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.

The paper asks how dissipation constrains the spectral response of a passive, finite-dimensional system at a full-order exceptional point (EP), where all $n$ modes form one Jordan chain. The response measure is $\xi=\|(H-E_{\mathrm{EP}}I)^{n-1}\|_2$, the norm of the highest-order resolvent-pole coefficient, which controls the leading perturbative eigenvalue-splitting bound. Writing $H=H_0-i\Gamma/2$ and $\beta=-2\operatorname{Im}E_{\mathrm{EP}}>0$, it proves the sharp passive bound $\xi\leq\beta^{n-1}$, with equality if and only if $\operatorname{rank}\Gamma=1$. At fixed total loss, this gives $\xi_{\max}=(\operatorname{tr}\Gamma/n)^{n-1}$. The bound improves the previous arbitrary-order estimate by a factor $(n/2)^{(n-1)/2}$ and establishes a previously numerically suggested third-order bound. A Schur decomposition expresses $\xi/\beta^{n-1}$ as a product of adjacent correlation magnitudes of a normalized positive-semidefinite decay matrix; Cauchy–Schwarz gives the bound and its equality condition. An explicit family saturates it for every order. Separately, a determinant-lemma argument establishes $\xi=(2|\operatorname{Im}E_{\mathrm{EP}}|)^{n-1}$ for rank-one loss or gain. Two passive trimers illustrate the result: rank-one loss gives $\xi=\beta^2$, while a uniformly loss-shifted parity-time-symmetric trimer with rank-two loss gives $\xi=\beta^2/2$. The theorem assumes EP order equals Hilbert-space dimension; embedded EPs and multiple Jordan chains remain outside its scope. It optimizes spectral response, not sensing performance, which also depends on perturbation orientation, excitation, readout, and noise.