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Optimal universal bounds for waves with varied coherence based on supremum and infimum coherence spectra

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Pith's one-line read For any set of partially coherent waves, two extremal coherence spectra—the set's supremum and infimum under the majorization order, which may lie outside the set—give optimal universal bounds: every measurement's achievable range is squeez

desk verdict The sup/inf construction is correct and the optimality argument holds; the paper's real value is turning per-measurement optimization into two universal spectra, with a few scoping and presentation issues to fix. read the letter →

arxiv 2601.10665 v1 pith:AKNXY23H submitted 2026-01-15 physics.optics cond-mat.mes-hall

Shiyu Li , Cheng Guo This is my paper · ORCID
classification physics.opticscond-mat.mes-hall
keywords coherencespectramajorizationuniversalboundspartialsupremumandinfimumwaveHermitianmeasurementscompletelattice
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 asks how to bound the physical response of a wave when its coherence can vary over a set of possible spectra. It shows that scalar measures like entropy fail for three or more modes, because the majorization order is only partial, so there is generally no single most- or least-coherent element. The key move is to relax the search to the whole order: the set's supremum and infimum—the least upper and greatest lower bounds under majorization—exist uniquely even when they are not themselves achievable spectra. For every linear measurement, the achievable range of any member of the set is contained in the supremum range and contains the infimum range; and any spectrum that claims to give universal bounds is dominated by (or dominates) these extrema, so the bounds are the tightest possible from one spectrum. A reader should care because this gives a measurement-independent certificate: two spectra summarise the worst- and best-case response of an entire family of waves.

What carries the argument

The central object is the majorization order on coherence spectra: for ordered probability vectors x,y ∈ Δ↓_n, x ≺ y iff all partial sums of x are ≤ those of y. The load-bearing identity is Eq. (11): λ↓(ρ1) ≺ λ↓(ρ2) iff, for every Hermitian O, the range of possible values of tr(ρ O) for ρ1 is a subset of that for ρ2. The paper exploits the fact that (Δ↓_n, ≺) is a complete lattice, so any subset Λ has a unique least upper bound (supremum) and greatest lower bound (infimum), even if these lie outside Λ. These two spectra then serve as single 'most coherent' and 'least coherent' stand-ins that dominate or are dominated by every element, allowing universal bounds for all measurements.

What would settle it

Perform a dense random search over Hermitian measurement operators O for the n=3 elliptical-disk set of Fig. 3, evaluating the exact achievable range {o}_k for every element k of a fine grid inside the set. If any measured value lies outside the supremum interval [λ_sup·λ↑(O), λ_sup·λ↓(O)]—or, for the inner bound, if the infimum interval fails to be contained in the intersection of all achievable ranges—then Eq. (27) is false. The paper's examples show agreement but do not exhaust the search space.

Watch

Extended reading notes

Core claim

The central claim is that, for any compact set Λ of coherence spectra (ordered eigenvalue vectors of density matrices), the supremum λ_sup and infimum λ_inf of Λ in the majorization order satisfy for every Hermitian measurement O: {o}_inf ⊆ ⋂_k {o}_k ⊆ ⋃_k {o}_k ⊆ {o}_sup, where {o}_k is the interval of achievable values for spectrum k. Moreover, if any spectrum λ_u yields universal outer bounds, then λ_sup ≺ λ_u, and if λ_l yields universal inner bounds, then λ_l ≺ λ_inf; hence no single spectrum can produce tighter universal constraints. The proof hinges on the equivalence between majorization and containment of achievable ranges for all measurements (Eq. 11), and on the complete-lattice p

Load-bearing premise

The whole construction rests on the premise that one coherence spectrum majorizes another exactly when every linear measurement's achievable range for the first is contained in that for the second; if that equivalence has any exception, the universal bounds and their optimality claim collapse.

Editorial extensions

If this is right

  • For any family of partially coherent waves, one can precompute the supremum and infimum spectra once and then bound any observable without re-optimising per measurement.
  • Scalar measures such as entropy are insufficient for n≥3; the majorization-based sup/inf replace them as the correct notion of 'most/least coherent' for bounding purposes.
  • The tightness result means the universal bounds cannot be improved by any single spectrum; improvements would require abandoning single-spectrum bounds or using more than one spectrum.
  • The geometric classification tells designers that for smooth sets of coherence spectra, the extremal bounding spectra are not physically realisable inputs—they are virtual limits—so exact worst/best cases are approached, not attained, within the set.
  • The algorithm gives a practical route to compute these bounds for closed infinite sets via convex hull and polygon approximations with O(N^-2) convergence.

Reading between the lines

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

  • The same machinery should apply to any resource theory where a partial order captures universal constraints on linear functions—for example, entanglement or thermodynamic resource theories—so the sup/inf of a set in the appropriate order could yield optimal universal bounds there too.
  • A testable extension: for a fixed measurement O, the gap between the supremum bound and the exact outer bound measures how much is lost by insisting on a single spectrum; the paper's examples suggest the gap can be small, and it would be interesting to bound this gap as a function of the geometry of Λ.
  • The paper's claim that sup/inf lie outside smooth Λ implies that for smooth families, no single physically realisable state attains the universal bound; experiments would need to approach it asymptotically, which could be probed by sampling dense subsets of Λ.
  • The dependence of the algorithm's convergence rate on boundary smoothness (O(N^-2) in examples) might be generalised to higher dimensions, where inscribing/circumscribing polytopes would have more complex convergence behaviour.
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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

0 major / 6 minor

Summary. The manuscript develops a majorization-based theory for bounding linear observables of partially coherent waves when the input coherence spectrum varies over a set Λ ⊆ Δ↓_n. For compact Λ it proves that the exact outer and inner bounds of an observable o = tr(ρO) over Λ are attained by maximal and minimal elements of Λ (Eqs. (21)–(22)). It then introduces the supremum λ_sup and infimum λ_inf of Λ in the majorization order, which exist because (Δ↓_n, ≺) is a complete lattice, and proves that these two spectra give universal, and optimal, single-spectrum bounds: for every Hermitian O, {o}_inf ⊆ ⋂_k {o}_k ⊆ ⋃_k {o}_k ⊆ {o}_sup, and any other spectrum with a universal outer (inner) bound lies above λ_sup (below λ_inf), so the bounds cannot be tightened (Eqs. (27)–(29)). An algorithm for computing λ_sup and λ_inf by polytope bracketing is presented, and the geometric location theorem states that for compact convex Λ with nonempty interior these extrema lie at singular boundary points or outside Λ. Numerical simulations for waveguide transport and n = 3, 4 examples illustrate the results.

Significance. The core optimality result is significant and, given the two cited inputs (Eq. (11) and completeness of the majorization lattice), rigorously established. It replaces scalar coherence measures, which fail for n ≥ 3, with a parameter-free, measurement-independent pair of spectral bounds. The proof of Eqs. (27)–(29) is clean and does not rely on fitting or ad hoc assumptions. The numerical demonstrations are consistent with the theory and include a physically relevant diffusive-waveguide setting. The paper's strengths are its conceptual clarity and the fact that the key optimality claims are checkable from standard majorization facts.

minor comments (6)
  1. [Abstract and Sec. VI] The location theorem is proved in Appendix G under the assumptions that Λ is compact, convex, and has nonempty interior (and n ≥ 3), but the abstract and introduction state it without these qualifications. Please restrict the claim or state the assumptions clearly where the theorem is advertised.
  2. [Sec. II, Eqs. (8)–(9)] The union and intersection of the intervals {o}_k are written as single intervals. This is true because every {o}_k contains tr(O)/n, so the union and intersection are themselves intervals with endpoints attained by compactness. A sentence explaining this would prevent an apparent gap.
  3. [Sec. V and Appendix F] The algorithm is called convergent with an O(N^{-2}) trend, but Appendix F only presents numerical fits; no proof is given for general compact convex sets. Please either provide a proof for the claimed convergence rate or soften the language to numerical evidence.
  4. [Sec. V] The algorithm is described and demonstrated for two-dimensional convex hulls in Δ↓_3. The paper's scope is n-mode waves; please clarify the extent to which the algorithm generalizes to n > 3.
  5. [Appendix G] In Proposition 2, 'singular boundary point' is defined via the normal cone after assuming convexity. Please make that definition and the convexity assumption explicit in the main-text statement of the location theorem.
  6. [Sec. III, Eq. (11)] Equation (11) is load-bearing for the central optimality argument. It is cited to Ref. [43]; a short proof or derivation in an appendix would make the paper more self-contained. This is a presentation issue rather than a correctness concern.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the sup/inf optimality proof is a corollary of an independent, parameter-free prior theorem (Eq. 11) and the external complete-lattice property; no fitted parameter is relabeled as a prediction.

full rationale

The derivation chain is: Eq. (6) gives the exact measurement range from a coherence spectrum; Eq. (11), cited to Ref. [43], states the equivalence between majorization and containment of all Hermitian-measurement ranges; Eqs. (23)-(24) define the supremum and infimum via the lattice property of majorization; Eqs. (27)-(29) then prove that these give optimal universal bounds. The only load-bearing assumptions are Eq. (11) and the completeness of (Delta_down_n, ≺). Eq. (11) is a parameter-free theorem with stated assumptions (n-dimensional density matrices and Hermitian measurements) and does not include the target result of this paper; it is independently derivable from the explicit interval formula in Eq. (6) by checking partial-sum inequalities. Although Ref. [43] shares an author with the present paper, the citation is real, checkable evidence rather than an unverified self-citation, so under the review rules it does not raise the circularity score. The complete-lattice property is cited to external Refs. [79-81] and is a standard mathematical fact, not an input tailored to the conclusion. The optimality proof is a direct translation of the defining universal property of a supremum/infimum through Eq. (11); it is a corollary, not an assumption of the result. No parameters are fitted to data and then announced as predictions: the numerical examples compare the computed sup/inf intervals with independently computed exact outer and inner bounds, and the geometric location theorem (Sec. VI, Appendix G) is proved from scratch. Any potential concern about Eq. (8)'s union-as-a-single-interval representation would be a correctness matter, not a circularity, and it does not affect the central optimality argument.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters were fitted: the theory introduces no constants, and the algorithm's tolerance ε=0.01 is a numerical control, not a fitted value. The axioms are the standard lattice/majorization facts and the physical modeling assumptions (unitary control, linear measurements) that the paper takes as input. The invented-entities list is empty because the supremum and infimum spectra are mathematical constructs, not new physical objects; they are allowed to lie outside the physically realizable set, so they are not postulated entities with independent evidence.

assumptions (5)
  • standard math The partially ordered set (Δ↓_n, ≺) is a complete lattice; every subset Λ has a unique supremum and infimum in Δ↓_n.
    Used in Sec. IV to define λ_sup and λ_inf; cited to Refs [79-81].
  • domain assumption Eq. (11): λ↓(ρ_1)≺λ↓(ρ_2) iff {o}_1 ⊆ {o}_2 for all Hermitian O.
    Central bridge between coherence order and observable ranges; cited to Ref. [43] by the same group; assumed without proof in this paper.
  • domain assumption Eq. (6): For a single ρ, the range of a Hermitian measurement O under unitary control is [λ↓(ρ)·λ↑(O), λ↓(ρ)·λ↓(O)].
    Used throughout; cited to Refs [36,37,43].
  • domain assumption The coherence spectrum set Λ is compact (and for the location theorem, convex with nonempty interior).
    Stated in Sec. III (compact) and Sec. VI (convex, nonempty interior). The convexity is used in Proposition 2 but omitted from the abstract.
  • domain assumption Unitary control can realize any U∈U(n) to generate all density matrices with the same spectrum.
    Physical realizability via spatial light modulators etc.; cited Refs [48-69].

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Pith. "Pith review of Optimal universal bounds for waves with varied coherence based on supremum and infimum coherence spectra." pith.science (2026). https://pith.science/paper/AKNXY23H

@misc{pith2026260110665,
  author       = {Pith},
  title        = {Pith review of: Optimal universal bounds for waves with varied coherence based on supremum and infimum coherence spectra},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AKNXY23H}},
  note         = {Machine review of arXiv:2601.10665}
}
read the original abstract

We establish a majorization-based theory for bounding observables of waves with varied coherence. For any measurement, exact bounds are attained by the maximal and minimal elements in the set of input coherence spectra. The set's supremum and infimum, which may lie outside the set, provide optimal universal bounds: any alternative spectrum yielding universal bounds produces weaker constraints. We present an algorithm to compute the supremum and infimum, and prove that they lie either at singular boundary points or strictly outside the set of coherence spectra.

Figures

Figures reproduced from arXiv: 2601.10665 by the authors.

Figure 1
Figure 1. FIG. 1. Transport of waves with varied coherence. (a) Par [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Schematic of a compact set Λ of coherence spectra. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Supremum and infimum of a compact set Λ [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Simulation setup for power delivery in diffusive waveguides. TM-polarized light at vacuum wavelength [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Coherence spectra that attain the outer bounds (filled circles) and inner bounds (empty circles) for (a) [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: shows a finite set Λ with n = 4 comprising partially coherent waves with coherence spectra λ ↓ (ρg) = (0.55, 0.35, 0.08, 0.02), λ ↓ (ρm) = (0.62, 0.17, 0.15, 0.06), λ ↓ (ρn) = (0.56, 0.23, 0.14, 0.07). (D1) The supremum and infimum of Λ are λ ↓ sup = (0.62, 0.28, 0.08,…
Figure 7
Figure 7. Figure 7: FIG. 7. Numerical example for a closed infinite set Λ with [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Achievable measured values [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Additional numerical example for a closed infinite set Λ with [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Geometric construction of inscribing and circumscribing polygons for a 2D convex set conv(Λ). For a given direction [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Inscribing (blue) and circumscribing (red) polygons for representative geometric shapes (black), constructed using [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Convergence of the supremum and infimum computations as a function of the number of polygon vertices [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]

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Works this paper leans on

90 extracted references · 1 linked inside Pith

  1. [1]

    Born and E

    M. Born and E. Wolf,Principles of Optics: Electromag- netic Theory of Propagation, Interference and Diffrac- tion of Light, 7th ed. (Cambridge University Press, Cam- 6 bridge, 1999)

    1999

  2. [2]

    J. W. Goodman,Statistical Optics, Wiley Classics Li- brary (Wiley, New York, 2000)

    2000

  3. [3]

    Mandel and E

    L. Mandel and E. Wolf,Optical Coherence and Quantum Optics(Cambridge University Press, Cambridge; New York, 1995)

    1995

  4. [4]

    E. L. O’Neill,Introduction to Statistical Optics(Dover Publications, Mineola, New York, 2003)

    2003

  5. [5]

    Wolf,Introduction to the Theory of Coherence and Polarization of Light(Cambridge University Press, Cam- bridge; England, 2007)

    E. Wolf,Introduction to the Theory of Coherence and Polarization of Light(Cambridge University Press, Cam- bridge; England, 2007)

    2007

  6. [6]

    R. J. Glauber, The quantum theory of optical coherence, Phys. Rev.130, 2529 (1963)

    1963

  7. [7]

    Mandel and E

    L. Mandel and E. Wolf, Coherence properties of optical fields, Rev. Mod. Phys.37, 231 (1965)

    1965

  8. [8]

    Peˇ rina,Coherence of Light, 2nd ed

    J. Peˇ rina,Coherence of Light, 2nd ed. (D. Reidel Pub- lishing Company, Dordrecht, Holland, 1985)

    1985

Show all 90 references
  1. [9]

    Korotkova,Theoretical Statistical Optics(World Sci- entific, Singapore, 2022)

    O. Korotkova,Theoretical Statistical Optics(World Sci- entific, Singapore, 2022)

    2022

  2. [10]

    Hariharan, Optical interferometry, Rep

    P. Hariharan, Optical interferometry, Rep. Prog. Phys. 54, 339 (1991)

    1991

  3. [11]

    T. Udem, R. Holzwarth, and T. W. H¨ ansch, Optical fre- quency metrology, Nature416, 233 (2002)

    2002

  4. [12]

    B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Aber- nathy, F. Acernese, K. Ackley,et al., Observation of grav- itational waves from a binary black hole merger, Phys. Rev. Lett.116, 061102 (2016)

    2016

  5. [13]

    Schnars and W

    U. Schnars and W. J¨ uptner, Direct recording of holo- grams by a CCD target and numerical reconstruction, Appl. Opt.33, 179 (1994)

    1994

  6. [14]

    Genevet and F

    P. Genevet and F. Capasso, Holographic optical meta- surfaces: A review of current progress, Rep. Prog. Phys. 78, 024401 (2015)

    2015

  7. [15]

    Javidi, A

    B. Javidi, A. Carnicer, A. Anand, G. Barbastathis, W. Chen, P. Ferraro,et al., Roadmap on digital hologra- phy [Invited], Opt. Express29, 35078 (2021)

    2021

  8. [16]

    Pierrottet, F

    D. Pierrottet, F. Amzajerdian, L. Petway, B. Barnes, G. Lockard, and M. Rubio, Linear FMCW laser radar for precision range and vector velocity measurements, MRS Online Proc. Libr.1076, 10760406 (2008)

    2008

  9. [17]

    C. V. Poulton, A. Yaacobi, D. B. Cole, M. J. Byrd, M. Raval, D. Vermeulen, and M. R. Watts, Coherent solid-state LiDAR with silicon photonic optical phased arrays, Opt. Lett.42, 4091 (2017)

    2017

  10. [18]

    Huang, E

    D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gre- gory, C. A. Puliafito, and J. G. Fujimoto, Optical coher- ence tomography, Science254, 1178 (1991)

    1991

  11. [19]

    A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, Optical coherence tomography—principles and applications, Rep. Prog. Phys.66, 239 (2003)

    2003

  12. [20]

    J. W. Goodman, Some fundamental properties of speckle*, J. Opt. Soc. Am.66, 1145 (1976)

    1976

  13. [21]

    Redding, M

    B. Redding, M. A. Choma, and H. Cao, Speckle-free laser imaging using random laser illumination, Nat. Photonics 6, 355 (2012)

    2012

  14. [22]

    Lin, Partially coherent imaging in two dimensions and the theoretical limits of projection printing in microfab- rication, IEEE Trans

    B. Lin, Partially coherent imaging in two dimensions and the theoretical limits of projection printing in microfab- rication, IEEE Trans. Electron Devices27, 931 (1980)

    1980

  15. [23]

    Y. Luo, D. Mengu, N. T. Yardimci, Y. Rivenson, M. Veli, M. Jarrahi, and A. Ozcan, Design of task-specific opti- cal systems using broadband diffractive neural networks, Light Sci. Appl.8, 112 (2019)

    2019

  16. [24]

    Q. Jia, B. Shi, Y. Zhang, H. Li, X. Li, R. Feng,et al., Par- tially coherent diffractive optical neural network, Optica 11, 1742 (2024)

    2024

  17. [25]

    Zernike, The concept of degree of coherence and its application to optical problems, Physica5, 785 (1938)

    F. Zernike, The concept of degree of coherence and its application to optical problems, Physica5, 785 (1938)

    1938

  18. [26]

    Karczewski, Degree of coherence of the electromag- netic field, Phys

    B. Karczewski, Degree of coherence of the electromag- netic field, Phys. Lett.5, 191 (1963)

    1963

  19. [27]

    Tervo, T

    J. Tervo, T. Set¨ al¨ a, and A. T. Friberg, Degree of coher- ence for electromagnetic fields, Opt. Express11, 1137 (2003)

    2003

  20. [28]

    R´ efr´ egier and F

    P. R´ efr´ egier and F. Goudail, Invariant degrees of coher- ence of partially polarized light, Opt. Express13, 6051 (2005)

    2005

  21. [29]

    Set¨ al¨ a, J

    T. Set¨ al¨ a, J. Tervo, and A. T. Friberg, Contrasts of Stokes parameters in Young’s interference experiment and elec- tromagnetic degree of coherence, Opt. Lett.31, 2669 (2006)

    2006

  22. [30]

    Laue, Die entropie von partiell koh¨ arenten strahlenb¨ undeln, Ann

    M. Laue, Die entropie von partiell koh¨ arenten strahlenb¨ undeln, Ann. Phys. (Berlin)328, 1 (1907)

    1907

  23. [31]

    Gamo, Thermodynamic entropy of partially coherent light beams, J

    H. Gamo, Thermodynamic entropy of partially coherent light beams, J. Phys. Soc. Jpn.19, 1955 (1964)

    1955

  24. [32]

    H. Cao, A. P. Mosk, and S. Rotter, Shaping the prop- agation of light in complex media, Nat. Phys.18, 994 (2022)

    2022

  25. [33]

    Shaughnessy, R

    L. Shaughnessy, R. E. McIntosh, A. Goetschy, C. W. Hsu, N. Bender, H. Yılmaz, A. Yamilov, and H. Cao, Multiregion light control in diffusive media via wavefront shaping, Phys. Rev. Lett.133, 146901 (2024)

    2024

  26. [34]

    Guo and S

    C. Guo and S. Fan, Majorization theory for unitary con- trol of optical absorption and emission, Phys. Rev. Lett. 130, 146202 (2023)

    2023

  27. [35]

    C. Guo, D. A. B. Miller, and S. Fan, Unitary control of multiport wave transmission, Phys. Rev. A111, 023507 (2025)

    2025

  28. [36]

    Guo and S

    C. Guo and S. Fan, Unitary control of partially coherent waves. I. Absorption, Phys. Rev. B110, 035430 (2024)

    2024

  29. [37]

    Guo and S

    C. Guo and S. Fan, Unitary control of partially coherent waves. II. Transmission or reflection, Phys. Rev. B110, 035431 (2024)

    2024

  30. [38]

    M. A. Nielsen, Conditions for a class of entanglement transformations, Phys. Rev. Lett.83, 436 (1999)

    1999

  31. [39]

    G. Gour, M. P. M¨ uller, V. Narasimhachar, R. W. Spekkens, and N. Yunger Halpern, The resource theory of informational nonequilibrium in thermodynamics, Phys. Rep.583, 1 (2015)

    2015

  32. [40]

    Luis, Coherence for vectorial waves and majorization, Opt

    A. Luis, Coherence for vectorial waves and majorization, Opt. Lett.41, 5190 (2016)

    2016

  33. [41]

    Bengtsson and K

    I. Bengtsson and K. ˙Zyczkowski,Geometry of Quantum States: An Introduction to Quantum Entanglement, 2nd ed. (Cambridge University Press, Cambridge; England, 2017)

    2017

  34. [42]

    G. Gour, D. Jennings, F. Buscemi, R. Duan, and I. Mar- vian, Quantum majorization and a complete set of en- tropic conditions for quantum thermodynamics, Nat. Commun.9, 5352 (2018)

    2018

  35. [43]

    C. Guo, D. A. Miller, and S. Fan, Transport measure- ments of majorization order for wave coherence, Phys. Rev. Lett.135, 053801 (2025)

    2025

  36. [44]

    Wolf and G

    P.-E. Wolf and G. Maret, Weak localization and coher- ent backscattering of photons in disordered media, Phys. Rev. Lett.55, 2696 (1985)

    1985

  37. [45]

    Yamazoe, Coherency matrix formulation for partially coherent imaging to evaluate the degree of coherence for 7 image, J

    K. Yamazoe, Coherency matrix formulation for partially coherent imaging to evaluate the degree of coherence for 7 image, J. Opt. Soc. Am. A29, 1529 (2012)

    2012

  38. [46]

    Okoro, H

    C. Okoro, H. E. Kondakci, A. F. Abouraddy, and K. C. Toussaint Jr, Demonstration of an optical-coherence con- verter, Optica4, 1052 (2017)

    2017

  39. [47]

    B. E. A. Saleh,Quantum Photonics: Bimodes, Qubits, and Biphotons, Graduate Texts in Physics (Springer Na- ture Switzerland, Cham, 2025)

    2025

  40. [48]

    I. M. Vellekoop and A. P. Mosk, Focusing coherent light through opaque strongly scattering media, Opt. Lett.32, 2309 (2007)

    2007

  41. [49]

    S. M. Popoff, A. Goetschy, S. Liew, A. D. Stone, and H. Cao, Coherent control of total transmission of light through disordered media, Phys. Rev. Lett.112, 133903 (2014)

    2014

  42. [50]

    H. Yu, K. Lee, and Y. Park, Ultrahigh enhancement of light focusing through disordered media controlled by mega-pixel modes, Opt. Express25, 8036 (2017)

    2017

  43. [51]

    M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, Experimental realization of any discrete unitary opera- tor, Phys. Rev. Lett.73, 58 (1994)

    1994

  44. [52]

    D. A. Miller, Establishing optimal wave communication channels automatically, J. Lightwave Technol.31, 3987 (2013)

    2013

  45. [53]

    D. A. Miller, Self-aligning universal beam coupler, Opt. Express21, 6360 (2013)

    2013

  46. [54]

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

    2013

  47. [55]

    Carolan, C

    J. Carolan, C. Harrold, C. Sparrow, E. Mart ´ ın-L´ opez, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Mat- suda, M. Oguma, M. Itoh,et al., Universal linear optics, Science349, 711 (2015)

    2015

  48. [56]

    D. A. Miller, Perfect optics with imperfect components, Optica2, 747 (2015)

    2015

  49. [57]

    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

  50. [58]

    Ribeiro, A

    A. Ribeiro, A. Ruocco, L. Vanacker, and W. Bogaerts, Demonstration of a 4×4-port universal linear circuit, Op- tica3, 1348 (2016)

    2016

  51. [59]

    C. M. Wilkes, X. Qiang, J. Wang, R. Santagati, S. Paesani, X. Zhou, D. A. Miller, G. D. Marshall, M. G. Thompson, and J. L. O’Brien, 60 dB high- extinction auto-configured Mach–Zehnder interferome- ter, Opt. Lett.41, 5318 (2016)

    2016

  52. [60]

    Annoni, E

    A. Annoni, E. Guglielmi, M. Carminati, G. Fer- rari, M. Sampietro, D. A. Miller, A. Melloni, and F. Morichetti, Unscrambling light—automatically undo- ing strong mixing between modes, Light Sci. Appl.6, e17110 (2017)

    2017

  53. [61]

    Perez, I

    D. Perez, I. Gasulla, F. J. Fraile, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, Silicon photonics rectan- gular universal interferometer, Laser Photonics Rev.11, 1700219 (2017)

    2017

  54. [62]

    D. A. Miller, Setting up meshes of interferometers– reversed local light interference method, Opt. Express25, 29233 (2017)

    2017

  55. [63]

    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)

    2018

  56. [64]

    S. Pai, B. Bartlett, O. Solgaard, and D. A. B. Miller, Ma- trix optimization on universal unitary photonic devices, Phys. Rev. Appl.11, 064044 (2019)

    2019

  57. [65]

    Morizur, L

    J.-F. Morizur, L. Nicholls, P. Jian, S. Armstrong, N. Treps, B. Hage, M. Hsu, W. Bowen, J. Janousek, and H.-A. Bachor, Programmable unitary spatial mode ma- nipulation, J. Opt. Soc. Am. A27, 2524 (2010)

    2010

  58. [66]

    Labroille, B

    G. Labroille, B. Denolle, P. Jian, P. Genevaux, N. Treps, and J.-F. Morizur, Efficient and mode selective spatial mode multiplexer based on multi-plane light conversion, Opt. Express22, 15599 (2014)

    2014

  59. [67]

    Kupianskyi, S

    H. Kupianskyi, S. A. R. Horsley, and D. B. Phillips, High- dimensional spatial mode sorting and optical circuit de- sign using multi-plane light conversion, APL Photonics 8, 026101 (2023)

    2023

  60. [68]

    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, Phys. Rev. Appl.19, 054002 (2023)

    2023

  61. [69]

    Zhang and N

    Y. Zhang and N. K. Fontaine, Multi-plane light con- version: A practical tutorial (2023), arXiv:2304.11323 [physics.optics]

    2023 arXiv

  62. [70]

    A. W. Marshall, I. Olkin, and B. C. Arnold,Inequalities: Theory of Majorization and Its Applications, 2nd ed., Springer Series in Statistics (Springer Science+Business Media, LLC, New York, 2011)

    2011

  63. [71]

    B. A. Davey and H. A. Priestley,Introduction to Lattices and Order, 2nd ed. (Cambridge university press, Cam- bridge, England, 2002)

    2002

  64. [72]

    Cicalese and U

    F. Cicalese and U. Vaccaro, Supermodularity and sub- additivity properties of the entropy on the majorization lattice, IEEE Trans. Inf. Theory48, 933 (2002)

    2002

  65. [73]

    F. D. Cunden, P. Facchi, G. Florio, and G. Gramegna, Generic aspects of the resource theory of quantum coher- ence, Phys. Rev. A103, 022401 (2021)

    2021

  66. [74]

    V. Jain, M. Kwan, and M. Michelen, Entangled states are typically incomparable, arXiv:2406.03335 (2024)

    2024

  67. [75]

    Harling, V

    M. Harling, V. A. Kelkar, K. C. Toussaint, and A. F. Abouraddy, Isoentropic partially coherent optical fields that cannot be interconverted unitarily, Phys. Rev. A 110, 013505 (2024)

    2024

  68. [76]

    B. S. W. Schr¨ oder,Ordered Sets(Birkh¨ auser Boston, Boston, MA, 2003)

    2003

  69. [77]

    Boyd and L

    S. Boyd and L. Vandenberghe,Convex Optimization, 1st ed. (Cambridge University Press, Cambridge, 2004)

    2004

  70. [78]

    Roman,Lattices and Ordered Sets, 1st ed

    S. Roman,Lattices and Ordered Sets, 1st ed. (Springer, New York, NY, 2008)

    2008

  71. [79]

    P. M. Alberti and A. Uhlmann,Stochasticity and Par- tial Order: Doubly Stochastic Maps and Unitary Mixing, Mathematics and Its Applications, Vol. 9 (D. Reidel Pub- lishing Co., Dordrecht, Holland, 1982)

    1982

  72. [80]

    Bapat, Majorization and singular values

    R. Bapat, Majorization and singular values. III, Linear Algebra Appl.145, 59 (1991)

    1991

  73. [81]

    G. M. Bosyk, G. Bellomo, F. Holik, H. Freytes, and G. Sergioli, Optimal common resource in majorization- based resource theories, New J. Phys.21, 083028 (2019)

    2019

  74. [82]

    P. M. Gruber,Convex and Discrete Geometry, Grundlehren Der Mathematischen Wissenschaften (Springer Berlin Heidelberg, Berlin; New York, 2007)

    2007

  75. [83]

    H.-C. Lin, Z. Wang, and C. W. Hsu, MESTI,https: //github.com/complexphoton/MESTI.m

  76. [84]

    H.-C. Lin, Z. Wang, and C. W. Hsu, Fast multi-source nanophotonic simulations using augmented partial fac- torization, Nat. Comput. Sci.2, 815 (2022)

    2022

  77. [85]

    Prince and A

    J. Prince and A. Willsky, Reconstructing convex sets from support line measurements, IEEE Trans. Pattern Anal. Mach. Intell.12, 377 (1990). 8 Appendix A: Simulation of power delivery in a diffusive waveguide In this section, we provide computational details for the power deliv...

    1990

  78. [86]

    Data for Fig. 1 Forn= 2, the eigenvalues ofI A andI B are λ↓(IA) = (1.47,0.53),λ ↓(IB) = (1.36,0.64).(B1) The eigenvalues of inputs{ρ}={ρ a, ρb, ρc}are λ↓(ρa) = (0.81,0.19),λ ↓(ρb) = (0.71,0.29),λ ↓(ρc) = (0.61,0.39).(B2) 9 Forn= 3, the eigenvalues ofI A andI B are λ↓(IA) = (1...

  79. [87]

    permutation

    Data for Fig. 3 The set Λ in Fig. 3(a) is an elliptical disk in ∆ ↓ 3 centered at (0.55,0.32,0.13), with major axis of length 0.32 along the (0.05,0.04,−0.09) direction and minor axis of length 0.12. Its supremum and infimum are λ↓ sup = (0.65,0.29,0.06),λ ↓ inf = (0.45,0.36,0...

  80. [88]

    Finite set Figure 6 shows a finite set Λ withn= 4 comprising partially coherent waves with coherence spectra λ↓(ρg) = (0.55,0.35,0.08,0.02),λ ↓(ρm) = (0.62,0.17,0.15,0.06),λ ↓(ρn) = (0.56,0.23,0.14,0.07).(D1) The supremum and infimum of Λ are λ↓ sup = (0.62,0.28,0.08,0.02),λ ↓...

  81. [89]

    The set Λ is a line segment in ∆ ↓ 2 with endpoints (0.90,0.10) and (0.60,0.40)

    Closed infinite set Figure 7 shows a closed infinite set withn= 2. The set Λ is a line segment in ∆ ↓ 2 with endpoints (0.90,0.10) and (0.60,0.40). The supremum and infimum are its endpoints: λ↓ sup = (0.90,0.10),λ ↓ inf = (0.60,0.40).(D6) The outer and inner bounds (light and...

  82. [90]

    Inscribing polygon.—We construct the inscribing polygon by identifying boundary points of conv(Λ) that are tangent to a series of supporting lines

    We focus on the cases where the convex hull conv(Λ) is a 2D region embedded in ∆ ↓ 3. Inscribing polygon.—We construct the inscribing polygon by identifying boundary points of conv(Λ) that are tangent to a series of supporting lines. For each angleθ i =i(2π/N) withi= 1, . . . ...

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