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arxiv: 2509.11856 · v3 · submitted 2025-09-15 · 🪐 quant-ph · cond-mat.mes-hall

Multi-block exceptional points in open quantum systems

Aysel Shiralieva , Grigory A. Starkov , Bj\"orn Trauzettel This is my paper

Pith reviewed 2026-05-18 17:03 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hall
keywords exceptional pointsopen quantum systemsnon-Hermitian HamiltoniansLiouvillian superoperatorsquantum jumpsquantum geometric tensorqubitsqutrits
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The pith

Exceptional points from non-Hermitian Hamiltonians correspond to multi-block exceptional points in the no-jump Liouvillian of open quantum systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper connects exceptional points that arise in non-Hermitian Hamiltonian approximations of open quantum systems to a new class of multi-block exceptional points that appear in the Liouvillian superoperator when quantum jump terms are omitted. It shows that these multi-block structures emerge directly from the separation between coherent evolution and jump processes, and that including the jump terms alters or lifts the block degeneracies. A reader would care because the mapping clarifies how environmental monitoring affects eigenvalue coalescences that control decay rates and coherence loss in quantum devices. Concrete examples of qubits and qutrits coupled to sink levels illustrate how physical parameters can be tuned to move through different orders of these points and change the resulting population dynamics. The quantum geometric tensor is identified as a diagnostic that registers both the non-Hermitian points and the multi-block points in the Liouvillian.

Core claim

Exceptional points in non-Hermitian Hamiltonians map onto a novel multi-block structure of exceptional points in the no-jump Liouvillian; quantum jump terms modify this block structure. In qubit and qutrit systems coupled to additional ground-state sinks, variation of physical parameters navigates the block structure, changes the order of the exceptional points, and alters the population dynamics of the states; the quantum geometric tensor detects the points of both kinds.

What carries the argument

The multi-block exceptional points of the no-jump Liouvillian superoperator, which inherit their degeneracy pattern from the exceptional points of the corresponding non-Hermitian Hamiltonian.

If this is right

  • Tuning system parameters moves the open-system dynamics through different orders of multi-block exceptional points.
  • Population decay rates and steady-state distributions change according to the order of the multi-block points.
  • The quantum geometric tensor registers both non-Hermitian exceptional points and the associated multi-block points in the Liouvillian.
  • Including quantum jump terms lifts or splits the block degeneracies that exist in the no-jump limit.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The block structure may provide a route to engineer controlled degeneracies in open-system spectra without requiring exact PT symmetry.
  • In regimes where the Markovian separation weakens, the multi-block signature should disappear, offering a diagnostic for non-Markovian effects.
  • Similar block patterns could appear in other superoperator descriptions that omit dissipative channels, such as in cascaded quantum systems.

Load-bearing premise

The standard Markovian approximation must hold so that coherent evolution remains separable from the quantum jump terms.

What would settle it

Compute or measure the eigenvalues and eigenvectors of the no-jump Liouvillian for a driven qubit coupled to a sink and check whether the predicted block degeneracies appear exactly where the non-Hermitian Hamiltonian shows exceptional points.

Figures

Figures reproduced from arXiv: 2509.11856 by Aysel Shiralieva, Bj\"orn Trauzettel, Grigory A. Starkov.

Figure 1
Figure 1. Figure 1: , while the rest describe the dissipation from the excited states into the ground state denoted by Lˆ l+k−1 = |1⟩⟨k|, for 2 ⩽ k ⩽ N. (5) To distinguish the corresponding terms, let us denote γk = Γl+k−1 for 2 ⩽ k ⩽ N. For simplicity, we as￾sume that there is no coherent drive between the excited states and the ground state. This way, we can write the system’s Hamiltonian explicitly as Hˆ = X i>1 ∆i |i⟩⟨i| … view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Realization of a non-Hermitian qubit formed by [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Formation of a non-Hermitian qutrit via a [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Evolution of the population of the [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Evolution of the population of the [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Evolution of the population of the [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Evolution of the population of the [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Quantum metric of an effective qubit with phase and bit-flip errors. (a)–(c) Dependence of the quantum [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Quantum metric of an effective qutrit with [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Real and imaginary parts of the spectrum of [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Newton polygon constructed from the [PITH_FULL_IMAGE:figures/full_fig_p013_13.png] view at source ↗
read the original abstract

Open quantum systems can be approximately described by non-Hermitian Hamiltonians (NHHs) and Liouvillian superoperators. The two approaches differ by quantum jump terms corresponding to a measurement of the system by its environment. We analyze the emergence of exceptional points (EPs) in NHHs and Liouvillian superoperators. In particular, we show how EPs in NHHs relate to a novel type of EPs -- multi-block EPs -- in the no-jump Liouvillian, i.e. the Liouvillian superoperator in absence of quantum jump terms. We further analyze how quantum jump terms modify the multi-block structure. To illustrate our general findings, we present two prime examples: qubits and qutrits coupled to additional ground state levels that serve as sinks of the population. In those examples, we can navigate through the EP block structure by a variation of physical parameters. We analyze how the dynamics of the population of the states is affected by the order of the EPs. Additionally, we demonstrate that the quantum geometric tensor serves as a sensitive indicator of EPs of different kinds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript claims that exceptional points (EPs) of non-Hermitian Hamiltonians (NHHs) map onto a novel multi-block EP structure in the no-jump Liouvillian (the Liouvillian superoperator with quantum jump terms removed). It derives how the addition of jump terms modifies this block structure, illustrates the construction with explicit qubit and qutrit models coupled to sink levels, shows that physical parameters can be tuned to traverse the EP blocks, examines the resulting population dynamics as a function of EP order, and demonstrates that the quantum geometric tensor detects EPs of different kinds.

Significance. If the central mapping holds under the stated approximations, the work supplies a concrete bridge between the NHH and full-Lindblad descriptions of open systems and introduces a new organizing principle (multi-block coalescence) for EPs in dissipative generators. The low-dimensional sink models are analytically tractable and allow explicit navigation of the EP structure; the use of the quantum geometric tensor as a diagnostic adds a geometric observable that may be measurable. These elements together provide a useful framework for analyzing and potentially controlling exceptional-point physics in Markovian open quantum systems.

major comments (2)
  1. [§3] §3, Eq. (8)–(10): the derivation of the multi-block Jordan structure in the no-jump Liouvillian follows directly from excising the jump dissipators, but the manuscript must explicitly verify that the block-diagonal form and eigenvalue coalescence survive the re-introduction of the jump terms for the parameter values used in the qubit and qutrit examples; otherwise the claimed modification of the multi-block EP by jumps rests on an untested perturbation assumption.
  2. [§4.2] §4.2, Fig. 3 and surrounding text: the navigation of the EP block structure by varying the sink coupling strength is presented for the qutrit model, yet no quantitative check is given that the Born-Markov or weak-coupling conditions remain satisfied across the plotted parameter range; if those conditions are violated, the separation into no-jump and jump parts (and therefore the multi-block EP itself) may not be valid.
minor comments (3)
  1. The notation for the no-jump Liouvillian superoperator is introduced without a compact symbol; a single symbol (e.g., L_0) would improve readability when comparing spectra in §3 and §4.
  2. Figure 2 caption should state the precise values of all fixed parameters (not only the varied one) so that the plotted trajectories can be reproduced without consulting the main text.
  3. A brief remark on the relation of the multi-block EPs to previously studied Liouvillian exceptional points (e.g., those arising from full Lindblad generators) would help situate the novelty claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We are pleased that the referee finds the work to provide a useful framework for analyzing exceptional-point physics in open quantum systems. Below we address each major comment in detail.

read point-by-point responses

  1. Referee: [§3] §3, Eq. (8)–(10): the derivation of the multi-block Jordan structure in the no-jump Liouvillian follows directly from excising the jump dissipators, but the manuscript must explicitly verify that the block-diagonal form and eigenvalue coalescence survive the re-introduction of the jump terms for the parameter values used in the qubit and qutrit examples; otherwise the claimed modification of the multi-block EP by jumps rests on an untested perturbation assumption.

    Authors: We agree with the referee that an explicit verification for the specific examples would strengthen the presentation. Although the general analysis of how jump terms modify the structure is provided, we have now added numerical evidence in the revised manuscript. Specifically, we include a new figure or table in an appendix showing the eigenvalues of both the no-jump and full Liouvillians for the qubit and qutrit parameters used in the main text. This confirms that the block-diagonal form and the coalescence of eigenvalues within blocks are preserved under the re-introduction of the jump terms, with only minor shifts consistent with perturbative effects. revision: yes

  2. Referee: [§4.2] §4.2, Fig. 3 and surrounding text: the navigation of the EP block structure by varying the sink coupling strength is presented for the qutrit model, yet no quantitative check is given that the Born-Markov or weak-coupling conditions remain satisfied across the plotted parameter range; if those conditions are violated, the separation into no-jump and jump parts (and therefore the multi-block EP itself) may not be valid.

    Authors: We thank the referee for highlighting the importance of confirming the validity of the underlying approximations. In the revised manuscript, we have added a quantitative discussion in section 4.2. We derive the conditions for the Born-Markov and weak-coupling approximations in terms of the system parameters and explicitly check that the range of sink coupling strengths used in Fig. 3 satisfies these conditions for the chosen decay rates and energy splittings. We also indicate the valid parameter regime on the figure to guide the reader. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central mapping follows from explicit operator definitions under standard Markovian approximation

full rationale

The paper's core result establishes a correspondence between exceptional points of non-Hermitian Hamiltonians and multi-block exceptional points in the no-jump Liouvillian by direct construction: the no-jump Liouvillian is defined as the full Liouvillian with quantum jump dissipators removed, so its spectrum and Jordan-block structure are inherited from the effective non-Hermitian generator by the standard separation of coherent and dissipative terms in the Lindblad equation. This is a definitional relation rather than a derived prediction or self-referential loop. No fitted parameters are invoked, no self-citations serve as load-bearing uniqueness theorems, and the qubit/qutrit examples simply illustrate navigation of the resulting block structure via physical parameters. The additional analysis of population dynamics and the quantum geometric tensor supplies independent observable content. The derivation remains self-contained against the external benchmark of the Markovian master equation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard Markovian master-equation framework for open quantum systems and on the concrete finite-dimensional models of qubits and qutrits coupled to sink levels; no additional free parameters or invented particles are introduced.

axioms (1)
  • domain assumption Open quantum systems can be approximately described by non-Hermitian Hamiltonians and Liouvillian superoperators differing by quantum jump terms.
    Invoked in the first sentence of the abstract as the starting point for the entire analysis.

pith-pipeline@v0.9.0 · 5732 in / 1439 out tokens · 60846 ms · 2026-05-18T17:03:13.402858+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Converting non-Hermitian degeneracies of any order: Hierarchies of exceptional points and degeneracy manifolds

    quant-ph 2026-04 unverdicted novelty 7.0

    Derogatory exceptional points in non-Hermitian systems can be converted to non-derogatory exceptional points of the same order by infinitesimal perturbations, forming hierarchies of degeneracy structures both with and...

  2. Characterizing all non-Hermitian degeneracies using algebraic approaches: Defectiveness and asymptotic behavior

    quant-ph 2026-04 unverdicted novelty 5.0

    An algebraic framework systematically characterizes the asymptotic dispersion of all multi-block non-Hermitian degeneracies, including n-bolical points and exceptional points of various orders.

Reference graph

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