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arxiv: 2607.00837 · v1 · pith:GVWRCLVSnew · submitted 2026-07-01 · 🪐 quant-ph

Twisted Gaussian Schell States in Quantum Optics: Twist-Assisted Nonclassicality and Entanglement

Fabricio Toscano , G. Ca\~nas , A. Z. Khoury , P. H. Souto Ribeiro , S. P. Walborn This is my paper

Pith reviewed 2026-07-02 12:09 UTC · model grok-4.3

classification 🪐 quant-ph
keywords twisted Gaussian Schell statenonclassicalityGaussian entanglementquantum opticstwist phasebeam splitterphoton number distributionseparability
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The pith

A separable two-mode Gaussian state gains nonclassical global squeezing and entanglement-generating power from a nonzero twist parameter.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines the twisted Gaussian Schell state as the quantum counterpart to a classical paraxial beam that carries a twist phase. This state is produced by sending an asymmetric two-mode thermal state through phase shifters and a beam splitter, then applying local squeezing. Although the two modes remain separable, the twist allows certain joint quadratures to drop below the vacuum noise level. When each mode is mixed with an ancillary vacuum on balanced beam splitters, the resulting four-mode state develops entanglement in chosen two-by-two partitions, and raising the twist turns on or strengthens that entanglement for any fixed amount of input squeezing. The work thereby shows how a parameter borrowed from classical beam shaping can control quantum nonclassicality and entanglement.

Core claim

The twisted Gaussian Schell state is obtained by applying local squeezing to an asymmetric two-mode thermal state after phase shifters and a beam splitter. Although separable in the natural bipartition, a nonzero twist parameter produces squeezing of global quadratures below the shot-noise limit. The joint photon-number distribution is derived in closed form. Mixing each mode with an ancillary vacuum on balanced beam splitters creates a four-mode state whose local description consists of two TGS states; this state exhibits entanglement in selected 2x2 bipartitions and in all 1x3 bipartitions. For fixed input squeezing, larger twist activates entanglement where it would otherwise be absent an

What carries the argument

The twist parameter, generated by the sequence of phase shifters, beam splitter, and local squeezing applied to an asymmetric two-mode thermal state, which produces nonclassical global squeezing and controllable entanglement without direct two-mode entanglement.

If this is right

  • Global quadratures exhibit squeezing below the shot-noise limit whenever the twist parameter is nonzero.
  • The joint photon-number distribution of the TGS state takes a closed-form expression.
  • The four-mode state obtained by coupling each mode to vacuum displays entanglement in select 2x2 bipartitions whose local description is two TGS states, plus entanglement in all 1x3 bipartitions.
  • For any fixed input squeezing, increasing the twist activates entanglement in bipartitions that would otherwise be separable and deepens entanglement that is already present.

Where Pith is reading between the lines

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

  • The twist could function as an independent control knob to turn entanglement on or off without altering the strength of the local squeezing operations.
  • The construction suggests that other parameters from classical paraxial optics might be imported to generate or tune specific forms of multimode quantum entanglement.
  • An optical experiment could test the activation of entanglement by varying only the twist while holding squeezing fixed, providing a direct check of the reported activation threshold.
  • The coincidence of classical and quantum physicality bounds may allow classical beam-propagation calculations to predict the allowable range of twist in quantum implementations.

Load-bearing premise

The TGS state is produced exactly by local squeezing of an asymmetric two-mode thermal state after phase shifters and a beam splitter, with the classical physicality bound on the twist matching the quantum physicality condition.

What would settle it

Measure the variance of a global quadrature formed from both modes and check whether it falls below the shot-noise limit exactly when the twist parameter is nonzero but below the physicality bound; or verify the appearance of entanglement witnesses in the 2x2 bipartitions of the four-mode state for twist values that the formulas predict will activate entanglement.

Figures

Figures reproduced from arXiv: 2607.00837 by A. Z. Khoury, Fabricio Toscano, G. Ca\~nas, P. H. Souto Ribeiro, S. P. Walborn.

Figure 1
Figure 1. Figure 1: FIG. 1. Scheme to generate the TGS state from two indepen [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Density plots of the mutual information in Eq. (38) [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Scheme to activate entanglement from the TGS state. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Joint photon number distribution [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Entangled regions of the ( [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The sign of [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
read the original abstract

We introduce the Twisted Gaussian Schell (TGS) state, a two-mode mixed Gaussian state defined as the quantum-optical analog of the Twisted Gaussian Schell-model beam of classical paraxial optics, characterized by the so-called twist phase. In the TGS state, the twist parameter arises when an asymmetric two-mode thermal state is subject to local squeezing after the action of phase shifters and a beam splitter. Its defining quantum feature is nonclassicality: although the state is separable in its natural bipartition, when the twist parameter is nonzero there are global quadratures that can be squeezed below the shot-noise limit. The nonclassicality has also a direct signature in the joint photon-number distribution, which we obtain in closed form. Moreover, coupling each mode to an ancillary vacuum at a balanced beam splitter yields a four-mode state with entanglement in select $2\times2$ bipartitions, with local description given by two TGS states, and all $1\times3$ bipartitions. For fixed input squeezing, increasing the twist parameter activates entanglement where the state is otherwise separable and deepens it where already present. The classical physicality bound on the twist parameter coincides with the quantum physicality condition. These results advance the two-way bridge between classical beam engineering and quantum information.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper introduces the Twisted Gaussian Schell (TGS) state as the quantum-optical analog of the classical twisted Gaussian Schell-model beam. The state is constructed from an asymmetric two-mode thermal state via phase shifters, a beam splitter, and local squeezing; the twist parameter is introduced by definition. Central claims include: (i) twist-assisted nonclassicality, in which the state remains separable in its natural bipartition yet exhibits global quadrature squeezing below the shot-noise limit for nonzero twist; (ii) a closed-form joint photon-number distribution; (iii) entanglement activation in a four-mode extension (each mode coupled to an ancillary vacuum at a balanced beam splitter), where increasing twist activates or deepens entanglement in selected 2x2 bipartitions while all 1x3 bipartitions are entangled; and (iv) exact coincidence between the classical physicality bound on the twist parameter and the quantum physicality condition.

Significance. If the construction and bound coincidence are verified, the work supplies an explicit Gaussian-state bridge between classical beam engineering and quantum information, demonstrating how a single twist parameter can induce nonclassicality without local squeezing in the natural modes and can control entanglement generation after ancillary coupling. The closed-form photon-number distribution is a concrete, potentially testable signature.

major comments (1)
  1. [Construction of the TGS state and physicality bounds] The assertion that the classical physicality bound on the twist parameter coincides exactly with the quantum physicality condition is load-bearing for the allowable range of the twist and therefore for both the nonclassicality claim and the entanglement-activation results. The manuscript states the coincidence but provides no explicit covariance-matrix verification under the symplectic transformations (phase shifters, beam splitter, local squeezing applied to the asymmetric thermal state). Any mismatch in the off-diagonal blocks arising from the quantum structure would restrict the twist range and weaken the central claims.
minor comments (1)
  1. [Abstract] The abstract states that closed-form results are obtained, yet the provided text does not include the explicit derivations or error analysis for the photon-number distribution or the covariance-matrix elements; these should be supplied or referenced to section numbers for reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need for explicit verification of the physicality bounds. We address this point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Construction of the TGS state and physicality bounds] The assertion that the classical physicality bound on the twist parameter coincides exactly with the quantum physicality condition is load-bearing for the allowable range of the twist and therefore for both the nonclassicality claim and the entanglement-activation results. The manuscript states the coincidence but provides no explicit covariance-matrix verification under the symplectic transformations (phase shifters, beam splitter, local squeezing applied to the asymmetric thermal state). Any mismatch in the off-diagonal blocks arising from the quantum structure would restrict the twist range and weaken the central claims.

    Authors: We agree that the manuscript would be strengthened by an explicit derivation. In the revision we will add a dedicated calculation (new subsection or appendix) that starts from the covariance matrix of the asymmetric two-mode thermal state and applies the sequence of symplectic transformations corresponding to the phase shifters, beam splitter, and local squeezing. We will show that the resulting 4×4 covariance matrix satisfies the quantum physicality condition (positive symplectic eigenvalues) if and only if the twist parameter lies inside the classical bound, with the off-diagonal blocks transforming in a manner that introduces no additional restriction. This verification will be performed by direct matrix multiplication under each symplectic map and will confirm that the classical and quantum bounds coincide exactly, thereby supporting the reported range of the twist parameter and the associated nonclassicality and entanglement results. revision: yes

Circularity Check

0 steps flagged

No circularity: TGS state constructed explicitly from standard operations; physicality coincidence derived, not assumed

full rationale

The TGS state is defined via an explicit sequence of standard quantum-optical operations (asymmetric two-mode thermal state, phase shifters, beam splitter, then local squeezing), so the twist parameter enters by construction rather than by fitting or self-definition. The claimed coincidence between classical and quantum physicality bounds on the twist is presented as a calculational result on the resulting covariance matrix, not presupposed. Nonclassicality (global quadrature squeezing while separable) and entanglement activation under ancillary coupling are shown by direct computation on this constructed state. No load-bearing self-citations or uniqueness theorems from prior author work appear in the provided derivation chain; the central claims retain independent content from the explicit construction and calculations.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central contribution is a new state definition resting on standard quantum optics operations and one new parameter; no external benchmarks or machine-checked proofs are mentioned.

free parameters (1)
  • twist parameter
    Introduced to quantify the asymmetry-induced phase in the state construction; its value controls nonclassicality and entanglement depth.
axioms (1)
  • domain assumption TGS state arises from asymmetric two-mode thermal state after phase shifters, beam splitter, and local squeezing
    This construction is the defining premise stated in the abstract.
invented entities (1)
  • twist parameter no independent evidence
    purpose: Characterizes nonclassical global squeezing and activates entanglement in the four-mode extension
    New parameter introduced in the state definition with no independent evidence outside the construction

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Reference graph

Works this paper leans on

76 extracted references · 6 canonical work pages · 4 internal anchors

  1. [1]

    [48]: a two-mode Gaussian state with block-structured covariance matrix V= A C C T B (15) is separable iff detC≥0

    The symplectic eigenvalues of a 2n×2nco- variance matrix ˜Vare the moduli of the eigenvalues of ı˜VJ[4, 5], with J=⊕ n j=1Jj ;J j = 0 1 −1 0 .(14) However, for the TGS state it suffices to invoke the lemma of Ref. [48]: a two-mode Gaussian state with block-structured covariance matrix V= A C C T B (15) is separable iff detC≥0 . Inspecting the TGS covarian...

  2. [2]

    the bosonic entropy in bits.I G is independent of the squeez- ing Ω, vanishes atv= 0 and increases monotonically withv. B. Nonclassicality Separability does not, however, imply classicality. For a Gaussian state the relevant notion of classicality is the positivity of the Glauber–SudarshanPfunction [52], which holds if and only if all quadrature variances...

  3. [3]

    The same crossing turns on sub-shot- noise squeezing here and, after the four-mode embedding of Sec

    Squeezing (Ω̸= 1) is required for any nonclassicality, but at fixed squeezing the twist is the trigger that carriesλ min across 5 the 1 2 threshold. The same crossing turns on sub-shot- noise squeezing here and, after the four-mode embedding of Sec. IV, entanglement, because both are monotone in λmin. See section IV and Fig. 5 for further discussion on no...

  4. [4]

    (31) imposes that 1< η <∞ while−∞ ≤A, B≤ ∞

    Γ(k+ 1 2) π ,(33a) Sn(k, x) = 1 (n−k)!x! 2F1 −(n−k),−x; 1 2; A 4 , (33b) where Γ is the gamma function, 2F1(−p,−q; 1 2; A 4 ) =Pmin(p,q) j=0 (−p)j(−q)j ( 1 2)j j! A 4 j is the Gauss hypergeometric polynomial withpandqnon-negative integers, and we define η=b− c2 4a ,(34) B= 4ab−8b−c 2 4ab−8a−c 2 ,(35) A=B 64c2 (4ab−8b−c 2)2 .(36) The conditions in Eqs. (31...

  5. [5]

    Across the four squeezing values shown in Fig

    As can be observed from the grayscale on the right of the plots in Figure 2, for any fixed value ofκandv, the mutual information is greater for weaker squeezing (larger Ω). Across the four squeezing values shown in Fig. 2, ranging from Ω = 0.3 to Ω = 0.9, this trend is monotonic:Igrows as Ω approaches 1. The behavior of the joint photon number distributio...

  6. [6]

    The solid black curve is the entanglement boundaryκF=Tof Eq. (48). Since f− < 1 2 ⇔λ min < 1 2, this boundary is identical to the non- classicality thresholdλ min = 1 2 of the input TGS state (see Eq. (19)). At fixed thermal noiseκand squeezing Ω (F), increasing the twistvdrives the state across the boundary. The shaded blue scale corresponds to the logar...

    2014

  7. [7]

    As its roots are real and non-negative,Phas the standard shape: it rises from P(0) =a 0 =− 1 4(κ2 −v 2)2 <0, crosses its smallest rootz 0, peaks, dips throughz 1, and rises throughz 2 (Fig. 6). EvaluatingPat the thresholdz= 1 4 gives the factorization P 1 4 = 1 64 4(κ2 −v 2)−1 2κF−4(κ 2 −v 2)−1 . (F5) Physicality (κ≥ 1 2 +v,v≥0) givesκ 2 −v 2 = (κ−v)(κ+ v...

  8. [8]

    We claim thatz min < 1 4 if and only ifP( 1 4)>0

    is that of 2κF−4(κ 2 −v 2)−1. We claim thatz min < 1 4 if and only ifP( 1 4)>0. If P( 1 4)>0, then sinceP(0)<0 the cubic crosses zero in (0, 1 4), soz min < 1 4 [Fig. 6(a)]. For the converse we must exclude the only failure mode: thattworoots lie below 1 4 whileP( 1

  9. [9]

    double dip

    is again negative [the “double dip” of Fig. 6(b)]. A direct computation froma 1 anda 2 gives P ′ 1 4 = 3−4κ 2 16 + κF 4 (2s−1), s≡κ 2 −v 2,(F6) 14 z P(z) 1 4 z0 z1 z2 (a) z P(z) 1 4 z0 z1 z2 (b) FIG. 6. The sign ofP( 1

  10. [10]

    double dip

    decides entanglement, provided the situation in panel (b) cannot occur. Panel (a): the honest case, a single crossing before 1 4, withP( 1 4)>0. Panel (b): the “double dip”, in which the curve rises above the axis and back below it before 1 4, hiding two roots there whileP( 1 4)<0. The text shows panel (b) never happens, because the peak of Pcannot lie to...

  11. [11]

    The peak therefore sits at or beyond 1 4, soPis nondecreasing on [0, 1 4] and, withP(0)<0, stays≤0 there: the cubic has no root below 1 4, i.e.z min ≥ 1

    HenceP ′( 1 4)≥0 places 1 4 outside the interval, and it can only be to theleft, 1 4 ≤c 1 (it cannot be to the right, since that side begins beyond the midpoint, already≥ 1 4). The peak therefore sits at or beyond 1 4, soPis nondecreasing on [0, 1 4] and, withP(0)<0, stays≤0 there: the cubic has no root below 1 4, i.e.z min ≥ 1

  12. [12]

    This excludes the double dip and establishes the equivalence. The bipartition 1|234 is therefore entangled if and only if κF > 1 2 + 2(κ2 −v 2) = 1 2 1 + 1 β =T,(F9) which is precisely the entanglement condition (48) ob- tained for the bipartition 14|23. By the beam-splitter symmetry, the fourW|XY Zbipartitions are entangled over the same region of parame...

  13. [13]

    Simon, E

    R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A36, 3868 (1987)

    1987

  14. [14]

    Simon, E

    R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A37, 3028 (1988)

    1988

  15. [15]

    Weedbrook, S

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

    2012

  16. [16]

    Serafini,Quantum Continuous Variables: A Primer of Theoretical Methods, 1st ed

    A. Serafini,Quantum Continuous Variables: A Primer of Theoretical Methods, 1st ed. (CRC Press, Boca Raton, FL, 2017)

    2017

  17. [17]

    Continuous variable quantum information: Gaussian states and beyond

    G. Adesso, S. Ragy, and A. R. Lee, Open Syst. Inf. Dyn. 21, 1440001 (2014), arXiv:1401.4679

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  18. [18]

    Dutta, N

    Arvind, B. Dutta, N. Mukunda, and R. Simon, Pramana J. Phys.45, 471 (1995)

    1995

  19. [19]

    R. J. C. Spreeuw, Physical Review A63, 062302 (2001)

    2001

  20. [20]

    C. E. R. Souza, J. A. O. Huguenin, P. Milman, and A. Z. Khoury, Phys. Rev. Lett.99, 160401 (2007)

    2007

  21. [21]

    Aiello, F

    A. Aiello, F. T¨ oppel, C. Marquardt, E. Yao, and G. Leuchs, New Journal of Physics17, 043024 (2015)

    2015

  22. [22]

    Forbes, A

    A. Forbes, A. Aiello, and B. Ndagano (Elsevier, 2019) pp. 99–153

    2019

  23. [23]

    Shen and C

    Y. Shen and C. Rosales-Guzm´ an, Laser Photonics Rev. 16, 2100533 (2022), arXiv:2203.00994

    work page arXiv 2022

  24. [24]

    Z. Wang, Z. Zhan, A. N. Vetlugin, J.-Y. Ou, Q. Liu, Y. Shen, and X. Fu, Light: Science & Applications13, 297 (2024)

    2024

  25. [25]

    B. M. Rodr´ ıguez-Lara and F. E. Becerra, arXiv:2509.12426 [physics.optics] (2025), preprint

    work page arXiv 2025

  26. [26]

    R. M. Gomes, A. Salles, F. Toscano, P. H. S. Ribeiro, and S. P. Walborn, Phys. Rev. Lett.103, 033602 (2009)

    2009

  27. [27]

    D. S. Tasca, R. M. Gomes, F. Toscano, P. H. Souto Ribeiro, and S. P. Walborn, Phys. Rev. A83, 052325 (2011)

    2011

  28. [28]

    Boucher, T

    G. Boucher, T. Douce, D. Bresteau, S. P. Walborn, A. Keller, T. Coudreau, S. Ducci, and P. Milman, Phys. Rev. A92, 023804 (2015)

    2015

  29. [29]

    High-Dimensional Quantum Photonics: Roadmap

    M. Malik, M. Kues, T. Ikuta, H. Takesue, D. Bajoni, D. J. Moss, R. Morandotti, A. Forbes, S. Walborn, E. Karimi, Y. Ding, S. Paesani, C. Vigliar, B. Brecht, C. Silberhorn, F. Bouchard, M. Karpi´ nski, B. Sussman, J. M. Lukens, Y. Bromberg, R. Fickler, T. Giordani, F. Sciarrino, Y. Zheng, J. Wang, M. Huber, A. Tavakoli, R. Uola, N. Brunner, N. Friis, N. H....

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  30. [30]

    A. N. de Oliveira, S. P. Walborn, and C. H. Monken, 15 Journal of Optics B: Quantum and Semiclassical Optics 7, 288 (2005)

    2005

  31. [31]

    C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, Phys. Rev. A82, 033833 (2010)

    2010

  32. [32]

    Chowdhury, A

    P. Chowdhury, A. S. Majumdar, and G. S. Agarwal, Phys. Rev. A88, 013830 (2013)

    2013

  33. [33]

    W. F. Balthazar, C. E. R. Souza, D. P. Caetano, E. F. Galv˜ ao, J. A. O. Huguenin, and A. Z. Khoury, Opt. Lett. 41, 5797 (2016)

    2016

  34. [34]

    G. S. Agarwal, R. R. Puri, and R. P. Singh, Phys. Rev. A56, 4207 (1997)

    1997

  35. [35]

    G. S. Agarwal, New Journal of Physics13, 073008 (2011)

    2011

  36. [36]

    Puentes and A

    G. Puentes and A. Banerji, Frontiers in Physics9, 690721 (2021)

    2021

  37. [37]

    K. Zhu, S. Li, X. Zheng, and Y. Zhou, Journal of Modern Optics59, 873 (2012)

    2012

  38. [38]

    Kai-Cheng, L

    Z. Kai-Cheng, L. Shao-Xin, T. Ying, Z. Xiao-Juan, , and T. Hui-Qin, Chinese Physics B21, 084204 (2012)

    2012

  39. [39]

    Barral, J

    D. Barral, J. Li˜ nares, and D. Balado, J. Opt. Soc. Am. B33, 2225 (2016)

    2016

  40. [40]

    Y.-Z. Li, F. Jia, H.-L. Zhang, J.-H. Huang, and L.-Y. Hu, Laser Physics Letters12, 115203 (2015)

    2015

  41. [41]

    Bandyopadhyay, S

    A. Bandyopadhyay, S. Prabhakar, and R. P. Singh, Physics Letters A375, 1926 (2011)

    1926

  42. [42]

    Bandyopadhyay and R

    A. Bandyopadhyay and R. P. Singh, Optics Communica- tions284, 256 (2011)

    2011

  43. [43]

    V. V. Dodonov, Journal of Physics A: Mathematical and Theoretical48, 435303 (2015)

    2015

  44. [44]

    Reb´ on and R

    L. Reb´ on and R. Rossignoli, Phys. Rev. A84, 052320 (2011)

    2011

  45. [45]

    Simon and N

    R. Simon and N. Mukunda, J. Opt. Soc. Am. A10, 95 (1993)

    1993

  46. [46]

    Simon and N

    R. Simon and N. Mukunda, J. Opt. Soc. Am. A15, 2373 (1998)

    1998

  47. [47]

    A. T. Friberg, E. Tervonen, and J. Turunen, J. Opt. Soc. Am. A11, 1818 (1994)

    1994

  48. [48]

    Serna and J

    J. Serna and J. M. Movilla, Opt. Lett.26, 405 (2001)

    2001

  49. [49]

    Second-order moment equivalence of twisted Gaussian Schell model beams and orbital angular momentum eigenmodes

    T. Ferreira, G. Santos, S. Ayala, L. Hutter, E. S. G´ omez, G. Lima, G. Ca˜ nas, and S. P. Walborn, Second- order moment equivalence of twisted gaussian schell model beams and orbital angular momentum eigenmodes (2026), arXiv:2605.15408 [physics.optics]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  50. [50]

    Wang and Y

    F. Wang and Y. Cai, Opt. Express18, 24661 (2010)

    2010

  51. [51]

    F. Wang, Y. Cai, H. T. Eyyubo˘ glu, and Y. Baykal, Opt. Lett.37, 184 (2012)

    2012

  52. [52]

    C. Zhao, Y. Cai, and O. Korotkova, Opt. Express17, 21472 (2009)

    2009

  53. [53]

    Y. Cai, Q. Lin, and O. Korotkova, Opt. Express17, 2453 (2009)

    2009

  54. [54]

    Tong and O

    Z. Tong and O. Korotkova, Opt. Lett.37, 2595 (2012)

    2012

  55. [55]

    M. G. de Oliveira, A. Santos, A. Barbosa, B. P. da Silva, G. dos Santos, G. Ca˜ nas, P. S. Ribeiro, S. Walborn, and A. Khoury, Optics & Laser Technology176, 110983 (2024)

    2024

  56. [56]

    Hutter, G

    L. Hutter, G. Lima, and S. P. Walborn, Phys. Rev. Lett. 125, 193602 (2020)

    2020

  57. [57]

    Hutter, E

    L. Hutter, E. S. Gomez, G. Lima, and S. P. Walborn, AVS Quantum Sci.3, 031401 (2021)

    2021

  58. [58]

    G. H. dos Santos, A. G. de Oliveira, N. R. da Silva, G. Ca˜ nas, E. S. G´ omez, S. Joshi, Y. Ismail, P. H. S. Ribeiro, and S. P. Walborn, Nanophotonics11, 763 (2022)

    2022

  59. [59]

    S. A. Ponomarenko, Opt. Lett.46, 5958 (2021)

    2021

  60. [60]

    Simon, Phys

    R. Simon, Phys. Rev. Lett.84, 2726 (2000)

    2000

  61. [61]

    Simon, K

    R. Simon, K. Sundar, and N. Mukunda, J. Opt. Soc. Am. A10, 2008 (1993)

    2008

  62. [62]

    H. Wang, X. Peng, L. Liu, F. Wang, Y. Cai, and S. A. Ponomarenko, Opt. Lett.44, 3709 (2019)

    2019

  63. [63]

    Williamson, Am

    J. Williamson, Am. J. Math.58, 141 (1936)

    1936

  64. [64]

    Vogel, Phys

    W. Vogel, Phys. Rev. Lett.84, 1849 (2000)

    2000

  65. [65]

    M. S. Kim, W. Son, V. Buˇ zek, and P. L. Knight, Phys. Rev. A65, 032323 (2002)

    2002

  66. [66]

    J. K. Asb´ oth, J. Calsamiglia, and H. Ritsch, Phys. Rev. Lett.94, 173602 (2005)

    2005

  67. [67]

    M. M. Wolf, J. Eisert, and M. B. Plenio, Phys. Rev. Lett. 90, 047904 (2003)

    2003

  68. [68]

    Groenewold, Physica12, 405 (1946)

    H. Groenewold, Physica12, 405 (1946)

    1946

  69. [69]

    Lebedev,Special Functions and Their Applications- N.N

    N. Lebedev,Special Functions and Their Applications- N.N. Lebedev(PRENTICE-HALL, INC., Englewood Cliffs, N.J., 1965)

    1965

  70. [70]

    V. V. Dodonov, O. V. Man’ko, and V. I. Man’ko, Physical Review A49, 2993 (1994)

    1994

  71. [71]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information, 10th ed. (Cambridge Univer- sity Press, Cambridge, 2010)

    2010

  72. [72]

    Toscano, A

    F. Toscano, A. Saboia, A. T. Avelar, and S. P. Walborn, Phys. Rev. A92, 052316 (2015)

    2015

  73. [73]

    Unitarily localizable entanglement of Gaussian states

    A. Serafini, G. Adesso, and F. Illuminati, Phys. Rev. A 71, 032349 (2005), arXiv:quant-ph/0411109

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  74. [74]

    R. F. Werner and M. M. Wolf, Physical Review Letters 86, 3658–3661 (2001)

    2001

  75. [75]

    The factor 2 2n is missing in the formula from page 251 of the book [64], but using Mathematica it seems that it is there

  76. [76]

    Magnus, F

    W. Magnus, F. Oberhettinger, and R. P. Soni,Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed. (Springer, Berlin, 1966)

    1966