The Schmidt Concentration Index: Enhanced Fidelity and Efficiency for Many-Body Localization Diagnostics

Disorder-averaged observables in many-body localization (MBL) transitions suffer from high sample-to-sample variance, requiring thousands of disorder realizations for reliable phase boundary determination. We introduce the Schmidt Concentration Index (K_c), defined as the inverse participation ratio of Schmidt coefficients, achieving simultaneous advantages in both statistical efficiency and structural fidelity. Systematic benchmarking in the random-field Heisenberg chain demonstrates 50-80× variance reduction in strongly localized phases with 10-15× gains near criticality, enabling phase boundary mapping with ~100 disorder samples versus ~5000 for conventional von Neumann entropy measurements. Correlation analysis against the inverse participation ratio reveals that K_c exhibits 32% stronger correlation with localization (ρ = 0.984) compared to von Neumann entropy (ρ = 0.747), establishing superiority on both efficiency and fidelity metrics. This dual advantage stems from quartic weighting that suppresses spectral tail noise while amplifying dominant localization signals—a mechanism distinct from recent symmetry-resolved approaches. Critical exponents satisfy the Harris criterion (ν ≈ 2), providing a computationally efficient diagnostic for quantum simulator experiments operating under sampling constraints. The methodology is immediately applicable to exact diagonalization studies, DMRG simulations, and experimental quantum platforms (cold atoms, trapped ions, superconducting circuits). Keywords: many-body localization, quantum phase transitions, entanglement entropy, Schmidt decomposition, disorder averaging, quantum simulators