Architecture-agnostic analysis of partially coherent light with programmable photonics

Jonathan Friedman; Kevin Zelaya; Matthew Markowitz; Mohammad-Ali Miri

REVIEW 3 major objections 6 minor 24 references

Architecture-agnostic analysis of partially coherent light with programmable photonics

T0 review · 3 major / 6 minor · reviewed 2026-07-31 · deepseek-v4-flash

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Pith's one-line read This paper claims that the eigenvalues of a light field's spatial coherence matrix can be read directly from output power meters after a blind optimization of any programmable photonic network, bypassing interferometry and knowledge of the

desk verdict A genuinely useful black-box trick for extracting coherence eigenvalues from any universal unitary PIC, but the loss-resilience claim is mathematically wrong and the Schur-Horn objective as written misses a sorting step. read the letter →

arxiv 2607.24104 v1 pith:WHYH3U5S submitted 2026-07-27 physics.optics math-phmath.MP

Kevin Zelaya , Matthew Markowitz , Jonathan Friedman , Mohammad-Ali Miri This is my paper · ORCID

classification physics.opticsmath-phmath.MP

keywords spatialcoherencematrixSchur-Horntheoremprogrammablephotonicspowermeasurementsblack-boxoptimizationeigenvalueextractionpartiallycoherentlight

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 tries to establish that the eigenvalues of the spatial coherence matrix of a partially coherent light field can be extracted from simple power measurements at the output of any programmable photonic circuit, without knowing the circuit's internal design. The trick is a scalar optimization, derived from the Schur-Horn theorem, that drives the output coherence matrix to diagonal form by maximizing cumulative output powers. If true, this removes the need for phase-sensitive interferometry and per-topology calibration, making coherence analysis practical on integrated platforms. The authors back the claim with numerical tests across several universal topologies and show graceful degradation for non-universal and lossy networks.

What carries the argument

The central mechanism is the Schur-Horn theorem, which says that the diagonal entries of a Hermitian matrix are majorized by its eigenvalues: sorted cumulative sums of the diagonal never exceed the corresponding cumulative eigenvalue sums, with equality only for diagonal matrices. The paper packages this into the objective ‖eP(Φ)‖², where eP collects the deficits (1 minus cumulative normalized output power) at each output port; minimizing this drives the device to the point where output powers saturate the Schur-Horn bound, i.e., the output coherence matrix is diagonal. Because the cost uses only power readings, the internal topology of the photonic circuit never needs to be known; the devic

What would settle it

Pick an 8-port universal network, feed a partially coherent state with known eigenvalues, and run the power-only optimization from many random initializations. If the best-found objective value is nonzero, or if the recovered eigenvalues differ from the known ones beyond numerical precision, then the equivalence between minimizing ‖eP‖² and diagonalizing the output coherence matrix does not hold for that architecture.

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Extended reading notes

Core claim

The paper's central claim is that for any programmable unitary photonic network — triangular, rectangular, or interlaced multiport-coupler — one can extract the eigenvalues of an unknown input coherence matrix by minimizing a scalar cost built from output power measurements alone. The cost is the squared norm of the vector of tail sums of normalized output powers. By the Schur-Horn theorem, a Hermitian matrix's sorted diagonal is majorized by its eigenvalues, and equality of the cumulative sums holds only when the matrix is diagonal; hence pushing the cumulative output powers to their maximum forces the output coherence matrix to diagonalize, making the measured powers equal to the input's c

Load-bearing premise

The load-bearing premise is that the global minimum of the cumulative-power cost is always zero and is reached only when the output coherence matrix is diagonal, and that gradient descent reliably finds that minimum; for lossy networks this exact equivalence cannot hold because a non-unitary transform changes the eigenvalues.

Editorial extensions

If this is right

Spatial coherence analysis becomes a plug-and-play task: any programmable photonic chip, regardless of internal layout, can act as a coherence meter after a blind optimization run.
The same circuit can be re-programmed back to the identity after analysis, so the light field is available downstream unaffected.
Lower-depth and under-parameterized networks are sufficient for useful accuracy, shrinking device footprint and fabrication cost.
Component-level losses and phase errors are absorbed by the black-box optimization, relaxing fabrication tolerances for integrated coherence analyzers.

Reading between the lines

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

The identical cost function could be pushed further to reconstruct the full coherence matrix, not just its eigenvalues, by combining power readings from multiple random unitary settings — the paper deliberately stops at eigenvalues.
Since the method never inspects the device, it should transfer as-is to other programmable unitary platforms, such as free-space spatial light modulators or fiber-based meshes, and possibly to classical wave systems that obey the same algebra.
The loss-error curves in the paper suggest a quantitative design rule: if the minimum per-component transmission stays above some threshold, reconstruction error stays below tolerance; deriving that threshold analytically would turn the numerical observation into an engineering bound.

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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

Zelaya et al. propose treating a programmable photonic circuit as a black box and using output power measurements only to recover the eigenvalues of the input coherence matrix. The core move is to drive the circuit until the output coherence matrix is diagonal, which they formulate via Schur-Horn as minimizing cumulative power shortfalls. That's a neat reformulation: it removes the need to know the circuit topology or to do layer-by-layer tuning as in Ref. [19]. The simulations on Reck, Clements, and interlaced networks are clean and the accuracy looks good for universal, lossless circuits. I believe the ideal-unitary part of the claim.

The paper has two problems.

First, the loss-resilience sales pitch is not supported. For a non-unitary transfer matrix A, the output coherence matrix is AρA†, whose eigenvalues are not the eigenvalues of ρ except in trivial cases. The optimization can at best diagonalize the output; the powers you read are the eigenvalues of the lossy output field, not the input. Their own Figure 2b shows the error growing as γ_min shrinks—they attribute that to convergence, but it's a fundamental information loss. The abstract's 'inherently resilient to arbitrary optical losses' is an overclaim. The method is robust to component deviations that are still unitary, but not to amplitude losses that scramble the spectrum.

Second, the Schur-Horn objective as written in Eq. (1) uses the output powers in fixed port order. But Schur-Horn majorization applies to the sorted diagonal entries. To know whether you've diagonalized, you need to compare the cumulative sums of the sorted powers, not the natural port order. Without sorting, minimizing ||eP||² could get stuck or might not correspond to diagonalization. The text says 'normalized power measurements will never exceed the normalized coherence matrix eigenvalues'—that's only true after sorting. If the authors intend to sort but didn't state it, fine; but as written, the math doesn't line up.

The equivalence between the scalar minimization and actual diagonalization is also asserted without proof. For universal networks the numerics are convincing, but for under-parameterized networks the claim that the objective recovers eigenvalues is heuristic. A proof or at least a convergence analysis would be needed.

The paper is worth a serious referee—the core idea is pragmatic and could be a useful tool for integrated photonics. But the loss-resilience claim should be toned down or reframed as characterizing the output field, and the sorting issue and equivalence need clarification. I'd recommend conditional acceptance after revision, not desk rejection.

Referee Report

3 major / 6 minor

Summary. The paper proposes an architecture-agnostic method for extracting the eigenvalues of the spatial coherence matrix of a partially coherent field using a programmable photonic network treated as a black box. The method is based on the Schur-Horn theorem: by defining a scalar objective from cumulative output power measurements (Eq. (2)) and minimizing it with respect to the network's phase parameters, the authors claim that the output coherence matrix becomes diagonal and the measured powers directly yield the coherence eigenvalues. The approach is validated numerically on Reck, Clements, and interlaced multiport-coupler topologies for 500 random fully coherent, partially coherent, and fully incoherent coherence matrices. The paper also reports performance for truncated (under-parameterized) interlaced networks and for networks with random layer-wise losses, claiming robustness and only minor precision loss.

Significance. If the ideal-unitary core of the method is correct, the proposal is practically significant: it replaces phase-sensitive interferometric coherence characterization with output power measurements from a programmable photonic circuit, independent of the circuit's internal architecture. The numerical study across three topologies and random ensembles is a strength, as is the absence of fitted parameters. The Schur-Horn connection is elegant and the black-box treatment is appealing for integration with imperfect devices. However, the central theoretical equivalence behind Eq. (2) is not proven, and the loss-resilience claim is mathematically overbroad. These issues must be corrected before the paper can be recommended for publication.

major comments (3)

[Results, Eq. (2)] The claim that minimizing ||eP(Phi)||^2 is 'equivalent' to diagonalizing rho_out is not established. First, eP is defined using P_k in fixed output-port order, whereas the Schur-Horn theorem applies to sorted diagonal entries. If the protocol sorts the measured powers before forming cumulative sums, this must be stated; otherwise the objective is not permutation-invariant and the theorem does not directly apply. Second, the sentence 'normalized power measurements will never exceed the normalized coherence matrix eigenvalues' is incorrect as written: for a 2x2 matrix with eigenvalues 0.8, 0.2 and diagonal entries 0.6, 0.4, the second power measurement exceeds the second eigenvalue. The correct statement is majorization of the sorted diagonal vector. Third, even accepting the objective, the paper provides no proof that gradient descent on this non-convex landscape reaches a global minimum,
[Results, loss paragraph and Fig. 2b] The claim of resilience to 'arbitrary optical losses' is unsupported and, as stated, cannot hold. For A in GL(N) non-unitary, rho_out = A rho_in A† does not preserve eigenvalues. The minimization in Eq. (2) operates on output powers and can at best diagonalize rho_out; the eigenvalues it returns are those of the field after the lossy transformation, not the input field. For example, take A = diag(gamma_1,...,gamma_N) with unequal gamma_n and rho = I/N. Then rho_out has eigenvalues gamma_n^2 / sum_k gamma_k^2, not 1/N, and no unitary post-processing can restore the equal input spectrum. The growing error in Fig. 2b is therefore a fundamental spectral change, not a precision or convergence artifact. The abstract's statement that the approach is 'inherently resilient to arbitrary optical losses' and the conclusion's 'highly adaptable... in the presence of arbitrary optical losses' should be
[Results, under-parameterized networks] The paper claims that under-parameterized, non-universal architectures achieve coherence analysis with 'only minor loss in precision' and that the objective can extract eigenvalues with as few as three layers. For M < N+1 the feasible set of unitaries is a proper subset, and there is no argument that the minimizer of Eq. (2) yields a diagonal rho_out or eigenvalues close to the input eigenvalues. The numerical averages over 500 random samples are useful, but no worst-case or theoretical bound is provided. Since the abstract and conclusions highlight non-universal architectures as a selling point, this should be either supported analytically or explicitly framed as an empirical observation with stated limitations.

minor comments (6)

[Results, under-parameterized networks] 'as few as three layers for PI light' appears to be a typo; the text elsewhere defines FC, PC, and FI ensembles. Please clarify which ensemble is meant, and whether the singular 'eigenvalue' should be 'eigenvalues'.
[Fig. 1b] 'Clemments' should be 'Clements'.
[Results, interlaced network] The phrase 'which is an splits the light among its output ports' contains a grammatical error; should be 'which splits the light'.
[Eq. (1)] 'power measuresd' should be 'power measured'.
[References] References [14] and [17] are identical (Markowitz, Zelaya, Miri, Opt. Express 31, 37673 (2023)); they should be merged to avoid duplicate citation.
[Results, numerical validation] The statement that 'exact gradients of the objective function can be readily obtained' is only true in simulation where the network transfer matrix is known. In an experimental black-box setting, gradient-free optimization would be needed; this should be clarified so the reader is not misled.

Circularity Check

0 steps flagged · score 2.0 of 10

No substantive circularity: the Schur-Horn minimization is self-contained and not fitted; the only self-citation is a non-load-bearing testbed reference, and the loss-resilience overclaim is a correctness issue, not a circular reduction.

full rationale

The paper's central derivation chain is self-contained. It defines the output coherence matrix as ρ_out = AρA†, notes that for unitary A the eigenvalues are preserved, invokes the Schur-Horn theorem (an external mathematical result, Horn 1954), and defines the objective min_Φ ||eP(Φ)||² from output power measurements only. The theorem supplies the equivalence between minimizing this objective and making the output powers the sorted eigenvalues of ρ_out; no parameter is fitted to the target eigenvalues, and the numerical validation uses randomly generated coherence matrices. Nothing in this chain reduces to its own inputs by construction. The only self-citation is Ref. [17], which duplicates Ref. [14] (Markowitz, Zelaya, Miri, Opt. Express 31, 37673) and is used solely to set the interlaced-network layer count M=N+1 for a simulation testbed; the architecture-agnostic method does not depend on that universality result, so it is not load-bearing. The abstract's claim of inherent resilience to arbitrary optical losses is a separate correctness concern: for non-unitary A the algorithm diagonalizes AρA† and returns the eigenvalues of the lossy output matrix, not the input coherence eigenvalues, and the paper's own Fig. 2b shows growing error. This is an overclaim or missing support, not a circular reduction. Accordingly, no circular step rises above score 2.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the Schur-Horn theorem (standard), the stationarity/PSD assumption for the coherence matrix, the assumed transfer-matrix model for the tested networks, and the unproved equivalence between the proposed minimization and diagonalization. No free parameters are fitted to data; simulation settings (N, M, γ_min, sample counts) are test choices. No invented entities are introduced.

assumptions (4)

standard math Schur-Horn theorem
External theorem from Horn (1954) used to justify that minimizing Eq. (2) diagonalizes the output coherence matrix; not proved in the paper.
domain assumption Statistical stationarity and Hermitian positive-semidefinite coherence matrix
The definition ρ = ⟨x x†⟩ and the eigen-decomposition require stationary fields; standard in optical coherence theory, but made without discussion.
domain assumption Network transfer-matrix model and universality of interlaced network
Simulations rely on parameterized unitary models (Reck, Clements, interlaced) and on the interlaced network being universal with M=N+1 layers, cited from the authors' own Ref. [17].
ad hoc to paper Global equivalence between Eq. (2) minimization and diagonalization
The sentence 'This is, in turn, equivalent to the minimization problem' (after Eq. 2) is asserted without proof; it is load-bearing for the method.

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Cite this review

Pith. "Pith review of Architecture-agnostic analysis of partially coherent light with programmable photonics." pith.science (2026). https://pith.science/paper/WHYH3U5S

@misc{pith2026260724104,
  author       = {Pith},
  title        = {Pith review of: Architecture-agnostic analysis of partially coherent light with programmable photonics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WHYH3U5S}},
  note         = {Machine review of arXiv:2607.24104}
}

read the original abstract

The precise characterization of the spatial degree of coherence of a radiation field is important for assessing its suitability for specific applications in optical communications, advanced imaging, and quantum information processing. However, measuring the full coherence matrix traditionally requires complex, phase-sensitive interferometric setups that are highly susceptible to noise and difficult to scale on integrated platforms. To address this, we propose an architecture-agnostic approach for analyzing partially coherent light that is compatible with any universal programmable photonic unitary circuit, regardless of its internal topology. Leveraging the Schur-Horn theorem, our method diagonalizes the output coherence matrix, enabling direct extraction of its eigenvalues from output power measurements alone. We numerically validate this framework across various universal topologies and demonstrate its efficacy even in under-parameterized, non-universal architectures with only minor loss in precision. Finally, our black-box optimization approach proves inherently resilient to arbitrary optical losses and component deviations, paving the way for robust, lower-depth, and programmable spatial coherence analyzers.

Figures

Figures reproduced from arXiv: 2607.24104 by the authors.

**Figure 1.** Algorithmic approach and numerical performance. a Schematic of the architecture-agnostic coherence matrix analyzer. A programmable unitary device (black box) is controlled by the parameter set Φ. Power detectors (PD) record the output powers Pi , which are used to compute a figure of merit, L. A parameter driver iteratively updates Φ based on these measurements until L is minimized. The left dashed box depicts the m… view at source ↗

**Figure 2.** Performance against truncation and losses. a Nonuniversal interlaced network and b Impact of inherent arbitrary losses in the interlaced unitary network. The losses are randomly assigned in each layer by replacing the phase ϕ (m) n → ϕ (m) n − i ln γ (m) n , where the transmitances are randomly sampled from the interval γ (m) n ∈ (γmin, 1). experience losses, particularly during routing between components and aro… view at source ↗

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Reference graph

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Pith tools

Reviewed July 31, 2026 · model on record in the stance chip above.

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Roques-Carmes, S

C. Roques-Carmes, S. Fan, and D. A. Miller, Light. Sci. & Appl.13, 260 (2024)

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1996

[3] [2]

S. Lee, D. Kim, S.-W. Nam,et al., Sci. reports10, 18832 (2020)

2020

[4] [3]

M. Reck, A. Zeilinger, H. J. Bernstein, and P . Bertani, Phys. review letters73, 58 (1994)

1994

[5] [4]

de Guise, O

H. de Guise, O. Di Matteo, and L. L. Sánchez-Soto, Phys. Rev. A97, 022328 (2018)

2018

[6] [5]

D. A. Miller, Photonics Res.1, 1 (2013)

2013

[7] [6]

D. Yi, Y . Wang, and H. K. Tsang, APL Photonics6(2021)

2021

[8] [7]

W. R. Clements, P . C. Humphreys, B. J. Metcalf,et al., Optica3, 1460 (2016)

2016