Pith. sign in

REVIEW 2 major objections 3 minor 54 references

Efficient construction of witnesses of stellar rank of nonclassical states of light

T0 review · 2 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read This paper shows that every stellar-rank witness threshold reduces to a supremum over Gaussian unitaries of a largest eigenvalue of a projected witness, computable with a rank-independent number of parameters.

desk verdict A clean, exact reformulation of stellar-rank witness thresholds that is useful for the quantum-optics community, with a genuine caveat about the reliability of the numerical supremum. read the letter →

arxiv 2412.14356 v1 pith:32NMRMV6 submitted 2024-12-18 quant-ph

Jaromír Fiurášek This is my paper · ORCID
classification quant-ph MSC 81P1681V80
keywords stellarrankquantumnon-GaussianstateswitnessoperatorsGaussianunitaryoperationsFockstateprobabilitiesSchrödingercatphoton-number-resolvingdetectiontracedistancelowerbound
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 establishes a general recipe for witnessing the stellar rank of optical quantum states. Instead of optimizing directly over the coefficients of the core Fock superposition, one optimizes over the Gaussian unitary that dresses the state, and for each Gaussian operation the best core state is simply the dominant eigenvector of the projected witness operator. The result is Eq. (12), which expresses the threshold as a maximum over Gaussian unitaries of the maximal eigenvalue of the projected witness. Because the initial phase shift commutes with the Fock-subspace projector, the effective optimization for a single mode is over four real parameters no matter how large the targeted stellar rank is, and over $O(N^2)$ parameters for $N$ modes. This matters because stellar rank counts the Fock resources needed to build a state together with Gaussian operations, so the thresholds certify genuine $n$-photon quantum non-Gaussianity in experiments.

What carries the argument

The central object is the projected, Gaussian-conjugated witness operator $\hat W_{n,\hat U_G} = \hat\Pi_{n-1}\hat U_G \hat W \hat U_G^\dagger \hat\Pi_{n-1}$, a Hermitian operator on the $n$-dimensional Fock subspace spanned by $|0\rangle,\dots,|n-1\rangle$. Eq. (12) says that its maximal eigenvalue, maximized over $\hat U_G$, is the witness threshold. The argument is carried by the observation that the initial phase shift commutes with $\hat\Pi_{n-1}$, so the Gaussian optimization is over squeezing $r$, one phase $\vartheta$, and the complex displacement $\alpha$; for Fock-diagonal witnesses the phase drops out and only three parameters remain. Numerical evaluation rests on the analytic Fock-basis matrix elements of Gaussian unitaries expressed as finite sums of Hermite polynomials. In the multimode case the same construction uses the Bloch-Messiah decomposition, and the output interferometer drops out because it commutes with the total-photon-number projector.

What would settle it

Take a specific witness, compute its threshold by Eq. (12) with a dense multistart or grid search over the Gaussian parameters, then evaluate the same witness on a known state of stellar rank $n-1$, such as a squeezed Fock state. If the expectation value ever comes out larger than the computed threshold, the optimization missed the global supremum and the published threshold is invalid.

Watch

Extended reading notes

Core claim

The paper claims that for every Hermitian witness operator $\hat W$ and every target stellar rank $n$, the threshold separating rank at least $n$ from lower ranks is exactly $$W_n = \sup_{\hat U_G} \max \mathrm{eig}\left(\hat\Pi_{n-1}\hat U_G \hat W \hat U_G^\dagger \hat\Pi_{n-1}\right),$$ where $\hat\Pi_{n-1}$ projects onto photon numbers $0,\dots,n-1$. The supremum over all pure lower-rank states is absorbed in two steps: the Gaussian unitary is separated from the core state, and for a fixed unitary the optimal core state is the largest-eigenvalue eigenstate of the projected operator. Thus the number of optimized parameters is fixed by the Gaussian operation alone, and for $\hat W = |w\rangle\langle w|$ the formula reduces to the earlier fidelity-based witnesses. The new generality allows witnesses built from arbitrary linear combinations of data, such as pairs of Fock-state probabilities or fidelities with even and odd coherent states, and the paper demonstrates these examples numerically.

Load-bearing premise

The numerical search over the squeezing, phase, and displacement parameters is assumed to reach the true global maximum rather than a local one, and a local maximum would make the threshold too low and could falsely certify a lower-rank state.

Editorial extensions

If this is right

  • Every Hermitian observable can be converted into a stellar-rank certificate, extending certification beyond projectors onto a single reference state.
  • Thresholds for any chosen $n$ can be computed with the same bounded parameter count, so high-rank certification avoids optimizing over the core-state coefficients.
  • The witness family is closed under Gaussian unitary conjugation: if $\hat W$ certifies rank $n$, then $\hat V_G \hat W \hat V_G^\dagger$ certifies the same rank with the same threshold.
  • Bounded witnesses give a quantitative lower bound on the trace distance from the set of states of stellar rank at most $n-1$, so witness violations measure how far the state is from that set.
  • For experiments, only a few probabilities or fidelities are needed, and the examples show that pairs of Fock probabilities or cat-state fidelities already certify ranks up to at least $n=3$.

Reading between the lines

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

  • Beyond the paper: the formula can be inverted for witness design, optimizing $\hat W$ to maximize the gap $\mathrm{Tr}[\hat W\hat\rho]-W_n$ for a target state $\hat\rho$, which would trade the fixed parameter count for a witness-design search.
  • Beyond the paper: the same spectral construction, with $\mathrm{SU}(2)$ rotations in place of Gaussian unitaries, would yield a stellar-rank hierarchy for spin states, a direction the paper sketches but leaves open.
  • Beyond the paper: because multiplexed click statistics are linear in thermal-state overlaps, realistic detector response functions might be folded into $\hat W$ analytically, making threshold computation fully analytic for common detector models.
  • Beyond the paper: a practical protocol could certify the highest stellar rank by sweeping $n$ and checking inequality (4) on the same measured data, without full state tomography.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper presents a method for constructing witnesses of the stellar rank of nonclassical states of light. For a Hermitian witness operator W, the threshold W_n is expressed as a supremum over Gaussian unitaries of the largest eigenvalue of a finite-dimensional operator: Eq. (12) for single-mode states and Eq. (18) for multimode states. This reduces the optimization from a search over core-state coefficients (whose number grows with n) to a fixed-dimensional search over Gaussian parameters (4 for one mode, O(N^2) for N modes). The method is illustrated with witnesses based on pairs of Fock-state probabilities and pairs of fidelities with even/odd coherent states, with threshold boundaries shown in Figs. 1 and 2. The paper also gives a lower bound on the trace distance to lower-rank states in Eq. (22).

Significance. The central derivation is exact: Eq. (5) to Eq. (12) follows from the definition of W_n as a supremum, with the saturation of the eigenvalue inequality by the core-state eigenvector. The multimode extension via Bloch-Messiah decomposition is well justified, and the recovery of the fidelity-witness results of Ref. [25] provides a useful consistency check. If the numerical suprema displayed in Figs. 1 and 2 are correct, the method is a significant practical advance: it turns stellar-rank witness construction into a fixed-parameter optimization problem whose dimension does not grow with the targeted rank. The trace-distance bound in Eq. (22) adds further value. The main uncertainty is whether the reported numerical thresholds are indeed global suprema, a point that is not secured in the manuscript.

major comments (2)
  1. [Sec. 3, Eq. (12) and Figs. 1-2] The practical validity of the constructed witnesses depends on computing the supremum over Gaussian unitaries in Eq. (12) and its multimode analogue (18). The paper reports numerical optimization over three or four real parameters but provides no evidence that the global supremum is reached. The objective is not shown to be convex or unimodal, the parameter domain is non-compact (the squeezing parameter r can diverge), and the manuscript does not specify the algorithm, the number of starting points, or the tolerances used. If the optimizer returns a local maximum, the computed W_n is below the true threshold and inequality (4) can falsely certify states of lower stellar rank. This is load-bearing for the central claim that witness thresholds can be efficiently constructed. The authors should either supply a global-optimization procedure or a rigorous upper bound on W_n, or explicitly present the thresholds as candidate values requiring independent validation. Making the numerical code and data behind Figs. 1 and 2 available would considerably strengthen the paper.
  2. [Sec. 4] The statement in Sec. 4 that 'for higher stellar ranks fully numerical approaches seem unavoidable' is in tension with the presentation of the thresholds as exact witness values. The paper should clarify that the thresholds are exact only if the numerical supremum is exact, and should discuss the consequences of an approximate optimization for the soundness of the witness inequality (4). A concrete remedy would be to cross-check the reported thresholds with an independent global optimization method (e.g., differential evolution or a grid-based multistart) and to report the best values found and any observed plateaus.
minor comments (3)
  1. [Introduction; Sec. 2; Sec. 3.1; Sec. 4] There are several typos: 'probabilitiies' in the Introduction, 'characterisitc' in Sec. 2, 'qantum' in Sec. 3.1, and 'Gussian' twice in Sec. 4. These should be corrected.
  2. [Abstract and Sec. 2] The claim that 'the number of parameters that need to be optimized' is independent of the stellar rank is precise, but it can be misread as a claim of rank-independent overall complexity. The cost of evaluating the objective function—the largest eigenvalue of the truncated operator (11)—grows with n (matrix dimension n for single mode, and combinatorially with n and N for multimode). A sentence clarifying that the fixed scaling applies only to the number of optimization parameters would avoid overstatement.
  3. [Fig. 1] The label 'n=0' in Fig. 1 is somewhat ambiguous; it may be helpful to state explicitly that n=0 corresponds to Gaussian states (no non-Gaussianity certified) and that the areas indicate the minimal stellar rank certified by the witness.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the witness threshold identity (12) is derived directly from the defining supremum over lower-rank states, with no fitted parameter or self-referential premise.

full rationale

The central result, Eq. (12), is obtained from the definition (5) of W_n as the supremum of Tr[W rho] over states of stellar rank at most n-1. The derivation uses only the pure-state reduction (6), the core-state representation (1), and the projector identity (9); the maximum-eigenvalue bound (10) is saturated by choosing the core state as the corresponding eigenstate. Thus the threshold is defined as the same supremum being computed, not an independently fitted quantity. The fidelity-witness recovery in Eq. (14) is a limiting case of the general expression, and the numerical examples in Section 3 compute thresholds from Eq. (12) for chosen operators (25) and (31); the illustrated probability pairs are generated from the optimizing states, after the fact, so no prediction is being assembled from data that define it. Self-citations to the author's earlier works are used for context (e.g., Refs. [12,22,33,48]) and are not load-bearing: the derivation of Eq. (12) does not depend on those prior papers. The practical reliance on numerical optimization over Gaussian parameters reaching the global supremum is a correctness and robustness caveat, not circularity, because the formal supremum in Eq. (12) is well defined independently of how it is computed. Score 0.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central method rests on the stellar-rank definition from prior work, standard Gaussian unitary decomposition, and an unproven numerical global-optimization premise. No new physical entities are introduced. The free parameter beta is only for an illustrative example.

free parameters (1)
  • beta (cat-state amplitude) = 2
    Chosen by hand in Sec. 3.2 for the illustrative cat-state witness. It is a demonstration parameter and does not affect the general method or the central claim.
assumptions (5)
  • domain assumption A mixed state has stellar rank n if it is a mixture of pure states of stellar rank n but not lower; consequently S_{n-1} is the convex hull of pure states with stellar rank at most n-1.
    Used in Sec. 2 to reduce the supremum over mixed states (5) to a supremum over pure states (6). This follows from the definition in Ref. [19].
  • standard math Any N-mode Gaussian unitary admits a Bloch-Messiah decomposition as in Eq. (19).
    Used in Sec. 2 for multimode witnesses. This is a standard result from Ref. [28].
  • standard math The passive unitary V_II commutes with the photon-number-truncation projector Pi_{n,N}.
    Used to remove V_II from the optimization in Eq. (18). True because V_II preserves total photon number.
  • standard math The matrix elements of a Gaussian unitary in Fock basis are given by the Hermite-polynomial formula (26).
    Used in the numerical evaluations of Sec. 3. Standard result from Ref. [30].
  • ad hoc to paper Numerical global optimization over the 4-parameter (single-mode) Gaussian unitary manifold reliably locates the supremum in Eq. (12).
    The paper does not prove convexity or provide global-optimization guarantees; Sec. 4 admits fully numerical approaches are unavoidable for high ranks. This is a load-bearing practical assumption for constructing valid thresholds.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Efficient construction of witnesses of stellar rank of nonclassical states of light." pith.science (2026). https://pith.science/paper/32NMRMV6

@misc{pith2026241214356,
  author       = {Pith},
  title        = {Pith review of: Efficient construction of witnesses of stellar rank of nonclassical states of light},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/32NMRMV6}},
  note         = {Machine review of arXiv:2412.14356}
}
read the original abstract

The stellar hierarchy of quantum states of light classifies the states according to the Fock-state resources that are required for their generation together with unitary Gaussian operations. States with stellar rank n can be also equivalently referred to as genuinely n-photon quantum non-Gaussian states. Here we present an efficient method for construction of general witnesses of the stellar rank. The number of parameters that need to be optimized in order to determine the witness does not depend on the stellar rank and it scales quadratically with the number of modes. We illustrate the procedure by constructing stellar rank witnesses based on pairs of Fock state probabilities and also based on pairs of fidelities with superpositions of coherent states.

Figures

Figures reproduced from arXiv: 2412.14356 by the authors.

Figure 1
Figure 1. Visualization of stellar rank witnesses based on pairs of Fock state probabilities [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Certification of stellar rank based on fidelities [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

54 extracted references · 54 canonical work pages

  1. [25]

    Certification of Non-Gaussian States with Operational Measurements,

    U. Chabaud, G. Roeland, M. Walschaers, F.c Grosshans, V. Parigi, D. Markham, and N. Treps, “Certification of Non-Gaussian States with Operational Measurements,” PRX Quantum2(2), 020333 (2021)

    work page 2021

  2. [1]

    Quantum Statistics of Linear and Nonlinear Optical Phenomena

    J. Peřina, “Quantum Statistics of Linear and Nonlinear Optical Phenomena” (Dordrecht: Reidel, 1984)

    work page 1984

  3. [2]

    Measure of the nonclassicality of nonclassical states,

    C.T. Lee, “Measure of the nonclassicality of nonclassical states,” Phys. Rev. A44(5), R2775-R2778 (1991)

    work page 1991

  4. [3]

    Detecting Quantum States with a Positive Wigner Function beyond Mixtures of Gaussian States,

    R. Filip and L. Mišta, Jr., “Detecting Quantum States with a Positive Wigner Function beyond Mixtures of Gaussian States,” Phys. Rev. Lett.106(20), 200401 (2011)

    work page 2011

  5. [4]

    Non-Gaussian Quantum States and Where to Find Them,

    M. Walschaers, “Non-Gaussian Quantum States and Where to Find Them,” PRX Quantum2(3), 030204 (2021)

    work page 2021

  6. [5]

    Quantum non-Gaussianity of light and atoms,

    L. Lachman and R. Filip, “Quantum non-Gaussianity of light and atoms,” Progress in Quantum Electronics83, 100395 (2022)

    work page 2022

  7. [6]

    Detectingquantumnon-Gaussianity via the Wigner function,

    M.G.Genoni,M.L.Palma,T.Tufarelli,S.Olivares,M.S.Kim,andM.G.A.Paris,“Detectingquantumnon-Gaussianity via the Wigner function,” Phys. Rev. A87(6), 062104 (2013)

    work page 2013

  8. [7]

    Robustness of quantum nonclassicality and non-Gaussianity of single-photon states in attenuating channels,

    L. Lachman and R. Filip, “Robustness of quantum nonclassicality and non-Gaussianity of single-photon states in attenuating channels,” Phys. Rev. A88(6), 063841 (2013)

    work page 2013

Show all 54 references
  1. [8]

    Quantum non-Gaussianity witnesses in phase space,

    C. Hughes, M.G. Genoni, T. Tufarelli, M.G.A. Paris, and M.S. Kim, “Quantum non-Gaussianity witnesses in phase space,” Phys. Rev. A90(1), 013810 (2014)

    2014

  2. [9]

    Revealing nonclassicality beyond Gaussian states via a single marginal distribution,

    J. Park, Y. Lu, J. Lee, Y. Shen, K. Zhang, S. Zhang, M.S. Zubairy, K. Kim, and H. Nha, “Revealing nonclassicality beyond Gaussian states via a single marginal distribution,” PNAS114(5), 891-896 (2017)

    2017

  3. [10]

    Sufficientconditionforaquantumstatetobegenuinelyquantum non-Gaussian,

    L.Happ,M.A.Efremov,H.Nha,andW.P.Schleich,“Sufficientconditionforaquantumstatetobegenuinelyquantum non-Gaussian,” New J. Phys.20(2), 023046 (2018)

    2018

  4. [11]

    Quantumnon-Gaussianityandquantificationofnonclassicality,

    B.KühnandW.Vogel, “Quantumnon-Gaussianityandquantificationofnonclassicality,” Phys.Rev.A 97(5), 053823 (2018)

    2018

  5. [12]

    Quantum non-Gaussianity criteria based on vacuum probabilities of original and attenuated state,

    J. Fiurášek, L. Lachman and R. Filip, “Quantum non-Gaussianity criteria based on vacuum probabilities of original and attenuated state,” New J. Phys.23(7), 073005 (2021)

    2021

  6. [13]

    ExperimentalTestoftheQuantumNon-Gaussian Character of a Heralded Single-Photon State,

    M.Ježek,I.Straka,M.Mičuda,M.Dušek,J.Fiurášek,andR.Filip,“ExperimentalTestoftheQuantumNon-Gaussian Character of a Heralded Single-Photon State,” Phys. Rev. Lett.107(21), 213602 (2011)

    2011

  7. [14]

    Quantum non-Gaussian Depth of Single-Photon States,

    I. Straka, A. Predojevi˙c, T. Huber, L. Lachman, L. Butschek, M. Miková, M. Mičuda, G.S. Solomon, G. Weihs, M. Ježek, and R. Filip, “Quantum non-Gaussian Depth of Single-Photon States,” Phys. Rev. Lett.113(22), 223603 (2014)

    2014

  8. [15]

    Experimental test of the strongly nonclassical character of a noisy squeezed single-photon state,

    M. Ježek, A. Tipsmark, R. Dong, J. Fiurášek, L. Mišta, Jr., R. Filip, and U. L. Andersen. “Experimental test of the strongly nonclassical character of a noisy squeezed single-photon state,” Phys. Rev. A86(4), 043813 (2012)

    2012

  9. [16]

    Quantum non-Gaussianity of frequency up-converted single photons,

    C. Baune, A.l Schönbeck, A. Samblowski, J. Fiurášek, and R. Schnabel, “Quantum non-Gaussianity of frequency up-converted single photons,” Opt. Express22(19), 22808-22816 (2014)

    2014

  10. [17]

    Straka, L

    I. Straka, L. Lachman, J. Hloušek, M. Miková, Michal Mičuda, M. Ježek, and R. Filip, Quantum non-Gaussian multiphoton light,” npj Quant. Inf.4, 4 (2018)

    2018

  11. [18]

    Faithful Hierarchy of Genuine n-Photon Quantum Non-Gaussian Light,

    L. Lachman, I. Straka, J. Hloušek, M. Ježek, and R. Filip, “Faithful Hierarchy of Genuine n-Photon Quantum Non-Gaussian Light,” Phys. Rev. Lett.123(4), 043601 (2019)

    2019

  12. [19]

    Stellar Representation of Non-Gaussian Quantum States,

    U. Chabaud, D. Markham, and F. Grosshans, “Stellar Representation of Non-Gaussian Quantum States,” Phys. Rev. Lett. 124(6), 063605 (2020)

    2020

  13. [20]

    Gaussian-optimized preparation of non-Gaussian pure states,

    D. Menzies and R. Filip, “Gaussian-optimized preparation of non-Gaussian pure states,” Phys. Rev. A79(1), 012313 (2009)

    2009

  14. [21]

    Generation of arbitrary quantum states of traveling fields,

    M. Dakna, J. Clausen, L. Knöll, and D.-G. Welsch, “Generation of arbitrary quantum states of traveling fields,” Phys. Rev. A59(2), 1658-1661 (1999)

    1999

  15. [22]

    Conditional generation of arbitrary single-mode quantum states of light by repeated photon subtractions,

    J. Fiurášek, R. García-Patrón, and Nicolas J. Cerf, “Conditional generation of arbitrary single-mode quantum states of light by repeated photon subtractions,” Phys. Rev. A72(3), 033822 (2005)

    2005

  16. [23]

    Quantum entanglement,

    R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys.81(2), 865-942 (2009)

    2009

  17. [24]

    Entanglement witnesses: construction, analysis and classification,

    D. Chruściński and G. Sarbicki, “Entanglement witnesses: construction, analysis and classification,” J. Phys. A: Math. Theor.47(48), 483001 (2014)

    2014

  18. [26]

    Quantum Non-Gaussianity of Multiphonon States of a Single Atom,

    L. Podhora, L. Lachman, T. Pham, A. Lešundák, O. Číp, L. Slodička, and R. Filip, “Quantum Non-Gaussianity of Multiphonon States of a Single Atom,” Phys. Rev. Lett.129(1), 013602 (2022)

    2022

  19. [27]

    Holomorphic Quantum Computing,

    U. Chabaud and S. Mehraban, “Holomorphic Quantum Computing,” arXiv:2111.00117

    arXiv

  20. [28]

    Squeezing as an irreducible resource,

    S.L. Braunstein, “Squeezing as an irreducible resource,” Phys. Rev. A71(5), 055801 (2005)

    2005

  21. [29]

    Witnessing Wigner Negativity,

    U. Chabaud, P.-E. Emeriau, and F. Grosshans, “Witnessing Wigner Negativity,” Quantum5, 471 (2021)

    2021

  22. [30]

    Displaced and Squeezed Fock States,

    P. Král, “Displaced and Squeezed Fock States,” J. Mod. Opt.37(5), 889-917 (1990)

    1990

  23. [31]

    On the identification of the convex hull of a finite set of points in the plane,

    R. A. Jarvis, “On the identification of the convex hull of a finite set of points in the plane,” Inf. Process. Lett.2(1), 18–21 (1973)

    1973

  24. [32]

    Nonclassicality detection from few Fock-state probabilities,

    L. Innocenti, L. Lachman, and R. Filip, “Nonclassicality detection from few Fock-state probabilities,” npj Quantum Information 8, 30 (2022)

    2022

  25. [33]

    Hloušek, M

    J. Hloušek, M. Ježek, and J. Fiurášek, Direct Experimental Certification of Quantum Non-Gaussian Character and Wigner Function Negativity of Single-Photon Detectors, Phys. Rev. Lett.126(4), 043601 (2021)

    2021

  26. [34]

    Eaton, A

    M. Eaton, A. Hossameldin, R.J. Birrittella, P.M. Alsing, C.C. Gerry, C. Cuevas, H. Dong, and O. Pfister, “Resolving 100 photons and quantum generation of unbiased random numbers,’ arXiv:2205.01221

    arXiv

  27. [35]

    Counting near-infrared single-photons with 95% efficiency,

    A. E. Lita, A. J. Miller, and S. W. Nam, “Counting near-infrared single-photons with 95% efficiency,” Opt. Expr. 16(5), 3032–3040 (2008)

    2008

  28. [36]

    Precisely determining photon-number in real-time,

    L. A. Morais, T.Weinhold, M. P. de Almeida, A. Lita, T. Gerrits, S. W. Nam, A. G. White, and G. Gillett, “Precisely determining photon-number in real-time,” arXiv:2012.10158 (2020)

    2020

  29. [37]

    Single-Mode Parametric-Down- Conversion States with 50 Photons as a Source for Mesoscopic Quantum Optics,

    G. Harder, T.J. Bartley, A.E. Lita, S.W. Nam, T. Gerrits, and C. Silberhorn, “Single-Mode Parametric-Down- Conversion States with 50 Photons as a Source for Mesoscopic Quantum Optics,” Phys. Rev. Lett.116(14), 143601 (2016)

    2016

  30. [38]

    Photon Chopping: New Way to Measure the Quantum State of Light,

    H. Paul, P. Törmä, T. Kiss, and I. Jex, “Photon Chopping: New Way to Measure the Quantum State of Light,” Phys. Rev. Lett.76(14), 2464-2467 (1996)

    1996

  31. [39]

    True photocounting statistics of multiple on-off detectors,

    J. Sperling, W. Vogel, and G. S. Agarwal, “True photocounting statistics of multiple on-off detectors,” Phys. Rev. A 85(2), 023820 (2012)

    2012

  32. [40]

    Fiber-assisted detection with photon number resolution,

    D. Achilles, C. Silberhorn, C. Śliwa, K. Banaszek, and I. A. Walmsley, “Fiber-assisted detection with photon number resolution,” Opt. Lett.28(23), 2387–2389 (2003)

    2003

  33. [41]

    Photon-number resolution using time-multiplexed single-photon detectors,

    M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number resolution using time-multiplexed single-photon detectors,” Phys. Rev. A68(4), 043814 (2003)

    2003

  34. [42]

    Multiple-photon resolving fiber-loop detector,

    J. Řeháček, Z. Hradil, O. Haderka, J. J. Peřina, and M. Hamar, “Multiple-photon resolving fiber-loop detector,” Phys. Rev. A67(6), 061801(R) (2003)

    2003

  35. [43]

    A high dynamic range optical detector for measuring single photons and bright light,

    J. Tiedau, E. Meyer-Scott, T. Nitsche, S. Barkhofen, T. J. Bartley, and C. Silberhorn, “A high dynamic range optical detector for measuring single photons and bright light,” Opt. Express27(1), 1-15 (2019)

    2019

  36. [44]

    Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths,

    A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Minaeva, G. Gol’tsman, K. G. Lagoudakis, M. Benkhaoul, F. Lévy, and A. Fiore, “Superconducting nanowire photon-number-resolving detector at telecommunication wavelen...

    2008

  37. [45]

    Zhu, Q.-Y

    D. Zhu, Q.-Y. Zhao, H. Choi, T.-J. Lu, A. E. Dane, D. Englund, and K. K. Berggren, A scalable multi-photon coincidence detector based on superconducting nanowires, Nat. Nanotechnol.13(7), 596-601 (2018)

    2018

  38. [46]

    Accurate detection of arbitrary photon statistics,

    J. Hloušek, M. Dudka, I. Straka, and M. Ježek, “Accurate detection of arbitrary photon statistics,” Phys. Rev. Lett. 123(15), 153604 (2019)

    2019

  39. [47]

    Unveiling photon statistics with a 100-pixel photon-number-resolving detector

    R. Cheng, Y. Zhou, S. Wang, M. Shen, T. Taher, H.X. Tang, “Unveiling photon statistics with a 100-pixel photon-number-resolving detector”, arXiv:2206.13753

    arXiv

  40. [48]

    Witnessing negativity of Wigner function by estimating fidelities of catlike states from homodyne measurements,

    J. Fiurášek and M. Ježek, “Witnessing negativity of Wigner function by estimating fidelities of catlike states from homodyne measurements,” Phys. Rev. A87(6), 062115 (2013)

    2013

  41. [49]

    Atomi orientati in campo magnetico variabile,

    E. Majorana, “Atomi orientati in campo magnetico variabile,” Nuovo Cimento 9, 43–50 (1932)

    1932

  42. [50]

    I.BengtssonandK.Życzkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement(Cambridge University Press, 2006)

    2006

  43. [51]

    Quantum Geometric Phase in Majorana’s Stellar Representation: Mapping onto a Many-Body Aharonov- Bohm Phase,

    P. Bruno, “Quantum Geometric Phase in Majorana’s Stellar Representation: Mapping onto a Many-Body Aharonov- Bohm Phase,” Phys. Rev. Lett.108(24), 240402 (2012)

    2012

  44. [52]

    Extremal quantum states,

    A.Z. Goldberg, A.B. Klimov, M. Grassl, G. Leuchs, and L.L. Sánchez-Soto, “Extremal quantum states,” AVS Quantum Sci.2(4), 044701 (2020)

    2020

  45. [53]

    Classicality of spin states,

    O. Giraud, P. Braun, and D. Braun, “Classicality of spin states,” Phys. Rev. A78(4), 042112 (2008)

    2008

  46. [54]

    Quantum metrology at the limit with extremal Majorana constellations,

    F. Bouchard, P. de la Hoz, G. Björk, R. W. Boyd, M. Grassl, Z. Hradil, E. Karimi, A. B. Klimov, G. Leuchs, J. Řeháček, and L. L. Sánchez-Soto, “Quantum metrology at the limit with extremal Majorana constellations,” Optica 4(11), 1429–1432 (2017)

    2017

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.