pith. sign in

arxiv: 2503.12694 · v1 · submitted 2025-03-16 · 🪐 quant-ph

Noisy dynamics of Gaussian entanglement: a transient bound entangled phase before separability

Gurvir Singh , Saptarshi Roy , Arvind This is my paper

Pith reviewed 2026-05-22 23:57 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Gaussian entanglementbound entangled statescontinuous-variable systemsnoisy dynamicsthermal bathseparabilityfour-mode statestransient phases
0
0 comments X

The pith

A family of four-mode Gaussian states passes through bound entanglement under thermal noise before separating.

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

The paper examines how various initial Gaussian entangled states in four continuous-variable modes evolve when coupled to a thermal bath. Most states lose their entanglement directly to separability after some time. However, the generalized four-mode squeezed vacuum states first become bound entangled for a period before fully separating. This transient bound entangled phase is detected using semi-definite programming on the time-evolved covariance matrices. The result indicates that bound entanglement can emerge dynamically in noisy continuous-variable systems.

Core claim

Certain initial NPT-entangled Gaussian states, specifically the generalized four-mode squeezed vacuum states, when evolved under interaction with a thermal bath, transition into a bound entangled state that persists for a finite time window before the state becomes separable. This is verified by tracking the covariance matrix evolution and applying separability tests via semi-definite programming, revealing a dynamical onset of bound entanglement not seen in other studied states including Haar-random ones.

What carries the argument

The generalized four-mode squeezed vacuum (gFMSV) states, a three-parameter family of Gaussian states whose noise-induced evolution produces a temporary bound entangled regime.

If this is right

  • The entanglement robustness varies with the initial state parameters for gFMSV.
  • Most Gaussian states do not exhibit this transient bound phase.
  • Analysis of random states shows the phenomenon is not generic.
  • Bound entanglement appears as an intermediate phase in the decay of NPT entanglement.

Where Pith is reading between the lines

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

  • Experimental setups with optical modes might observe this transient phase if noise is controlled.
  • The finding may apply to other noise models beyond thermal baths.
  • It raises the question of whether similar transients occur in higher-mode systems.

Load-bearing premise

That the semi-definite programming applied to the evolved covariance matrices accurately identifies bound entanglement without misclassification errors.

What would settle it

Direct computation of the partial transpose and separability criteria on the covariance matrix of a gFMSV state at an intermediate evolution time showing positive partial transpose but detected entanglement.

Figures

Figures reproduced from arXiv: 2503.12694 by Arvind, Gurvir Singh, Saptarshi Roy.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic of the time evolution of the entanglement char [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic of the local bath acting on four mode states. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
read the original abstract

We discover a new class of Gaussian bound entangled states of four-mode continuous-variable systems. These states appear as a transient phase when certain NPT-entangled Gaussian states are evolved under a noisy environment. A thermal bath comprising of harmonic oscillators is allowed to interact with one or modes of the system and a wide variety of initial Gaussian entangled (NPT as well as PPT) states are studied. The robustness of entanglement is defined as the time duration for which the entanglement of the initial state is preserved under the noisy dynamics. We access the separability by utilizing standard semi-definite programming techniques. While most states lose their entanglement after a certain time across all bi-partitions, an exception is observed for a three-parameter family of states which we call the generalized four-mode squeezed vacuum (gFMSV) states, which transitions to a bound entangled state, and remains so for a finite window of time. This dynamical onset of bound entanglement in continuous-variable systems is the central observation of our work. We carry out the analysis for Haar-random four-mode states (both pure and mixed) to scan the state space for transient bound entangled phase

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

2 major / 2 minor

Summary. The manuscript reports the discovery of a transient bound-entangled phase for a three-parameter family of four-mode Gaussian states (generalized four-mode squeezed vacuum or gFMSV states) under thermal-bath noise. These states begin as NPT-entangled, evolve such that their covariance matrices enter a regime that is positive under partial transpose yet classified as entangled by semi-definite programming, and remain so for a finite time window before separability. The authors contrast this with the generic loss of entanglement across bipartitions for other initial states, including Haar-random four-mode pure and mixed Gaussian states, and position the dynamical onset of bound entanglement in continuous-variable systems as the central result.

Significance. If the numerical classification is robust, the work identifies a previously unreported dynamical mechanism for generating bound entanglement in Gaussian continuous-variable systems, which is rare in the static case. The use of SDP separability tests on evolved covariance matrices and the scan over random states provide a concrete, falsifiable observation that could stimulate further analytic work on open-system trajectories in covariance space.

major comments (2)
  1. [paragraph describing separability assessment and SDP usage] The central claim of a finite-time bound-entangled window for gFMSV states rests on SDP correctly identifying entanglement for evolved covariance matrices that remain PPT. The text invokes 'standard semi-definite programming techniques' but supplies no solver name, tolerance settings, or dual-gap verification. Near the PPT/separable boundary even modest numerical error can flip the classification, so the reported transient phase could be an artifact; explicit implementation details and robustness checks against tolerance variation are required to substantiate the observation.
  2. [section on the noisy dynamics and thermal-bath interaction] The thermal-bath model (harmonic-oscillator baths coupled to one or more modes) determines the precise trajectory through covariance space and thus the duration of any bound-entangled window. The manuscript does not provide the explicit master equation, coupling strengths, bath temperatures, or Markovian approximation details used for the gFMSV family, making it impossible to assess whether the reported window is generic or an artifact of the chosen parameters.
minor comments (2)
  1. [abstract] The abstract and introduction would benefit from an explicit statement of the bipartitions considered when declaring separability or bound entanglement.
  2. [definition of gFMSV states] Notation for the three parameters of the gFMSV family should be introduced with a clear covariance-matrix expression early in the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address each major comment below and will revise the manuscript to incorporate the requested details for improved clarity and reproducibility.

read point-by-point responses

  1. Referee: The central claim of a finite-time bound-entangled window for gFMSV states rests on SDP correctly identifying entanglement for evolved covariance matrices that remain PPT. The text invokes 'standard semi-definite programming techniques' but supplies no solver name, tolerance settings, or dual-gap verification. Near the PPT/separable boundary even modest numerical error can flip the classification, so the reported transient phase could be an artifact; explicit implementation details and robustness checks against tolerance variation are required to substantiate the observation.

    Authors: We agree that explicit SDP implementation details are required to substantiate the classification near the PPT boundary. In the revised manuscript we will name the solver (CVXOPT), report the tolerance (1e-8) and duality-gap threshold, and add a robustness section showing that the transient bound-entangled window for the gFMSV family persists when the tolerance is varied over two orders of magnitude. These additions will confirm that the reported phase is not a numerical artifact. revision: yes

  2. Referee: The thermal-bath model (harmonic-oscillator baths coupled to one or more modes) determines the precise trajectory through covariance space and thus the duration of any bound-entangled window. The manuscript does not provide the explicit master equation, coupling strengths, bath temperatures, or Markovian approximation details used for the gFMSV family, making it impossible to assess whether the reported window is generic or an artifact of the chosen parameters.

    Authors: We acknowledge the omission of explicit dynamical parameters. The evolution follows the standard Markovian Lindblad master equation for harmonic-oscillator baths in the weak-coupling limit. In the revision we will insert the explicit master equation, the coupling rates used for the gFMSV family (e.g., uniform damping rate 0.1), bath temperatures, and a brief justification of the Markovian approximation. This will allow readers to reproduce the trajectories and evaluate parameter dependence. revision: yes

Circularity Check

0 steps flagged

No circularity: numerical observation of transient bound entanglement

full rationale

The paper reports a numerical study: covariance matrices of initial Gaussian states (including the independently defined gFMSV family) are evolved under a thermal-bath model, then classified for separability/entanglement via standard SDP on the PPT criterion. The central claim is an observed finite-time window where gFMSV states become bound-entangled before full separability. No derivation chain exists that reduces by construction to its inputs; there are no fitted parameters renamed as predictions, no self-citation load-bearing uniqueness theorems, and no ansatz smuggled via prior work. The analysis of Haar-random states further confirms the result is data-driven rather than self-referential. This matches the default expectation of a self-contained numerical finding.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The claim rests on the definition of the three-parameter gFMSV family and the reliability of SDP separability tests under the thermal-bath model; no independent evidence for either is supplied in the abstract.

free parameters (1)
  • three parameters defining gFMSV states
    The family is introduced with three free parameters chosen to exhibit the transient bound-entangled behavior.
axioms (2)
  • domain assumption Gaussian states remain Gaussian under linear coupling to a thermal bath of harmonic oscillators
    Standard assumption in continuous-variable open quantum systems invoked to justify the evolution model.
  • domain assumption Semi-definite programming on covariance matrices can reliably detect bound entanglement versus separability for these states
    Methodological premise required to interpret the numerical results as bound entanglement.

pith-pipeline@v0.9.0 · 5729 in / 1359 out tokens · 43363 ms · 2026-05-22T23:57:38.757646+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

74 extracted references · 74 canonical work pages · 1 internal anchor

  1. [1]

    The Detection of Gaussian Entanglement via Solving Linear Matrix inequalities was developed by Ma et

    Detection using linear matrix inequalities (LMI) The method of constructing an entanglement witness for CV systems was originally introduced by Hyllus and Eis- ert [42]. The Detection of Gaussian Entanglement via Solving Linear Matrix inequalities was developed by Ma et. al. [45] and the method has since been extended to detect entangle- ment of unknown C...

    work page

  2. [2]

    While complete extendibility was originally discussed in [46], a proper framework was developed by Lami et

    SDP via Symmetric extension of Gaussian States We can verify the entanglement of Gaussian states via another SDP program which is based on the extension of Doherty-Parrilo-Spedalieri (DPS) hierarchy to Gaussian Sys- tems. While complete extendibility was originally discussed in [46], a proper framework was developed by Lami et. al. [55] who introduced the...

    work page

  3. [3]

    Consider the partial transposition and check its sign using the criterion in Eq. (9)

    work page

  4. [4]

    FMSV state

    Check whether the state is separable across the consid- ered cut using semidefinite programming techniques as laid out above. A given multi-mode Gaussian state is bound entangled across a considered bipartition if it has positive partial transposition (PPT) in that cut but it is inseparable across the same cut. Summing up, to check whether a Gaussian stat...

    work page

  5. [5]

    In this case, the time- evolved FMSV state after being NPT entangled for some time becomes PPT entangled at least in one bipartition

    Transient bound entanglement phase A closer analysis of time evolved FMSV state when the lo- cal noise acts on two next to next neighboring modes reveals a striking feature that there is a transient bound entangled phase before the entanglement disappears. In this case, the time- evolved FMSV state after being NPT entangled for some time becomes PPT entan...

    work page

  6. [6]

    τ ∈ [0, τBE): the time evolved state is NPT entangled at least across one bipartition

    work page

  7. [7]

    τ ∈ [τBE, τ ∗): the time evolved state is PPT (bound) entangled at least across one bipartition and is separable across the other cuts

    work page

  8. [8]

    7 This counterintuitive phase of bound entanglement occurs when the next to next modes such as {1, 3} or {2, 4} are sub- jected to a local noisy environment

    τ ∈ [τ ∗, 1): the time evolved state is separable across all bipartitions. 7 This counterintuitive phase of bound entanglement occurs when the next to next modes such as {1, 3} or {2, 4} are sub- jected to a local noisy environment. The two time scales τBE and τ ∗) for different average photon number are tabulated in Table II. Noisy Modes N τBE τ ∗ {1, 3}...

    work page

  9. [9]

    Weedbrook, S

    C. Weedbrook, S. Pirandola, R. Garc´ıa-Patr´on, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Reviews of Modern Physics 84, 621–669 (2012)

    work page 2012

  10. [10]

    Adesso, S

    G. Adesso, S. Ragy, and A. R. Lee, Open Systems & Informa- tion Dynamics 21, 1440001 (2014)

    work page 2014

  11. [11]

    Dutta, N

    Arvind, B. Dutta, N. Mukunda, and R. Simon, Pramana 45, 471–497 (1995)

    work page 1995

  12. [12]

    Dutta, N

    Arvind, B. Dutta, N. Mukunda, and R. Simon, Phys. Rev. A 52, 1609 (1995)

    work page 1995

  13. [13]

    L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 84, 2722 (2000)

    work page 2000

  14. [14]

    Simon, Phys

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

    work page 2000

  15. [15]

    Mukunda, Physics Letters A 259, 421 (1999)

    Arvind and N. Mukunda, Physics Letters A 259, 421 (1999)

    work page 1999

  16. [16]

    Kumar, G

    C. Kumar, G. Saxena, and Arvind, Phys. Rev. A 103, 042224 (2021)

    work page 2021

  17. [17]

    S. L. Braunstein and H. J. Kimble, Phys. Rev. A 61, 042302 (2000)

    work page 2000

  18. [18]

    T. C. Ralph and E. H. Huntington, Phys. Rev. A 66, 042321 (2002)

    work page 2002

  19. [19]

    S. L. Braunstein and H. J. Kimble, Phys. Rev. Lett. 80, 869 (1998)

    work page 1998

  20. [20]

    Pirandola and S

    S. Pirandola and S. Mancini, Laser Physics 16, 1418–1438 (2006)

    work page 2006

  21. [21]

    Patra, R

    A. Patra, R. Gupta, S. Roy, and A. Sen(De), Phys. Rev. A 106, 022433 (2022)

    work page 2022

  22. [22]

    L. Lami, A. Serafini, and G. Adesso, New Journal of Physics 20, 023030 (2018)

    work page 2018

  23. [23]

    Giorda and M

    P. Giorda and M. G. A. Paris, Phys. Rev. Lett. 105, 020503 (2010)

    work page 2010

  24. [24]

    H.-K. Lau, R. Pooser, G. Siopsis, and C. Weedbrook, Phys. Rev. Lett. 118, 080501 (2017)

    work page 2017

  25. [25]

    S. Das, G. Siopsis, and C. Weedbrook, Phys. Rev. A97, 022315 (2018)

    work page 2018

  26. [26]

    Killoran, T

    N. Killoran, T. R. Bromley, J. M. Arrazola, M. Schuld, N. Que- sada, and S. Lloyd, Phys. Rev. Res. 1, 033063 (2019)

    work page 2019

  27. [27]

    Continuous vari- able quantum teleportation, u(2) invariant squeezing and non- gaussian resource states,

    M. Sharma, C. Kumar, S. Arora, and Arvind, “Continuous vari- able quantum teleportation, u(2) invariant squeezing and non- gaussian resource states,” (2025), arXiv:2502.17182 [quant- ph]

    work page arXiv 2025

  28. [28]

    Laudenbach, C

    F. Laudenbach, C. Pacher, C. F. Fung, A. Poppe, M. Peev, B. Schrenk, M. Hentschel, P. Walther, and H. H¨ubel, Advanced Quantum Technologies 1 (2018), 10.1002/qute.201800011

    work page doi:10.1002/qute.201800011 2018

  29. [29]

    Pirandola and P

    S. Pirandola and P. Papanastasiou, Phys. Rev. Res. 6, 023321 (2024)

    work page 2024

  30. [30]

    Zhang, Y

    Y . Zhang, Y . Bian, Z. Li, S. Yu, and H. Guo, Applied Physics Reviews 11 (2024), 10.1063/5.0179566

    work page doi:10.1063/5.0179566 2024

  31. [31]

    S. L. Braunstein and P. van Loock, Rev. Mod. Phys. 77, 513 (2005)

    work page 2005

  32. [32]

    Horodecki and M

    P. Horodecki and M. Lewenstein, Physical Review Letters 85, 2657 (2000)

    work page 2000

  33. [33]

    Bound entanglement for continuous variables is a rare phenomenon

    P. Horodecki, J. I. Cirac, and M. Lewenstein, “Bound entangle- ment for continuous variables is a rare phenomenon,” (2001), arXiv:quant-ph/0103076 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  34. [34]

    R. F. Werner and M. M. Wolf, Phys. Rev. Lett.86, 3658 (2001)

    work page 2001

  35. [35]

    P. W. Shor, J. A. Smolin, and A. V . Thapliyal, Phys. Rev. Lett. 90, 107901 (2003)

    work page 2003

  36. [36]

    D. P. Chi, J. W. Choi, J. S. Kim, T. Kim, and S. Lee, Phys. Rev. A 75, 032306 (2007)

    work page 2007

  37. [37]

    Mishra, R

    M. Mishra, R. Sengupta, and Arvind, Phys. Rev. A102, 032415 (2020)

    work page 2020

  38. [38]

    T ´oth and T

    G. T ´oth and T. V´ertesi, Phys. Rev. Lett. 120, 020506 (2018)

    work page 2018

  39. [39]

    Chatziioannou, K

    M. Navascu ´es, J. Bae, J. I. Cirac, M. Lewestein, A. Sanpera, and A. Ac´ın, Physical Review Letters94 (2005), 10.1103/phys- revlett.94.010502

    work page doi:10.1103/phys- 2005

  40. [40]

    Navascu ´es and A

    M. Navascu ´es and A. Ac´ın, Phys. Rev. A72, 012303 (2005)

    work page 2005

  41. [41]

    Xiang, Shao-Hua and Song, Ke-Hui, Eur. Phys. J. D 67, 157 (2013)

    work page 2013

  42. [42]

    Marian, I

    P. Marian, I. Ghiu, and T. A. Marian, Physica Scripta 90, 074041 (2015)

    work page 2015

  43. [43]

    S. H. Xiang, K. H. Song, W. Wen, and Z. G. Shi, The European Physical Journal D 62, 289 (2011)

    work page 2011

  44. [44]

    Xiang, B

    S.-H. Xiang, B. Shao, and K.-H. Song, Phys. Rev. A 78, 052313 (2008)

    work page 2008

  45. [45]

    Barbosa, A

    F. Barbosa, A. De Faria, A. Coelho, K. Cassemiro, A. Vil- lar, P. Nussenzveig, and M. Martinelli, Physical Review A—Atomic, Molecular, and Optical Physics 84, 052330 (2011)

    work page 2011

  46. [46]

    Isar, Physica Scripta T143, 014012 (2011)

    A. Isar, Physica Scripta T143, 014012 (2011)

    work page 2011

  47. [47]

    Kumar, G

    Rishabh, C. Kumar, G. Narang, and Arvind, Phys. Rev. A 105, 042405 (2022). 12

    work page 2022

  48. [48]

    Tserkis and T

    S. Tserkis and T. C. Ralph, Phys. Rev. A 96, 062338 (2017)

    work page 2017

  49. [49]

    S. Roy, T. Das, and A. Sen(De), Phys. Rev. A 102, 012421 (2020)

    work page 2020

  50. [50]

    Hyllus and J

    P. Hyllus and J. Eisert, New Journal of Physics8, 51–51 (2006)

    work page 2006

  51. [51]

    Mihaescu, H

    T. Mihaescu, H. Kampermann, G. Gianfelici, A. Isar, and D. Bruß, New Journal of Physics 22, 123041 (2020)

    work page 2020

  52. [52]

    Giustino, Electron-phonon interactions from first prin- ciples, Reviews of Modern Physics 89, 10.1103/revmod- phys.89.015003 (2017)

    A. Tavakoli, A. Pozas-Kerstjens, P. Brown, and M. Ara ´ujo, Reviews of Modern Physics 96 (2024), 10.1103/revmod- phys.96.045006

    work page doi:10.1103/revmod- 2024

  53. [53]

    S. Ma, S. Xue, Y . Guo, and C.-C. Shu, Quantum Information Processing 19 (2020), 10.1007/s11128-020-02726-1

    work page doi:10.1007/s11128-020-02726-1 2020

  54. [54]

    B. V . Rajarama Bhat, K. R. Parthasarathy, and R. Sengupta, Reviews in Mathematical Physics 29, 1750012 (2017)

    work page 2017

  55. [55]

    Lepp ¨aj¨arvi, I

    L. Lepp ¨aj¨arvi, I. Nechita, and R. Sengupta, Journal of Mathe- matical Physics 65 (2024), 10.1063/5.0202147

    work page doi:10.1063/5.0202147 2024

  56. [56]

    Thomas, M

    P. Thomas, M. Bohmann, and W. V ogel, Phys. Rev. A 96, 042321 (2017)

    work page 2017

  57. [57]

    Williamson, American journal of mathematics 58, 141 (1936)

    J. Williamson, American journal of mathematics 58, 141 (1936)

    work page 1936

  58. [58]

    Horodecki, Physics Letters A 232, 333–339 (1997)

    P. Horodecki, Physics Letters A 232, 333–339 (1997)

    work page 1997

  59. [59]

    Horodecki, P

    M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. 80, 5239 (1998)

    work page 1998

  60. [60]

    Horodecki, P

    M. Horodecki, P. Horodecki, and R. Horodecki, Physics Letters A 223, 1–8 (1996)

    work page 1996

  61. [61]

    Giedke, B

    G. Giedke, B. Kraus, M. Lewenstein, and J. I. Cirac, Physical Review Letters 87 (2001), 10.1103/physrevlett.87.167904

    work page doi:10.1103/physrevlett.87.167904 2001

  62. [62]

    Serafini, G

    A. Serafini, G. Adesso, and F. Illuminati, Physical Review A 71 (2005), 10.1103/physreva.71.032349

    work page doi:10.1103/physreva.71.032349 2005

  63. [63]

    L. Lami, S. Khatri, G. Adesso, and M. M. Wilde, Phys. Rev. Lett. 123, 050501 (2019)

    work page 2019

  64. [64]

    C. W. Gardiner and M. J. Collett, Phys. Rev. A31, 3761 (1985)

    work page 1985

  65. [65]

    Olivares, The European Physical Journal Special Topics203, 3–24 (2012)

    S. Olivares, The European Physical Journal Special Topics203, 3–24 (2012)

    work page 2012

  66. [66]

    Ma and W

    X. Ma and W. Rhodes, Phys. Rev. A 41, 4625 (1990)

    work page 1990

  67. [67]

    T. Das, R. Prabhu, A. Sen(De), and U. Sen, Phys. Rev. A 93, 052313 (2016)

    work page 2016

  68. [68]

    S. Ma, L. Zhou, J. Ma, and S. Xue, in 2024 43rd Chinese Con- trol Conference (CCC) (2024) pp. 6777–6782

    work page 2024

  69. [69]

    Adesso, M

    G. Adesso, M. Ericsson, and F. Illuminati, Physical Review A 76 (2007), 10.1103/physreva.76.022315

    work page doi:10.1103/physreva.76.022315 2007

  70. [70]

    Simon, N

    R. Simon, N. Mukunda, and B. Dutta, Phys. Rev. A 49, 1567 (1994)

    work page 1994

  71. [71]

    Typical behaviour of genuine multimode entanglement of pure gaussian states,

    S. Roy, “Typical behaviour of genuine multimode entanglement of pure gaussian states,” (2024)

    work page 2024

  72. [72]

    S. Ma, M. J. Woolley, X. Jia, and J. Zhang, Physical Review A 100 (2019), 10.1103/physreva.100.022309

    work page doi:10.1103/physreva.100.022309 2019

  73. [73]

    X.-y. Chen, M. Miao, R. Yin, and J. Yuan, Phys. Rev. A 107, 022410 (2023)

    work page 2023

  74. [74]

    An and W.-M

    J.-H. An and W.-M. Zhang, Phys. Rev. A 76, 042127 (2007)

    work page 2007