Singular Value Decomposition Entanglement Entropy : A Random Matrix Theory Based Exploration

We investigate singular value decomposition entanglement entropy (SVDEE), a new measure that generalizes ordinary entanglement entropy by incorporating two distinct quantum states. Studies show that SVD EE increases when two states belong to different quantum phases, highlighting its potential for identifying quantum phase transitions in many-body systems [1]. However, this quantity remains largely unexplored in the quantum-information-theoretic context of random matrices and paradigmatic quantum chaotic systems. This motivates our investigation into its properties for eigenstates sampled from well-documented random matrix ensembles of Gaussian, Wishart, and Bures-Hall ensembles, extending beyond prior work on Haar-random matrices [1]. Our results demonstrate that while subsystem entropies for SVDEE vary, bounds on it are independent of the specific random matrix ensemble. Additionally, we find that the spread of SVDEE between subsystems depends on the symmetry class of the sampled eigenstates, characterized using individual subsystem entropies and related measures of subadditivity and Araki-Lieb inequality. We also investigate the SVDEE of coupled kicked tops, performing extensive numerical explorations for a wide range of coupling strengths and chaoticity parameters of the system.

[1] SVD Entanglement Entropy, A. J. Parzygnat, T. Takayanagi, Y. Taki & Z. Wei, JHEP 12 (2023) 123 (2024).