Joint control of coherent transmission, reflection, and absorption

Cheng Guo; Dongha Kim; Shanhui Fan; Shiyu Li

Joint control of coherent transmission, reflection, and absorption

T0 review · 0 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read A passive linear system's achievable transmission, reflection, and absorption triples are exactly the numerical range of a single composite matrix, and the paper provides an algorithm to realize any target within that set.

desk verdict A clean, useful mapping of joint coherent control onto numerical-range theory; the central theorem is definitionally immediate, but the paper packages it with enough imported machinery and honest numerics to deserve a serious referee. read the letter →

arxiv 2511.04788 v1 pith:KFPWMVHM submitted 2025-11-06 physics.optics cond-mat.mes-hall

Shiyu Li , Dongha Kim , Shanhui Fan , Cheng Guo This is my paper

classification physics.opticscond-mat.mes-hall

keywords coherentcontroltransmissionreflectionabsorptionnumericalrangewavefrontshapingscatteringmatrixnon-abelian

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves a complete answer to joint coherent control of wave transmission, reflection, and absorption in passive linear time-invariant systems. For any such system characterized by field transmission and reflection matrices t and r, the set of all attainable (τ, ρ, α) tuples is exactly the numerical range of the non-Hermitian matrix T+iR, where T=t†t and R=r†r (and analogously with T+iA or R+iA). This means the achievable trade-off surface is a convex set determined by a single well-studied matrix object. The paper further shows that when T, R, and A commute (abelian case) the set is just the convex hull of eigenvalue points, while non-commutativity extends it; it provides inner and outer bounds and a constructive algorithm to find the input wavefront for any target. If correct, this unifies previous single-property control results and opens joint control to systematic design.

What carries the argument

The numerical range W(M) = {a†Ma : ||a||=1} of the composite non-Hermitian matrix T+iR, where T=t†t, R=r†r, and A=I−T−R are the power matrices. The numerical range carries the argument: it is convex, compact, has eigenvalues as inner points, and its boundary can be computed by Johnson's algorithm; it also has known bounds via eigenvalue convex hulls and departure from normality. The non-commutativity of T and R is quantified by dep(T+iR), the departure from normality, which controls the gap between the inner bound (eigenvalue convex hull) and the true Ω.

What would settle it

Build a passive linear system with known t and r (e.g., a multimode waveguide with controlled scatterers), compute W(T+iR) via a boundary algorithm, then use an inverse numerical-range solver to generate inputs for several boundary points; if the measured (τ,ρ,α) ever falls outside W(T+iR) or fails to approach the boundary, the central theorem is wrong.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1: Ω = {(τ,ρ,1−τ−ρ) ∈ R³ | τ+iρ ∈ W(T+iR)}, with analogous expressions using T+iA and R+iA, where W(M) is the numerical range {z = a†Ma : a†a=1}. This is proved by noting that τ+iρ = a†(T+iR)a and energy conservation fixes α = 1−τ−ρ, and conversely any point in W(T+iR) gives a unit vector a. The same reasoning holds for the other two projections. Consequently, all achievable responses form a compact convex subset of the triangle τ+ρ+α=1, and its projections onto coordinate planes are numerical ranges. The paper also gives an inverse algorithm (based on inverse numerical range methods) to construct a for a specified (τ₀,ρ₀,α₀), demonstrated on a disordered multimo

Load-bearing premise

The paper assumes the field transmission and reflection matrices t and r fully capture all output channels, so that A = I − t†t − r†r is the complete absorption matrix; if a real system has unmodeled loss channels (e.g., leaky radiation not accounted for), the predicted attainable set will not match physical reality.

Editorial extensions

If this is right

For any passive system with known t and r, the full achievable trade-off surface for transmission, reflection, and absorption is computable from the numerical range of T+iR.
A single input wavefront can realize any point inside the achievable set; the inverse numerical range algorithm constructs it.
In abelian systems (T, R, A commute) Ω is a polygon equal to the convex hull of eigenvalue points; for non-abelian systems the set is larger, with the excess controlled by the departure from normality.
The shape of Ω is classified for n=2 (elliptic disk, line, point) and n=3 (seven shapes); the classification for n≥4 is open.
The theory applies to all wave types and can be extended to joint control of other physical quantities.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The result suggests that coherent perfect absorption (α=1) is achievable iff the corresponding point lies in W(T+iR), linking the numerical range to perfect-absorber existence and potentially unifying prior CPA conditions in terms of power matrices.
The framework could be tested experimentally in multimode fibers or scattering samples by measuring t and r, computing Ω, and then comparing random wavefront sampling against the predicted boundary.
Joint control of other pairs (e.g., phase or polarization alongside power) may be governed by numerical ranges of other composite matrices, possibly revealing similar non-abelian structure.
The open n≥4 shape classification might be resolved using recent matrix theory on numerical ranges of higher-order matrices; if the achievable set becomes increasingly polygon-like, design rules for multi-port systems could simplify.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper's core claim is true and easy to verify: for a passive linear time-invariant system, the set of attainable (τ, ρ, α) tuples is exactly the numerical range of T+iR (and the two analogous composites). The proof in Appendix B is a direct substitution into the definition of the numerical range, so there is no hidden step. That is also the paper's main weakness as a research contribution: Theorem 1 is essentially an observation. The novelty lives in recognizing that this object is the right one and in importing the rich, standard theory of numerical ranges—convexity, eigenvalue-hull bounds, polygon classification, inverse-range algorithms—to give a practical framework for wave control. That importation is done cleanly and with the right citations.

What the paper does well: it gives both inner and outer bounds, explains the abelian/non-abelian distinction via departure from normality, provides a constructive algorithm for reaching any target inside the achievable set, and backs it up with FDTD simulations where scatter plots over tens of thousands of random inputs match the predicted boundaries. No fitting, no post-hoc selection. The matrices in the appendices are explicit enough to reproduce the figures. The paper is honest about which parts are new: the numerical-range theorems are cited to Horn and Johnson, Henrici, and others.

The soft spots are proportionate. The physical interpretation of A = I − t†t − r†r as the absorption matrix requires that t and r capture every non-absorbed output port. In the open-waveguide example, leaky radiation is declared to be absorbed by an unshown cladding, so the reported Ω is technically the set of (τ, ρ, lost-power) tuples with lost power equal to absorption plus any radiation leaving the guided-mode basis. That is a modeling caveat, not an internal inconsistency; the theorem is conditional on t and r being complete. The description of the FDTD geometry is a little thin, but the matrices are given, so the numerics are reproducible in spirit. A second, minor caveat is that the “non-abelian effects” section is largely a repackaging of Henrici's bounds with the paper's physical interpretation; the mathematics there is not new, though the connection is useful.

Overall, this is a solid theory paper. It will be most valuable to people working on wavefront shaping, multimode fibers, and coherent perfect absorption who want a single geometric picture for simultaneous control. It deserves peer review. I would support sending it out with a request to clarify the port-completeness assumption in the applications section.

Would I cite it? Yes, if I were writing on coherent control. It is not groundbreaking, but it is a clean, correct framework that connects a well-studied mathematical object to a practical problem. I would bring it to a reading group as a good example of borrowing matrix-analysis results into optics. Serious thinker: yes—the paper is coherent on its own terms and engages the literature honestly.

Referee Report

0 major / 5 minor

Summary. The paper considers a passive (l+m)-port linear time-invariant system and an n-dimensional input subspace. For fixed field transmission and reflection matrices t and r, it defines the power matrices T=t†t, R=r†r, and A=I−T−R, and studies the set Ω of attainable triples (τ,ρ,α)=(†aTa, a†Ra, a†Aa) over unit input vectors a. Theorem 1 identifies Ω with the image of the numerical range W(T+iR) (and equivalently W(T+iA), W(R+iA)) under the affine map (τ,ρ)→(τ,ρ,1−τ−ρ). The paper gives inner and outer bounds using eigenvalues and Henrici's departure-from-normality bound, discusses abelian versus non-abelian cases, classifies possible shapes for n=2 and n=3, and solves the inverse numerical-range problem with a concrete disordered-waveguide example.

Significance. The mathematical core is correct: Theorem 1 follows directly from the definitions, and the applications of numerical-range theory (convexity, the elliptical-disk theorem for n=2, the Keeler–Rodman–Spitkovsky classification, Henrici bounds, and inverse-range algorithms) are standard and properly cited. The paper's contribution is the physical mapping: joint control of power transmittance, reflectance, and absorptance is reduced to control of the quadratic form of T+iR. The numerical examples are reproducible (full matrices are given in appendices), the scatter plots match the predicted boundaries with no fitting, and the inverse algorithm is demonstrated with an explicit target. This is a useful unifying framework for wavefront shaping. It is not a deep new theorem in matrix analysis, and the central equality is close to a definition, but for the optics audience this reformulation is significant and likely to be cited.

minor comments (5)

[Appendix D3, Eq. (D15)] The matrix T listed for Fig. 4(c) is not Hermitian: T12=0.14+0.06i while T21=0.14−0.05i, which is not its conjugate. Since T must be t†t as in Eq. (7), this matrix cannot arise from a physical t. Please correct the typo and regenerate the panel; the claimed flat-boundary shape and the seven-shape classification discussion may depend on this example.
[Waveguide example, Fig. 1(d)] The main text states that leaky radiation is 'assumed to be absorbed by an absorbing cladding outside the silica not shown'. This is an unmodeled output channel. Theorem 1 is conditional on t and r containing all non-absorbed output channels; otherwise α in Eq. (7) is not material absorption but total 'lost' power (absorption plus leakage). Please add an explicit completeness assumption to the statement of Theorem 1, or at least a sentence clarifying that α should be read as 'loss' if additional channels exist.
[Typos and notation] There are several minor typographical issues: 'non-abeliean' in the non-abelian-effects section; 'equialently' in Appendix G3; in Eq. (23) the notation should be σ_j^2 and |λ_j|^2 rather than the compressed forms. Also, 'Fig. (5)' should be 'Fig. 5'.
[Figure 5] The three regions are described as 'gray', 'pink', and 'brown'. If the journal does not use color, these shades may be difficult to distinguish. Consider adding hatching, labels, or different line styles to the boundary sets.
[Appendix G3] The criterion for n≥5 (normal M1 plus direct-sum condition with W(M2)⊆W(M1)) is stated in one sentence. Since this is a non-obvious use of Horn–Johnson Corollary 1.6.9, a brief explanation or a restatement of the relevant corollary would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 is a definitional reformulation, and the substantive content is imported from external numerical-range theory.

full rationale

The central identity, Theorem 1 (Eq. 11), is an immediate consequence of the definitions in Eqs. (2)-(7) and (14): for any unit a, tau + i rho = a^dag(T+iR)a, and alpha = 1 - tau - rho by definition of A = I - t^dag t - r^dag r. Thus the attainable set Omega is f(W(T+iR)) by construction. This is a reformulation, not a fitted prediction, and the paper does not use it as an empirical test in a way that hides an input in the output: the FDTD examples compute the scatter and the boundary from the same t,r matrices, and no parameter is fitted to the claimed attainable set. The independent content of the paper lies in applying external numerical-range machinery: convexity and compactness from Horn-Johnson, the eigenvalue convex-hull inner bound, Henrici's departure-from-normality outer bound, the n=3 shape classification of Keeler-Rodman-Spitkovsky, Johnson's boundary algorithm, and inverse-range algorithms of Uhlig/Carden/Bebiano et al. These results are standard, external, and not borrowed from the authors' own prior work. The self-citations ([68,79,91,99]) are contextual and not load-bearing in the proof. The only caveat is physical completeness: Eq. (7) counts all non-guided output as absorption, and the leaky-radiation cladding assumption is explicitly stated in the waveguide example; this affects applicability, not deductive circularity. No derivation step reduces to its own conclusion, so the paper has no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no free parameters fitted to data and postulates no new physical entities. Its load-bearing assumptions are standard numerical-range theorems and the physical modeling assumption that all output channels are captured by t and r. The theory itself is parameter-free; the numerical examples use chosen material parameters as inputs, not as fitted values.

assumptions (6)

standard math The numerical range of a matrix is convex and compact, and its eigenvalues lie in it.
Used in Appendix E to prove compactness and convexity of Ω and to establish the inner bound from eigenvalues.
standard math A matrix is normal iff its numerical range equals the convex hull of its eigenvalues; a numerical range is a polygon iff it equals the convex hull of the eigenvalues.
Used in Appendix G for the abelian case and for the polygon criterion Ωin = Ω.
standard math Henrici's bound: W(M) ⊆ conv(λ(M)) + disk of radius sqrt((1−1/n)/2) dep(M).
Used for the non-abelian outer bound in Eq. (24).
standard math Keeler–Rodman–Spitkovsky classification of numerical ranges of 3×3 matrices.
Used to enumerate the seven possible shapes of Ω for n=3.
domain assumption The field transmission and reflection matrices t and r are block submatrices of the scattering matrix of a passive linear time-invariant system, with T+R+A=I and A positive semidefinite.
Foundational physical premise stated in Eqs. (3), (7), and (8); required for the definition of Ω and energy conservation.
domain assumption In the waveguide example, leaky radiation is absorbed by an absorbing cladding, so all non-guided output is captured in A.
Stated near Fig. 1(d); needed for the numerical verification of Theorem 1.

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Cite this review

Pith. "Pith review of Joint control of coherent transmission, reflection, and absorption." pith.science (2026). https://pith.science/paper/KFPWMVHM

@misc{pith2026251104788,
  author       = {Pith},
  title        = {Pith review of: Joint control of coherent transmission, reflection, and absorption},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KFPWMVHM}},
  note         = {Machine review of arXiv:2511.04788}
}

read the original abstract

Controlling multiple wave properties simultaneously poses a key challenge in coherent control of wave transport. We present a theory for joint coherent control of transmission, reflection, and absorption in linear systems. We prove that the numerical range provides the mathematical structure governing achievable responses, and reveal non-abelian effects due to non-commutativity between transmission, reflection, and absorption matrices. We provide an algorithm to achieve arbitrary target responses. Our results establish a theoretical foundation for joint coherent control of waves.

Figures

Figures reproduced from arXiv: 2511.04788 by the authors.

**Figure 2.** FIG. 2. Attainable responses for a two-mode ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** FIG. 4. Possible shapes of Ω for [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: FIG. 5. Non-abelian bounds on the numerical range [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

**Figure 6.** Figure 6: FIG. 6. Structural details of the disordered waveguide. (a) [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

**Figure 7.** Figure 7: FIG. 7. The geometry for the dielectric slab example. The [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

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Reference graph

Works this paper leans on

126 extracted references · 1 linked inside Pith

[1]

What is the set of all attainable tuples: Ω :={(τ[a], ρ[a], α[a])∈R3 :a∈C n,a †a= 1}? (9)
[2]

We illustrate joint coherent control with a concrete ex- ample

How to find an input unit vectora 0 that realizes a given target (τ 0, ρ0, α0)∈Ω: τ[a 0] =τ 0, ρ[a 0] =ρ 0, α[a 0] =α 0? (10) This paper provides complete answers to both questions. We illustrate joint coherent control with a concrete ex- ample. Consider a silicon slab waveguide (refractive in- dexn i = 3.48) embedded in silica cladding (n 0 = 1.444) [Fig...
[3]

Matrices for Fig. 2 The two-mode disordered waveguide has transmission and reflection matrices: t= −0.24 + 0.06i0.15−0.14i 0.20−0.11i−0.02−0.58i ,(D1) r= −0.39−0.06i−0.05−0.20i −0.20 + 0.05i−0.13−0.25i .(D2) We calculateT,R, andAfromtandrusing Eq. (7): T= 0.12 0.01−0.09i 0.01 + 0.09i0.37 ,(D3) R= 0.20 0.05 + 0.13i 0.05−0.13i0.12 ,(D4) A= 0.69−0.06−0.04i −...
[4]

Matrices for Fig. 3 The three-mode disordered waveguide has transmis- sion and reflection matrices: t=   0.23 + 0.22i0.12−0.02i−0.05 + 0.03i 0.00−0.17i0.40 + 0.44i0.03−0.01i −0.07−0.13i−0.07−0.01i−0.43−0.07i   , (D6) r=   0.31 + 0.35i0.07−0.08i−0.01 + 0.25i −0.03−0.10i−0.37 + 0.01i0.03 + 0.07i 0.09−0.23i0.06−0.04i−0.05 + 0.04i   . (D7) We calculat...
[5]

4 Here we provide theTandRmatrices used to generate each panel in Fig

Matrices for Fig. 4 Here we provide theTandRmatrices used to generate each panel in Fig. 4. The correspondingAmatrices can be derived usingA=I−T−R. 7 (a) Triangular disk: T=   0.1 0 0 0 0.6 0 0 0 0.1   ,(D11) R=   0.7 0 0 0 0.3 0 0 0 0.1   .(D12) (b) Convex hull of an ellipse and a point: T=   0.2 0.15 + 0.05i0 0.15−0.05i0.6 0 0 0 0.1   ,(D13)...
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[102], p.8, Property 1.2.1.)

Compactness The numerical rangeW(T+iR) is compact (see Ref. [102], p.8, Property 1.2.1.). Sincefis continuous and the continuous image of a compact set is compact, Ω is compact
[7]

[102], p.8, Property 1.2.2.)

Convexity The numerical rangeW(T+iR) is convex (see Ref. [102], p.8, Property 1.2.2.). Since the image of a con- vex set under an affine function is convex (see Ref. [107], p.36.), Ω is convex
[8]

[102], p.12.)

Outer boundΩ⊆Ω out SinceTis Hermitian, the quadratic formτ=a †Ta takes all values satisfying λmin(T)≤τ≤λ max(T) (E2) asaranges over complex unit vectors (see Ref. [102], p.12.). The same holds forRandA, giving: λmin(R)≤ρ≤λ max(R),(E3) λmin(A)≤α≤λ max(A),(E4) and all the bounds are attainable. Therefore, Ω is in- scribed in the hexagon Ω out defined in Eq. (16)

Show all 126 references

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[102], p.10, Property 1.2.6.), givingτ ′ k+iρ′ k =λ k(T+ iR)∈W(T+iR)

Inner boundΩ in ⊆Ω The eigenvalues ofT+iRlie inW(T+iR) (see Ref. [102], p.10, Property 1.2.6.), givingτ ′ k+iρ′ k =λ k(T+ iR)∈W(T+iR). Therefore, (τ ′ k, ρ′ k,1−τ ′ k −ρ ′ k)∈Ω for allk= 1,2, . . . , n. Similarly, the points from Eqs. (18) and (19) belong to Ω. The convexity o...
[10]

G 2, Ω in ̸= Ω

Determining the system type fromΩ’s shape If Ω is not a polygon, then by the previous result in Sec. G 2, Ω in ̸= Ω. The contrapositive of the result in Sec. G 1 then implies that the system is non-abelian. If Ω is a polygon,W(T+iR) must be a polygon with C(T+iR) =W(T+iR). For...
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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

What is the set of all attainable tuples: Ω :={(τ[a], ρ[a], α[a])∈R3 :a∈C n,a †a= 1}? (9)

[2] [2]

We illustrate joint coherent control with a concrete ex- ample

How to find an input unit vectora 0 that realizes a given target (τ 0, ρ0, α0)∈Ω: τ[a 0] =τ 0, ρ[a 0] =ρ 0, α[a 0] =α 0? (10) This paper provides complete answers to both questions. We illustrate joint coherent control with a concrete ex- ample. Consider a silicon slab waveguide (refractive in- dexn i = 3.48) embedded in silica cladding (n 0 = 1.444) [Fig...

[3] [3]

Matrices for Fig. 2 The two-mode disordered waveguide has transmission and reflection matrices: t= −0.24 + 0.06i0.15−0.14i 0.20−0.11i−0.02−0.58i ,(D1) r= −0.39−0.06i−0.05−0.20i −0.20 + 0.05i−0.13−0.25i .(D2) We calculateT,R, andAfromtandrusing Eq. (7): T= 0.12 0.01−0.09i 0.01 + 0.09i0.37 ,(D3) R= 0.20 0.05 + 0.13i 0.05−0.13i0.12 ,(D4) A= 0.69−0.06−0.04i −...

[4] [4]

Matrices for Fig. 3 The three-mode disordered waveguide has transmis- sion and reflection matrices: t=   0.23 + 0.22i0.12−0.02i−0.05 + 0.03i 0.00−0.17i0.40 + 0.44i0.03−0.01i −0.07−0.13i−0.07−0.01i−0.43−0.07i   , (D6) r=   0.31 + 0.35i0.07−0.08i−0.01 + 0.25i −0.03−0.10i−0.37 + 0.01i0.03 + 0.07i 0.09−0.23i0.06−0.04i−0.05 + 0.04i   . (D7) We calculat...

[5] [5]

4 Here we provide theTandRmatrices used to generate each panel in Fig

Matrices for Fig. 4 Here we provide theTandRmatrices used to generate each panel in Fig. 4. The correspondingAmatrices can be derived usingA=I−T−R. 7 (a) Triangular disk: T=   0.1 0 0 0 0.6 0 0 0 0.1   ,(D11) R=   0.7 0 0 0 0.3 0 0 0 0.1   .(D12) (b) Convex hull of an ellipse and a point: T=   0.2 0.15 + 0.05i0 0.15−0.05i0.6 0 0 0 0.1   ,(D13)...

[6] [6]

[102], p.8, Property 1.2.1.)

Compactness The numerical rangeW(T+iR) is compact (see Ref. [102], p.8, Property 1.2.1.). Sincefis continuous and the continuous image of a compact set is compact, Ω is compact

[7] [7]

[102], p.8, Property 1.2.2.)

Convexity The numerical rangeW(T+iR) is convex (see Ref. [102], p.8, Property 1.2.2.). Since the image of a con- vex set under an affine function is convex (see Ref. [107], p.36.), Ω is convex

[8] [8]

[102], p.12.)

Outer boundΩ⊆Ω out SinceTis Hermitian, the quadratic formτ=a †Ta takes all values satisfying λmin(T)≤τ≤λ max(T) (E2) asaranges over complex unit vectors (see Ref. [102], p.12.). The same holds forRandA, giving: λmin(R)≤ρ≤λ max(R),(E3) λmin(A)≤α≤λ max(A),(E4) and all the bounds are attainable. Therefore, Ω is in- scribed in the hexagon Ω out defined in Eq. (16)