Detecting quantum chaos via pseudo-entropy

TL;DR

This paper investigates pseudo-entropy as a diagnostic tool for quantum chaos by analyzing variants of the SYK model. It establishes connections between pseudo-entropy and spectral form factor (SFF), both for the full two-sided system and for subsystems via reduced transition matrices, and shows how pseudo-Rényi entropy can robustly reflect chaotic versus integrable dynamics. The work uncovers subtleties arising from the multivalued nature of complex logarithms in the pseudo-entropy definition, motivating a shift toward pseudo-Rényi entropy to faithfully capture SFF features. It also connects these entanglement-based measures to scrambling through local-quench setups and OTOCs, and surveys chaos indicators in sparse SYK and SpinXY4 models, including a chaos–integrability transition near a critical sparseness. Overall, the study suggests pseudo-entropy and pseudo-Rényi entropy as complementary tools for diagnosing quantum chaos, with potential relevance to holography and broader chaotic dynamics.

Abstract

Quantum informatic quantities such as entanglement entropy are useful in detecting quantum phase transitions. Recently, a new entanglement measure called pseudo-entropy was proposed which is a generalization of the more well-known entanglement entropy. It has many nice properties and is useful in the study of post-selection measurements. In this paper, one of our goals is to explore the properties of pseudo-entropy and study its effectiveness as a quantum chaos diagnostic, i.e., as a tool to distinguish between chaotic and integrable systems. Using various variants of the SYK model, we study the signal of quantum chaos captured in the pseudo-entropy and relate it to the spectral form factor (SFF) and local operator entanglement (LOE).

Figures (11)

• Figure 1: The plot of the real part of the pseudo-entropy of the SYK$_4$ model with various values of $N$ and $\beta=0$.
• Figure 2: (a). The real part of the $n$-th pseudo-Rényi entropy. (b). Logarithm of $R_{2n}(t)$. $N=16$, $n=2,\cdots 8$ and $\beta=0$.
• Figure 3: Plot of the real part of the pseudo-Rényi entropy of the SYK$_4$ model with various values of $N$ and $\beta=0$.
• Figure 4: Real part of the pseudo-Rényi entropy of the binary sparse SYK model with different $N$ and $k_{cpl}$.
• Figure 5: Real part of the pseudo-Rényi entropy of the binary sparse spinXY$_4$ model with different $k_{cpl}$.