Non-Hermitian Quantum Metrology Enhancement and Skin Effect Suppression in PT-Symmetric Bardeen-Cooper-Schrieffer Chains

Abstract

We outline a theoretical framework for quantum metrology in non-Hermitian systems, demonstrating both significant failure and exceptional regimes in PT-symmetric Bardeen-Cooper-Schrieffer chains. Through biorthogonal quantum Fisher information analysis, we identify two distinct regimes: exponential sensitivity suppression in the non-Hermitian skin effect phase ($F_Q \propto N^3 e^{-2\kappa N}$) where eigenstates localize exponentially, and quadratic enhancement near PT-breaking exceptional points [1-4] ($F_Q \propto N^2/\delta$) achieving Heisenberg scaling. Our multiparameter analysis establishes optimal simultaneous estimation of chemical potential, Peierls phase, and gain/loss strength with quantum Fisher information matrix scaling as $N^2$, surpassing the standard quantum limit by factors exceeding $10^2$. For realistic parameters ($t/2\pi=10$ MHz, $\Delta/2\pi=1$ MHz, $N=50$), we predict enhancement factors $\eta_\mu \approx 20\sqrt{N}=141$ for chemical potential estimation and $\eta_\phi \approx t^2 \sqrt{3N/2}=100\sqrt{N}$ over classical sensing. These results are validated through exact finite-size calculations and provide concrete protocols for superconducting circuit implementations.We reveal a core dichotomy in non-Hermitian quantum metrology: NHSE suppresses sensitivity exponentially, while $\mathcal{PT}$-symmetry enables Heisenberg-limited enhancement -- each arising from distinct spectral and localization topologies.