Hybrid non-Hermitian singularities and mapped-state concurrence in a superconducting-circuit dynamical matrix

Non-Hermitian spectral singularities have attracted considerable interest, but how such structures emerge from a concrete microscopic system and connect to measurable observables remains an important question. Here, we study a coupled double-cavity system mediated by a superconducting qutrit and show that, although the effective two-mode Hamiltonian is Hermitian, the corresponding four-component Bogoliubov operator dynamics is governed by a non-Hermitian dynamical matrix. Starting from the microscopic circuit model, we derive the effective bosonic Hamiltonian and construct the dynamical matrix from the Heisenberg equations of motion. We show that this dynamical matrix hosts exceptional points, exceptional lines, and hybrid diabolical-exceptional points, also known as diabolically degenerate exceptional points. We further present a quadrature-based, model-assisted protocol for extracting the dynamical-matrix spectrum from measurable cavity-field observables. Finally, we introduce a virtual two-qubit representation of the four-component dynamical vector and use the mapped-state concurrence as a diagnostic of eigenvector nonseparability and its dynamical evolution. Our results provide a concrete route to connect Hermitian bosonic dynamics, non-Hermitian dynamical-matrix singularities, spectrum reconstruction, and virtual two-qubit diagnostics in a superconducting-circuit platform.