Converting non-Hermitian degeneracies of any order: Hierarchies of exceptional points and degeneracy manifolds
Grigory A. Starkov, Sharareh Sayyad
Abstract
The emergence of various types of degeneracies plays a crucial role in optimizing and engineering different physical phenomena in non-Hermitian physics. In our work, we focus on the derogatory Exceptional Points (EPs), which are characterized by multiple Jordan blocks corresponding to the same eigenvalue. We demonstrate that, under certain infinitesimal perturbations, a derogatory EP can be converted into an EP of different structure without varying the total order of degeneracy. In particular, such conversion can increase the size of the largest Jordan block and, hence, the sensitivity of the eigenspectrum to parameter variation, which is an important feature for practical applications. Furthermore, by analyzing all possible conversions, we introduce hierarchies of degeneracies of the same order that appear when perturbing non-Hermitian systems. We systematically explore hierarchies in the absence of any symmetry and when pseudo-Hermitian symmetry is present. Our study facilitates engineering various degeneracies of non-Hermitian systems, paving the way to extending the implications of non-Hermitian physics.
AI Impact Assessments
(3 models)Scientific Impact Assessment
Core Contribution
This paper addresses the conversion of derogatory exceptional points (EPs) — degeneracies characterized by multiple Jordan blocks for the same eigenvalue — into EPs of different internal structure while preserving the total degeneracy order. The central insight is that certain infinitesimal perturbations can restructure a derogatory EP, potentially increasing the size of the largest Jordan block and thereby enhancing the eigenspectral sensitivity (scaling as Δ^{1/m} for a block of size m). The authors formalize this through hierarchies of degeneracy manifolds: a type-A EP can be converted to type-B if and only if manifold M_A lies in the boundary of manifold M_B. They construct these hierarchies systematically for the symmetry-free case and for pseudo-Hermitian symmetry using the mathematical machinery of Young diagrams, dominance ordering, and signed Young diagrams (for the pseudo-Hermitian case via nilpotent orbit closures in classical Lie groups).
Methodological Rigor
The mathematical framework is well-grounded. The connection between EP conversion and manifold closure is established rigorously by leveraging classical results from matrix theory (Gerstenhaber's theorem on nilalgebras, Arnold's canonical form for matrix families, and Djokovic's work on conjugacy class closures in real Lie groups). The dominance ordering criterion (Eq. 59) via column sums of Young diagrams provides a clean, computationally verifiable condition.
The paper builds from simple 2×2 and 3×3 examples to the general n×n case, making the abstract mathematical concepts accessible. The visualization of manifold M_3 near a type-(2,1) EP (Fig. 2) is particularly instructive. The non-Hermitian Lieb lattice example (Section II.D) demonstrates the phenomenon in a physically motivated setting with explicit parameter conditions.
For the pseudo-Hermitian case, the authors correctly identify the relevant symmetry group U(p,q) and use Theorem 1 (from indefinite linear algebra) to classify degeneracies via signed Young diagrams. The dominance criterion (Eq. 84/87) is adapted from Djokovic's work on orbit closures in su(p,q). The treatment is mathematically sound, though the paper relies heavily on citing existing mathematical theorems rather than providing independent proofs.
One methodological limitation: the hierarchy analysis tells us which conversions are *topologically possible* (the manifold boundary condition is satisfied) but does not guarantee that a specific physical perturbation achieves the conversion. The authors acknowledge this explicitly in Section V when discussing the Liouvillian application — the quantum jump terms have restricted form and may not constitute a general perturbation.
Potential Impact
Practical relevance: The key practical implication is a novel mechanism for engineering higher-order EPs. Since EP-based sensing scales with block size (Δ^{1/m}), converting a derogatory EP with small blocks into one with a larger block could enhance sensor performance without changing the total system size. This is particularly relevant for Liouvillian superoperators, where the dimensionality scales as N² relative to the Hamiltonian dimension, offering potentially large Jordan blocks.
Theoretical utility: The hierarchy diagrams (Figs. 5-9, 11-13) serve as practical lookup tables for researchers working with non-Hermitian systems. The developed Mathematica notebook and Julia package lower the barrier for applying these results. The extension to pseudo-Hermitian symmetry is important given the prevalence of PT-symmetric systems.
Connections to adjacent fields: The work bridges non-Hermitian physics with algebraic geometry (nilpotent orbit closures) and representation theory (Young diagram lattices). The self-similar structures between hierarchies of different orders (Section III) hint at deeper algebraic properties that could be explored further.
Timeliness & Relevance
The paper addresses a timely gap. While non-derogatory EPs have been extensively studied, derogatory EPs are gaining attention due to their natural occurrence in Liouvillian superoperators and composite systems. Recent works (Refs. [32-35]) have begun exploring these multi-block EPs, and this paper provides the systematic framework that was missing for understanding their interconversion. The connection to EP-enhanced sensing — a very active experimental area — adds practical urgency.
Strengths
1. Systematic completeness: The paper provides a complete classification of all possible EP conversions for arbitrary order, not just specific examples.
2. Clean mathematical framework: The translation of the physical conversion problem into manifold closure analysis is elegant and makes the problem tractable.
3. Symmetry extension: The pseudo-Hermitian case is non-trivial and practically relevant; the signed Young diagram formalism captures how symmetry constrains the hierarchy (e.g., certain EP types become forbidden, as seen in the η_{3,1} case).
4. Computational tools: The accompanying software packages enable practical use.
5. Progressive exposition: Moving from 2×2 to general n×n builds understanding effectively.
Limitations
1. Gap between possibility and realizability: The hierarchy identifies necessary conditions for conversion but not sufficient ones for specific physical systems. The Liouvillian example (Section V) illustrates this limitation — the authors can only argue certain conversions are "theoretically possible."
2. PT-symmetry gap: The most physically relevant symmetry for Liouvillians (generalized PT-symmetry) is not fully treated; the pseudo-Hermitian analysis requires a parameter-independent pseudo-metric, which may not always exist.
3. No experimental protocol: The paper lacks concrete prescriptions for how to implement EP conversions in specific experimental platforms.
4. Limited physical examples: The Lieb lattice example is illustrative but relatively simple; the Liouvillian examples are analyzed only at the level of identifying possible conversions without completing the analysis.
5. Codimension analysis: While mentioned in passing, a more systematic discussion of codimensions of different manifolds would strengthen the practical guidance — higher-codimension conversions require more fine-tuning and are harder to achieve experimentally.
Overall Assessment
This is a mathematically rigorous and systematic contribution that establishes the theoretical foundation for understanding EP conversions. It successfully translates a physically important question into a well-posed mathematical framework with complete answers. The impact is primarily theoretical/structural, providing tools and classifications that will be useful for the growing community working on non-Hermitian degeneracies. The gap between the general mathematical framework and specific physical realizability somewhat limits immediate practical impact.
Generated Apr 20, 2026
Comparison History (39)
Paper 2 addresses exceptional points in non-Hermitian physics, a rapidly growing field with broad applications in photonics, sensing, and topological physics. The systematic framework for converting between different types of exceptional points and the introduction of degeneracy hierarchies provides practical tools for engineering enhanced sensitivity in sensors and other devices. Paper 1, while mathematically rigorous, extends coherence resource theory to continuous variables in a somewhat incremental way, with the notable limitation that the incoherent set contains no normal states, reducing practical applicability. Paper 2's broader cross-disciplinary relevance and timeliness give it higher impact potential.
Paper 2 bridges quantum information theory and computational complexity by introducing the concept of computational min-entropy. Its demonstration that computational limits fundamentally restrict observable quantum correlations has profound implications for quantum cryptography, computing, and thermodynamics. While Paper 1 provides valuable methods for engineering non-Hermitian systems, Paper 2 addresses a fundamental theoretical problem with broader, cross-disciplinary applicability and higher potential impact in the rapidly advancing field of quantum technologies.
Paper 2 presents a systematic framework for converting and engineering exceptional points in non-Hermitian systems, which has broad applications across photonics, sensing, and condensed matter physics. Its methodological contribution—hierarchies of degeneracies and their conversions—provides practical tools for optimizing sensitivity in sensors and other devices. Paper 1 offers an interesting conceptual reinterpretation of boson correlations via Simpson's paradox, but its impact is more niche, primarily reframing existing understanding rather than enabling new capabilities. Paper 2's broader applicability and practical engineering implications give it higher potential impact.
Paper 1 addresses the critical barren plateau problem in variational quantum algorithms by introducing a novel quantum sparsity principle grounded in topological entanglement entropy. It bridges concepts from classical machine learning, quantum information, and topological physics, deriving a quantum Nyquist-Shannon theorem and demonstrating practical numerical improvements. Its breadth of impact spans quantum computing, machine learning, and information theory, with immediate practical applications to VQA design. Paper 2, while mathematically rigorous in characterizing exceptional point hierarchies, addresses a more specialized topic in non-Hermitian physics with narrower immediate applications.
Paper 2 addresses a fundamental theoretical question in non-Hermitian physics—the conversion and hierarchy of exceptional points—with broad implications across photonics, condensed matter, and open quantum systems. Its systematic framework for engineering degeneracies is widely applicable and methodologically rigorous. Paper 1, while timely in combining PQC with quantum teleportation, addresses a relatively niche integration problem with results heavily dependent on specific parameter assumptions (e.g., 1 ms coherence). Paper 2's foundational nature and cross-disciplinary relevance give it higher potential for broad scientific impact.
Paper 2 addresses the fundamental problem of quantum Gibbs state preparation with rigorous complexity bounds, connecting quantum thermodynamics, open quantum systems, and quantum computing. Its result that KMS detailed balance overcomes Lamb shift issues is surprising and practically significant, with direct applications to quantum algorithms and quantum simulation. The O(ε⁻¹) complexity bound for Gibbs state preparation is a concrete, broadly applicable result. Paper 1, while mathematically rigorous in classifying exceptional point hierarchies, addresses a more specialized topic within non-Hermitian physics with narrower immediate applications.
Paper 2 addresses a critical bottleneck in near-term quantum computing by experimentally demonstrating an error mitigation technique on actual hardware. By enabling faster measurements while mitigating the resulting increased error rates, this work offers immediate, practical improvements for quantum algorithms and NISQ devices, leading to potentially broader and faster real-world impact compared to the theoretical advancements in non-Hermitian physics presented in Paper 1.
Paper 2 offers foundational theoretical advancements in non-Hermitian physics, applicable across various wave-physics domains like optics, acoustics, and quantum systems. Its focus on engineering exceptional points to enhance eigenspectrum sensitivity provides broad implications for next-generation sensors. In contrast, while Paper 1 demonstrates practical quantum circuit implementations, its impact is narrower, primarily localized to near-term quantum computing diagnostics and entanglement measurement.
Paper 1 addresses a central challenge in quantum optics—deterministic multiphoton emission—with a concrete, programmable scheme achieving orders-of-magnitude improvements in photon purity. It combines interference and interaction engineering in a unified framework with clear practical applications in quantum photonic devices. Paper 2, while mathematically rigorous in classifying exceptional point hierarchies, is more abstract and incremental within non-Hermitian physics. Paper 1's direct relevance to quantum technology development, experimental feasibility in cavity-QED systems, and quantitative performance gains suggest broader and more immediate scientific impact.
Paper 1 tackles a critical bottleneck in quantum computing—scalability and resource overhead—offering a novel framework for low-overhead hardware. The real-world applicability and timeliness of scalable quantum architectures provide a broader and more immediate potential impact across physics, computer science, and industry compared to the highly specialized, theoretical non-Hermitian physics explored in Paper 2.
Paper 2 has higher potential impact due to a clearer route to near-term quantum-technology applications (deterministic single-photon and photon-pair sources with simultaneously high purity and brightness), strong timeliness for quantum networking/photonic computing, and broader cross-field relevance (many-body physics, cavity QED, quantum photonics, quantum information). Its claims of large antibunching improvement and programmable switching suggest experimentally testable, scalable functionality. Paper 1 is novel and valuable for non-Hermitian theory and sensor engineering, but its impact is more specialized and may require additional steps to translate hierarchy/EP conversion results into widely adopted devices.
Paper 1 offers a highly novel, interdisciplinary approach combining quantum vacuum fluctuations, machine learning, and materials science. Its direct real-world application in characterizing thin-film permittivity and geometry over broad frequencies gives it broader potential impact across engineering and applied physics compared to the foundational, highly theoretical focus of Paper 2 on non-Hermitian exceptional points.
Paper 2 proposes a reusable, foundational framework for evaluating quantum distance-bounding protocols, addressing a critical gap in quantum cryptography. Its standardization of security models and fraud experiments provides a broadly applicable tool for future research in quantum communication and cybersecurity, offering clearer and more immediate real-world impact compared to the highly specialized, theoretical physics focus of Paper 1.
Paper 1 addresses a timely open question in stochastic thermodynamics — finite-time constraints on autonomous information engines — connecting information geometry, thermodynamic trade-offs, and practical optimization. It uncovers a novel synergistic regime and a fundamental trade-off relation, with broad relevance to nanoscale engines, biological systems, and information processing. Paper 2 makes a solid mathematical contribution to non-Hermitian degeneracy theory, but its scope is more specialized. Paper 1's bridging of information theory, thermodynamics, and optimization gives it broader interdisciplinary impact and stronger real-world applicability.
Paper 2 presents a fundamental theoretical advance in non-Hermitian physics by establishing a systematic framework for converting and classifying exceptional point degeneracies. This has broad implications across multiple fields (photonics, acoustics, quantum mechanics, sensing) where EPs are actively exploited. The introduction of degeneracy hierarchies and the demonstration that perturbations can increase Jordan block size offers both conceptual novelty and practical utility for enhancing sensor sensitivity. Paper 1, while practically valuable, is primarily a characterization study of existing commercial hardware in a specific deployment, with more limited generalizability and narrower impact scope.
Paper 2 offers direct, tangible real-world applications by bridging quantum sensing with power systems (smart grids). Its proposed Fabry-Pérot cavity-enhanced configuration and rigorous Fisher Information framework provide clear pathways for high-precision metrology. While Paper 1 presents valuable fundamental theoretical advancements in non-Hermitian physics, Paper 2's immediate interdisciplinary impact and practical engineering solutions give it a higher potential for broad scientific and technological impact.
Paper 1 focuses on non-Hermitian exceptional points, which have broad and immediate practical applications in enhanced sensing, photonics, and acoustics. The ability to engineer these degeneracies offers clear pathways for optimizing physical phenomena. Paper 2, while significant for foundational quantum information and many-body dynamics, is more theoretical and currently lacks the immediate cross-disciplinary real-world applications present in Paper 1.
Paper 2 likely has higher impact due to strong timeliness and clear real-world applicability to quantum networking/Internet development, linking physical entanglement dynamics with queueing-based congestion control and adaptive policies. The methodology is rigorous and system-oriented (stochastic traffic models, stability/delay/fidelity analysis, multi-user extension), and the ideas transfer broadly to networking, control, and quantum engineering. Paper 1 is novel in non-Hermitian degeneracy theory and could influence sensing/EP engineering, but it is more specialized and its application pathway is less direct than quantum network resource-control frameworks.
Paper 1 addresses a fundamental and broadly applicable topic in non-Hermitian physics—the systematic conversion and classification of exceptional points of arbitrary order. This provides a general mathematical framework with wide-ranging implications across photonics, acoustics, and sensing. The systematic hierarchy of degeneracies is novel and practically relevant for EP-based sensing applications. Paper 2, while elegant in connecting geometric concepts to open quantum thermodynamics, is more niche and builds on established geometric phase/Berry curvature frameworks. Paper 1's broader applicability across multiple experimental platforms gives it higher impact potential.
Paper 1 is likely to have higher scientific impact due to strong timeliness and broad, cross-cutting applicability: quantum simulators underpin much of near-term quantum computing research, and a large empirical bug study (394 bugs across 12 tools) can directly change testing practices, reliability standards, and downstream results in many papers and products. Its methodological contribution (systematic taxonomy and evidence about silent correctness failures) is actionable for both academia and industry. Paper 2 is intellectually novel within non-Hermitian physics, but its impact is more specialized and may translate to applications more slowly.