Faithfulness of the singular-value decomposition for normal non-Hermitian generators

Establish whether the singular-value decomposition approach remains faithful for diagnosing spectral statistics when a non-Hermitian generator is normal in the integrable limit.

Background

The paper analyzes dissipative Sachdev–Ye–Kitaev physics using the Liouvillian spectrum of cavity-QED systems. In the integrable corners of the single-mode spontaneous-emission and multimode cavity-loss setups, the effective Liouvillian is normal and can be diagonalized in a basis of fermionic occupation operators. Consequently, its singular values are obtained by applying the pointwise map from each centered Liouvillian eigenvalue to its absolute value, without eigenvalue mixing.

The authors argue that this structure avoids the blind spot previously identified for singular-value diagnostics of non-Hermitian random free-fermion systems, where Hermitianization can generate effective interactions and alter spectral statistics. They conjecture that this favorable behavior extends more generally to normal non-Hermitian generators in the integrable limit, but do not establish the claim beyond the setting analyzed in the paper.