Singular Value Decomposition and Its Blind Spot for Quantum Chaos in Non-Hermitian Sachdev-Ye-Kitaev Models

Abstract: The study of chaos and complexity in non-Hermitian quantum systems poses significant challenges due to the emergence of complex eigenvalues in their spectra. Recently, the singular value decomposition (SVD) method was proposed to address these challenges. In this work, we identify two critical shortcomings of the SVD approach when analyzing Krylov complexity and spectral statistics in non-Hermitian settings. First, we show that SVD fails to reproduce conventional eigenvalue statistics in the Hermitian limit for systems with non-positive definite spectra, as exemplified by a variant of the Sachdev-Ye-Kitaev (SYK) model. Second, and more fundamentally, Krylov complexity and spectral statistics derived via SVD cannot distinguish chaotic from integrable non-Hermitian dynamics, leading to results that conflict with complex spacing ratio analysis. Our findings reveal that SVD is inadequate for probing quantum chaos in non-Hermitian systems, and we advocate employing more robust methods, such as the bi-Lanczos algorithm, for future research in this direction.