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REVIEW 4 major objections 5 minor 56 references

Unveiling Hierarchical Invariants in Multiphoton Linear Optics

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Three conserved purity parts govern multiphoton evolution in linear optical networks.

desk verdict A clean, careful experimental paper whose three 'invariants' reduce by its own equations to one known invariant; the formalism is useful, but the novelty claims overshoot. read the letter →

arxiv 2506.12857 v2 pith:DPNQMWPV submitted 2025-06-15 quant-ph

Baichuan Yang , Hao Zhan , Minghao Mi , Aonan Zhang , Liang Xu , Lijian Zhang This is my paper · ORCID
classification quant-ph
keywords linearopticalnetworksmultiphotoninterferencestatepurityinvariantsHermitiantransfermatrixJordan-Schwingermaptwo-photonstatesreachability
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

Linear optical networks cannot perform arbitrary multiphoton unitary evolutions, because a scattering matrix has far fewer parameters than the multiphoton Fock space. This paper argues that the missing structure is captured by a Hermitian transfer matrix: under any lossless linear-optical unitary, the multiphoton density matrix splits into three orthogonal components, and the squared length of each component is individually conserved. The total state purity is the sum of these three conserved purity-like invariants, so each subspace contributes a fixed amount of purity no matter how the photons are scrambled. The authors verify the conservation for two-photon, two-mode states using both full tomography and a direct measurement that needs only three settings. If correct, the invariants provide necessary conditions for state reachability and a practical way to certify multiphoton probes.

What carries the argument

The Hermitian transfer matrix (HTM) is the central object: an orthogonal matrix R_V whose entries are Hilbert-Schmidt overlaps tr(H_i V H_j V†), the multiphoton analogue of the Pauli transfer matrix. The Jordan-Schwinger map sends scattering generators h ∈ H(m) to multiphoton generators H ∈ H(M), splitting H(M) into the tangent space T (single-photon-like generators, dimension m²) and its orthogonal complement P (genuinely multiphoton generators). For any LON-realizable V, the HTM is block-diagonal in this split, and the single-photon block equals the HTM of the scattering matrix S. This block-diagonality is what turns the previously known tangent invariant into three separate purity-like invariants.

What would settle it

Prepare a two-photon state whose traceless tangent invariant I_t′ is known, apply the eight U(2) unitaries used in the paper, and measure I_t′ after each; if the spread among the measured values scales with the reported 5% phase error and 1.5% polarization impurity in a way that the error model cannot explain, the claimed conservation is an artifact. A sharper test is to compute the off-block entries of the HTM for a randomly chosen S: any nonzero entry (R_V)_{ij} with i in the tangent subspace and j in the perpendicular subspace would directly falsify the block-diagonal structure.

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Extended reading notes

Core claim

The central claim is that the evolution of an n-photon, m-mode state through a lossless linear optical network is governed by a Hermitian transfer matrix with a fixed block-diagonal form, 1 ⊕ R(T′)_V ⊕ R(P)_V, where the first block is the trivial photon-number direction, the middle block is the single-photon (traceless tangent) subspace, and the third block is the multi-photon (perpendicular) subspace. Because the blocks never mix, the squared norm of the density-vector component in each subspace is conserved, giving the photon-number invariant I_n = 1/M, the traceless tangent invariant I_t′, and the perpendicular invariant I_p. Their sum equals the total purity tr(ρ²), so the purity contribution of each subspace is a separate invariant of linear-optical evolution. The paper derives this structure from the Jordan-Schwinger map and confirms it experimentally for two-photon two-mode states, observing that I_t′, I_t and I_p stay at their theoretical values under eight different U(2) scatterings.

Load-bearing premise

The experimental confirmation assumes the wave-plate groups and the non-polarizing beam splitter behave as ideal lossless linear optics acting on near-ideal two-photon states, so that measured deviations from the predicted invariant values are just small errors rather than genuine violations.

Editorial extensions

If this is right

  • The total purity of any n-photon m-mode state splits into three individually conserved parts, so no linear-optical network can transfer purity between the single-photon and multi-photon sectors of the state.
  • The invariants provide necessary conditions for reachability: two states can be connected by a LON only if their I_t′, I_p and I_n match, ruling out many pairs even when total photon number matches.
  • The direct-measurement method obtains I_t′ from m²−1 observables, avoiding the combinatorially growing cost of full tomography and enabling invariant verification for larger systems.
  • The framework extends to quantum processes on multiphoton states and could bound the expressivity of photonic neural networks and assist metrological probe certification.

Reading between the lines

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

  • A natural extension the paper does not pursue is applying the same block-decomposition to other homomorphic encodings of single-particle unitaries, such as fermionic or qudit linear optics, where the tangent space would be the appropriate Lie subalgebra.
  • Because I_t′ is a quadratic function of mode-operator expectation values, the direct-measurement protocol could be adapted as a fast certification tool for boson-sampling devices, where full tomography is intractable.
  • The reported 96.8% HOM visibility implies a small but nonzero part of the state leaves the ideal two-photon subspace; tracking how loss and mode mismatch move purity among the invariant sectors could quantify device non-ideality.
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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 / 5 minor

Summary. The paper introduces a Hermitian transfer matrix (HTM) formalism for n-photon m-mode linear optical networks. It shows that the n-photon unitary's adjoint action is block-diagonal with blocks 1, R_S (single-photon dynamics), and R_V^(P) (multi-photon interference), so the norms of the density-vector components in the traceless tangent and perpendicular subspaces are conserved. The authors call these purity-like invariants I_t', I_p, and I_n. They report an experiment with two-photon two-mode polarization-encoded states in which eight input states are subjected to eight U(2) unitaries, and the invariants are extracted by quantum state tomography and by a direct measurement protocol. The measured values cluster around the theoretical predictions.

Significance. The HTM construction is a clean and parameter-free derivation of known block structure, and the paper provides a direct-measurement protocol for the traceless tangent invariant with only m^2-1 observables, which could be useful for larger systems. The experimental data are consistent with the conservation law and the paper honestly reports preparation imperfections. However, the central conceptual claim that the purity decomposes into 'three distinct invariants' is overstated: I_n is a constant, and I_p is determined by total purity and I_t'. Moreover, SM Theorem 2 shows I_t' is an affine function of the observable invariant I_o already measured by Rodari et al. (Ref. [45]). The genuinely new content is therefore the HTM interpretation and the efficient measurement scheme, not a new hierarchy of independent invariants or a high-order generalization supported by the two-photon experiment. Significance as a verification experiment is moderate.

major comments (4)
  1. [Theoretical framework, Eqs. (3)-(5)] The three quantities I_n, I_t', and I_p are not independent. Since I_n = 1/M for every n-photon state and tr(rho^2) is invariant under all unitary evolutions, Eq. (5) implies I_p = tr(rho^2) - 1/M - I_t'; conservation of I_p follows from conservation of I_t' plus unitary invariance of the purity. The abstract and introduction should not describe the purity as decomposing into 'three distinct invariants' or a 'hierarchy of invariants' without clarifying that only the traceless tangent invariant is a new LON-specific constraint, while I_p is a derived quantity. This also affects the interpretation of Fig. 3(f), which cannot independently confirm a perpendicular invariant.
  2. [SM Theorem 2 and Note on Ref. [45]] SM Theorem 2 establishes I_t' = I_o/((m+n choose m+1)) - n^2/(m (m+n choose m+1)) up to normalization, so the traceless tangent invariant carries exactly the same experimental content as the observable invariant I_o. Since Ref. [45] already reported an experimental observation of I_o, the present experiment re-confirms a known quantity in a degenerate two-mode, two-photon setting. The paper must clearly state that the novelty is the HTM interpretation and the direct-measurement method, not the discovery of a new invariant, and it should temper the abstract's claim to 'high-order invariants.'
  3. [Experiments, SM Sec. VII C and Fig. 3] The paper does not provide a systematic uncertainty budget for the direct-measurement method. The SM reports a pre-compensation phase error of about 5% and a residual polarization impurity of about 1.5% that are reduced to below 0.1% by QHQ compensation, but the uncertainties of the compensation angles are not propagated to the invariants, and the error bars in Fig. 3(d-e) reflect only Poissonian counting statistics. Without this budget, the agreement between the measured and theoretical invariants cannot be distinguished from mitigation of implementation artifacts, especially for the states with large alpha where the invariant is small (e.g., Table S1, alpha = 90 deg, I_t' = 0).
  4. [Experiments, Fig. 3(f) and Conclusion] The experimental verification is restricted to n=2, m=2, and pure states. In this setting the perpendicular space has dimension 5, but the purity constraint fixes I_p = 2/3 - I_t' for pure states, so the measurement of I_p provides no evidence about the multi-photon interference block R_V^(P) beyond what is already contained in I_t'. The conclusion's statement that the experiment confirms 'the conservation of these invariants' should be qualified to the two-photon, two-mode, pure-state case, and the proposed applications to state reachability and metrology are not demonstrated by the data.
minor comments (5)
  1. [Abstract] The phrase 'high-order invariants' is undefined; for n=2 the invariants are quadratic functions of the density matrix, and the experiments do not probe higher photon numbers.
  2. [SM Sec. II] The trace formulas in Eqs. (S5)-(S7) use inconsistent summation index i in Eq. (S7) with a free parameter in the binomial; these should be rewritten with different indices for clarity.
  3. [SM Sec. VI, Eq. (S45)] The proof of Lemma 1 uses an unusual bra-ket notation for expectation values that is difficult to parse; a standard notation such as tr(rho O)^2 would improve readability.
  4. [Experimental setup, Sec. Experiments] There are typographical errors in the text, including 'comsisting' for 'consisting' and in the SM 'donate' for 'denote' and 'T raceless' for 'Traceless'.
  5. [Fig. 3] The figure caption does not state the number of experimental runs or the method used to compute the standard deviation in panels (a) and (d); this information should be added for reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the invariant conservation law is a derived theorem and the experiment tests, rather than fits, the theory.

full rationale

The derivation chain is self-contained and non-circular. The invariants in Eqs. (3)-(5) are defined as squared norms of the components of the density vector in a basis obtained from the Jordan-Schwinger map; their conservation is proven from the block-diagonal structure of the Hermitian transfer matrix (SM Sec. III), which follows from Lie-algebra homomorphism properties and the independence of the tangent/perpendicular decomposition. This invariance result is cited to Ref. [17], an independent prior work, and is not smuggled in as an ansatz. No parameter is fitted to the experimental data and then relabeled as a prediction: the theoretical values of I_t, I_t', and I_p are computed directly from the prepared state |ψ_α⟩, and the measured values are compared against these fixed curves. The relation I_p = tr(ρ²) − I_n − I_t' is a mathematical consequence of basis completeness and is explicitly visible from Eqs. (3)-(5); it means the perpendicular invariant is not an independent addition, but this is a redundancy among derived corollaries, not a circular use of the conclusion. Likewise, SM Theorem 2 and the Note on Ref. [45] explicitly give the affine relation between I_t' and the earlier observable invariant I_o, so the paper does not conceal the connection to prior work. Experimental imperfections are characterized separately (HOM visibility 96.8%, residual polarization and phase errors) and are not used to force agreement. Accordingly, no step in the paper's derivation reduces to its own input, and no load-bearing self-citation occurs.

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

The central theoretical claim rests on standard quantum mechanics and the Jordan-Schwinger map being a Lie algebra homomorphism, both standard tools. The experiment assumes faithful wave-plate unitaries and post-selected two-photon states. No free parameters are fitted to make the invariant theory work; the only curve fitting (HOM dip) is calibration. No new entities are introduced.

assumptions (5)
  • domain assumption Lossless linear optical network implements any unitary S in U(m) on single-photon modes (Reck decomposition, Ref [23]).
    Used to define LON-realizable multi-photon unitaries V = phi(S) in the theoretical framework section.
  • standard math The Jordan-Schwinger map dphi is a Lie algebra homomorphism, satisfying dphi(Ad_S(h)) = Ad_phi(S)(dphi(h)) (SM Eq. S21).
    Core of the proof that R_V^T = R_S in SM Sec. III.
  • standard math The multi-photon unitary is the permanent homomorphism phi: U(m) to U(M) (Ref [12]).
    Defines the evolution V under which invariants are conserved.
  • standard math The chosen Gell-Mann-style basis {H_i} forms an orthonormal Hermitian basis for H(M); completion of the perpendicular space uses Gram-Schmidt.
    Needed for the density-vector expansion and for the norms I_t', I_p to be well-defined.
  • domain assumption Post-selected HOM interference prepares the ideal state |psi_alpha> = cos alpha |2H,0> + sin alpha |1H,1V>.
    Directly used to compute theoretical invariant values in Table S1; deviations quantified by HOM visibility.

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Cite this review

Pith. "Pith review of Unveiling Hierarchical Invariants in Multiphoton Linear Optics." pith.science (2026). https://pith.science/paper/DPNQMWPV

@misc{pith2026250612857,
  author       = {Pith},
  title        = {Pith review of: Unveiling Hierarchical Invariants in Multiphoton Linear Optics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DPNQMWPV}},
  note         = {Machine review of arXiv:2506.12857}
}
read the original abstract

Linear optical networks driven by quantum states of light are important building blocks of photonic quantum technologies. They access large bosonic Hilbert spaces through multiphoton interference. At the same time, their dynamics are generated by single-particle mode transformations, thereby defining a highly structured subset of multiphoton unitaries and setting boundary on linear optics capability. To elucidate this boundary, we reveal an underlying fine-grained symmetry structure that partitions the multiphoton operator space into invariant subspaces and generates a hierarchy of invariants. We experimentally confirm the conservation of high-order invariants and demonstrate their operational utility in characterizing state reachability and the metrological capability of multiphoton probes. Our framework provides a symmetry-based perspective for understanding and harnessing structured multiphoton dynamics across photonic quantum technologies.

Figures

Figures reproduced from arXiv: 2506.12857 by the authors.

Figure 1
Figure 1. FIG. 1. Theoretical framework of multi-photon linear optics. (a) The scattering unitary [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Experimental setup. (a) State Preparation: Photon pairs are generated via spontaneous parametric down-conversion [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Experimental observation of invariants under LON unitaries. (a, d) Average experimental invariant [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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Works this paper leans on

56 extracted references · 40 canonical work pages

  1. [45]

    Observation of lie al- gebraic invariants in quantum linear optics,

    G. Rodari, T. Francalanci, E. Caruccio, F. Hoch, G. Car- vacho, T. Giordani, N. Spagnolo, R. Albiero, N. D. Gi- ano, F. Ceccarelli, G. Corrielli, A. Crespi, R. Osellame, U. Chabaud, and F. Sciarrino, “Observation of lie al- gebraic invariants in quantum linear optics,” (2025), arXiv:2505.03001 [quant-ph]

    arXiv 2025

  2. [1]

    Slussarenko and G

    S. Slussarenko and G. J. Pryde, Applied Physics Reviews 6, 041303 (2019)

    2019

  3. [2]

    Flamini, N

    F. Flamini, N. Spagnolo, and F. Sciarrino, Reports on Progress in Physics82, 016001 (2018)

    work page 2018

  4. [3]

    Knill, R

    E. Knill, R. Laflamme, and G. J. Milburn, Nature409, 46 (2001)

    2001

  5. [4]

    Peruzzo, J

    A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien, Nature Communications5, 4213 (2014)

    2014

  6. [5]

    L. Xiao, T. Deng, K. Wang, Z. Wang, W. Yi, and P. Xue, 6 Phys. Rev. Lett.126, 230402 (2021)

    work page 2021

  7. [6]

    W. Luo, L. Cao, Y. Shi, L. Wan, H. Zhang, S. Li, G. Chen, Y. Li, S. Li, Y. Wang, S. Sun, M. F. Karim, H. Cai, L. C. Kwek, and A. Q. Liu, Light: Science & Applications12, 175 (2023)

    work page 2023

  8. [7]

    W. Ge, K. Jacobs, Z. Eldredge, A. V. Gorshkov, and M. Foss-Feig, Phys. Rev. Lett.121, 043604 (2018)

    work page 2018

Show all 56 references
  1. [8]

    Nagata, R

    T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, Science316, 726 (2007)

    2007

  2. [9]

    Hou, J.-F

    Z. Hou, J.-F. Tang, J. Shang, H. Zhu, J. Li, Y. Yuan, K.- D. Wu, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, Nature Communications9, 1414 (2018)

    2018

  3. [10]

    Takeda and A

    S. Takeda and A. Furusawa, APL Photonics4, 060902 (2019)

    2019

  4. [11]

    P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Rev. Mod. Phys.79, 135 (2007)

    2007

  5. [12]

    Aaronson and A

    S. Aaronson and A. Arkhipov, inProceedings of the Forty-Third Annual ACM Symposium on Theory of Com- puting, STOC ’11 (Association for Computing Machin- ery, New York, NY, USA, 2011) p. 333–342

    2011

  6. [13]

    C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, Phys. Rev. Lett.119, 170501 (2017)

    2017

  7. [14]

    Zhong, H

    H.-S. Zhong, H. Wang, Y.-H. Deng, M.-C. Chen, L.-C. Peng, Y.-H. Luo, J. Qin, D. Wu, X. Ding, Y. Hu, P. Hu, X.-Y. Yang, W.-J. Zhang, H. Li, Y. Li, X. Jiang, L. Gan, G. Yang, L. You, Z. Wang, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, Science370, 1460 (2020)

    2020

  8. [15]

    A. W. Harrow and A. Montanaro, Nature549, 203 (2017)

    2017

  9. [16]

    J. J. Moyano-Fern ˜A¡ndez and J. C. Garcia-Escartin, Op- tics Communications382, 237 (2017)

    2017

  10. [17]

    P. V. Parellada, V. Gimeno i Garcia, J. J. Moyano- Fern˜A¡ndez, and J. C. Garcia-Escartin, Results in Physics54, 107108 (2023)

    2023

  11. [18]

    Lie algebraic invariants in quantum linear optics,

    P. V. Parellada, V. G. i Garcia, J. J. Moyano-Fern´ andez, and J. C. Garcia-Escartin, “Lie algebraic invariants in quantum linear optics,” (2024), arXiv:2409.12223 [quant-ph]

    2024 arXiv

  12. [19]

    See Supplemental Material for the detailed theoretical and experimental information

  13. [20]

    R. A. Bertlmann and P. Krammer, Journal of Physics A: Mathematical and Theoretical41, 235303 (2008)

    2008

  14. [21]

    B. C. Hall,Lie Groups, Lie Algebras, and Representa- tions: An Elementary Introduction, Graduate Texts in Mathematics, Vol. 222 (Springer International Publish- ing, Cham, 2015)

    2015

  15. [22]

    Zyczkowski and M

    K. Zyczkowski and M. Kus, Journal of Physics A: Math- ematical and General27, 4235 (1994)

    1994

  16. [23]

    M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, Phys. Rev. Lett.73, 58 (1994)

    1994

  17. [24]

    Bouland and S

    A. Bouland and S. Aaronson, Phys. Rev. A89, 062316 (2014)

    2014

  18. [25]

    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, G. D. Marshall, M. G. Thomp- son, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, Science349, 711 (2015)

    2015

  19. [26]

    W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walmsley, Optica3, 1460 (2016)

    2016

  20. [27]

    Permanents in linear optical networks,

    S. Scheel, “Permanents in linear optical networks,” (2004), arXiv:quant-ph/0406127 [quant-ph]

    2004 arXiv

  21. [28]

    J. C. Garcia-Escartin, V. Gimeno, and J. J. Moyano- Fern´ andez, Phys. Rev. A100, 022301 (2019)

    2019

  22. [29]

    G. Park, I. Matsumoto, T. Kiyohara, H. F. Hofmann, R. Okamoto, and S. Takeuchi, Science Advances9, eadj8146 (2023)

    2023

  23. [30]

    J. C. Garcia-Escartin, V. Gimeno, and J. J. Moyano- Fern´ andez, Optics Communications430, 434 (2019)

    2019

  24. [31]

    Jordan, Zeitschrift f¨ ur Physik94, 531 (1935)

    P. Jordan, Zeitschrift f¨ ur Physik94, 531 (1935)

    1935

  25. [32]

    Schwinger,ON ANGULAR MOMENTUM, Tech

    J. Schwinger,ON ANGULAR MOMENTUM, Tech. Rep. (Harvard Univ., Cambridge, MA (United States); Nu- clear Development Associates, Inc. (US), 1952)

    1952

  26. [33]

    A. Y. Kitaev, A. Shen, and M. N. Vyalyi,Classical and quantum computation, 47 (American Mathematical Soc., 2002)

    2002

  27. [34]

    Introduction to quantum gate set to- mography,

    D. Greenbaum, “Introduction to quantum gate set to- mography,” (2015), arXiv:1509.02921 [quant-ph]

    2015 arXiv

  28. [35]

    Banchi, W

    L. Banchi, W. S. Kolthammer, and M. S. Kim, Phys. Rev. Lett.121, 250402 (2018)

    2018

  29. [36]

    R. A. Campos, B. E. A. Saleh, and M. C. Teich, Phys. Rev. A40, 1371 (1989)

    1989

  30. [37]

    C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987)

    1987

  31. [38]

    Hong-ou-mandel interference,

    A. M. Bra´ nczyk, “Hong-ou-mandel interference,” (2017), arXiv:1711.00080 [quant-ph]

    2017 arXiv

  32. [39]

    R. B. A. Adamson, L. K. Shalm, M. W. Mitchell, and A. M. Steinberg, Phys. Rev. Lett.98, 043601 (2007)

    2007

  33. [40]

    Piani, D

    M. Piani, D. Pitkanen, R. Kaltenbaek, and N. L¨ utkenhaus, Phys. Rev. A84, 032304 (2011)

    2011

  34. [41]

    T. Ono, W. Roga, K. Wakui, M. Fujiwara, S. Miki, H. Terai, and M. Takeoka, Phys. Rev. Lett.131, 013601 (2023)

    2023

  35. [42]

    B. Y. Gan, D. Leykam, and D. G. Angelakis, EPJ Quan- tum Technology9, 16 (2022)

    2022

  36. [43]

    Ferdous, M

    J. Ferdous, M. Hong, R. B. Dawkins, F. Mostafavi, A. Oktyabrskaya, C. You, R. d. J. Le´ on-Montiel, and O. S. Magana-Loaiza, ACS Photonics11, 3197 (2024)

    2024

  37. [44]

    Bae and L.-C

    J. Bae and L.-C. Kwek, Journal of Physics A: Mathemat- ical and Theoretical48, 083001 (2015)

    2015

  38. [46]

    Simon and N

    R. Simon and N. Mukunda, Physics Letters A138, 474 (1989). 7 SUPPLEMENT AR Y MA TERIAL: EXPERIMENT AL OBSER V A TION OF PURITY-LIKE INV ARIANTS OF MUL TIPHOTON ST A TES IN LINEAR OPTICS I. Generalized Gell-Mann matrices The Generalized Gell-Mann matrices (GGM) [20] generalize ...

    1989

  39. [47]

    symmetric matrix:λ (m) i = 1√ 2 (Ekj +E jk ), k < j

  40. [48]

    antisymmetric matrix:λ (m) i =− i√ 2 (Ekj −E jk ), k > j

  41. [49]

    diagonal matrix:λ (m) i = 1√ l(l+1) Pl j=1 Ejj −lE l+1,l+1 ,1≤l≤m−1 In this work, we adopt a convention where each GGM element is normalized by a factor of 1/ √

  42. [50]

    tr λ(m) i tr λ(m) j − X k λ(m) i kk λ(m) j kk # (k=l, s=t, k̸=t) + m+n m+ 1

    Hence, unless otherwise specified, all references to GGM herein pertain to the normalized form. The GGM exhibit the following fundamental properties: •hermitian: λ(m) i † =λ (m) i . •traceless: tr λ(m) i = 0. •orthonormal: tr λ(m) i λ(m) j =δ i,j. In the case of 2-level system...

  43. [51]

    Reality: The entries ofR V are real as: tr HiV HjV † † = tr V HjV †Hi = tr HiV HjV † .(S15)

  44. [52]

    Using the Hilbert-Schmidt inner product: ⟨ ⟨V†HiV|V †HjV⟩ ⟩= X k ⟨ ⟨V†HiV|k⟩ ⟩⟨ ⟨k|V†HjV⟩ ⟩ = X k tr HkV †HiV tr HkV †HjV = X k (RV )ki(RV )kj

    Orthogonality: The matrixR V is orthogonal. Using the Hilbert-Schmidt inner product: ⟨ ⟨V†HiV|V †HjV⟩ ⟩= X k ⟨ ⟨V†HiV|k⟩ ⟩⟨ ⟨k|V†HjV⟩ ⟩ = X k tr HkV †HiV tr HkV †HjV = X k (RV )ki(RV )kj . (S16) Meanwhile, the orthonormality of{H i}gives: ⟨ ⟨V†HiV|V †HjV⟩ ⟩= tr(HiHj) =δ ij.(S1...

  45. [53]

    Block-Diagonality: Due to the limited degrees of freedomm2 in a LON,R V cannot fully achieve the orthogonal group O(M 2). However, the invariance of the tangentTand perpendicularPspaces—i.e.,V †HiV∈ Tif and only if Hi ∈ T, and likewise forP—ensures thatR V is block-diagonal: (...

  46. [54]

    The following derivation relies on the adjoint representation and the consistency condition for Lie algebra homomorphisms from the group theory

    Equivalence ofR(T) V andR S The matrixR S represent the HTM of scattering matrix, define in the hermitian basis{h i}with entries given by (R S)ij = tr hiShjS† . The following derivation relies on the adjoint representation and the consistency condition for Lie algebra homomorp...

  47. [55]

    For any multiphoton quantum stateρ, ⟨ ⟨0|ρ⟩ ⟩= tr(ρH0) = 1√ M tr(ρ) = 1√ M .(S25) This implies thatR V acts as the identity on|0⟩ ⟩

    F urther Block-Diagonality: Within tangent spaceT, the basis elementH 0 =I(M)/ √ Mis the only non- traceless operator. For any multiphoton quantum stateρ, ⟨ ⟨0|ρ⟩ ⟩= tr(ρH0) = 1√ M tr(ρ) = 1√ M .(S25) This implies thatR V acts as the identity on|0⟩ ⟩. The entries (RV )0j = tr ...

  48. [56]

    The tangent traceless invariantI t′ ofρ α corresponding to the squared distance from the origin. θ α|ψ α⟩I t Ip It′ Io 0◦ 0◦ |2H ,0 V ⟩0.833 0.167 0.5 4 7.5◦ 10.7◦ 0.983|2 H ,0 V ⟩+ 0.186|1 H ,1 V ⟩0.833 0.167 0.500 3.998 11.25◦ 16.3◦ 0.960|2 H ,0 V ⟩+ 0.281|1 H ,1 V ⟩0.830 0....

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Reviewed August 7, 2026 · model on record in the stance chip above.