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Universality of Photonic Interlacing Architectures for Learning Discrete Linear Unitaries

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves that every N×N unitary matrix can be approximated by interlacing N layers of diagonal phase masks with the propagator of a single tridiagonal lattice Hamiltonian, and reports that N layers suffice numerically.

desk verdict A genuine formal-proof attempt for finite-layer universality, with a real gap in the Lie-group step and an honest, unproven empirical M=N claim. read the letter →

arxiv 2506.08454 v2 pith:K7P5PXWP submitted 2025-06-10 quant-ph math-phmath.MPphysics.optics

Matthew Markowitz , Mohammad-Ali Miri , Alexey Ovchinnikov , Kevin Zelaya This is my paper
classification quant-phmath-phmath.MPphysics.optics MSC 22E7015A2381V80
keywords unitarymatricesphotoniccircuitsphaseshifterswaveguidearraysLiegroupsuniversalityfactorizationopticallogicgates
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

The paper claims to close a gap in the theory of programmable photonic circuits: no formal proof existed that interlacing diagonal phase operations with a fixed unitary operator can represent the full unitary group U(N). It proves that for any N, any U(N) matrix, and any tolerance, there is a finite number of layers M such that a product of N-parameter phase diagonals and 1-parameter propagators $e^{{iℓH}}$ of a fixed tridiagonal lattice Hamiltonian approximates U to that tolerance. The proof works by showing the alternating product closes as a Lie group and, under a generic coupling condition on H, equals U(N). Numerical experiments with Haar-random targets show the error drops sharply at M=N, the minimum allowed by parameter counting, although the exact value of M is not established by the proof. A single passive lossless three-port circuit is designed and simulated as an all-optical logic gate performing AND, OR, NAND, and XOR operations.

What carries the argument

The central object is the alternating product G = T_N · G_H · T_N · G_H · ... (M factors), where T_N = {$e^{{i diag(ϕ_1,...,ϕ_N)}}$} is the compact group of diagonal phase matrices and G_H = {$e^{{iℓH}}$ : ℓ ∈ R} is the one-parameter unitary group generated by a fixed tridiagonal Hamiltonian H, interpreted as the propagator of a coupled waveguide array. The proof's load-bearing mechanism is the interplay between this group and the Lie algebra generated by H together with the Cartan subalgebra of T_N: when that algebra contains a matrix conjugate to a single Jordan block of size N, Lemma II.2 forces the complexified group to be the full general linear group, hence the compact group to be U(N). The physical counterpart is that $e^{{iℓH}}$ is exactly the evolution of light through a nearest-neighbor coupled waveguide lattice, so each factor is an elementary optical component.

What would settle it

For a fixed tridiagonal H with all nearest-neighbor couplings nonzero and a fixed small N (say N=4), generate many Haar-random unitary targets and globally minimize the error norm L(x) for M=N. If any target's minimum error remains bounded well above machine precision, the claimed transition at M=N fails. Algebraically, compute the real Lie algebra generated by H together with the diagonal Cartan subalgebra of T_N; if for some N this is a proper subalgebra of u(N), then Lemma II.2 cannot be correct.

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Extended reading notes

Core claim

The central claim is that the factorization U(x) = $e^{{iQ^{(M)}}$} $e^{{iℓ_{M-1}}$H} ... $e^{{iℓ_1 H}}$ $e^{{iQ^{(1)}}$} is universal: for every positive integer N, every U ∈ U(N), and every ε > 0, there exist a finite number of layers M, coupling lengths ℓ_p, and phase parameters $ϕ_j^{{(n)}}$ such that the product approximates U within ε in the Frobenius norm, with exact equality whenever the nonzero eigenvalues of H have rational ratios. The proof is organized through two lemmas: Lemma II.1 establishes that the alternating product T_N · G_H · T_N · G_H ... forms a real Lie subgroup of U(N), and Lemma II.2 shows that if H lies in a class J (satisfied by any tridiagonal H with all nearest-neighbor couplings nonzero) then this subgroup is all of U(N). The argument passes through the complexification of the group, uses maximal-rank and centralizer structure theorems, and reduces the group to a direct product of general linear groups, which the presence of a single Jordan block in the Lie algebra forces to be just GL(N,C).

Load-bearing premise

The proof rests on a chain of cited Lie-group theorems asserting that the alternating product closes as a Lie subgroup and that, when the Lie algebra contains a single Jordan block, this subgroup is the full unitary group; these theorems are not proved in the paper, and the Borel-de Siebenthal step in Lemma II.2 is garbled, so a gap in that chain would leave the finite-layer universality claim unsupported.

Editorial extensions

If this is right

  • Any N×N unitary operation can be implemented by a fixed, passively manufactured waveguide lattice with only the diagonal phase layers programmable, which removes the need to tune the coupling structure of the device.
  • The proof provides a termination criterion for parameter-search algorithms: one can start at M=N layers and only increase M if the target is not reached, because a finite M is guaranteed to exist.
  • For Hamiltonians whose nonzero eigenvalues have rational ratios, the reconstruction is exact rather than approximate, which singles out lattices such as the J_x lattice as exact universal building blocks.
  • With M=N the architecture uses N(M+1)-1 parameters, matching the N^2 parameter count of U(N) up to a global phase, so the layer count is minimal in the information-theoretic sense.
  • The demonstrated three-port logic gate shows that a single passive lossless circuit can encode several Boolean functions by switching only input phases and reading power thresholds at the outputs.

Reading between the lines

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

  • If the numerically observed transition at M=N is generic for every tridiagonal H with all couplings nonzero, then the alternating product may already be dense in U(N) after exactly N factors, which would upgrade the existence theorem to a sharp quantitative bound.
  • The rational-versus-irrational eigenvalue dichotomy suggests a deeper connection to Diophantine approximation: for irrational ratio spectra the approximation error should decay with the number of layers at a rate governed by how well integer combinations of the eigenvalues approximate integers, a rate the paper does not analyze.
  • The passive logic gate could be extended to larger Boolean functions by cascading several fixed unitary stages with threshold readouts, an avenue the paper mentions only implicitly.
  • Because the proof treats H only through its Lie-algebraic class J, any physical system with the same alternating structure—not only waveguide arrays—should inherit the universality guarantee, which would generalize the result to other platforms such as coupled resonators or atomic ensembles.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper claims a formal proof that any element of U(N) can be approximated to arbitrary precision by an interlaced product of diagonal phase matrices e^{iQ(n)} and one-parameter propagators e^{i\ell H} of a fixed tridiagonal Hamiltonian H, with a finite number of layers M (Corollary II.1). The proof is based on Lie-group arguments: Lemma II.1 asserts that a finite alternating product T_N G_H T_N G_H ... is a real Lie subgroup, and Lemma II.2 asserts that under a Jordan-block condition on the Lie algebra this subgroup equals U(N). The paper further reports numerical evidence, for N=4, 6, 8, that the minimal number of layers is exactly M=N, and it demonstrates a lossless passive photonic logic-gate circuit for N=3 with full-wave simulations. The central theorem is independent of the numerical optimization, but the exact-M=N claim is presented as empirical rather than proven.

Significance. If the Lie-group argument can be made rigorous, the paper would close a notable gap in the photonic-unitary-decomposition literature: previous work [12-14] gave strong numerical evidence but no proof of finite-layer universality for a fixed tridiagonal Hamiltonian. The factorized form (1) is physically natural, since it corresponds to interlacing phase-shifter layers with coupled-waveguide-array propagators, and the proposed logic-gate device is a concrete application. The strength of the paper is its clear identification of the mathematical question and its use of compact Lie-group theory to attack it. However, the current manuscript does not fully support the main theorem: the two key lemmas contain an incomplete cited-theorem application and a missing computational verification, and the 'exactly N layers' headline claim is only numerically observed for three small dimensions. These issues are repairable but are load-bearing for the paper's central claims.

major comments (4)
  1. [Section II, Lemma II.2] The sentence 'By [32, Théorème 5], G^C is the connected component of the normalizer of and therefore is the centralizer of Z(G^C)⊂T^C in GL(N,C)' is incomplete: the object of 'normalizer of' is missing, and no statement of the Borel-de Siebenthal theorem is supplied. This is the step that converts the single-Jordan-block condition into G^C = GL(N,C) and hence G = U(N), so it is load-bearing. The authors should state the theorem precisely, verify its hypotheses (including the maximal-rank and maximal-torus conditions for G^C), and show explicitly how the conclusion follows. Without this, the main universality proof cannot be checked.
  2. [Section II, Lemma II.1] The lemma asserts that the finite alternating product T_N · G_H · T_N · G_H · ... is a real Lie subgroup of U(N), citing [31, Section 7.5]. This does not follow from the cited reference as written: finite products of closed subgroups of a Lie group are not generally closed, and no argument is given that this particular alternating product is closed under multiplication and inversion. If the intended statement is that the closure of the subgroup generated by T_N and G_H is a Lie subgroup, then the exact finite-layer conclusion in Corollary II.1 does not follow without an additional argument that the finite alternating product saturates that subgroup. This issue directly affects the claimed termination criterion at M=N.
  3. [Abstract, Section IV, Remark III.1, Figure 2] The claim that the required number of layers is 'exactly N' is not proven. Corollary II.1 only establishes existence of a finite M, and Remark III.1 explicitly states that the exact value of M is 'yet to be found' and that parameter counting gives only the lower bound M≥N. The numerical experiments in Figure 2 cover N=4, 6, 8 and show a sharp drop in error at M=N, but that is empirical evidence, not a theorem. The abstract and Section IV should present M=N as a numerically supported conjecture, not as part of the formal result, and the proposed algorithm's termination at M=N should be discussed accordingly.
  4. [Section II, Example II.1] The claim that H∈J for every tridiagonal H with nonzero nearest-neighbor couplings is supported only by the phrase 'A direct computation shows...'. This computation is load-bearing because Corollary II.1 applies to all such H. The authors should provide the explicit Lie-bracket iteration and demonstrate that the resulting matrix is conjugate to a single Jordan block of size N, or give a precise reference where this computation appears.
minor comments (6)
  1. [Section II heading] The heading 'PROOF OF UIVERSALITY' contains a typo; it should read 'PROOF OF UNIVERSALITY'. There is also a typo 'thee-diagonal' in Section II.A.
  2. [Section II.A, Eq. (2)] The index range 'p, q ∈ {1, ..., N−1}' for the N×N matrix H should be 'p, q ∈ {1, ..., N}'; the coupling coefficients are κ_p for p=1,...,N−1.
  3. [Section II.A] The reference list entry '[25?]' appears in the sentence 'literature of lossless photonic architectures [12, 13, 25?]'; this should be corrected to a proper citation, presumably [25].
  4. [Section II, Lemma II.1] The definition of G_H is ambiguous: 'G_H := {e^{i\ell H} | \ell∈R}, the closure as a real Lie group.' It is unclear whether G_H denotes the one-parameter subgroup, its closure, or both; the proof distinguishes these cases only later, and the notation should be clarified.
  5. [Section IV] The statement 'yielding a total of N parameters per layer combination e^{i\ell_p}F, with p∈{1,...,N}' appears inconsistent with Eq. (1), where there are M phase layers and M−1 length parameters. The parameter count should be reconciled with the definition of the architecture.
  6. [Section III.B and Remark III.2] The numerical section would benefit from stating the optimization tolerance δ and the iteration limit N_it used in Figure 2, since these directly affect the interpretation of 'error norm drastically drops at M=N'.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity: the universality proof uses external Lie-group theorems and is not derived from the numerical optimizer; self-citations are motivational only.

full rationale

The central derivation chain (Lemma II.1, Lemma II.2, Corollary II.1) does not reduce to any fitted parameter or to the numerical optimization. The factorization (1) is an ansatz, but universality is proved by showing that the alternating product T_N G_H T_N G_H ... generates U(N); the proof rests on standard Lie-group references [30-33] that the paper does not derive and, crucially, does not use the paper's own numerical outputs as premises. Self-citations [12-14] are used only as prior numerical evidence and implementation context, not as load-bearing steps in the proof. The M=N claim is explicitly presented as a numerical finding (Remark II.1, Section III.B), not as a consequence of the theorem; this is an empirical observation rather than a renamed fit. The garbled Borel-de Siebenthal sentence in Lemma II.2 and the unproved closure claim in Lemma II.1 are significant correctness risks because the proof depends on precise applications of those external theorems, but these are gaps in external-theorem verification, not circularity: no equation in the proof is equivalent by construction to an input of the optimization, and no 'prediction' is defined in terms of the fitted phases/lengths.

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

The central proof relies on standard Lie-group theorems from Onishchik-Vinberg, Humphreys, and Borel-de Siebenthal, plus the physical coupled-mode model. No ad hoc fitted constants are introduced; the phases and lengths in factorization (1) are existential variables with unspecified values. The numerical M=N observation is an empirical extrapolation, not a fitted parameter.

assumptions (6)
  • standard math Closedness criterion for one-parameter subgroups: e^{iℓ diag(a_i)} is closed iff ratios of nonzero a_i are rational (Onishchik-Vinberg Problem 14).
    Used in Lemma II.1 to identify G_H with the one-parameter subgroup for rational eigenvalue ratios, and to ensure G_H is a compact Lie subgroup.
  • standard math A theorem that the group generated by two connected subgroups is realized as a finite alternating product (Humphreys Section 7.5).
    Used in Lemma II.1 to assert G = T_N G_H T_N G_H ... is a real Lie subgroup for some finite M.
  • standard math Borel-de Siebenthal theorem: a closed maximal-rank subgroup of a compact Lie group is the identity component of the centralizer of its center, and its complexification is a product of GL(k_i,C) factors.
    Used in Lemma II.2 to conclude G^C is conjugate to a direct product of general linear groups.
  • standard math A single Jordan block of size N is a regular nilpotent; if it lies in a block-diagonal reductive subalgebra of gl(N,C), the subalgebra must be a single factor GL(N,C).
    Used in Lemma II.2 to force s=1 and k=N, completing the proof that G^C = GL(N,C).
  • domain assumption Coupled-mode theory models wave propagation through a waveguide array as e^{iℓH} with H a tridiagonal Hermitian matrix.
    Basis for the physical photonic implementation in Section III.A and the choice of H in Eq. (2).
  • standard math Parameter counting: the map from M(N+1)-1 real parameters to U(N) (dimension N^2) requires M≥N for a surjective smooth parametrization.
    Remark III.1 uses this to establish the lower bound M≥N and to justify that M=N is the minimal layer count.

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Pith. "Pith review of Universality of Photonic Interlacing Architectures for Learning Discrete Linear Unitaries." pith.science (2026). https://pith.science/paper/K7P5PXWP

@misc{pith2026250608454,
  author       = {Pith},
  title        = {Pith review of: Universality of Photonic Interlacing Architectures for Learning Discrete Linear Unitaries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K7P5PXWP}},
  note         = {Machine review of arXiv:2506.08454}
}
abstract

Recent investigations suggest that the discrete linear unitary group $U(N)$ can be represented by interlacing a finite sequence of diagonal phase operations with an intervening unitary operator. However, despite rigorous numerical justifications, no formal proof has been provided. Here, we show that elements of $U(N)$ can be decomposed into a sequence of $N$-parameter phases alternating with $1$-parameter propagators of a lattice Hamiltonian. The proof is based on building a Lie group by alternating these two operators and showing its completeness to represent $U(N)$ for a finite number of layers, which is numerically found to be exactly $N$. This architecture can be implemented using elementary optical components and can successfully reconstruct arbitrary unitary matrices. We propose example devices such as optical logic gates, which perform logic gate operations using a single-layer lossless and passive optical circuit design.

Figures

Figures reproduced from arXiv: 2506.08454 by the authors.

Figure 1
Figure 1. FIG. 1. Sketch of the proposed factorization and its physical [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Block diagram and photonic circuit sketch for the proposed logic gate operation. A single input field [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. (a) Density plot of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

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Works this paper leans on

42 extracted references · 36 canonical work pages

  1. [1]

    M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, Experimental realization of any discrete unitary opera- tor, Physical review letters73, 58 (1994)

    work page 1994

  2. [2]

    N. C. Harris, J. Carolan, D. Bunandar, M. Prabhu, M. Hochberg, T. Baehr-Jones, M. L. Fanto, A. M. Smith, C. C. Tison, P. M. Alsing,et al., Linear programmable nanophotonic processors, Optica5, 1623 (2018)

    work page 2018

  3. [3]

    Bogaerts, D

    W. Bogaerts, D. P´ erez, J. Capmany, D. A. Miller, J. Poon, D. Englund, F. Morichetti, and A. Melloni, Pro- grammable photonic circuits, Nature586, 207 (2020)

    2020

  4. [4]

    Bogaerts and A

    W. Bogaerts and A. Rahim, Programmable photonics: An opportunity for an accessible large-volume pic ecosys- tem, IEEE Journal of Selected Topics in Quantum Elec- tronics26, 1 (2020)

    work page 2020

  5. [5]

    N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. Wong, T. Baehr- Jones, M. Hochberg,et al., Quantum transport simula- tions in a programmable nanophotonic processor, Nature Photonics11, 447 (2017)

    work page 2017

  6. [6]

    Pelucchi, G

    E. Pelucchi, G. Fagas, I. Aharonovich, D. Englund, E. Figueroa, Q. Gong, H. Hannes, J. Liu, C.-Y. Lu, 8 N. Matsuda,et al., The potential and global outlook of integrated photonics for quantum technologies, Nature Reviews Physics4, 194 (2022)

    work page 2022

  7. [7]

    W. Luo, L. Cao, Y. Shi, L. Wan, H. Zhang, S. Li, G. Chen, Y. Li, S. Li, Y. Wang,et al., Recent progress in quantum photonic chips for quantum communication and internet, Light: Science & Applications12, 175 (2023)

    2023

  8. [8]

    W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walmsley, Optimal design for uni- versal multiport interferometers, Optica3, 1460 (2016)

    2016

Show all 42 references
  1. [9]

    V. L. Pastor, J. Lundeen, and F. Marquardt, Arbitrary optical wave evolution with Fourier transforms and phase masks, Optics Express29, 38441 (2021)

    2021

  2. [10]

    R. Tang, R. Tanomura, T. Tanemura, and Y. Nakano, Ten-port unitary optical processor on a silicon photonic chip, ACS Photonics8, 2074 (2021)

    2021

  3. [11]

    Taballione, R

    C. Taballione, R. van der Meer, H. J. Snijders, P. Hooi- jschuur, J. P. Epping, M. de Goede, B. Kassenberg, P. Venderbosch, C. Toebes, H. van den Vlekkert,et al., A universal fully reconfigurable 12-mode quantum photonic processor, Materials for Quantum Technology1, 035002 (2021)

    2021

  4. [12]

    Markowitz and M.-A

    M. Markowitz and M.-A. Miri, Universal unitary photonic circuits by interlacing discrete fractional fourier transform and phase modulation (2023), arXiv:2307.07101 [physics.optics]

    2023 arXiv

  5. [13]

    Markowitz, K

    M. Markowitz, K. Zelaya, and M.-A. Miri, Auto- calibrating universal programmable photonic circuits: hardware error-correction and defect resilience, Optics Express31, 37673 (2023)

    2023

  6. [14]

    Zelaya, M

    K. Zelaya, M. Markowitz, and M.-A. Miri, The goldilocks principle of learning unitaries by interlacing fixed oper- ators with programmable phase shifters on a photonic chip, Scientific Reports14, 10950 (2024)

    2024

  7. [15]

    D. A. Miller, Self-configuring universal linear optical component, Photonics Research1, 1 (2013)

    2013

  8. [16]

    Markowitz, K

    M. Markowitz, K. Zelaya, and M.-A. Miri, Learning ar- bitrary complex matrices by interlacing amplitude and phase masks with fixed unitary operations, Physical Re- view A110, 033501 (2024)

    2024

  9. [17]

    X. Meng, G. Zhang, N. Shi, G. Li, J. Aza˜ na, J. Cap- many, J. Yao, Y. Shen, W. Li, N. Zhu,et al., Compact optical convolution processing unit based on multimode interference, Nature Communications14, 3000 (2023)

    2023

  10. [18]

    Zelaya and M.-A

    K. Zelaya and M.-A. Miri, Photonic matrix multi- plier makes a direction-finding sensor, arXiv preprint arXiv:2411.06731 (2024)

    2024 arXiv

  11. [19]

    D. A. Miller, Communicating with waves between vol- umes: evaluating orthogonal spatial channels and limits on coupling strengths, Applied optics39, 1681 (2000)

    2000

  12. [20]

    D. A. Miller, Waves, modes, communications, and optics: a tutorial, Advances in Optics and Photonics11, 679 (2019)

    2019

  13. [21]

    Tanomura, R

    R. Tanomura, R. Tang, T. Umezaki, G. Soma, T. Tane- mura, and Y. Nakano, Scalable and robust photonic inte- grated unitary converter based on multiplane light con- version, Physical Review Applied17, 024071 (2022)

    2022

  14. [22]

    Taguchi, Y

    Y. Taguchi, Y. Wang, R. Tanomura, T. Tanemura, and Y. Ozeki, Iterative configuration of programmable uni- tary converter based on few-layer redundant multiplane light conversion, Physical Review Applied19, 054002 (2023)

    2023

  15. [23]

    Tanomura, R

    R. Tanomura, R. Tang, S. Ghosh, T. Tanemura, and Y. Nakano, Robust integrated optical unitary converter using multiport directional couplers, Journal of Light- wave Technology38, 60 (2020)

    2020

  16. [24]

    Tanomura, T

    R. Tanomura, T. Tanemura, and Y. Nakano, Multi- wavelength dual-polarization optical unitary processor using integrated multi-plane light converter, Japanese Journal of Applied Physics , SC1029 (2023)

    2023

  17. [25]

    Saygin, I

    M. Saygin, I. Kondratyev, I. Dyakonov, S. Mironov, S. Straupe, and S. Kulik, Robust Architecture for Pro- grammable Universal Unitaries, Physical Review Letters 124, 010501 (2020), publisher: American Physical Soci- ety

    2020

  18. [26]

    Skryabin, I

    N. Skryabin, I. Dyakonov, M. Y. Saygin, and S. Ku- lik, Waveguide-lattice-based architecture for multichan- nel optical transformations, Optics Express29, 26058 (2021)

    2021

  19. [27]

    Williams, X

    D. Williams, X. Bao, and L. Chen, All-optical nand/not/and/or logic gates based on combined brillouin gain and loss in an optical fiber, Applied Optics52, 3404 (2013)

    2013

  20. [28]

    B. Real, C. Cantillano, D. L´ opez-Gonz´ alez, A. Szameit, M. Aono, M. Naruse, S.-J. Kim, K. Wang, and R. A. Vicencio, Flat-band light dynamics in stub photonic lat- tices, Scientific reports7, 15085 (2017)

    2017

  21. [29]

    Huang, Coupled-mode theory for optical waveg- uides: an overview, JOSA A11, 963 (1994)

    W.-P. Huang, Coupled-mode theory for optical waveg- uides: an overview, JOSA A11, 963 (1994)

    1994

  22. [30]

    Onishchik and E

    A. Onishchik and E. Vinberg,Lie Groups and Algebraic Groups(Springer, 1990)

    1990

  23. [31]

    Humphreys,Linear Algebraic Groups(Springer, 1975)

    J. Humphreys,Linear Algebraic Groups(Springer, 1975)

    1975

  24. [32]

    Borel and J

    A. Borel and J. De Siebenthal, Les sous-groupes ferm´ es de rang maximum des groupes de Lie clos, Commentarii mathematici Helvetici23, 200 (1949)

    1949

  25. [33]

    Goren, Algebraic Groups

    E. Goren, Algebraic Groups. Lecture Notes. Part V (2011)

    2011

  26. [34]

    R ´ ıos, Q

    C. R ´ ıos, Q. Du, Y. Zhang, C.-C. Popescu, M. Y. Shalagi- nov, P. Miller, C. Roberts, M. Kang, K. A. Richardson, T. Gu,et al., Ultra-compact nonvolatile phase shifter based on electrically reprogrammable transparent phase change materials, PhotoniX3, 26 (2022)

    2022

  27. [35]

    Youngblood, C

    N. Youngblood, C. A. R ´ ıos Ocampo, W. H. Pernice, and H. Bhaskaran, Integrated optical memristors, Nature Photonics , 1 (2023)

    2023

  28. [36]

    N. M. Atakishiyev and K. B. Wolf, Fractional Fourier– Kravchuk transform, J. Opt. Soc. Am. A14, 1467 (1997)

    1997

  29. [37]

    Weimann, A

    S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Gr¨ afe, R. Heilmann, S. Nolte, H. Moya- Cessa, G. Weihs,et al., Implementation of quantum and classical discrete fractional fourier transforms, Nature Communications7, 1 (2016)

    2016

  30. [38]

    Honari-Latifpour and M.-A

    M. Honari-Latifpour and M.-A. Miri, Mapping thexy hamiltonian onto a network of coupled lasers, Phys. Rev. Research2, 043335 (2020)

    2020

  31. [39]

    D. N. Christodoulides, F. Lederer, and Y. Silberberg, Discretizing light behaviour in linear and nonlinear waveguide lattices, Nature424, 817 (2003)

    2003

  32. [40]

    Guerrero and H

    J. Guerrero and H. M. Moya-Cessa, Coherent states for equally spaced, homogeneous waveguide arrays (2021), arXiv:2112.01673 [quant-ph]

    2021 arXiv

  33. [41]

    Mezzadri, How to generate random matrices from the classical compact groups (2007), arXiv:math- ph/0609050

    F. Mezzadri, How to generate random matrices from the classical compact groups (2007), arXiv:math- ph/0609050

    2007

  34. [42]

    Yariv and P

    A. Yariv and P. Yeh,Photonics: optical electronics in modern communications(Oxford university press, 2007)

    2007

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