Rank-One Loss Maximizes the Spectral Response Strength of Passive Full-Order Exceptional Points

Abstract

We consider finite-dimensional non-Hermitian systems in which all modes coalesce at a single exceptional point. For a rank-one anti-Hermitian contribution, the spectral response strength is computed exactly. For passive systems, we then derive a sharp upper bound at fixed exceptional-point decay rate and show that it is reached only when the decay operator has rank one, corresponding to a single independent dissipative direction. This establishes rank-one loss as the optimal configuration for enhancing the response of a full-order exceptional point. The bound holds for arbitrary order and improves on earlier passive estimates beyond second order. Two third-order trimers illustrate the result: a rank-one-loss system saturates the bound, whereas a passive rank-two-loss realization remains below it. The result turns dissipation into a design resource: concentrating the loss into a single effective dissipative direction maximizes the highest-order spectral response of a passive full-order exceptional-point device.