Pith. sign in

REVIEW 5 minor 57 references

Fully-connected three-mode squeezed vacuum: Gaussian entanglement, steering, and collective photon subtraction

T0 review · 0 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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

Pith's one-line read A fully-connected three-mode squeezed vacuum is genuinely tripartite entangled for any nonzero squeezing, yet shows no pairwise Gaussian steering — the steering resource is collective.

desk verdict Exactly solvable CV example where pairwise entanglement and pairwise Gaussian steering decouple, with steering only in collective cuts; solid work worth refereeing despite the unquantified symmetry assumption. read the letter →

arxiv 2607.17629 v1 pith:CH55MD4U submitted 2026-07-20 quant-ph

Manjia Mai , Jifeng Sun , Teng Zhao , Ming Zhang , Liyun Hu This is my paper · ORCID
classification quant-ph
keywords continuous-variablequantuminformationthree-modesqueezedvacuumGaussiansteeringmultipartiteentanglementphotonsubtractionWignernegativitycomplete-graphtopologycovariancematrix
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 studies a three-mode squeezed vacuum in which every pair of modes interacts with equal strength (a triangle, or complete-graph, topology). It shows that the state is genuinely tripartite entangled for any nonzero squeezing, but that no physical pair of modes exhibits two-mode Gaussian steering: the conditional covariance is pinned exactly at 1/2, so measurements on one mode never reduce another mode's uncertainty below the vacuum level. The nonzero steering resource is instead collective — any single mode steers the joint state of the other two, and that collective steering grows with squeezing and is independent of the squeezing phase. The same pairwise-versus-collective split governs photon subtraction: subtracting a photon from a single physical mode cannot create Wigner negativity in another physical mode, while subtracting from the symmetric collective mode (b+c)/√2 does. A sympathetic reader would care because this cleanly separates pairwise entanglement, collective Gaussian steering, and collective non-Gaussianity in one exactly solvable continuous-variable state, with a concrete heralded protocol for remote Wigner-negative state preparation.

What carries the argument

The load-bearing object is the complete-graph (K3) topology with equal couplings χ12=χ13=χ23. It diagonalizes through a Fourier matrix into one symmetric mode squeezed at 2r and two antisymmetric modes squeezed at −r; the Schur complement of any two-mode reduction then saturates the Heisenberg bound, enforcing ν̄_{2|1}=1/2. The carrier of the argument is the covariance matrix derived from the Bogoliubov transformation, plus the normalized phase-space Wigner-function calculation that identifies the collective mode d=(b+c)/√2 as the one whose subtraction produces negativity.

What would settle it

Measure the conditional covariance between any two physical modes in a symmetrically coupled FC-C3MSV at, say, r=0.5; if the conditional symplectic eigenvalue ν̄_{2|1} is found below 1/2 (equivalently, if the Schur complement has product <1/4), then the central claim of zero two-mode Gaussian steering is refuted. A simpler cross-check: apply unequal couplings and observe whether two-mode steering becomes nonzero, as the paper itself predicts.

Watch

Extended reading notes

Core claim

The central discovery is the exact pinning of the two-mode conditional symplectic eigenvalue at ν̄_{2|1}=1/2 for all r>0 and all squeezing phase θ. Because the Gaussian steering criterion requires ν̄<1/2, this pins every physical two-mode Gaussian steering measure to zero even while the same two-mode reductions are entangled (PPT eigenvalue <1/2). In the Fourier basis, the symmetric mode is squeezed at 2r and the two antisymmetric modes at −r; their contributions to the Schur complement balance identically. Steering survives only at the collective level: the one-mode-versus-two-mode conditional eigenvalue ν̄_c = 3e^{3r}/[2√(e^{6r}+2√(2e^{6r}+1))] < 1/2 yields G^{i→jk} = −ln(2ν̄_c), which is

Load-bearing premise

The exact symmetric-coupling condition χ12=χ13=χ23 is load-bearing: if the three pair-coupling strengths are unequal, the pinning of the conditional variance at 1/2 is lifted and pairwise Gaussian steering generically reappears, so the zero-pairwise-steering claim collapses.

Editorial extensions

If this is right

  • All three Armstrong-type variance-product witnesses are violated simultaneously in the window 0.2668<r<0.9084, giving a finite-window experimental certification that no chain-type topology provides.
  • Collective steering G^{i→jk} is phase-independent and grows linearly with r at large squeezing, so the complete-graph state offers a permutation-symmetric multipartite steering resource that does not depend on locking the squeezing phase.
  • Under full symmetric loss at r=0.5, one-to-two collective steering has a sudden-death threshold η≈0.58, while reverse two-to-one steering survives to η≈0.502, creating a loss window in which the joint subsystem still steers one mode.
  • Physical pairwise PPT entanglement survives all the way to η→0 along the symmetric loss path at r=0.5, even though all pairwise Gaussian steering is zero, so entanglement and steerability are sharply decoupled in this state.
  • Subtracting a photon from the collective mode (b+c)/√2 produces a Wigner-negative state on the remaining mode with threshold η_c≈0.667, enabling a heralded remote-preparation protocol with percent-level click probability under weak tapping.

Reading between the lines

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

  • If the equal-coupling condition is slightly broken, the exact pinning ν̄_{2|1}=1/2 lifts and small two-mode steering generically reappears; the zero-steering claim is therefore a symmetry-protected feature, and a quantitative tolerance analysis for coupling imbalance would be the natural next step.
  • An energy-normalized comparison with the chain-type state might change the apparent resource advantage: the fully-connected state has higher mean photon number at the same r, so the collective-steering gap could shrink or reverse when compared at fixed energy.
  • The same Fourier-diagonalization argument suggests a natural generalization to an N-mode complete-graph squeezer: for all-to-all equal couplings, one could expect all two-mode Gaussian steering to vanish while (N−1)-modal collective steering carries the resource.
  • The protocol's loss threshold η_WN≈0.667 translates to roughly 12 km of telecom fiber at 0.2 dB/km, so a tabletop multi-pump four-wave-mixing experiment with balanced channels should be able to test the pairwise-zero/collective-nonzero steering split directly.
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

0 major / 5 minor

Summary. This paper analyzes the fully-connected three-mode squeezed vacuum (FC-C3MSV), with all three modes pairwise coupled through a K3 interaction. For symmetric couplings it derives the Bogoliubov transformation, the normal-ordered squeezing operator via IWOP, and the exact covariance matrix. From the CM the authors show that the pure state is genuinely tripartite entangled for every r>0, with a finite-window Armstrong-type witness (0.2668<r<0.9084), while all physical two-mode Gaussian steering measures G^{i→j} vanish identically; the steering resource is instead collective, G^{i→jk}, which is θ-independent and grows with r. They then model independent vacuum losses and compute loss thresholds: at r=0.5 and full symmetric loss, G^{i→jk} disappears at η≈0.58, the reverse G^{jk→i} at η≈0.502, and pairwise PPT entanglement survives for all η>0. Finally, using a normalized phase-space derivation, they show that photon subtraction on a single physical mode cannot generate Wigner negativity in another physical mode, whereas subtraction from the collective mode (b+c)/√2 can, with threshold η≈0.667.

Significance. If correct, the manuscript establishes a clean exactly solvable continuous-variable example where pairwise bipartite entanglement coexists with exactly zero pairwise Gaussian steering, while nonzero steering and photon-subtraction-induced Wigner negativity are collective resources. The central claims are not fitted: the pinning ν̄_{2|1}=1/2, the collective steering eigenvalues, and the loss thresholds are analytic or roots of analytic expressions, and Appendices A-D give self-contained derivations. I spot-checked the CM, the Schur complement in Eq. (27), the collective-mode subtraction coefficient in Eq. (52), and the threshold values; they are consistent. The concrete thresholds and the collective-subtraction protocol are falsifiable predictions. The main limitations—exact coupling symmetry and independent-reservoir losses—are explicitly acknowledged and are not internal inconsistencies, although they bound the direct experimental reach.

minor comments (5)
  1. [Sec. II.E.3, Eq. (30)] The analytic expression for ν̄_c is typeset ambiguously; it should read ν̄_c = 3e^{3r}/[2√(e^{6r}+2)√(2e^{6r}+1)] or equivalent. As printed, the denominator could be misread, and the correct reading is needed to reproduce G^{i→jk}.
  2. [Sec. II.D] The Armstrong-witness window is computed at θ=0. A sentence stating whether the finite violation window is θ-independent or giving its θ dependence would remove ambiguity, especially since the following section emphasizes θ-independence of several other quantities.
  3. [Sec. III.A / Table IV] The absence of pairwise steering under single- and two-mode loss is verified only at r=0.5. The text is careful about this, but a one-sentence reminder in the loss-threshold paragraph would prevent over-reading of the no-pairwise-steering claim.
  4. [Sec. III.C and Sec. V] The zero-pairwise-steering phenomenon is pinned to exact equality χ12=χ13=χ23. A quantitative estimate of how ν̄_{2|1} and G^{i→jk} respond to a small coupling imbalance would substantially strengthen the experimental discussion; this is a limitation, not an error.
  5. [Throughout] Please correct typographical/formatting artifacts (e.g., 'notwo-mode' in the abstract, missing spaces/operators in displayed formulas) in the final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all central results are derived analytically from the Hamiltonian and covariance matrix, with no fitted parameters or load-bearing self-citations.

full rationale

The paper's central claims follow by closed-form analytic derivation from the interaction Hamiltonian of Eq. (1). The Bogoliubov coefficients, Eqs. (3)-(6), and the covariance matrix, Eqs. (12)-(14), are obtained through standard IWOP and similarity-transformation identities from the cited literature; no parameter is adjusted to any data set. The headline result of zero pairwise Gaussian steering is an algebraic identity, ¯ν_{2|1}=1/2 in Eq. (27), derived from the Schur complement of the covariance matrix, not imposed as an input. Collective steering, Armstrong-witness violations, loss thresholds, and Wigner-negativity thresholds are analytic roots or closed-form values (e.g., ¯ν_c, C_d(η_c)=0), with r, θ, and η treated as free physical parameters. The paper explicitly acknowledges its two idealizations—exact symmetric coupling and independent vacuum losses—and identifies robustness to asymmetric coupling as future work (Sec. III.C); these are physical limitations, not circular reductions. There are no load-bearing self-citations and no imported uniqueness theorems. The derivation chain is therefore self-contained and non-circular.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central claims are derived, not postulated: no numbers are fitted to data and no new physical entities are introduced. The model assumptions are the symmetric equal-coupling Hamiltonian, independent amplitude-damping loss, and standard Gaussian-state criteria, all explicitly stated.

free parameters (3)
  • r (squeezing strength) = 0.5 in numerical tables; otherwise a free variable
    Central claims hold for all r>0; numerical loss thresholds and Wigner-negativity values are reported at representative r=0.5.
  • θ (squeezing phase) = 0 for tables; invariance argued for all θ
    All steering and photon-subtraction conclusions are θ-independent by local rotation symmetry; tables set θ=0.
  • η (vacuum channel transmittance) = swept 0–1; thresholds computed at r=0.5
    Loss parameter in the independent-amplitude-damping model; critical transmittances are roots of analytic equations, not fits.
assumptions (7)
  • standard math Canonical commutation relations [â_i, â_j†]=δ_ij and quadrature convention [x,p]=i
    Used throughout for Bogoliubov transformations and covariance matrices.
  • standard math Gaussian states are fully characterized by their covariance matrix, with steering determined by Schur-complement symplectic eigenvalues
    Invoked in Sections II.E and III for Gaussian steering measures.
  • standard math The IWOP normal-ordering theorem of Fan and Zou (Ref. [38]) applies to the three-mode squeezing operator
    Used in Appendix B to derive the normal product form of S.
  • domain assumption The Hamiltonian in Eq. (1) is an effective Gaussian-resource model for phase-matched multi-pump FWM
    Section III.C states the three modes may be spatial, frequency, or polarization degrees of freedom provided pair-generation channels are phase matched and phase locked.
  • domain assumption Independent amplitude-damping loss model: Vloss = T V T^T + Nloss with Nloss = ⊕ (1-η_j)/2 I2
    Section III explicitly models losses as independent reservoirs and defers correlated losses to future work.
  • domain assumption Exact symmetric coupling χ12 = χ13 = χ23 is assumed for all central results
    The zero pairwise steering and collective steering results are derived under this equal-coupling condition; the paper itself notes symmetry breaking would lift the exact pinning.
  • standard math Ideal photon subtraction is described by n̂_m with Weyl symbol (x_m^2+p_m^2-1)/2
    Used in Appendix D to compute conditional Wigner functions after single- and collective-mode subtraction.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Fully-connected three-mode squeezed vacuum: Gaussian entanglement, steering, and collective photon subtraction." pith.science (2026). https://pith.science/paper/CH55MD4U

@misc{pith2026260717629,
  author       = {Pith},
  title        = {Pith review of: Fully-connected three-mode squeezed vacuum: Gaussian entanglement, steering, and collective photon subtraction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CH55MD4U}},
  note         = {Machine review of arXiv:2607.17629}
}
abstract

We investigate a fully-connected three-mode squeezed vacuum (FC-C3MSV) state, where all three modes are pairwise coupled through nonlinear interactions in a triangle ($K_3$) topology. Using the integration-within-ordered-product technique, we derive the normal product form of the squeezing operator and obtain the covariance matrix directly from the Bogoliubov transformation. Under symmetric coupling, the physical state is genuinely tripartite entangled for any nonzero squeezing, while the three Armstrong-type witnesses provide a finite-window sufficient experimental test; in the chain-type C3MSV only one of these witnesses is violated. We find that, despite two-mode entanglement, the fully-connected topology admits \emph{no} two-mode Gaussian steering ($\mathcal{G}^{i\to j}=0$) between any pair of physical modes; the steering resource is instead collective one-mode-versus-two steering $\mathcal{G}^{i\to jk}$, which is $\theta$-independent and grows with $r$. We analyze independent vacuum losses and obtain critical transmittances for steering survival: under full symmetric loss at $r=0.5$, one-to-two collective steering disappears at $\eta\approx0.58$, whereas reverse two-to-one collective steering survives down to $\eta\approx0.502$ and the underlying two-mode entanglement persists for all $\eta>0$. Finally, we revisit photon subtraction using a normalized phase-space derivation. A photon subtraction on a single physical mode does not generate Wigner negativity on another single mode, consistent with the absence of two-mode steering. Wigner negativity can instead be generated when Bob subtracts from the collective mode $(b+c)/\sqrt{2}$, with a loss threshold $\eta_c\approx0.667$ at $r=0.5$. These results distinguish pairwise and collective nonclassical resources in the FC-C3MSV and clarify the operational role of the complete-graph topology.

Figures

Figures reproduced from arXiv: 2607.17629 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Chain-type coupling ( [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (Color online) (a) Collective steering [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (Color online) One-to-two collective steering under [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (Color online) Loss robustness of pairwise and col [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (Color online) Photon subtraction on physical and collective modes at [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

57 extracted references · 3 linked inside Pith

  1. [1]

    The Schur complementσ 2|1 = A−BA −1Bis diagonal forθ= 0 with elementsa x−b2 x/ax anda p −b 2 p/ap

    Two-mode steering For any pair of modes, the reduced two-mode CM isσ 12 = A B B A . The Schur complementσ 2|1 = A−BA −1Bis diagonal forθ= 0 with elementsa x−b2 x/ax anda p −b 2 p/ap. Using Eqs. (13)–(14), a remarkable sim- plification occurs: ax − b2 x ax = e−2r(2e6r + 1) 2(e6r + 2) , a p − b2 p ap = e2r(e6r + 2) 2(2e6r + 1) . (26) Their product yields th...

  2. [2]

    The symmetric modeb 1 (squeezed by 2r) and the two antisymmetric modesb 2,3 (squeezed by−r) contribute to the reduced two-mode state with weights determined byQ

    Physical origin of the vanishing two-mode steering The vanishing of two-mode steering can be understood in the Fourier basis. The symmetric modeb 1 (squeezed by 2r) and the two antisymmetric modesb 2,3 (squeezed by−r) contribute to the reduced two-mode state with weights determined byQ. The Schur complement el- ements satisfy (a x −b 2 x/ax)(ap −b 2 p/ap)...

  3. [3]

    The collective steering is Gi→jk =−ln(2¯νc), (30) which is independent ofθby symmetry and identi- cal for all three modes

    Collective steering For one mode versus the remaining two, the 4×4 Schur complementσ 23|1 has symplectic eigenvalues ¯νc (which depends onr) and 1/2 (inherited from the degenerate antisymmetric subspace). The collective steering is Gi→jk =−ln(2¯νc), (30) which is independent ofθby symmetry and identi- cal for all three modes. Here ¯ν c <1/2 is the non- tr...

  4. [4]

    The useful conclusion is instead structural: Gaussian steerability is not stored on any physical two-mode edge of the com- plete graph

    Pairwise-versus-collective separation WithG i→j = 0, the usual Gaussian-steering monogamy constraints [43] impose no non-trivial resid- ual constraint on physical two-mode edges. The useful conclusion is instead structural: Gaussian steerability is not stored on any physical two-mode edge of the com- plete graph. It is delocalized across the network and a...

  5. [5]

    [32], not the microscopic coupling rate in the Hamiltonian

    Thisris the effective squeezing strength used in Ref. [32], not the microscopic coupling rate in the Hamiltonian. The steering numbers are evaluated in the real-CM Kogias-Adesso convention used throughout this paper. In this convention, the center-to-edge steering isG 2→1 =−ln[(c 2 −s 2 cos 2ϕ)/(c2 +s 2)] = 0.194 atr= 0.5. The chain-type Gaussian loss thr...

    1996

  6. [6]

    (57) For reference, Table VI collects the full-symmetric-loss thresholds over representative squeezing values

    SolvingC d(ηc) = 0 atr= 0.5 gives ηc = 0.6671605. . . .(57) For reference, Table VI collects the full-symmetric-loss thresholds over representative squeezing values. The one- to-two thresholds decrease slowly withr, and throughout this range the ordering remainsη WN > ηcrit i→jk . For the representative pointr= 0.5 used in Fig. 4, physical pair- wise PPT ...

    1996

  7. [7]

    Thex-p cross-terms vanish since all quantities are real atθ= 0

    atθ= 0: ⟨ˆxi ˆxj⟩= (v †v)ij + (uvT )ij,(C10) using the expansion ˆxi ˆxj = 1 2 (ˆai +ˆa† i )(ˆaj +ˆa† j) and noting that cross terms⟨ˆa iˆa† j⟩combine with (v †v)ij. Thex-p cross-terms vanish since all quantities are real atθ= 0. 14 Step 6: AssemblingAandB.The single-mode block Ahas diagonal entriesA 11 =⟨ˆx 2 1⟩,A 22 =⟨ˆp 2 1⟩(off- diagonals vanish atθ= ...

  8. [8]

    (13)–(14) is:a x =A 11 = 1/2 + (v†v)11 + (uvT )11, bx =B 11 = (v†v)12+(uv T )12, with analogous expressions for thep-quadrature entries

    For reference, the correspondence between the second- moment notation used above and the CM-block entries of Eqs. (13)–(14) is:a x =A 11 = 1/2 + (v†v)11 + (uvT )11, bx =B 11 = (v†v)12+(uv T )12, with analogous expressions for thep-quadrature entries. Appendix D: Normalized Wigner function after photon subtraction We derive the conditional Wigner function ...

    1996

Show all 57 references
  1. [9]

    Einstein, B

    A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev.47, 777 (1935)

    1935

  2. [10]

    D. M. Greenberger, M. A. Horne, and A. Zeilinger, in Bell’s Theorem, Quantum Theory and Conceptions of the Universe, edited by M. Kafatos (Kluwer Academic, Dor- drecht, 1989), pp. 69–72

    1989

  3. [11]

    D. M. Greenberger, M. A. Horne, and A. Zeilinger, arXiv:0712.0921 (2007)

    2007 arXiv

  4. [12]

    N. D. Mermin, Am. J. Phys.58, 731 (1990)

    1990

  5. [13]

    D¨ ur, G

    W. D¨ ur, G. Vidal, and J. I. Cirac, Phys. Rev. A62, 062314 (2000)

    2000

  6. [14]

    H. J. Briegel and R. Raussendorf, Phys. Rev. Lett.86, 910 (2001)

    2001

  7. [15]

    Bouwmeesteret al., Phys

    D. Bouwmeesteret al., Phys. Rev. Lett.82, 1345 (1999)

    1999

  8. [16]

    Panet al., Nature403, 515 (2000)

    J.-W. Panet al., Nature403, 515 (2000)

    2000

  9. [17]

    Wehner, D

    S. Wehner, D. Elkouss, and R. Hanson, Science362, eaam9288 (2018)

    2018

  10. [18]

    Zeilinger, Nat

    A. Zeilinger, Nat. Phys.14, 3 (2018)

    2018

  11. [19]

    Buhrmanet al., Rev

    H. Buhrmanet al., Rev. Mod. Phys.82, 665 (2010)

    2010

  12. [20]

    McCutcheonet al., Nat

    W. McCutcheonet al., Nat. Commun.7, 13251 (2016)

    2016

  13. [21]

    H. P. Yuen and J. H. Shapiro, Opt. Lett.4, 334 (1979)

    1979

  14. [22]

    R. S. Bondurantet al., inCoherence and Quantum Optics V(Springer, 1984), pp. 767–774

    1984

  15. [23]

    Kumar and J

    P. Kumar and J. H. Shapiro, Phys. Rev. A30, 1568 (1984)

    1984

  16. [24]

    M. D. Reid and D. F. Walls, Phys. Rev. A31, 1622 (1985)

    1985

  17. [25]

    C. C. Gerry and P. L. Knight,Introductory Quantum Optics(Cambridge University Press, 2005)

    2005

  18. [26]

    C. F. McCormicket al., Opt. Lett.32, 178 (2007)

    2007

  19. [27]

    Guoet al., Phys

    M. Guoet al., Phys. Rev. A89, 033813 (2014)

    2014

  20. [28]

    Guptaet al., J

    P. Guptaet al., J. Mod. Opt.63, 185 (2016)

    2016

  21. [29]

    J. D. Swaim and R. T. Glasser, Phys. Rev. A96, 033818 (2017)

    2017

  22. [30]

    Jiaet al., Opt

    J. Jiaet al., Opt. Lett.42, 4024 (2017)

    2017

  23. [31]

    Caiet al., Opt

    Y. Caiet al., Opt. Express28, 25278 (2020)

    2020

  24. [32]

    A. Dong, K. Zhang, and J. Jing, Opt. Commun.521, 128470 (2022)

    2022

  25. [33]

    H. Wang, C. Fabre, and J. Jing, Phys. Rev. A95, 051802(R) (2017)

    2017

  26. [34]

    J. D. Swaimet al., Opt. Lett.43, 2716 (2018)

    2018

  27. [35]

    Wanget al., Opt

    H. Wanget al., Opt. Express24, 23459 (2016)

    2016

  28. [36]

    H. Wang, L. Cao, and J. Jing, Sci. Rep.7, 40410 (2017)

    2017

  29. [37]

    Caoet al., Phys

    L. Caoet al., Phys. Rev. A95, 023803 (2017)

    2017

  30. [38]

    Caiet al., Phys

    Y. Caiet al., Phys. Rev. A91, 013843 (2015)

    2015

  31. [39]

    Zhang and R

    W. Zhang and R. T. Glasser, arXiv:2002.00323v2 (2022)

    2002 arXiv

  32. [40]

    Z.-W. Zhan, B. Lan, J. Wang, and X.-X. Xu, Phys. Rev. A108, 012436 (2023)

    2023

  33. [41]

    G.-W. Mi, X. Huang, and T. Zhang, arXiv:2606.29144 (2026)

    2026 arXiv

  34. [42]

    R. Y. Tehet al., Phys. Rev. A105, 012202 (2022)

    2022

  35. [43]

    Q. Y. He, Q. H. Gong, and M. D. Reid, Phys. Rev. Lett. 114, 060402 (2015)

    2015

  36. [44]

    H. Y. Fan and N. Q. Jiang, Chin. Phys. Lett.19, 1403 (2002)

    2002

  37. [45]

    H.-Y. Fan, H. R. Zaidi, and J. R. Klauder, Phys. Rev. D 35, 1831 (1987)

    1987

  38. [46]

    Fan and H

    H.-Y. Fan and H. Zou, Phys. Lett. A252, 281 (1999)

    1999

  39. [47]

    H.-Y. Fan, J. Opt. B5, R147 (2003)

    2003

  40. [48]

    Cariolaro and G

    G. Cariolaro and G. Pierobon, J. Math. Phys.56, 122109 (2015)

    2015

  41. [49]

    Armstronget al., Nat

    S. Armstronget al., Nat. Phys.11, 167 (2015)

    2015

  42. [50]

    Kogiaset al., Phys

    I. Kogiaset al., Phys. Rev. Lett.114, 060403 (2015)

    2015

  43. [51]

    M. D. Reid, Phys. Rev. A88, 062108 (2013)

    2013

  44. [52]

    Ferraro, S

    A. Ferraro, S. Olivares, and M. G. A. Paris,Gaus- sian States in Continuous Variable Quantum Informa- tion(Bibliopolis, Napoli, 2005)

    2005

  45. [53]

    Walschaers, V

    M. Walschaers, V. Parigi, and N. Treps, Phys. Rev. Lett. 124, 150501 (2020)

    2020

  46. [54]

    Walschaers, Quantum7, 1038 (2023)

    M. Walschaers, Quantum7, 1038 (2023)

    2023

  47. [55]

    M. S. Kimet al., Phys. Rev. A71, 043805 (2005)

    2005

  48. [56]

    Kenfack and K

    A. Kenfack and K. ˙Zyczkowski, J. Opt. B6, 396 (2004)

    2004

  49. [57]

    Liuet al., Phys

    S. Liuet al., Phys. Rev. Lett.128, 200401 (2022)

    2022

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.