Pith. sign in

REVIEW 3 major objections 4 minor 89 references

Stellar rank is not about photon addition; it witnesses particle entanglement once superselection rules are taken into account.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Quadrature non-Gaussianity and nonzero stellar rank are shown to be witnesses of particle entanglement rather than photon addition, with a generalized basis-dependent stellar rank proposed.

T0 review reviewed 2026-08-02 challenge →

load-bearing objection A genuinely new interpretation of stellar rank as a particle-entanglement witness, but the key CV-limit theorem is asserted rather than proved and deferred to missing supplementary material; worth refereeing, not yet citable. the 3 major comments →

arxiv 2603.20810 v1 pith:BYYPJR34 submitted 2026-03-21 quant-ph

Non-Gaussianity from superselection rules

Nicolas Moulonguet , Eloi Descamps , Jos\'e Lorger\'e , Astghik Saharyan , Arne Keller , P\'erola Milman This is my paper
classification quant-ph
keywords stellar rankMajorana polynomialsnon-Gaussianitysuperselection rulesparticle entanglementBargmann functioncontinuous-variable limitphase reference
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 replace the usual story that stellar rank counts how many photons were added to a Gaussian state. Treating the phase reference as a quantum degree of freedom, the authors show that the continuous-variable limit of the Majorana polynomial exists only for states whose normalization mass is concentrated in a small disk; in that limit a bounded number of roots survive and become the zeros of the Bargmann function, i.e., the stellar rank. Because symmetric N-boson states that are not Fock states are particle-entangled, any state with a nonzero stellar rank is necessarily particle-entangled in the superselection-respecting description. The connection is also basis-dependent: changing the computational basis changes what counts as Gaussian, so the stellar rank is only one instance of a generalized, basis-dependent witness.

Core claim

On the paper's own terms, the central discovery is that the stellar rank — the number of zeros of the Bargmann function, usually read as the number of displaced photon additions — is, in the proper superselection-rule framework, a witness of particle entanglement. The argument runs through the Majorana polynomial P_N(z) that represents a pure state of N photons plus a quantum phase-reference mode. Normalizing this polynomial and taking the limit N→∞ to recover the Bargmann function forces the condition that the normalization mass lies in |z|≲√K with K ≪ √N; only roots inside this region survive, so the stellar rank is bounded by K, not by N. Consequently r*≠0 implies the original SSRC state

What carries the argument

The Majorana polynomial P_N(z) = Σ √(C(N,n)) c_n z^n — the analytic representation of a symmetric N-boson state that includes a quantized phase-reference mode — is the main object. Its normalization integral over the complex plane with the measure (1+|z|^2/N)^{-(N+2)} reduces, under the change of variable z→z/√N, to the Gaussian measure that defines the Bargmann function. The paper's limiting argument (Stirling approximation for coefficients, then convergence of truncated polynomials on compact disks and tracking of their zeros) shows exactly which Majorana roots survive and become the stellar rank, and which are pushed outside the physical region.

Load-bearing premise

The proof that the limiting zeros of the truncated Majorana polynomials coincide with the zeros of the Bargmann function relies on an unproved uniform-convergence/growth condition on the state coefficients; if that condition fails, the stellar rank might not be readable from the Majorana roots.

What would settle it

Construct a normalized sequence of SSRC states for which the integral (4) satisfies I_D→1 on a disk D but the truncated polynomials P_{N,K} do not converge uniformly on D (e.g., coefficients with slow decay that still pass the bound). If the limit function's zeros differ from the Bargmann function's zeros, the claimed CV-limit equivalence and the entanglement-witness interpretation collapse. Such a counterexample only needs one family of states, and can be checked numerically by computing zeros of P_N(z/√N) and B(z).

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • A nonzero stellar rank becomes a sufficient condition for particle entanglement in the superselection-respecting picture, so entanglement witnesses can be built from stellar-rank witnesses.
  • The stellar rank of any physical continuous-variable state is effectively bounded (r* ≪ √N), which limits how much of the Majorana constellation can be accessed in the CV regime.
  • Quadrature non-Gaussianity is not a basis-independent resource; a state Gaussian in one phase space can be non-Gaussian in another, so claims of quantum advantage must specify the computational basis.
  • The infinite-dimensional CV Hilbert space is an effective description: only SSRC states with energy-constrained coefficients admit a normalized Bargmann representation.
  • Generalized stellar rank defined in arbitrary phase-space representations is proposed as a basis-dependent witness of potential quantum advantage.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same limiting structure should apply to other zero-based nonclassicality measures: if their zeros are defined by a holomorphic function, they will inherit the same boundedness and basis dependence; one could test this by rederiving Wigner-negativity thresholds in the SSRC picture.
  • The bound r* ≪ √N suggests a direct experimental test: prepare families of SSRC states with known N and extract the stellar rank of their CV limit; the rank should saturate well below N.
  • If the basis-dependence is taken seriously, quantum-advantage experiments in CV systems should be re-examined: a state that is non-Gaussian in the quadrature basis might become Gaussian in a rotated phase space, implying the advantage may be an artifact of the chosen encoding.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes that stellar rank, a continuous-variable (CV) non-Gaussianity hierarchy based on zeros of the Bargmann function, emerges as a limit of the roots of Majorana polynomials associated with finite-dimensional states that respect the photon-number superselection rule (SSR). The authors argue that this limit is not the naive N→∞ qudit-to-oscillator transition but requires energy constraints, yielding a scaling bound r⋆ ≤ K ≪ √N. They further claim that nonzero stellar rank acts as a witness of particle entanglement once the phase reference is quantized, and they propose a basis-dependent generalization of stellar rank to arbitrary computational bases using unitarily equivalent phase spaces.

Significance. If the central argument holds, the paper provides a physically motivated interpretation of the stellar rank and connects single-mode non-Gaussianity to particle entanglement, with potential implications for bosonic resource theories and quantum computation. The explicit analytic treatment of Fock and cat states usefully illustrates the relevant scaling issues, and the authors are careful to frame stellar rank as a witness rather than a necessary condition. However, the main mathematical bridge from Majorana polynomials to the Bargmann function is asserted rather than proved, and key technical steps are deferred to a Supplementary Material that is not included in the submission. The generalization section also contains an overstatement that is in tension with the paper's own squeezed-state example. These issues prevent acceptance in the present form.

major comments (3)
  1. [Results (paragraph after Eq. (4))] The Montel–Hurwitz argument is not justified as written. Montel's theorem only guarantees a subsequential limit for a locally bounded family; it does not identify the limit as the Bargmann function B. To recover the full stellar hierarchy, the authors must specify a diagonal sequence K=K(N) with K(N)→∞ and K(N)=o(√N), and prove uniform convergence of P_{N,K(N)} to B on compact sets. If K is fixed while N→∞, the limit is a polynomial of degree K, so infinitely many zeros—as in the odd cat state—cannot be recovered. Hurwitz's theorem requires uniform convergence of the actual sequence of functions, not merely existence of a subsequential limit. This step is load-bearing because the revised interpretation of stellar rank as a particle-entanglement witness relies on connecting the finite-N Majorana roots to the zeros of the Bargmann function.
  2. [Results and Supplementary Material [56]] Key proofs are deferred to a Supplementary Material that is not provided: the claim that normalization in C implies the existence of the truncated polynomial P_{N,K}, the localization argument leading to I_D→1, and the precise conditions under which I_D→1 are all in [56]. Since the necessity of the CV limit is a central claim, the manuscript cannot be fully verified without this material. The authors should include the Supplementary Material or fold these arguments into the main text.
  3. [Generalization (third paragraph)] The statement that if a unitary Û is not a rotation then it 'can, in the CV limit, be associated with r⋆(ψ)≠0' is too strong and is contradicted by the squeezed-state example. Squeezed states arise as CV limits of SSRC states generated by non-rotations (Ref. [50]) yet have r⋆=0, as the paper itself notes. The generalization to arbitrary computational bases needs a more precise criterion for which unitary transformations yield nonzero generalized stellar rank, rather than the current 'if—then' implication.
minor comments (4)
  1. [Methods (definition of Majorana polynomial)] The expression for P(z) contains 'z^*' before the inner product, which is inconsistent with the subsequent expansion in z^n. Please check whether this is a typesetting artifact or an intended complex conjugation.
  2. [Reference [56]] Reference [56] is given as 'See supplementary material [url]' with no URL or attachment. The actual supplementary material should be supplied with the submission.
  3. [Results (Fock-state example)] The notation for the roots of the Fock-state Majorana polynomial switches between z_j and z_i in consecutive sentences; please standardize.
  4. [Throughout] The condition 'K≪√N' is used without a precise asymptotic definition. Since the diagonalization K=K(N) is essential, the authors should specify the exact scaling, e.g., K(N)=o(√N), and state whether R is fixed or also scales with N.

Circularity Check

0 steps flagged

No significant circularity; the central derivation is self-contained.

full rationale

The paper's main chain of derivation — showing that the Majorana polynomial P_N(z/√N) tends to the Bargmann function B(z) under a constrained CV limit, and that the stellar rank emerges from the limiting roots — is a mathematical limit argument, not a circular reduction. The coefficients of the truncated Majorana polynomial are shown via Stirling's approximation to coincide with those of the truncated Bargmann–Fock expansion, and the limiting identification is made by comparing coefficients. No parameter is fitted to data and then renamed as a prediction. The claim that nonzero stellar rank witnesses particle entanglement is a one-way implication derived from the fact that separable SSRC states are Fock states whose CV limits are coherent states with zero stellar rank. This is not circular. The generalization to arbitrary computational bases invokes the authors' prior work (Ref. [39]), but this is an extension of the framework rather than a load-bearing reduction of the central result; self-citation alone does not constitute circularity. The Montel–Hurwitz passage identified by the skeptic is a potential rigor gap concerning the double limit N→∞, K→∞, and the identification of the subsequential limit, but a gap in proof is not an equivalence between the paper's inputs and its outputs. Therefore, no specific circular step can be exhibited, and the appropriate score is 0.

Axiom & Free-Parameter Ledger

1 free parameters · 3 axioms · 0 invented entities

The paper does not introduce new physical entities. The main assumptions are mathematical and domain-specific, revolving around the CV limit and the truncation of polynomials.

free parameters (1)
  • Truncation degree K = K << sqrt(N)
    The paper assumes a truncation degree K of the Majorana polynomial that is much smaller than sqrt(N) to achieve normalization in the CV limit. This is a crucial assumption that is not derived from first principles.
axioms (3)
  • domain assumption The CV limit is defined by |z|^2 << sqrt(N).
    This is an assumption about the regime where the Majorana polynomial converges to the Bargmann function, and it is central to the paper's claims.
  • domain assumption The Majorana polynomial can be truncated to degree K without significant error on compact sets.
    The paper assumes that for any fixed domain and epsilon, there exists a K << sqrt(N) such that the truncation error is bounded. This is not proved in detail.
  • standard math Montel's theorem and Hurwitz's theorem apply to the sequence of truncated polynomials.
    These theorems are invoked to show the limit function is holomorphic and its zeros are limits of the polynomial zeros, but the conditions for this are not fully verified.

reviewed 2026-08-02 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Non-Gaussianity from superselection rules." pith.science (2026). https://pith.science/paper/BYYPJR34

@misc{pith2026260320810,
  author       = {Pith},
  title        = {Pith review of: Non-Gaussianity from superselection rules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BYYPJR34}},
  note         = {Machine review of arXiv:2603.20810}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

The quantum theory of the electromagnetic field enables the description of multiphoton states exhibiting nonclassical statistical properties, often reflected in non-Gaussian phase-space distributions. While non-Gaussianity alone does not fully characterize quantum states, several classifications have been proposed to hierarchize non-Gaussian states according to physically or informationally relevant resources. Here, we provide a physical interpretation of non-Gaussianity and connect it to a computational perspective by showing how a prominent classification-the stellar rank-emerges as a limiting case of the roots of polynomials that univocally represent bosonic states defined with a quantized phase reference, namely the Majorana polynomials. A direct consequence of our results is a revised interpretation of both the stellar rank and non-Gaussianity itself: when superselection rules are properly taken into account, quadrature non-Gaussianity - and nonzero stellar rank - act as witnesses of particle entanglement, rather than being linked with photon addition to Gaussian states as previously assumed. In addition, we show that because the stellar rank depends on a specific choice of coherent states, its relation to computational resources and potential quantum advantage is inherently basis-dependent, being naturally tied to quadrature eigenstates as the computational basis. Motivated by this observation, we generalize the notion of stellar rank to arbitrary computational bases, thereby establishing it as a genuine witness of bosonic resources that may enable quantum advantage.

Figures

Figures reproduced from arXiv: 2603.20810 by Arne Keller, Astghik Saharyan, Eloi Descamps, Jos\'e Lorger\'e, Nicolas Moulonguet, P\'erola Milman.

Figure 1
Figure 1. Figure 1: FIG. 1: (Color online) Principles of the inverse [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

89 extracted references · 2 canonical work pages

  1. [1]

    Wigner, On the quantum correction for thermody- namic equilibrium, Phys

    E. Wigner, On the quantum correction for thermody- namic equilibrium, Phys. Rev.40, 749 (1932)

    1932

  2. [2]

    Weedbrook, S

    C. Weedbrook, S. Pirandola, R. Garc ´ ıa-Patr´ on, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaussian quantum information, Rev. Mod. Phys.84, 621 (2012)

    2012

  3. [3]

    Nichols, P

    R. Nichols, P. Liuzzo-Scorpo, P. A. Knott, and G. Adesso, Multiparameter gaussian quantum metrology, Phys. Rev. A98, 012114 (2018)

    2018

  4. [4]

    Maccone and A

    L. Maccone and A. Riccardi, Squeezing metrology: a uni- fied framework, Quantum4, 292 (2020)

    2020

  5. [5]

    Mari and J

    A. Mari and J. Eisert, Positive Wigner functions ren- der classical simulation of quantum computation efficient, Phys. Rev. Lett.109, 230503 (2012)

    2012

  6. [6]

    Veitch, C

    V. Veitch, C. Ferrie, D. Gross, and J. Emerson, Negative quasi-probability as a resource for quantum computation, New Journal of Physics14, 113011 (2012)

    2012

  7. [7]

    Veitch, S

    V. Veitch, S. H. Mousavian, D. Gottesman, and J. Emer- son, The resource theory of stabilizer quantum computa- tion, New Journal of Physics16, 013009 (2014). 6

    2014

  8. [8]

    S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, Efficient classical simulation of continuous variable quantum information processes, Physical Review Letters88, 097904 (2002)

    2002

  9. [9]

    S. D. Bartlett and B. C. Sanders, Efficient classical simu- lation of optical quantum circuits, Physical Review Let- ters89, 207903 (2002)

    2002

  10. [10]

    Crespi, R

    A. Crespi, R. Osellame, R. Ramponi, D. Brod, E. Galv˜ ao, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, Integrated multimode interferometers with arbitrary designs for photonic boson sampling, Nature Photonics7, 545 (2013)

    2013

  11. [11]

    Gross, Hudson’s theorem for finite-dimensional quan- tum systems, Journal of Mathematical Physics47, 122107 (2006)

    D. Gross, Hudson’s theorem for finite-dimensional quan- tum systems, Journal of Mathematical Physics47, 122107 (2006)

    2006

  12. [12]

    Hudson, When is the wigner quasi-probability density non-negative?, Reports on Mathematical Physics6, 249 (1974)

    R. Hudson, When is the wigner quasi-probability density non-negative?, Reports on Mathematical Physics6, 249 (1974)

    1974

  13. [13]

    Soto and P

    F. Soto and P. Claverie, When is the Wigner function of multidimensional systems nonnegative?, J. Math. Phys. 24, 97 (1983)

    1983

  14. [14]

    Knill, R

    E. Knill, R. Laflamme, and G. Milburn, A scheme for efficient quantum computation with linear optics, Nature 409, 46 (2001)

    2001

  15. [15]

    P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Linear optical quantum computing with photonic qubits, Rev. Mod. Phys.79, 135 (2007)

    2007

  16. [16]

    H. J. Kimble, M. Dagenais, and L. Mandel, Photon anti- bunching in resonance fluorescence, Phys. Rev. Lett.39, 691 (1977)

    1977

  17. [17]

    Gilchrist, K

    A. Gilchrist, K. Nemoto, W. J. Munro, T. C. Ralph, S. Glancy, S. L. Braunstein, and G. J. Milburn, Schr¨ odinger cats and their power for quantum informa- tion processing, Journal of Optics B: Quantum and Semi- classical Optics6, S828 (2004)

    2004

  18. [18]

    Li, C.-L

    L. Li, C.-L. Zou, V. V. Albert, S. Muralidharan, S. M. Girvin, and L. Jiang, Cat codes with optimal decoherence suppression for a lossy bosonic channel, Phys. Rev. Lett. 119, 030502 (2017)

    2017

  19. [19]

    Omanakuttan, V

    S. Omanakuttan, V. Buchemmavari, J. A. Gross, I. H. Deutsch, and M. Marvian, Fault-tolerant quantum com- putation using large spin-cat codes, PRX Quantum5, 020355 (2024)

    2024

  20. [20]

    Gottesman, A

    D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in an oscillator, Phys. Rev. A64, 012310 (2001)

    2001

  21. [21]

    Calcluth, A

    C. Calcluth, A. Ferraro, and G. Ferrini, Efficient simula- tion of Gottesman-Kitaev-Preskill states with Gaussian circuits, Quantum6, 867 (2022)

    2022

  22. [22]

    Chabaud, G

    U. Chabaud, G. Ferrini, F. Grosshans, and D. Markham, Classical simulation of Gaussian quantum circuits with non-Gaussian input states, Phys. Rev. Res.3, 033018 (2021)

    2021

  23. [23]

    Calcluth, O

    C. Calcluth, O. Hahn, J. Bermejo-Vega, A. Ferraro, and G. Ferrini, Classical simulation of circuits with real- istic odd-dimensional Gottesman-Kitaev-Preskill states, Phys. Rev. Lett.135, 010601 (2025)

    2025

  24. [24]

    Dias and R

    B. Dias and R. K¨ onig, Classical simulation of non- gaussian bosonic circuits, Phys. Rev. A110, 042402 (2024)

    2024

  25. [25]

    O. Hahn, R. Takagi, G. Ferrini, and H. Yamasaki, Clas- sical simulation and quantum resource theory of non- Gaussian optics, Quantum9, 1881 (2025)

    2025

  26. [26]

    Podhora, L

    L. Podhora, L. Lachman, T. Pham, A. Leˇ sund´ ak, O.ˇC ´ ıp, L. Slodiˇ cka, and R. Filip, Quantum non-gaussianity of multiphonon states of a single atom, Phys. Rev. Lett. 129, 013602 (2022)

    2022

  27. [27]

    Lachman, I

    L. Lachman, I. Straka, J. Hlouˇ sek, M. Jeˇ zek, and R. Filip, Faithful hierarchy of genuinen-photon quantum non- gaussian light, Phys. Rev. Lett.123, 043601 (2019)

    2019

  28. [28]

    Descamps, N

    E. Descamps, N. Fabre, A. Saharyan, A. Keller, and P. Milman, Superselection rules and bosonic quantum computational resources, Phys. Rev. Lett.133, 260605 (2024)

    2024

  29. [29]

    Chabaud and M

    U. Chabaud and M. Walschaers, Resources for bosonic quantum computational advantage, Phys. Rev. Lett. 130, 090602 (2023)

    2023

  30. [30]

    Popescu, Knill-laflamme-milburn linear optics quan- tum computation as a measurement-based computation, Phys

    S. Popescu, Knill-laflamme-milburn linear optics quan- tum computation as a measurement-based computation, Phys. Rev. Lett.99, 250501 (2007)

    2007

  31. [31]

    Chabaud, D

    U. Chabaud, D. Markham, and F. Grosshans, Stellar rep- resentation of non-Gaussian quantum states, Phys. Rev. Lett.124, 063605 (2020)

    2020

  32. [32]

    Provazn ´ ık,ˇSimon Br¨ auer, V

    J. Provazn ´ ık,ˇSimon Br¨ auer, V. Kala, J. Fiur´ aˇ sek, and P. Marek, Witnesses of non-gaussian features as lower bounds of stellar rank (2026), arXiv:2603.03185 [quant- ph]

    arXiv 2026

  33. [33]

    Fiur´ aˇ sek, Efficient construction of witnesses of the stel- lar rank of nonclassical states of light, Opt

    J. Fiur´ aˇ sek, Efficient construction of witnesses of the stel- lar rank of nonclassical states of light, Opt. Express30, 30630 (2022)

    2022

  34. [34]

    Chabaud, G

    U. Chabaud, G. Roeland, M. Walschaers, F. Grosshans, V. Parigi, D. Markham, and N. Treps, Certification of non-gaussian states with operational measurements, PRX Quantum2, 020333 (2021)

    2021

  35. [35]

    Bargmann, On a hilbert space of analytic functions and an associated integral transform

    V. Bargmann, On a hilbert space of analytic functions and an associated integral transform. part i, Communica- tions on Pure and Applied Mathematics14, 187 (1961)

    1961

  36. [36]

    I. E. Segal,Mathematical Problems of Relativistic Physics, Lectures in Applied Mathematics, Vol. 2 (Amer- ican Mathematical Society, Providence, RI, 1963)

    1963

  37. [37]

    Amiet and S

    J.-P. Amiet and S. Weigert, Contracting the wigner ker- nel of a spin to the wigner kernel of a particle, Phys. Rev. A63, 012102 (2000)

    2000

  38. [38]

    Ricci, A contraction of s u (2) to the heisenberg group., Monatshefte f¨ ur Mathematik101, 211 (1986)

    F. Ricci, A contraction of s u (2) to the heisenberg group., Monatshefte f¨ ur Mathematik101, 211 (1986)

    1986

  39. [39]

    Descamps, A

    E. Descamps, A. Saharyan, A. Keller, and P. Milman, Heisenberg-weyl bosonic phase spaces: emergence, con- straints and quantum informational resources (2025), arXiv:2512.05603 [quant-ph]

    arXiv 2025

  40. [40]

    B. C. Sanders, S. D. Bartlett, T. Rudolph, and P. L. Knight, Photon-number superselection and the entangled coherent-state representation, Phys. Rev. A68, 042329 (2003)

    2003

  41. [41]

    Mølmer, Optical coherence: A convenient fiction, Phys

    K. Mølmer, Optical coherence: A convenient fiction, Phys. Rev. A55, 3195 (1997)

    1997

  42. [42]

    optical coherence: A convenient fiction

    J. Gea-Banacloche, Comment on “optical coherence: A convenient fiction”, Phys. Rev. A58, 4244 (1998)

    1998

  43. [43]

    comment on ‘optical coherence: A convenient fiction’

    K. Mølmer, Reply to “comment on ‘optical coherence: A convenient fiction’ ”, Phys. Rev. A58, 4247 (1998)

    1998

  44. [44]

    S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Dialogue concerning two views on quantum co- herence: Factist and fictionist, International Journal of Quantum Information04, 17 (2006), https://doi.org/10.1142/S0219749906001591

    work page doi:10.1142/s0219749906001591 2006

  45. [45]

    B. C. Sanders, Review of entangled coherent states, Jour- nal of Physics A: Mathematical and Theoretical45, 244002 (2012)

    2012

  46. [46]

    Aharonov and L

    Y. Aharonov and L. Susskind, Charge superselection rule, 7 Phys. Rev.155, 1428 (1967)

    1967

  47. [47]

    S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Refer- ence frames, superselection rules, and quantum informa- tion, Rev. Mod. Phys.79, 555 (2007)

    2007

  48. [48]

    Schwinger, On angular momentum 10.2172/4389568

    J. Schwinger, On angular momentum 10.2172/4389568

    doi:10.2172/4389568

  49. [49]

    Saharyan, E

    A. Saharyan, E. Descamps, A. Keller, and P. Milman, Resources for bosonic metrology: quantum-enhanced precision from a superselection rule perspective (2025), arXiv:2507.13245 [quant-ph]

    arXiv 2025

  50. [50]

    Descamps, A

    E. Descamps, A. Saharyan, A. Chivet, A. Keller, and P. Milman, Unified framework for bosonic quantum in- formation encoding, resources and universality from su- perselection rules (2025), arXiv:2501.03943 [quant-ph]

    arXiv 2025

  51. [51]

    Vourdas, Quantum systems with finite hilbert space, Reports on Progress in Physics67, 267 (2004)

    A. Vourdas, Quantum systems with finite hilbert space, Reports on Progress in Physics67, 267 (2004)

    2004

  52. [52]

    Vourdas, Analytic representations in quantum me- chanics, Journal of Physics

    A. Vourdas, Analytic representations in quantum me- chanics, Journal of Physics. A, Mathematical and Gen- eral39, p. R65–R141 (2006)

    2006

  53. [53]

    Majorana, Atomi orientati in campo magnetico vari- abile, Il Nuovo Cimento (1924-1942)9, 43 (1932)

    E. Majorana, Atomi orientati in campo magnetico vari- abile, Il Nuovo Cimento (1924-1942)9, 43 (1932)

    1924

  54. [54]

    Chryssomalakos, E

    C. Chryssomalakos, E. Guzm´ an-Gonz´ alez, and E. Serrano-Ens´ astiga, Geometry of spin coherent states, Journal of Physics A: Mathematical and Theoretical51, 165202 (2018)

    2018

  55. [55]

    Holtz and J

    R. Holtz and J. Hanus, On coherent spin states, Journal of Physics A: Mathematical, Nuclear and General7, L37 (1974)

    1974

  56. [56]

    See supplementary material [url]

  57. [57]

    J. B. Conway,Functions of One Complex Variable I (Springer-Verlag, New York, 1978)

    1978

  58. [58]

    Arzani, R

    F. Arzani, R. I. Booth, and U. Chabaud, Effective de- scriptions of bosonic systems can be considered complete, Nature Communications16, 9744 (2025)

    2025

  59. [59]

    Upreti, D

    V. Upreti, D. Rudolph, and U. Chabaud, Bounding the computational power of bosonic systems (2025), arXiv:2503.03600 [quant-ph]

    Pith/arXiv arXiv 2025

  60. [60]

    Marshall and N

    J. Marshall and N. Anand, Simulation of quantum optics by coherent state decomposition, Optica Quantum1, 78 (2023)

    2023

  61. [61]

    V. A. Orlov, L. A. Markovich, A. N. Rubtsov, and V. I. Man’ko, From discrete to continuous-variable systems via jordan-schwinger tomographic transformation (2025), arXiv:2510.21476 [quant-ph]

    arXiv 2025

  62. [62]

    Maltesson, L

    A. Maltesson, L. Rodung, N. Budinger, G. Ferrini, and C. Calcluth, Equivalence of continuous- and discrete- variable gate-based quantum computers with finite en- ergy (2025), arXiv:2510.08546 [quant-ph]

    Pith/arXiv arXiv 2025

  63. [63]

    Walschaers, Non-gaussian quantum states and where to find them, PRX Quantum2, 030204 (2021)

    M. Walschaers, Non-gaussian quantum states and where to find them, PRX Quantum2, 030204 (2021)

    2021

  64. [64]

    A. S. Sørensen and K. Mølmer, Entanglement and ex- treme spin squeezing, Phys. Rev. Lett.86, 4431 (2001)

    2001

  65. [65]

    B. J. Dalton, L. Heaney, J. Goold, B. M. Garraway, and T. Busch, New spin squeezing and other entanglement tests for two mode systems of identical bosons, New Jour- nal of Physics16, 013026 (2014)

    2014

  66. [66]

    B. J. Dalton, J. Goold, B. M. Garraway, and M. D. Reid, Quantum entanglement for systems of identical bosons: Ii. spin squeezing and other entanglement tests, Physica Scripta92, 023005 (2017)

    2017

  67. [67]

    Fadel and M

    M. Fadel and M. Gessner, Relating spin squeezing to mul- tipartite entanglement criteria for particles and modes, Phys. Rev. A102, 012412 (2020)

    2020

  68. [68]

    Gutman, A

    N. Gutman, A. Gorlach, O. Tziperman, R. Ruimy, and I. Kaminer, Universal control of symmetric states using spin squeezing, Phys. Rev. Lett.132, 153601 (2024)

    2024

  69. [69]

    Gross, Non-negative wigner functions in prime dimen- sions, Applied Physics B86, 367 (2007)

    D. Gross, Non-negative wigner functions in prime dimen- sions, Applied Physics B86, 367 (2007)

    2007

  70. [70]

    Cormick, E

    C. Cormick, E. F. Galv˜ ao, D. Gottesman, J. P. Paz, and A. O. Pittenger, Classicality in discrete wigner functions, Phys. Rev. A73, 012301 (2006)

    2006

  71. [71]

    E. F. Galv˜ ao, Discrete wigner functions and quantum computational speedup, Phys. Rev. A71, 042302 (2005)

    2005

  72. [72]

    Chabaud and S

    U. Chabaud and S. Mehraban, Holomorphic representa- tion of quantum computations, Quantum6, 831 (2022)

    2022

  73. [73]

    Pezz` e, A

    L. Pezz` e, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Quantum metrology with nonclassical states of atomic ensembles, Rev. Mod. Phys.90, 035005 (2018)

    2018

  74. [74]

    F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Atomic coherent states in quantum optics, Phys. Rev. A 6, 2211 (1972)

    1972

  75. [75]

    Wang and B

    X. Wang and B. C. Sanders, Relations between bosonic quadrature squeezing and atomic spin squeezing, Phys. Rev. A68, 033821 (2003)

    2003

  76. [76]

    Giraud, P

    O. Giraud, P. Braun, and D. Braun, Quantifying quan- tumness and the quest for queens of quantum, New Jour- nal of Physics12, 063005 (2010)

    2010

  77. [77]

    Bastin, S

    T. Bastin, S. Krins, P. Mathonet, M. Godefroid, L. Lamata, and E. Solano, Operational families of en- tanglement classes for symmetricn-qubit states, Phys. Rev. Lett.103, 070503 (2009)

    2009

  78. [78]

    Ribeiro and R

    P. Ribeiro and R. Mosseri, Entanglement in the sym- metric sector ofnqubits, Phys. Rev. Lett.106, 180502 (2011)

    2011

  79. [79]

    AULBACH, Classification of entanglement in symmetric states, International Journal of Quantum Information10, 1230004 (2012), https://doi.org/10.1142/S0219749912300045

    M. AULBACH, Classification of entanglement in symmetric states, International Journal of Quantum Information10, 1230004 (2012), https://doi.org/10.1142/S0219749912300045

    work page doi:10.1142/s0219749912300045 2012

  80. [80]

    Wang and D

    Z. Wang and D. Markham, Nonlocality and entanglement for symmetric states, Phys. Rev. A87, 012104 (2013)

    2013

Showing first 80 references.

This paper was first reviewed by deepseek-v4-flash on August 2, 2026.