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

The gist The rank-one loss operator maximizes the highest-order spectral response strength of a passive full-order exceptional point.

Introduction to Exceptional Points and Response Strength

Exceptional points (EPs) are non-Hermitian degeneracies where eigenvalues and eigenvectors coalesce, leading to eigenvalue splitting proportional to the nth root of the perturbation strength [1]. The spectral response strength ξ is defined as the norm of the highest-order pole coefficient of the Green’s function and serves as a measure of eigenvalue splitting under perturbation [3]. For full-order EPs, where order equals Hilbert space dimension, an upper bound on ξ for passive systems is established as ξ ≤ √2n Im EEP(n−1) [1]. This bound is shown to be strictly tighter than earlier estimates for n > 2, and equality holds when the decay operator has rank one [3].

Analytical Derivation of Rank-One Response

The exact response strength for a rank-one anti-Hermitian contribution is derived using the matrix determinant lemma [32] to establish an identity relating the Green’s function elements to the decay parameter. Expanding this expression yields that the coefficient of the highest-order pole is D v N n−1 v E, which equals β(n−1) = (2Im EEP)(n−1) [30]. This result is further refined by relating it to the right and left EP eigenvectors, showing that D v N n−1 v E = ξ/n [36]. Consequently, the exact response strength for a rank-one anti-Hermitian contribution is found to be ξ = β(n−1) = (2Im EEP)(n−1) [37].

Optimality in Passive Systems

When passivity is imposed, the analysis utilizes a Schur decomposition where the normalized decay operator becomes a correlation matrix [36]. The spectral response strength is then expressed as ξ = β(n−1)/j=1 t j for the first-superdiagonal elements t j of the strictly upper-triangular operator Tˆ [43]. By applying the Cauchy–Schwarz inequality, it is shown that every passive full-order EPn satisfies ξ ≤ β(n−1) [49]. The equality condition requires ⟨u j u j+1⟩ = 1 for all j, which implies the correlation matrix Cˆ has rank one [53].

Design Principle and Comparison with Other Orders

The paper demonstrates that the bound is attainable for arbitrary order by constructing a specific rank-one loss system where Tˆn−1 = 0, Tˆn−1!= 0, and Tˆn−12 = β(n−1) [58]. This construction yields a maximal response of ξmax = γtot / n(n-1) [60]. The rank-one loss system saturates this bound, whereas a passive rank-two-loss realization falls below it by a factor of two in the trimer comparison [73]. The universal passive-system estimate for eigenvalue displacement is Ej (ε) − EEP(n ≤ εVˆ 2 β(n−1) [64].

Physical Implications and Conclusion

The result establishes that concentrating dissipation into a single effective dissipative direction maximizes the highest-order spectral response of a passive full-order exceptional-point device [8]. This principle suggests that loss-channel engineering is a practical resource for optimizing coupled resonators and integrated photonic circuits operating near EPs [8]. The analysis confirms that the EP order and decay rate alone do not determine the response strength; rather, the rank and correlation structure of the loss operator are decisive [73].

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

I. INTRODUCTION EPs are non-Hermitian degeneracies at which eigenvalues and the corresponding eigenvectors coalesce At an EP of order n, the Hamiltonian possesses a Jordan chain of length n, and the resolvent contains poles up to order n A generic perturbation therefore produces an eigenvalue splitting proportional to the nth root of the perturbation strength [1], which, for sufficiently small perturbations, is significantly larger than for conventional degeneracies (diabolic points) The prefactor of this fractional-power law is not universal It depends on the nonnormal structure of the Hamiltonian [2] and can be characterized by the spectral response strength [3]. EPs have been investigated in a broad range of wave and quantum systems, with particularly extensive activity in optics and photonics [4–6] Proposed and demonstrated applications include mode selection [7] and directional lasing [8], asymmetric state conversion [9, 10], coherent perfect absorption [11], control of light transport [12], non-Hermitian topology [13], and sensing [14–17]. Rank-one loss operators arise naturally when a finite set of internal states couples coherently to a single external decay channel In projection-operator and effectiveHamiltonian descriptions, the anti-Hermitian part of the Hamiltonian represents decay and is given by a sum of outer products of the channel-coupling amplitudes; the associated decay matrix therefore has rank no greater than the number of open channels [18–20] Hence, a single open channel yields a rank-one anti-Hermitian part, or equivalently, a rank-one decay matrix Conversely, several physical decay channels can also produce a rank-one decay matrix if their coupling vectors are linearly dependent, in which case they constitute a single independent dissipative direction This rank-one dissipative structure is relevant, for example, to nuclear and mesoscopic resonances with only one energetically open reaction channel, to wave resonators coupled to a single effective port, and to collective emission through one common reservoir channel The same factorized structure underlies superradiant width segregation in open quantum systems [21] The corresponding loss vector is generally a coherent superposition of internal modes and need not coincide with one basis state The spectral response strength ξ is the norm of the highest-order pole coefficient of the Green’s function and is a measure of the eigenvalue splitting under perturbation [3] Later work connected it to the distance from diabolic points, Petermann factors, phase rigidities, and EPs embedded in larger Hilbert spaces [22–27] For fullorder EPs, where the EP order equals the Hilbert-space dimension, Ref. [22] established the following upper bound on the spectral response strength for passive systems (i.e., systems with loss but no gain): ξ ≤ √ 2n Im EEP(n−1) (1) where n is the order and EEP the eigenvalue of the EP The bound is not sharp for n > 2 For n = 3, numerical results in Ref. [3] suggested the sharp upper bound ξ ≤ (2Im EEP)(n−1) (2) Equality holds for this bound, as well as for inequality (1) in the case n = 2, whenever the decay operator has rank one Thus, rank-one losses maximize the spectral response strength for a fixed EP decay rate 2Im EEP, which is potentially useful for sensing applications [28] In this paper, we prove analytically for full-order EPs in passive systems the sharp bound ξ ≤ (2Im EEP)(n−1) (3) which is strictly tighter than Eq. (1) for n > 2 and proves the bound previously suggested by numerical results in arXiv:2610.31268v1 [quant-ph] 22 Oct 2026

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A. Determinant identity We now specialize to the rank-one Hermitian operator Γˆ in Eq. (12) The matrix determinant lemma [32] states that, for an invertible matrix Aˆ and vectors u⟩ and w⟩, det Aˆ + u⟩⟨w = det(Aˆ) 1 + ⟨wAˆ−1 u⟩ For Eq. (12), Hˆ † = Hˆ + iγv⟩⟨v, (20) and therefore E1 − Hˆ † = E1 − Hˆ − iγv⟩⟨v> Apply Eq.

1. Precision Loss Engineering for Passive Devices: The improved system can design passive wave and photonic systems where concentrating dissipation into a single independent coupling direction maximizes highest-order spectral response of an exceptional-point device.

2. Sharp Upper Bound Calculation: The system can derive a sharp upper bound for the spectral response strength in passive full-order exceptional points, improving on earlier estimates beyond second order by proving that rank-one loss as the optimal configuration for enhancing the response of a full-order exceptional point.

3. Optimal Loss Operator Identification: The AI can determine if a given dissipation operator is rank one, as this condition is reached only when the decay operator has rank one, which is crucial for maximizing spectral response strength.

4. Performance Comparison and Design Guidance: The improved system can compare passive rank-one-loss trimers with passive rank-two-loss realizations to show that the rank-one example reaches the universal maximum in Eq. (49), whereas the passive version of the paritytime trimer with rank-two loss falls below it by a factor of two.

5. Perturbative Scaling Analysis: The system can provide bounds on eigenvalue displacement for perturbed Hamiltonians, showing that The largest possible prefactor is realized by rank-one loss, thus guiding perturbation analysis in sensing applications.