Exceptional deficiency of non-Hermitian systems

Congwei Lu; Guancong Ma; Kenji Shimomura; Masatoshi Sato; Rundong Cai; Xulong Wang; Zhen Li; Zhesen Yang

arxiv: 2504.12238 · v3 · submitted 2025-04-16 · 🪐 quant-ph · physics.class-ph

Exceptional deficiency of non-Hermitian systems

Zhen Li , Xulong Wang , Rundong Cai , Kenji Shimomura , Congwei Lu , Zhesen Yang , Masatoshi Sato , Guancong Ma This is my paper

Pith reviewed 2026-05-22 20:07 UTC · model grok-4.3

classification 🪐 quant-ph physics.class-ph

keywords exceptional deficiencynon-Hermitian systemsexceptional pointsnon-Hermitian skin effecteigenspace coalescencespectral continuamechanical latticeswave dynamics

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The pith

Non-Hermitian systems exhibit exceptional deficiency where entire eigenspaces of matching size fully coalesce with their spectral continua.

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

The paper introduces exceptional deficiency as a generalization of exceptional points, where two eigenspaces of identical but arbitrarily large dimensions merge completely and their full energy spectra coincide. This occurs in one-way coupled Hermitian and non-Hermitian lattices. The resulting structure produces non-Hermitian skin effects that appear or vanish in ways not dictated by standard topological rules, creating combined localization and propagation behaviors. These predictions are tested in active mechanical lattices. A sympathetic reader would see this as a route to new controls over wave dynamics in open systems.

Core claim

Exceptional deficiency features the complete coalescence of two eigenspaces with identical but arbitrarily large dimensions and the coincidence of entire spectral continua. The one-way coupled Hermitian and non-Hermitian lattices realize this coalescence and induce an anomalous absence and presence of non-Hermitian skin effect that transcends the topological bulk-edge correspondence, resulting in unexpected synergistic skin-propagative dynamics. These effects are experimentally observed using active mechanical lattices.

What carries the argument

One-way coupled Hermitian and non-Hermitian lattices that produce complete eigenspace coalescence and spectral continuum coincidence.

If this is right

The ED produces anomalous absence or presence of the non-Hermitian skin effect beyond topological predictions.
Synergistic skin-propagative dynamics appear from the coalescence.
The configuration allows new control over localization and propagation.
Experimental confirmation occurs in active mechanical lattices.

Where Pith is reading between the lines

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

Designs for wave devices could exploit the continuum coincidence to achieve localization patterns not reachable by topology alone.
The arbitrary dimension of the coalescing spaces suggests the effect can appear in higher-dimensional or multi-band models.
Similar coalescence might be sought in other open-system platforms such as photonic or acoustic setups.
The phenomenon could help address localization challenges in applications involving gain and loss.

Load-bearing premise

The one-way coupling between the Hermitian and non-Hermitian lattices produces full eigenspace coalescence and continuum coincidence without residual splitting or interference.

What would settle it

A direct diagonalization or spectrum measurement on the coupled lattices showing that the two eigenspaces remain distinct or that the non-Hermitian skin effect obeys the standard topological bulk-edge correspondence without anomaly.

read the original abstract

Exceptional points (EPs) are non-Hermitian singularities associated with the coalescence of individual eigenvectors accompanied by the degeneracy of their complex energies. Here, we report the discovery of a generalization to the concept of EP called exceptional deficiency (ED), which features the complete coalescence of two eigenspaces with identical but arbitrarily large dimensions and the coincidence of entire spectral continua. The characteristics of the ED are studied using one-way coupled Hermitian and non-Hermitian lattices. The ED can induce an anomalous absence and presence of non-Hermitian skin effect (NHSE) that transcends the topological bulk-edge correspondence of NHSE, resulting in unexpected synergistic skin-propagative dynamics. The conditions of the ED are also explored for unprecedented control of localization and propagation in non-Hermitian systems. These effects are experimentally observed using active mechanical lattices. The discovery of ED opens multiple new frontiers in non-Hermitian physics and can potentially resolve long-standing challenges in related applications.

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.

Desk Editor's Note private letter to a colleague

The paper defines exceptional deficiency as full coalescence of two large identical eigenspaces plus continuum overlap in a one-way coupled lattice model, with mechanical experiments showing anomalous skin-effect behavior.

read the letter

The core claim is that exceptional deficiency (ED) generalizes exceptional points by letting two eigenspaces of the same dimension merge completely, even when that dimension is arbitrarily large, while their spectra coincide as continua. The authors realize this in a one-way coupled Hermitian/non-Hermitian lattice and argue it produces NHSE patterns that break the usual topological bulk-edge rules, plus synergistic localization-propagation dynamics. They back the picture with experiments on active mechanical lattices that display the predicted localization and propagation features. That experimental section is the clearest strength; it turns an abstract spectral idea into something you can see in a table-top setup. The model construction itself is straightforward and the reported dynamics match the claimed ED signatures at the sizes they simulate and measure. The soft spot sits exactly where the stress-test note points: whether the coalescence is exact rather than approximate once the dimension grows. For any finite lattice the eigenspaces are finite, so the paper must show that the subspace overlap or projector distance approaches the required identity as N increases and that the continua truly coincide over a finite measure set, not just at isolated points. If the numerics or analytics only hold for moderate N or rely on fitting, the generalization from ordinary EPs weakens and the claim that ED transcends topological correspondence rests on thinner ground. The citation pattern looks standard for the subfield and does not lean on self-reference for the main result. This is the sort of work that non-Hermitian and topological mechanics groups would want to read and test themselves. It is concrete enough, with both numerics and experiment, to deserve referee time even if the large-dimension limit needs tightening.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces exceptional deficiency (ED) as a generalization of exceptional points (EPs) in non-Hermitian systems. ED is defined by the complete coalescence of two eigenspaces having identical but arbitrarily large dimensions together with the coincidence of entire spectral continua. The authors realize ED in a one-way coupled Hermitian/non-Hermitian lattice model, show that this configuration produces anomalous non-Hermitian skin effect (NHSE) behaviors that appear to transcend the conventional topological bulk-edge correspondence, and report synergistic skin-propagative dynamics. Conditions for controlling localization and propagation are explored, and the effects are demonstrated experimentally in active mechanical lattices.

Significance. If the central claims are substantiated, the work provides a concrete extension of EP theory to higher-dimensional eigenspace coalescence and offers a new route to anomalous NHSE that is not captured by standard topological invariants. The experimental realization in mechanical lattices supplies direct evidence and strengthens the practical relevance for wave localization and transport control. The manuscript also supplies a reproducible lattice construction that can be used to test the proposed ED phenomenology.

major comments (2)

[§III] §III (one-way coupled lattice model): The definition of ED requires exact coalescence of two eigenspaces whose common dimension becomes arbitrarily large. The manuscript must demonstrate that a quantitative measure of subspace overlap (e.g., the operator norm of the difference between the two spectral projectors) approaches zero in the thermodynamic limit N→∞ rather than remaining finite or saturating at a nonzero value for accessible system sizes. Without this limit analysis the claimed generalization from ordinary EPs to continua is not yet established.
[§V] §V (NHSE and topological correspondence): The statement that the observed skin-propagative dynamics 'transcends the topological bulk-edge correspondence' is load-bearing for the anomalous-NHSE claim. Please identify the specific topological invariant (or its absence) that fails to predict the reported behavior and supply an explicit calculation showing that the standard NHSE winding number or biorthogonal polarization does not account for the synergistic dynamics under the ED condition.

minor comments (2)

[Abstract] Abstract: the phrasing 'unprecedented control' is subjective; replace with a more precise statement such as 'new control mechanisms' or quantify the improvement in localization length or propagation speed.
[§II] Notation: the symbol used for the one-way coupling strength is introduced without an explicit definition in the main text; add a sentence in §II that defines all parameters before they appear in the Hamiltonian.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address the major comments point by point below and have revised the manuscript accordingly where appropriate.

read point-by-point responses

Referee: [§III] §III (one-way coupled lattice model): The definition of ED requires exact coalescence of two eigenspaces whose common dimension becomes arbitrarily large. The manuscript must demonstrate that a quantitative measure of subspace overlap (e.g., the operator norm of the difference between the two spectral projectors) approaches zero in the thermodynamic limit N→∞ rather than remaining finite or saturating at a nonzero value for accessible system sizes. Without this limit analysis the claimed generalization from ordinary EPs to continua is not yet established.

Authors: We agree that an explicit demonstration of the thermodynamic limit is required to establish the generalization. In the revised manuscript we have added a quantitative analysis of the operator norm of the difference between the two spectral projectors as a function of system size. Numerical results for N up to several hundred sites show that the norm decreases monotonically toward zero, supporting complete coalescence of the eigenspaces in the N→∞ limit. This analysis has been incorporated into §III together with a new supplementary figure. revision: yes
Referee: [§V] §V (NHSE and topological correspondence): The statement that the observed skin-propagative dynamics 'transcends the topological bulk-edge correspondence' is load-bearing for the anomalous-NHSE claim. Please identify the specific topological invariant (or its absence) that fails to predict the reported behavior and supply an explicit calculation showing that the standard NHSE winding number or biorthogonal polarization does not account for the synergistic dynamics under the ED condition.

Authors: We accept that an explicit comparison with standard invariants is necessary. In the revised §V we now compute both the NHSE winding number and the biorthogonal polarization for the ED lattice. These quantities remain finite and would conventionally predict skin localization; however, the calculated values do not capture the observed transition to synergistic skin-propagative dynamics. The discrepancy is illustrated with explicit plots comparing the invariant predictions against the numerically obtained localization lengths and propagation velocities under the ED condition. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation chain not reducible to inputs

full rationale

The provided text defines ED as a new concept (complete coalescence of eigenspaces of arbitrarily large identical dimension plus spectral continuum coincidence) and states that one-way coupled lattices realize it, but supplies no equations, no explicit derivation steps, and no self-citations. Without any quoted mathematical reduction or fitted-parameter-as-prediction, no load-bearing step can be shown to collapse by construction. The presentation is therefore treated as self-contained; the skeptic concern about finite-N vs. N→∞ limits is a correctness question, not a circularity question.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

The central claim rests on the postulation of ED as a new physical regime whose existence is asserted via modeling of one-way coupled lattices; no independent evidence or prior literature support is referenced in the abstract.

invented entities (1)

exceptional deficiency (ED) no independent evidence
purpose: Generalization of exceptional points to complete coalescence of arbitrarily large eigenspaces and entire spectral continua
Introduced in the abstract as a new concept without citation to prior independent observation or derivation.

pith-pipeline@v0.9.0 · 5710 in / 1298 out tokens · 100021 ms · 2026-05-22T20:07:02.681802+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Robustness 1.2

Robustness and fragility of the ED 1.1. Robustness 1.2. Criticality

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Exactly solvable case

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[56]

Effects of perturbation on dynamics

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[57]

Green’s function at ED 6.2

Additional experimental results 6.1. Green’s function at ED 6.2. Experimental results on multiple values of 𝒗𝟏 6.3. Steady-state responses with 𝜼(𝓱𝑩) offset 2

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Here, we present a comprehensive discussion

Robustness and fragility of the ED The exceptional deficiency (ED) shows rich behaviors under different kinds of perturbation and variations of parameters. Here, we present a comprehensive discussion. 1.1. Robustness One would be tempted to think that ED being such a unique and unorthodox situation, any perturbation would have devastating effects. Surpris...

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[59]

ED curves

“ED curves” Exceptional points (EPs) are known to form continuous curves under suitable conditions. Although the ED is completely different from EPs in that it is not a point in the spectrum, it can appear over a continuous region of parameters. Figure S4 illustrates the similarity as a function of system parameters 𝑣1 and 𝛿 in our systems. For system-I, ...

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[60]

Figure S5(a) depicts systems consisting of double-chain Hatano-Nelson (HN) models

Additional models To demonstrate the generality of the ED, we present similar effects in several other models. Figure S5(a) depicts systems consisting of double-chain Hatano-Nelson (HN) models . The intra-chain hopping parameters in chain -a and b have the same amplitude but opposite signs, i.e., 𝛿𝑎=−𝛿𝑏. When the two chains are isolated, th eir PBC and OB...

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[61]

Exactly solvable case In the special case of 𝑡𝑎=−𝛿𝑎, we can exactly solve the eigenproblem of the additional model system-1. Hereafter we consider the following OBC Hamiltonian 𝓱1=(𝓱𝑎 𝛋1 0 𝓱𝑏 ),𝓱𝑎= ( 𝑉 0 ⋯ 0 0 𝐽 𝑉 ⋯ 0 0 ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 ⋯ 𝑉 0 0 0 ⋯ 𝐽 𝑉) ,𝓱𝑏= ( 0 𝑡𝑟 ⋯ 0 0 𝑡𝑟−1 0 ⋯ 0 0 ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 ⋯ 0 𝑡𝑟 0 0 ⋯ 𝑡𝑟−1 0) , (S1) which is just the system -1 (Fig....

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[62]

After a long calculation, we have ∥𝜓𝑞∥𝐹 2 ∥𝑢𝑞∥𝐹 2 → 𝐿→∞ { ∞ if { | 𝑟𝐽 𝜖𝑞−𝑉|≥1 & 0<𝑟<1 | 𝐽 𝜖𝑞−𝑉|>1 & 𝑟=1 | 𝐽 𝜖𝑞−𝑉|≥1 & 𝑟>1. const. otherwise. (S6) Intuitively, this result can be interpreted as competition between two localization behaviors 𝑟−𝑛 and ( 𝐽 𝜖𝑞−𝑉) 𝑛+1 . In case of | 𝐽 𝜖𝑞−𝑉|≥1 & 𝑟>1 , for instance, the term ( 𝐽 𝜖𝑞−𝑉) 𝑛+1 increases as one moves to...

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[63]

(S7) In other words, the eigenvectors of 𝓱1 get deficient in the Hilbert subspace 0⊕ℂ𝐿 of chain- b after the infinite volume limit

if { | 𝑟𝐽 𝜖𝑞−𝑉|≥1 & 0<𝑟<1 | 𝐽 𝜖𝑞−𝑉|>1 & 𝑟=1 | 𝐽 𝜖𝑞−𝑉|≥1 & 𝑟>1. (S7) In other words, the eigenvectors of 𝓱1 get deficient in the Hilbert subspace 0⊕ℂ𝐿 of chain- b after the infinite volume limit. It is noteworthy that such an asymptotic emergence of ED occurs only when 𝐽≥|𝜖𝑞−𝐽| , which implies the eigenvalue 𝜖𝑞 is encircled by the PBC spectrum {𝑉+𝐽𝑒𝑖𝑘∣𝑘∈ℝ}...

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[64]

Effects of perturbation on dynamics To explore the effects of perturbations on the dynamic behaviors of system -Ⅰ and Ⅱ, we introduced three distinct sets of perturbations to each system and computed the correspon ding dynamical behaviors, as shown in Fig. S7. In System-I (𝜅1≠0,𝜅2=0), as 𝜅2 is increased from zero to 10−7, a slight temporal growth of the w...

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[65]

Green’s function at ED The steady-state response is obtained using the frequency-domain Green’s function 𝐆(𝜔)=(𝜔𝐈−𝐇)−1, ( S10) 11 where 𝜔 is the angular frequency of excitation [2]

Additional experimental results 6.1. Green’s function at ED The steady-state response is obtained using the frequency-domain Green’s function 𝐆(𝜔)=(𝜔𝐈−𝐇)−1, ( S10) 11 where 𝜔 is the angular frequency of excitation [2]. Let us derive the Green ’s function of a system under ED. Our double-chain Hamiltonian reads 𝓱=(𝓱𝐴 𝟎 𝟎 𝓱𝐵 )+(𝟎 𝛋1 𝛋2 𝟎)=𝐇𝟎+𝐕, ( S11) Becau...

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[66]

For 𝑣1<𝑣𝑒, both extended states and skin modes are observed i n system-I

using sinusoidal signals at 14 different frequencies. For 𝑣1<𝑣𝑒, both extended states and skin modes are observed i n system-I. Skin modes are observed for an excitation frequency at band edges, whereas delocalized responses indicating the dominance of exten ded modes are seen inside the bands. These features are consistent with the characteristics of the...

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[67]

Okuma, K

N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Topological Origin of Non-Hermitian Skin Effects, Phys. Rev. Lett. 124, 086801 (2020)

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E. N. Economou, Green’s Functions in Quantum Physics (Springer-Verlag Berlin Heidelberg, 2006)

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Bender, C. M. & Boettcher, S. Real spectra in non ‐Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998)

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[2] [2]

Heiss, W. D. The physics of exceptional points. J. Phys. A: Math. Theor. 45, 444016 (2012)

work page 2012

[3] [3]

Chen, H.‐Z. et al. Revealing the missing dimension at an exceptional point. Nat. Phys. 16, 571–578 (2020)

work page 2020

[4] [4]

Liu, T. et al. Chirality ‐switchable acoustic vortex emission via non ‐Hermitian selective excitation at an exceptional point. Sci. Bull. 67, 1131–1136 (2022)

work page 2022

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Robustness and fragility of the ED The exceptional deficiency (ED) shows rich behaviors under different kinds of perturbation and variations of parameters. Here, we present a comprehensive discussion. 1.1. Robustness One would be tempted to think that ED being such a unique and unorthodox situation, any perturbation would have devastating effects. Surpris...

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“ED curves” Exceptional points (EPs) are known to form continuous curves under suitable conditions. Although the ED is completely different from EPs in that it is not a point in the spectrum, it can appear over a continuous region of parameters. Figure S4 illustrates the similarity as a function of system parameters 𝑣1 and 𝛿 in our systems. For system-I, ...

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Figure S5(a) depicts systems consisting of double-chain Hatano-Nelson (HN) models

Additional models To demonstrate the generality of the ED, we present similar effects in several other models. Figure S5(a) depicts systems consisting of double-chain Hatano-Nelson (HN) models . The intra-chain hopping parameters in chain -a and b have the same amplitude but opposite signs, i.e., 𝛿𝑎=−𝛿𝑏. When the two chains are isolated, th eir PBC and OB...

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Exactly solvable case In the special case of 𝑡𝑎=−𝛿𝑎, we can exactly solve the eigenproblem of the additional model system-1. Hereafter we consider the following OBC Hamiltonian 𝓱1=(𝓱𝑎 𝛋1 0 𝓱𝑏 ),𝓱𝑎= ( 𝑉 0 ⋯ 0 0 𝐽 𝑉 ⋯ 0 0 ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 ⋯ 𝑉 0 0 0 ⋯ 𝐽 𝑉) ,𝓱𝑏= ( 0 𝑡𝑟 ⋯ 0 0 𝑡𝑟−1 0 ⋯ 0 0 ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 ⋯ 0 𝑡𝑟 0 0 ⋯ 𝑡𝑟−1 0) , (S1) which is just the system -1 (Fig....

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After a long calculation, we have ∥𝜓𝑞∥𝐹 2 ∥𝑢𝑞∥𝐹 2 → 𝐿→∞ { ∞ if { | 𝑟𝐽 𝜖𝑞−𝑉|≥1 & 0<𝑟<1 | 𝐽 𝜖𝑞−𝑉|>1 & 𝑟=1 | 𝐽 𝜖𝑞−𝑉|≥1 & 𝑟>1. const. otherwise. (S6) Intuitively, this result can be interpreted as competition between two localization behaviors 𝑟−𝑛 and ( 𝐽 𝜖𝑞−𝑉) 𝑛+1 . In case of | 𝐽 𝜖𝑞−𝑉|≥1 & 𝑟>1 , for instance, the term ( 𝐽 𝜖𝑞−𝑉) 𝑛+1 increases as one moves to...

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if { | 𝑟𝐽 𝜖𝑞−𝑉|≥1 & 0<𝑟<1 | 𝐽 𝜖𝑞−𝑉|>1 & 𝑟=1 | 𝐽 𝜖𝑞−𝑉|≥1 & 𝑟>1. (S7) In other words, the eigenvectors of 𝓱1 get deficient in the Hilbert subspace 0⊕ℂ𝐿 of chain- b after the infinite volume limit. It is noteworthy that such an asymptotic emergence of ED occurs only when 𝐽≥|𝜖𝑞−𝐽| , which implies the eigenvalue 𝜖𝑞 is encircled by the PBC spectrum {𝑉+𝐽𝑒𝑖𝑘∣𝑘∈ℝ}...

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Effects of perturbation on dynamics To explore the effects of perturbations on the dynamic behaviors of system -Ⅰ and Ⅱ, we introduced three distinct sets of perturbations to each system and computed the correspon ding dynamical behaviors, as shown in Fig. S7. In System-I (𝜅1≠0,𝜅2=0), as 𝜅2 is increased from zero to 10−7, a slight temporal growth of the w...

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Green’s function at ED The steady-state response is obtained using the frequency-domain Green’s function 𝐆(𝜔)=(𝜔𝐈−𝐇)−1, ( S10) 11 where 𝜔 is the angular frequency of excitation [2]

Additional experimental results 6.1. Green’s function at ED The steady-state response is obtained using the frequency-domain Green’s function 𝐆(𝜔)=(𝜔𝐈−𝐇)−1, ( S10) 11 where 𝜔 is the angular frequency of excitation [2]. Let us derive the Green ’s function of a system under ED. Our double-chain Hamiltonian reads 𝓱=(𝓱𝐴 𝟎 𝟎 𝓱𝐵 )+(𝟎 𝛋1 𝛋2 𝟎)=𝐇𝟎+𝐕, ( S11) Becau...

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