Notes on Pseudo Entropy Amplification

TL;DR

This work defines and analyzes pseudo entropy $S(\tau^{\psi|\varphi}_A)$ for a linear overlap of states and investigates when amplification, i.e. $S(\tau^{\psi|\varphi}_A) > \log\dim\mathcal{H}_A$, occurs across qubit systems, 2D free CFT, and 2D holographic CFT. Using explicit qubit Bell-state perturbations, replica-trick calculations in 2D free CFT, and heavy BTZ-like states in holographic CFT, the authors show amplification in the qubit and 2D free CFT cases, with $S^{(2)}(\tau_A^{\varphi|\psi})=\log\left(\dfrac{2\epsilon^2}{1+\epsilon^2}\right)$ during the signal window, but no amplification in the holographic CFT within the perturbative regime where $\langle\varphi|\psi\rangle$ dominates. They argue that off-diagonal contributions to the area operator are exponentially suppressed in holographic theories (ETH-like behavior), so amplification would require non-perturbative effects $\sim e^{-\mathcal{O}(c)}$, providing a diagnostic for holographic vs non-holographic regimes. The results clarify when pseudo entropy can exceed entropic bounds and offer potential paths to experimental probing via pseudo Rényi entropies.

Abstract

We study pseudo entropy for a particular linear combination of entangled states in qubit systems, two-dimensional free conformal field theories (CFT), and two-dimensional holographic CFT. We observe phenomena that the pseudo entropy can be parametrically large compared with the logarithm of the dimension of Hilbert space. We call these phenomena pseudo entropy amplification. The pseudo entropy amplification is analogous to the amplification of the weak value. In particular, our result suggests the holographic CFT does not lead the amplification as long as the non-perturbative effects are negligible. We also give a heuristic argument when such (non-)amplification can occur.

Figures (3)

• Figure 1: Amplification of pseudo entropy for the states \ref{['eq:psi']} and \ref{['eq:varphi']}. Here we show the case $\epsilon$ is (a) real or (b) pure imaginary. In the case (b), the real-valuedness comes from a pair of eigenvalues that are complex conjugates of each other. It is useful to note that the lower or upper bound of the real part is given by $\log2$ as $|\epsilon|\rightarrow\infty$ limit corresponds to the limit $\psi\rightarrow\varphi$.
• Figure 2: Amplification of the second pseudo Rényi entropy for the states \ref{['eq:psi']} and \ref{['eq:varphi']}. Again we show the case $\epsilon$ is (a) real or (b) pure imaginary. In the case (b) with $|\epsilon|<1$, we have $\mathrm{Im}S^{(2)}=\pi$.
• Figure 3: Euclidean path integral on $\Sigma_2$ for calculating the numerator of the correlation function in \ref{['eq:4pt']}.