Schmidt Concentration as Lock-In Amplifier for Classical Phase Verification in Monitored Quantum Circuits

Noisy intermediate-scale quantum (NISQ) devices require efficient diagnostics for verifying state preparation and quantum Zeno dynamics. Conventional approaches using von Neumann entropy S_vN exhibit divergent variance in the measurement-dominated classical phase, precisely where verification is most critical. We demonstrate that the Schmidt concentration index κ = Σλᵢ² provides phase-dependent complementary information with superior stability in the classical regime. Through numerical simulations of Haar-random circuits with projective measurements (L=12 qubits, 100 trajectories per measurement rate), we find that while κ exhibits higher variance than S_vN in the quantum phase (coefficient of variation ratio CV(κ)/CV(S) = 5.1), it achieves 4-fold lower variance in the classical phase (CV(κ)/CV(S) = 0.25). This crossover occurs at measurement rate p ≈ 0.27, establishing κ as a lock-in amplifier for classical phase characterization. The enhanced stability translates to a 75% reduction in required measurement shots for equivalent verification confidence, offering immediate practical value for NISQ state preparation protocols and error correction threshold detection.