSR framework to study monitored classical stochastic processes and the informationLearnability transitions they exhibit. By conditioning on measurement records, it derives effective inter-replica field theories that reveal a charge-fuzzy fixed point with emergent relativistic invariance (z=1) under weak monitoring, and demonstrates that diffusive and KPZ transport flow to the same monitored fixed point, while non-Abelian symmetries produce a new strongly coupled phase with 1<z<2. It provides concrete analytic results for charge correlators and the Shannon entropy of measurement records, and supports them with extensive numerics (SSEP, TASEP, gradient monitoring, Burgers/KPZ) and replica-field-theory analyses. The framework also covers extensions to gradient observables and discrete charges, and discusses a charge-sharpening transition via vortex proliferation in the compactified boson description. Overall, the work offers a unifying classical stochastic perspective on measurement-induced learnability and sharpening phenomena with clear connections to information theory and dynamical criticality.

Abstract

We introduce a hydrodynamic framework for describing monitored classical stochastic processes. We study the conditional ensembles for these monitored processes -- i.e., we compute spacetime correlation functions conditioned on a fixed, typical measurement record. In the presence of global symmetries we show that these conditional ensembles can undergo measurement-induced "sharpening" phase transitions as a function of the monitoring rate; moreover, even weak monitoring can give rise to novel critical phases, derived entirely from a classical perspective. We give a simple hydrodynamic derivation of the known "charge-fuzzy phase" for weakly monitored diffusive many-body quantum systems. We show that although the unmonitored symmetric and asymmetric exclusion processes are in different universality classes of transport, the fluctuations in their conditional ensembles flow to the same fixed point with emergent relativistic invariance under monitoring. On the other hand, weakly monitored systems with non-Abelian symmetries enter a novel strongly coupled fixed point with non-trivial dynamical exponent, which we characterize. Our formalism naturally accounts for monitoring general observables, such as currents or density gradients, and allows for a direct calculation of information-theoretic diagnostics of sharpening transitions, including the Shannon entropy of the measurement record.

Figures (4)

• Figure 1: Monitored SSEP: measurement of local occupation at various space-time coordinates constrain the possible state trajectories; particle trajectories are forced to avoid unoccupied pins ($\times$) and to pass through occupied pins ($\bullet$).
• Figure 2: Linear diffusion with gradient monitoring. (Left) The rescaled dynamical correlator $\sqrt{t}\, C(x,t)$ plotted as a function of $x/\sqrt{t}$ for monitoring strengths $\gamma= 0,\,0.05,\,0.2$ in the monitored noisy linear diffusive system described in Sec. \ref{['gradient-monitoring-numerics']}. Consistent with the field-theory prediction Eq. \ref{['eqn:field-theory-grad-monitoring-corr']}, the correlators collapse onto a Gaussian diffusive scaling form, with monitoring increasing the effective diffusion constant and therefore broadening the profile. The solid lines show the theoretical scaling function. (Right) The autocorrelator $C(0,t)$ for the same values of $\gamma$, shown on a log--log scale. In all cases the decay follows the diffusive form $C(0,t)\sim t^{-1/2}$, with a $\gamma$-dependent prefactor $D_{\textrm{eff}}=\sqrt{D^2 + c\,\gamma\,\sigma^2}$ with $c\approx 1.4$, consistent with the renormalized diffusivity in Eq. \ref{['eqn:field-theory-grad-monitoring-corr']}. Numerical data is obtained using the Doob-tilted two-replica stochastic evolution described in Appendix \ref{['Appendix B']}.
• Figure 3: Burgers' equation with weak monitoring. (a) The proposed renormalization-group flow shows the unstable fixed points D (diffusion) and K (KPZ), which flow to the stable free-boson fixed point M (monitored) at weak monitoring. For simplicity, we have not shown the strong monitoring phase and the KT-like phase transition here. We are also not depicting the full replica structure -- the diagram depicts only the inter-replica degrees of freedom, the replica symmetric degrees of freedom are un-modified by monitoring. (b) The connected equal-time density-density correlator $C(x)\equiv \mathbb{E}_{\boldsymbol{m}}\left[\langle \rho_x \rho_0\rangle_{\boldsymbol{m}} - \langle \rho_x \rangle_{\boldsymbol{m}} \langle \rho_0\rangle_{\boldsymbol{m}}\right]$ (conditional on a typical measurement , i.e., drawn according to its likelihood) as a function of distance: consistent with the field theory prediction, this correlation function follows a power-law with exponent $\alpha =2 \Delta_\rho$ close to $2$ for weak measurement rate $p$ (main and inset), while for stronger measurements the system enters the sharp phase and these correlations decay exponentially (main). The numerical data are on systems of size $L = 80$ and averaged over $N\approx 1000$ samples (measurement trajectories) data. For each measurement trajectory, the probability state vector is evolved using matrix product state methods with two different maximum bond dimension cutoffs $\chi=200,300$, shown with circle ($\bullet$) and cross ($\times$) markers, to verify convergence. The values of the measurement rate for the various curves are $p \in \{0.025, 0.0375, 0.05, 0.075, 0.1, 0.15, 0.175, 0.2, 0.225\}$ (light to dark). (c) Dynamical two-point function $C(x,t)$ for TASEP under weak monitoring ($p=0.025$). The autocorrelator $C(0,t)$ shows a clear crossover from the KPZ scaling $C(0,t)\sim t^{-2/3}$ at early times to the predicted $C(0,t)\sim t^{-2}$ decay at late times (main), consistent with flow to the stable monitored free-boson fixed point. Furthermore, we observe a scaling collapse of the full dynamical correlation function $C(x,t)$ when plotted as a function of $x/t$ (inset), consistent with the $z=1$ scaling of the free-boson fixed point and showing qualitative agreement with the universal scaling function of Eq. \ref{['eq:free-boson-two-point-func']}. These simulations sample posteriors for typical measurement histories using a time-evolving block decimation (TEBD) algorithm with measurement rate $p=0.025$, system size $L=140$, and bond dimension $\chi=150$.
• Figure 4: Non-Abelian monitoring. (a) Matrix product state numerics for the entropy of the posterior charge distribution (see main text) $S(t)$ at time $t = L$ for $L=16,\cdots, 36$ (light to dark) in the three-state stochastic automaton with strong $U(1) \rtimes Z_2$ symmetry. The simulations required bond dimension $\chi = 50$ and are averaged over $N=250$ measurement trajectories. The flow reversal suggests a sharpening transition at $p_{\#}\approx 0.48$, a fuzzy phase ($p < 0.48$) with a sharpening time greater than $L$, i.e., a dynamical exponent $z>1$, and a replica-locked/sharp phase ($p>0.48$). In the replica-locked phase, inter-replica fluctuations are gapped and correlation functions should decay exponentially, as observed for the autocorrelation function of squared local charge density $C(0,t) \equiv \mathbb{E}_{_{\boldsymbol{m}}}\left[\langle \rho^2_0(t) \rho^2_0(0) \rangle_{\boldsymbol{m}} - \langle \rho^2_0(t)\rangle_{\boldsymbol{m}} \langle \rho^2_0(0) \rangle_{\boldsymbol{m}}\right]$ at $p=0.525$ in the sharp phase (inset). (b) The autocorrelator $C(0,t)$ at $p=0.3$ in the fuzzy phase, showing a algebraic decay $C(0,t)\sim t^{-1.76}$. This scaling sits between the abelian $z=1$ fuzzy phase (directly monitoring charge) which has $C(0,t)\sim 1/t^4$ and standard diffusion (the unmonitored $z=2$ phase) which has $C(0,t)\sim 1/t$. These simulations used system size $L=80$, bond dimensions $\chi=50, 100$ to verify convergence, and are averaged over $N=40000$ measurement trajectories. The data is smoothed (with a Gaussian kernel) over a window of size $\Delta t=2$ to remove persistent oscillations due to the even/odd layer structure of the circuit implementation of the model and the exponent $\alpha=1.76(1)$ is fit over the second half of the evolution time, $t\in [40,80]$. (c) The full dynamical correlation function $C(x,t) \equiv \mathbb{E}_{_{\boldsymbol{m}}}\left[\langle \rho^2_x(t) \rho^2_0(0) \rangle_{\boldsymbol{m}} - \langle \rho^2_x(t)\rangle_{\boldsymbol{m}} \langle \rho^2_0(0) \rangle_{\boldsymbol{m}}\right]$ at $p=0.3$ for times $t=25,\cdots,75$ (light to dark), showing a scaling collapse with non-trivial dynamical exponent $1/z\approx 0.57$ (inset). These simulations used system size $L=80$, bond dimension $\chi=100$, and are averaged over $N=40000$ samples.