Shannon entropy of the measurement record at measurement-dominated criticality and RG flow: A c-theorem for effective central charge and a g-theorem for effective boundary entropy

TL;DR

The paper proves two nonperturbative monotonicity theorems for information content in measurement-record Shannon entropy under RG flow: a $c_{ ext{eff}}$-theorem showing $c_{ ext{eff}}<c$ for $(1+1)$D/$(2+0)$ replica field theories in the $R o1$ limit, and a $g_{ ext{eff}}$-theorem showing $s_{ ext{eff}}<0$ (hence $g_{ ext{eff}}<1$) for defect replica theories in the same limit. The approach maps measurement-driven problems to replica field theories, then uses Zamolodchikov’s $C$-function and a gradient-like boundary-entropy framework, together with the representation of key quantities as randomness-averaged squares of correlators, to establish monotonic decreases of these universal information-theoretic measures along RG flow. The results connect to Shannon-entropy scaling in 2D Bayes-conditioned classical critical systems and to universal finite-size scaling of Shannon entropy for 1D quantum critical ground states under weak measurements, with a complementary discussion of possible $R o0$ behavior under uncorrelated disorder. The work provides a principled, nonperturbative link between RG flow and information content in measurement-induced critical phenomena, offering a foundation for further information-theoretic approaches and potential extensions to broader replica-encoded systems.

Abstract

We present two theorems demonstrating non-perturbatively the decrease under relevant renormalization group (RG) flow of two quantities, $c_{\text{eff}}$ and $g_{\text{eff}}$ characterizing, respectively, the universal information content of the Shannon entropy of the measurement record for two different types of measurement-dominated criticality. First, we demonstrate the decrease of the "effective central charge" $c_{\text{eff}}$ of $2D$ replica field theories in the $R\rightarrow1$ replica limit that govern the long-distance physics of weakly monitored $2D$ classical critical systems (Baysian inference problems) studied recently in the literature [arXiv:2504.01264; arXiv:2504.12385; arXiv:2504.08888]. In particular, we show that $c_{\text{eff}}$ is $\textit{less}$ than the central charge $c$ of the unmeasured critical system. We refer to this result as the "$c$-effective theorem''. In addition, we present an analogous "$g$-effective theorem" demonstrating the decrease under RG flow of the effective "Affleck-Ludwig'' boundary entropy $\ln g_\text{eff}$, quantifying a corresponding contribution to the Shannon entropy for analogous $2D$ $\textit{defect}$ replica field theories in the $R\rightarrow1$ replica limit, which govern the long-distance physics in the problem of performing weak $\textit{quantum}$ measurements on one-dimensional quantum critical ground states. Lastly, we discuss a possible consequence of our theorems for classical systems with generic uncorrelated impurity-type quenched disorder, according to which, under a certain assumption, and as opposed to problems with measurement-induced randomness, the corresponding universal quantities $c_{\text{eff}}^{(R\rightarrow0)}$ and $g_{\text{eff}}^{(R\rightarrow0)}$ in the $R\rightarrow0$ replica limit would $\textit{increase}$ under RG flow.