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arxiv: 1907.05222 · v1 · pith:TULSFGUXnew · submitted 2019-07-11 · 🪐 quant-ph

Quantum non-Gaussianity and secure quantum communication

Jaehak Lee , Jiyong Park , Hyunchul Nha This is my paper

Pith reviewed 2026-05-24 23:14 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum non-Gaussianityno-cloning boundcontinuous-variable communicationWigner negativityteleportationsecure communicationFock statescat states
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The pith

Non-Gaussian states require more resources to exceed the no-cloning bound for secure continuous-variable communication.

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

The paper calculates no-cloning bounds for sets of unknown non-Gaussian inputs including Fock states, their superpositions, and Schrödinger-cat states. It reports that these bounds typically fall as quantum non-Gaussianity rises. For mixed-state inputs, however, the Wigner negativity that marks stronger non-Gaussianity forces higher output fidelity thresholds in continuous-variable teleportation before security is guaranteed. This matters for protocol design because it shows that greater non-Gaussianity does not ease the path to secure communication. A reader would care because practical CV channels must meet these fidelity conditions with finite resources.

Core claim

The central claim is that the no-cloning bound decreases with quantum non-Gaussianity for the examined pure states, yet when inputs are mixed the same measure of non-Gaussianity (Wigner negativity) increases the resources needed to reach output fidelities above the bound during continuous-variable teleportation, so that more non-Gaussian states make secure communication harder.

What carries the argument

The no-cloning bound computed for chosen sets of unknown non-Gaussian states, compared against the output fidelity achievable in continuous-variable teleportation and quantified by Wigner negativity.

If this is right

  • Secure CV protocols that employ highly non-Gaussian inputs must achieve higher channel fidelity or expend extra resources to surpass the bound.
  • Gaussian coherent states can reach the security threshold with lower resource cost than states with large Wigner negativity.
  • The pattern holds across Fock states, their superpositions, and cat states when the inputs are mixed.
  • Resource accounting in teleportation must include the cost of preparing or preserving Wigner negativity if non-Gaussian inputs are used.

Where Pith is reading between the lines

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

  • Hybrid protocols that combine Gaussian and non-Gaussian resources could balance security margins against preparation cost.
  • Channel noise models not examined here might alter the relative resource demand between Gaussian and non-Gaussian inputs.
  • Experimental tests could fix the input state ensemble and measure the actual fidelity needed to exceed the calculated bound.

Load-bearing premise

The no-cloning bound obtained from abstract sets of states translates directly to the security condition for practical continuous-variable teleportation channels.

What would settle it

Compute the minimum teleportation fidelity required for a mixed Fock-state superposition with high Wigner negativity and check whether that fidelity lies above the no-cloning bound by a larger margin than for a less non-Gaussian mixed state.

Figures

Figures reproduced from arXiv: 1907.05222 by Hyunchul Nha, Jaehak Lee, Jiyong Park.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) A quantum state [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Teleportation fidelity using TMSV as a resource [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) QNG in terms of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Critical squeezing [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. No-cloning bound of DFS for [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Profiles of the functions [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Illustration of [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Wigner negativity [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Teleportation fidelity as a function of average photon number using non-Gaussian resource states with [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Teleportation fidelity with PNES. Symbols represent the fidelity for [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Values of Tr [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
read the original abstract

No-cloning theorem, a profound fundamental principle of quantum mechanics, also provides a crucial practical basis for secure quantum communication. The security of communication can be ultimately guaranteed if the output fidelity via communication channel is above the no-cloning bound (NCB). In quantum communications using continuous-variable (CV) systems, Gaussian states, more specifically, coherent states have been widely studied as inputs, but less is known for non-Gaussian states. We aim at exploring quantum communication covering CV states comprehensively with distinct sets of unknown states properly defined. Our main results here are (i) to establish the NCB for a broad class of quantum non-Gaussian states including Fock states, their superpositions and Schrodinger-cat states and (ii) to examine the relation between NCB and quantum non-Gaussianity (QNG). We find that NCB typically decreases with QNG. Remarkably, this does not mean that quantum non-Gaussian states are less demanding for secure communication. By extending our study to mixed-state inputs, we demonstrate that QNG specifically in terms of Wigner negativity requires more resources to achieve output fidelity above NCB in CV teleportation. The more non-Gaussian, the harder to achieve secure communication, which can have crucial implications for CV quantum communications.

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 / 2 minor

Summary. The manuscript establishes no-cloning bounds (NCB) for a broad class of continuous-variable quantum non-Gaussian states (Fock states, superpositions, and Schrödinger-cat states) and examines their relation to quantum non-Gaussianity (QNG). It reports that NCB typically decreases with QNG, yet for mixed-state inputs Wigner negativity increases the resources needed to reach output fidelity above NCB in CV teleportation, leading to the conclusion that greater non-Gaussianity makes secure communication harder.

Significance. If the derivations and resource comparisons hold, the work supplies concrete NCB values for non-Gaussian ensembles and identifies a counter-intuitive resource penalty arising from Wigner negativity, which could inform protocol design in CV quantum communication.

major comments (1)
  1. [Abstract] Abstract (and the section presenting the teleportation fidelity comparison): the claim that Wigner negativity requires more resources to exceed NCB rests on the unexamined assumption that the ideal-cloning NCB directly supplies the operative security threshold once states enter a realistic CV teleportation channel. No explicit model for loss, excess noise, or finite squeezing of the shared EPR resource is supplied, so it is unclear whether the fidelity expression contains additional degradation terms that scale differently with negativity than the ideal NCB.
minor comments (2)
  1. [Results] Define the precise ensembles of unknown states used for each NCB calculation and state whether the bounds are obtained analytically or numerically.
  2. [Methods] Clarify the precise measure of QNG employed (e.g., Wigner negativity volume versus another quantifier) when comparing across pure and mixed states.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the substantive comment on the abstract and teleportation analysis. We address the point directly below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the section presenting the teleportation fidelity comparison): the claim that Wigner negativity requires more resources to exceed NCB rests on the unexamined assumption that the ideal-cloning NCB directly supplies the operative security threshold once states enter a realistic CV teleportation channel. No explicit model for loss, excess noise, or finite squeezing of the shared EPR resource is supplied, so it is unclear whether the fidelity expression contains additional degradation terms that scale differently with negativity than the ideal NCB.

    Authors: We thank the referee for highlighting this modeling assumption. Our analysis employs the standard ideal CV teleportation fidelity formula with a two-mode squeezed vacuum resource of finite squeezing and compares the resulting fidelity directly to the no-cloning bound derived for the input state. The NCB itself is a channel-independent fundamental limit set by the no-cloning theorem; any additional loss, excess noise, or finite squeezing beyond the EPR resource would only further degrade the achievable fidelity. Consequently, the demonstrated increase in required squeezing for states with stronger Wigner negativity remains a lower-bound resource penalty. We agree that the abstract and teleportation section would benefit from an explicit statement of these modeling choices and a brief remark on the effect of realistic imperfections. We will therefore revise the manuscript accordingly. revision: partial

Circularity Check

0 steps flagged

No circularity; NCB derived independently for state ensembles and compared to QNG measures

full rationale

The paper computes the no-cloning bound (NCB) directly from the no-cloning theorem applied to explicitly defined ensembles of non-Gaussian states (Fock states, superpositions, cat states) and then compares the resulting NCB values to independent QNG quantifiers such as Wigner negativity. No equations or text in the abstract indicate that NCB is obtained by fitting to the same data used for the QNG comparison, nor is any uniqueness theorem or ansatz imported via self-citation. The extension to mixed-state teleportation fidelity likewise rests on separate resource counting rather than re-using the NCB definition itself. The derivation chain is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms or invented entities listed. The no-cloning theorem itself is treated as background.

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discussion (0)

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Reference graph

Works this paper leans on

61 extracted references · 61 canonical work pages · 1 internal anchor

  1. [1]

    & Hillery, M

    Buˇ zek, V. & Hillery, M. Quantum copying: Beyond the no-cloning theorem, Phys. Rev. A 54, 1844-1888 (1996)

    work page 1996

  2. [2]

    L., Cerf, N

    Braunstein, S. L., Cerf, N. J., Iblisdir, S., van Loock, P. & Massar, S. Optimal Cloning of Coherent States with a Linear Amplifier and Beam Splitters, Phys. Rev. Lett. 86, 4938-4941 (2001)

    work page 2001

  3. [3]

    Optical Implementation of Continuous- Variable Quantum Cloning Machines, Phys

    Fiur´ aˇ sek, J. Optical Implementation of Continuous- Variable Quantum Cloning Machines, Phys. Rev. Lett. 86, 4942-4945 (2001)

    work page 2001

  4. [4]

    & Ac´ ın, A

    Scarani, V., Iblisdir, S., Gisin, N. & Ac´ ın, A. Quantum cloning, Rev. Mod. Phys. 77, 1225-1256 (2005)

    work page 2005

  5. [5]

    & Massar, S

    Gisin, N. & Massar, S. Optimal Quantum Cloning Ma- chines, Phys. Rev. Lett. 79, 2153-2156 (1997)

    work page 1997

  6. [6]

    Bruß, D., Cinchetti, M., D’Ariano, G. M. & Macchiavello, C. Phase-covariant quantum cloning, Phys. Rev. A 62, 012302 (2000)

    work page 2000

  7. [7]

    J., Ipe, A

    Cerf, N. J., Ipe, A. & Rottenberg, X. Cloning of Contin- uous Quantum Variables, Phys. Rev. Lett. 85, 1754-1757 (2000)

    work page 2000

  8. [8]

    Cerf, N. J. & Iblisdir, S. Optimal N-to-M cloning of con- jugate quantum variables, Phys. Rev. A 62, 040301(R) (2000)

    work page 2000

  9. [9]

    J., Kr¨ uger, O., Navez, P., Werner, R

    Cerf, N. J., Kr¨ uger, O., Navez, P., Werner, R. F. & Wolf, M. M. Non-Gaussian Cloning of Quantum Coherent States is Optimal, Phys. Rev. Lett. 95, 070501 (2005)

    work page 2005

  10. [10]

    & Scarani, V

    Ac´ ın, A., Gisin, N. & Scarani, V. Coherent-pulse im- plementations of quantum cryptography protocols resis- tant to photon-number-splitting attacks, Phys. Rev. A 69, 012309 (2004)

    work page 2004

  11. [11]

    & Grangier, P

    Grosshans, F. & Grangier, P. Continuous Variable Quan- tum Cryptography Using Coherent States, Phys. Rev. Lett. 88, 057902 (2002). 7

    work page 2002

  12. [12]

    Cerf, N. J. & Grangier, P. From quantum cloning to quantum key distribution with continuous variables: a re- view, J. Opt. Soc. Am. B 24, 324-334 (2007)

    work page 2007

  13. [13]

    Braunstein, S. L. & Kimble, H. J. Teleportation of Con- tinuous Quantum Variables, Phys. Rev. Lett. 80, 869 (1998)

    work page 1998

  14. [14]

    Giovannetti, V., Garc´ ıa-Patr´ on, R., Cerf, N. J. & Holevo, A. S. Ultimate classical communication rates of quantum optical channels, Nat. Photon. 8, 796-800 (2014)

    work page 2014

  15. [15]

    & Holevo, A

    Mari, A., Giovannetti, V. & Holevo, A. S. Quantum state majorization at the output of bosonic Gaussian channels, Nat. Commun. 5, 3826 (2014)

    work page 2014

  16. [17]

    Gaussian Transformations and Distillation of Entangled Gaussian States, Phys

    Fiur´ aˇ sek, J. Gaussian Transformations and Distillation of Entangled Gaussian States, Phys. Rev. Lett. 89, 137904 (2002)

    work page 2002

  17. [18]

    Menicucci, N. C. et al. Universal Quantum Computation with Continuous-Variable Cluster States, Phys. Rev. Lett. 97, 110501 (2006)

    work page 2006

  18. [19]

    & Cerf, N

    Niset, J., Fiur´ aˇ sek, J. & Cerf, N. J. No-Go Theorem for Gaussian Quantum Error Correction, Phys. Rev. Lett. 102, 120501 (2009)

    work page 2009

  19. [20]

    Faithful measure of Quantum non-Gaussianity via quantum relative entropy

    Park, J., Lee, J., Baek, K., Ji, S.-W. & Nha, H. Faithful measure of Quantum non-Gaussianity via quantum rela- tive entropy, Preprint at arXiv:1809.02999

    work page internal anchor Pith review Pith/arXiv arXiv

  20. [21]

    F., Furusawa, A

    Ide, T., Hofmann, H. F., Furusawa, A. & Kobayashi, T. Gain tuning & fidelity in continuous-variable quantum teleportation, Phys. Rev. A 65, 062303 (2002)

    work page 2002

  21. [22]

    & Illuminati, F

    Dell´Anno, F., De Siena, S., Albano, L. & Illuminati, F. Continuous-variable quantum teleportation with non- Gaussian resources, Phys. Rev. A 76, 022301 (2007)

    work page 2007

  22. [23]

    Farias L. A. & Stephany, J. Optimization of the trans- mission of observable expectation values and observable statistics in continuous-variable teleportation, Phys. Rev. A 82, 062322 (2010)

    work page 2010

  23. [24]

    & Paris, M

    Allegra, M., Giorda, P. & Paris, M. G. A. Role of Initial Entanglement and Non-Gaussianity in the Decoherence of Photon-Number Entangled States Evolving in a Noisy Channel, Phys. Rev. Lett. 105, 100503 (2010)

    work page 2010

  24. [25]

    Role of Initial Entanglement and Non-Gaussianity in the Deco- herence of Photon-Number Entangled States Evolving in a Noisy Channel

    Lee, J., Kim, M. S. & Nha, H. Comment on “Role of Initial Entanglement and Non-Gaussianity in the Deco- herence of Photon-Number Entangled States Evolving in a Noisy Channel”, Phys. Rev. Lett. 107, 238901 (2011)

    work page 2011

  25. [26]

    K., Ivan, J

    Sabapathy, K. K., Ivan, J. S. & Simon, R. Robustness of Non-Gaussian Entanglement against Noisy Amplifier and Attenuator Environments, Phys. Rev. Lett. 107, 130501 (2011)

    work page 2011

  26. [27]

    & Nha, H

    Lee, J. & Nha, H. Entanglement distillation for continu- ous variables in a thermal environment: Effectiveness of a non-Gaussian operation, Phys. Rev. A 87, 032307 (2013)

    work page 2013

  27. [28]

    & Welsch, D.-G

    Opatrn´ y, T., Kurizki, G. & Welsch, D.-G. Improvement on teleportation of continuous variables by photon sub- traction via conditional measurement, Phys. Rev. A 61, 032302 (2000)

    work page 2000

  28. [29]

    & Nha, H

    Lee, S.-Y., Ji, S.-W., Kim, H.-J. & Nha, H. Enhancing quantum entanglement for continuous variables by a co- herent superposition of photon subtraction and addition, Phys. Rev. A 84, 012302 (2011)

    work page 2011

  29. [30]

    & Kim, M

    Nha, H., Lee, S.-Y., Ji, S.-W. & Kim, M. S. Efficient En- tanglement Criteria beyond Gaussian Limits Using Gaus- sian Measurements, Phys. Rev. Lett. 108, 030503 (2012)

    work page 2012

  30. [31]

    & Nha, H

    Lee, J., Park, J. & Nha, H. Optimal continuous-variable teleportation under energy constraint, Phys. Rev. A 95, 052343 (2017)

    work page 2017

  31. [32]

    G., Paris, M

    Albarelli, F., Genoni, M. G., Paris, M. G. A. & Fer- raro, A. Resource theory of quantum non-Gaussianity and Wigner negativity, Phys. Rev. A 98, 052350 (2018)

    work page 2018

  32. [33]

    & Zhuang, Q

    Takagi, R. & Zhuang, Q. Convex resource theory of non- Gaussianity, Phys. Rev. A 97, 062337 (2018)

    work page 2018

  33. [34]

    & Adesso, G

    Takagi, R., Regula, B., Bu, K., Liu, Z.-W. & Adesso, G. Operational advantage of quantum resources in subchan- nel discrimination, Phys. Rev. Lett. 122, 140402 (2019)

    work page 2019

  34. [35]

    & Reid, M

    He, Q., Rosales-Z´ arate, L., Adesso, G. & Reid, M. D. Secure Continuous Variable Teleportation and Einstein- Podolsky-Rosen Steering, Phys. Rev. Lett. 115, 180502 (2015)

    work page 2015

  35. [36]

    & Adesso, G

    Liuzzo-Scorpo, P. & Adesso, G. Optimal secure quantum teleportation of coherent states of light, Proc. SPIE Opt. Photon. 10358, 103580V (2017)

    work page 2017

  36. [37]

    Kr¨ uger, O.Quantum information theory with Gaussian systems (Ph. D. thesis, Technische Universit¨ at Braun- schweig, 2006)

    work page 2006

  37. [38]

    M., Radmore, P

    Barnett, S. M., Radmore, P. M., Methods in Theoret- ical Quantum Optics, Oxford University Press (2003)

    work page 2003

  38. [39]

    & Nha, H

    Lee, J., Ji, S.-W., Park, J. & Nha, H. Monogamy rela- tion in multipartite continuous-variable quantum telepor- tation, Phys. Rev. A 94, 062318 (2016)

    work page 2016

  39. [40]

    & Marian, T

    Marian, P. & Marian, T. A. Continuous-variable tele- portation in the characteristic-function description, Phys. Rev. A 74, 042306 (2006)

    work page 2006

  40. [41]

    and Takei, N

    Furusawa, A. and Takei, N. Quantum teleportation for continuous variables and related quantum information processing, Physics Reports 443, 97 (2007)

    work page 2007

  41. [42]

    Genoni, M. G. & Paris, M. G. A. Quantifying non- Gaussianity for quantum information, Phys. Rev. A 82, 052341 (2010)

    work page 2010

  42. [43]

    & Marian, T

    Marian, P. & Marian, T. A. Relative entropy is an ex- act measure of non-Gaussianity, Phys. Rev. A 88, 012322 (2013)

    work page 2013

  43. [44]

    B., Polzik, E

    Owari, M., Plenio, M. B., Polzik, E. S., Serafini, A., Wolf, M. M., Squeezing the limit: quantum benchmarks for the teleportation and storage of squeezed states, New J. Phys. 10, 113014 (2008)

    work page 2008

  44. [45]

    R., Preiss, P

    Islam. R., Preiss, P. M., Tai, M. E., Lukin, A., Rispoli, M., Greiner, M., Measuring entanglement entropy in a quantum many-body system, Nature 528, 77 (2015)

    work page 2015

  45. [46]

    & Miˇ sta, Jr., L

    Filip, R. & Miˇ sta, Jr., L. Detecting Quantum States with a Positive Wigner Function beyond Mixtures of Gaussian States, Phys. Rev. Lett. 106, 200401 (2011)

    work page 2011

  46. [47]

    Experimental Test of the Quantum Non- Gaussian Character of a Heralded Single-Photon State, Phys

    Jeˇ zek, M.et al. Experimental Test of the Quantum Non- Gaussian Character of a Heralded Single-Photon State, Phys. Rev. Lett. 107, 213602 (2011)

    work page 2011

  47. [48]

    Genoni, M. G. et al. Detecting quantum non-Gaussianity via the Wigner function, Phys. Rev. A 87, 062104 (2013)

    work page 2013

  48. [49]

    G., Tufarelli, T., Paris, M

    Hughes, C., Genoni, M. G., Tufarelli, T., Paris, M. G. A. & Kim, M. S. Quantum non-Gaussianity witnesses in phase space, Phys. Rev. A 90, 013810 (2014)

    work page 2014

  49. [50]

    Park, J. et al. Testing Nonclassicality and Non- Gaussianity in Phase Space, Phys. Rev. Lett. 114, 190402 (2015)

    work page 2015

  50. [51]

    and Nha, H

    Park, J. and Nha, H. Demonstrating nonclassicality and non-Gaussianity of single-mode fields: Bell-type tests us- ing generalized phase-space distributions, Phys. Rev. A 92, 062134 (2015). 8

    work page 2015

  51. [52]

    A., Nha, H

    Happ, L., Efremov, M. A., Nha, H. & Schleich, W. P. Sufficient condition for a quantum state to be genuinely quantum non-Gaussian, New J. Phys. 20, 023046 (2018)

    work page 2018

  52. [53]

    Straka, I. et al. Quantum non-Gaussian multiphoton light, npj Quantum Inf. 4, 4 (2018)

    work page 2018

  53. [54]

    T., Ralph, T

    Cochrane, P. T., Ralph, T. C. & Doli´ nska, A. Optimal cloning for finite distributions of coherent states, Phys. Rev. A 69, 042313 (2004). Supplemental Information S1. FIDELITY OF THE COVARIANT CLONER For a general input state ρin = ˆD†(α)ρ ˆD(α), the phase-space overlap between the input and the ith output of a covariant cloner, F (i) = Tr [ T (ρin)ρ(i) ...

    work page 2004

  54. [55]

    By choosing ρT =|0⟩⟨0|, we find that the classical bound can be achieved with Gaussian schemes

    and the bound is achieved with ρT =|0⟩⟨0| or|2n⟩⟨2n|. By choosing ρT =|0⟩⟨0|, we find that the classical bound can be achieved with Gaussian schemes. S5. PLOT OF RESOURCE REQUIREMENT FOR VARIOUS PURE INPUT STATES We here provide the plot showing the resource requirementrc against the Wigner negativity WN≡ log[ ∫ d2α|W (α)|] of input state, which correspond...

    work page

  55. [56]

    M., Radmore, P

    Barnett, S. M., Radmore, P. M., Methods in Theoretical Quantum Optics, Oxford University Press (2003)

    work page 2003

  56. [57]

    K. S. K¨ olbig and H. Scherb, On a Hankel transform integral containing an exponential function and two Laguerre polyno- mials, J. Comp. Appl. Math. 71, 357 (1996)

    work page 1996

  57. [58]

    W. N. Bailey, Some integrals involving Hermite polynomials. J. London Math. Soc. 23, 291-297, (1948)

    work page 1948

  58. [59]

    Kr¨ uger, Quantum information theory with Gaussian systems, Ph

    O. Kr¨ uger, Quantum information theory with Gaussian systems, Ph. D. thesis, Technische Universit¨ at Braunschweig (2006)

    work page 2006

  59. [60]

    J., Ralph, T

    Weedbrook, C., Pirandola, S., Garcia-Patron, R., Cerf, N. J., Ralph, T. C., Shapiro, J. H., Lloyd, S., Gaussian quantum information, Rev. Mod. Phys. 84, 621 (2012)

    work page 2012

  60. [61]

    J. Lee, J. Park, and H. Nha, Optimal continuous-variable teleportation under energy constraint, Phys. Rev. A 95, 052343 (2017)

    work page 2017

  61. [62]

    & Kim, M

    Nha, H., Lee, S.-Y., Ji, S.-W. & Kim, M. S. Efficient Entanglement Criteria beyond Gaussian Limits Using Gaussian Measurements, Phys. Rev. Lett. 108, 030503 (2012)

    work page 2012