pith. sign in

arxiv: 2605.18605 · v1 · pith:CTUKMQJLnew · submitted 2026-05-18 · 🪐 quant-ph

Non-Gaussian Entanglement Hierarchy Based on the Schmidt Number

Jiajie Guo , Shuheng Liu , Matteo Fadel , Qiongyi He This is my paper

Pith reviewed 2026-05-20 10:09 UTC · model grok-4.3

classification 🪐 quant-ph
keywords non-Gaussian entanglementSchmidt numberGaussian operationsentanglement witnesscontinuous-variable systemsNOON statesbosonic systemsquantum optics
0
0 comments X

The pith

The witness E_NG equals 1 exactly for Gaussian-entanglable states and its ceiling lower-bounds the Schmidt number that Gaussian operations cannot reduce.

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

The paper introduces a quantitative witness E_NG for non-Gaussian entanglement in bipartite bosonic systems. This witness is constructed to equal 1 for every state that can be obtained by applying Gaussian operations to separable inputs. Any value strictly greater than 1 certifies entanglement that lies outside this Gaussian-entanglable set. The ceiling of E_NG then supplies a lower bound on the Schmidt number that remains after all possible Gaussian transformations have been applied. This bound creates a hierarchy of non-Gaussian entanglement levels, which the authors show is sharp for pure states and directly reflects the complexity of learning such states.

Core claim

The central claim is that the witness E_NG satisfies E_NG = 1 precisely on the set of states obtainable by Gaussian operations applied to separable inputs, while E_NG > 1 certifies non-Gaussian entanglement; its ceiling d = ceil(E_NG) then provides a lower bound on the Schmidt number irreducible by Gaussian transformations, thereby defining a natural hierarchy of non-Gaussian entanglement.

What carries the argument

The quantitative witness E_NG, built so that its value equals exactly 1 on all Gaussian-entanglable states and whose ceiling supplies the lower bound on the Gaussian-irreducible Schmidt number.

If this is right

  • E_NG greater than 1 directly certifies non-Gaussian entanglement.
  • ceil(E_NG) gives a concrete lower bound on the minimal Schmidt number left after Gaussian processing.
  • For pure states the bound is tight and tracks the number of independent parameters needed to learn the state.
  • The framework applies to standard non-Gaussian resources such as NOON states and squeezed Kerr states.
  • A practical NOON-type witness can be realized with only four projective measurements.

Where Pith is reading between the lines

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

  • The hierarchy may help experimenters select the minimal non-Gaussianity level required for a given continuous-variable protocol.
  • Robustness results against loss suggest the measure can be used to quantify resource degradation in realistic channels.
  • Similar witnesses could be constructed for other classes of operations beyond Gaussian ones.

Load-bearing premise

The witness E_NG is defined such that it equals exactly 1 for every state reachable by Gaussian operations from separable inputs.

What would settle it

An explicit state that can be produced by Gaussian operations on separable inputs yet yields E_NG greater than 1, or a state with E_NG equal to 1 that cannot be produced that way.

Figures

Figures reproduced from arXiv: 2605.18605 by Jiajie Guo, Matteo Fadel, Qiongyi He, Shuheng Liu.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (b) and (c) present the entanglement for a superposi￾tion of two-mode squeezed vacuum (TMSV) states, |ψsTMSV⟩ = c  Sˆ AB(r) + S AB(−r)  |0, 0⟩, (11) and for states generated by a three-photon spontaneous para￾metric down-conversion (SPDC) process [40], |ψ3-SPDC⟩ = e g(aˆ †bˆ†2−aˆbˆ2 ) |0, 0⟩, (12) respectively. Here, c is a normalization constant, r denotes the squeezing strength in two-mode squeezing op… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The complexity of pure state learning in a finite-energy truncation with local dimension [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
read the original abstract

Non-Gaussian entanglement is a promising resource in various quantum tasks. A recently defined class identifies entanglement that cannot be generated by applying Gaussian operations to separable inputs. To further explore the entanglement in this context, we introduce a quantitative witness $E_{\rm NG}$ in bipartite bosonic systems, which satisfies $E_{\rm NG}=1$ for all Gaussian-entanglable states, while $E_{\rm NG}>1$ certifies non-Gaussian entanglement. Its ceiling $d=\lceil E_{\rm NG}\rceil$ provides a lower bound on the Schmidt number irreducible by Gaussian transformations, thereby defining a natural hierarchy of non-Gaussian entanglement. For pure states, the condition is sharp and the hierarchy reflects the complexity of state learning. We benchmark the framework with some paradigmatic non-Gaussian states, such as NOON states and squeezed Kerr states, and analyze its robustness against loss. Moreover, we construct an experimentally economical NOON-type witness requiring only four projective measurements, where an analytical Gaussian-entanglable threshold is derived. These results establish an operationally meaningful and experimentally accessible framework for identifying non-Gaussian entanglement resources in continuous-variable quantum platforms.

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 introduces a quantitative witness E_NG for non-Gaussian entanglement in bipartite bosonic systems. It claims that E_NG = 1 for all Gaussian-entanglable states (obtainable via Gaussian operations on separable inputs), while E_NG > 1 certifies non-Gaussian entanglement. The ceiling d = ceil(E_NG) supplies a lower bound on the Schmidt number that remains irreducible under Gaussian transformations, thereby establishing a hierarchy. For pure states the bound is asserted to be sharp and to reflect state-learning complexity. The framework is benchmarked on NOON states and squeezed Kerr states, loss robustness is analyzed, and an economical NOON-type witness requiring only four projective measurements is constructed with an analytical Gaussian-entanglable threshold.

Significance. If the central properties of E_NG hold with the claimed tightness, the work would supply an operationally meaningful, experimentally accessible hierarchy for classifying non-Gaussian entanglement resources in continuous-variable platforms, together with a direct link to Schmidt-number complexity and state learning.

major comments (2)
  1. [Definition of E_NG] Definition of E_NG (abstract and the paragraph introducing the witness): the load-bearing claim that E_NG saturates exactly at 1 for every Gaussian-entanglable state requires explicit verification that the infimum (or optimization) over Gaussian operations is attained in infinite-dimensional bosonic Hilbert space. The abstract states the property but supplies no derivation or general argument showing the bound is tight without residual gap; specific examples (NOON, squeezed Kerr) do not substitute for this general step.
  2. [Hierarchy and Schmidt-number bound] Hierarchy and Schmidt-number bound (section on the ceiling d = ceil(E_NG)): the assertion that d furnishes a lower bound irreducible by Gaussian transformations rests on the exact saturation E_NG = 1; without a proof that the witness evaluates to precisely 1 on the entire set, the hierarchy claim cannot be fully supported.
minor comments (2)
  1. [Experimental witness construction] The abstract states that an analytical Gaussian-entanglable threshold is derived for the four-measurement NOON witness; the main text should present the explicit derivation and the resulting threshold value.
  2. [Notation] Notation for the optimization defining E_NG should be introduced with a clear equation early in the manuscript to avoid ambiguity in later sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. We appreciate the opportunity to clarify and strengthen the presentation of our results. Below, we provide point-by-point responses to the major comments. We will incorporate the suggested clarifications and proofs in the revised manuscript.

read point-by-point responses

  1. Referee: [Definition of E_NG] Definition of E_NG (abstract and the paragraph introducing the witness): the load-bearing claim that E_NG saturates exactly at 1 for every Gaussian-entanglable state requires explicit verification that the infimum (or optimization) over Gaussian operations is attained in infinite-dimensional bosonic Hilbert space. The abstract states the property but supplies no derivation or general argument showing the bound is tight without residual gap; specific examples (NOON, squeezed Kerr) do not substitute for this general step.

    Authors: We agree with the referee that a general argument for the exact saturation E_NG = 1 is essential for the claims. In the original manuscript, the witness is constructed such that for Gaussian-entanglable states, which are those obtainable from separable states via Gaussian unitaries and displacements, the optimization defining E_NG reaches its minimum value of 1 by definition, as the witness quantifies deviation from Gaussian-entanglability. However, to rigorously address the infinite-dimensional nature of bosonic systems, we will add a dedicated paragraph or appendix in the revision. This will include a proof sketch demonstrating that the infimum over the set of Gaussian operations is attained or can be approached arbitrarily closely for any such state, leveraging the continuity properties of Gaussian maps and the compactness in appropriate topologies on the space of states. This ensures no residual gap and confirms the bound is tight. Specific examples serve as illustrations, but the general case will now be covered. revision: yes

  2. Referee: [Hierarchy and Schmidt-number bound] Hierarchy and Schmidt-number bound (section on the ceiling d = ceil(E_NG)): the assertion that d furnishes a lower bound irreducible by Gaussian transformations rests on the exact saturation E_NG = 1; without a proof that the witness evaluates to precisely 1 on the entire set, the hierarchy claim cannot be fully supported.

    Authors: The referee correctly identifies that the hierarchy and the irreducible Schmidt-number bound depend on the precise evaluation of E_NG = 1 for all Gaussian-entanglable states. With the addition of the general verification as outlined in response to the first comment, this foundation will be solidified. The ceiling d = ceil(E_NG) then provides a lower bound on the Gaussian-irreducible Schmidt number, as any Gaussian transformation cannot reduce the witness value below 1, hence cannot decrease d. For pure states, we maintain that the bound is sharp, reflecting the minimal number of non-Gaussian resources needed, which aligns with state-learning complexity. We will also ensure the section explicitly references the new proof to support the hierarchy claim. revision: yes

Circularity Check

1 steps flagged

E_NG threshold set by definition on Gaussian-entanglable set

specific steps
  1. self definitional [Abstract]
    "we introduce a quantitative witness $E_{rm NG}$ in bipartite bosonic systems, which satisfies $E_{rm NG}=1$ for all Gaussian-entanglable states, while $E_{rm NG}>1$ certifies non-Gaussian entanglement. Its ceiling $d=ceil(E_{rm NG})$ provides a lower bound on the Schmidt number irreducible by Gaussian transformations"

    The paper defines the witness E_NG such that its value is exactly 1 on the set of Gaussian-entanglable states (Gaussian operations on separable inputs). The equality E_NG=1 is therefore true by the way the witness is introduced, not obtained as a prediction or theorem from prior independent assumptions. The hierarchy and certification claims inherit this definitional property.

full rationale

The central witness is introduced with the explicit property that E_NG equals 1 exactly on states obtainable by Gaussian operations on separable inputs. This threshold is therefore satisfied by construction of the definition rather than derived from an independent optimization or external principle. The subsequent hierarchy (ceiling bound on Schmidt number) and certification claims rest on this definitional equality holding tightly in infinite dimensions. No self-citation chain or fitted-parameter renaming is visible in the abstract or stated claims; the construction may still be non-circular if an explicit formula is later shown to achieve the bound without residual gap. Overall partial circularity because the load-bearing saturation claim reduces to how E_NG is defined.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard definition of Gaussian operations and separable states in bosonic systems plus the mathematical properties of the Schmidt number; no free parameters or new invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Gaussian operations are the set of transformations generated by quadratic Hamiltonians on bosonic modes.
    Invoked when defining the class of Gaussian-entanglable states.
  • standard math The Schmidt number is a well-defined entanglement monotone for bipartite pure states.
    Used to interpret the ceiling of E_NG as a lower bound.

pith-pipeline@v0.9.0 · 5740 in / 1411 out tokens · 39643 ms · 2026-05-20T10:09:51.531043+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.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    E_NG := inf_{Û_G ∈ G_2} ||(Û_G ρ Û_G†)^{T_A}||_1 ... satisfies E_NG = 1 for all GE states, while E_NG > 1 certifies non-Gaussian entanglement. Its ceiling d = ⌈E_NG⌉ provides a lower bound on the Schmidt number irreducible by Gaussian transformations

  • IndisputableMonolith/Foundation/BranchSelection.lean branch_selection echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    E_NG(ρ) ≥ 1 for all physical states ... E_NG(ρ) = 1 for ρ ∈ GE ... invariant under any Gaussian unitaries

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

53 extracted references · 53 canonical work pages · 2 internal anchors

  1. [1]

    Horodecki, P

    R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys.81, 865 (2009)

    work page 2009

  2. [2]

    Gisin and R

    N. Gisin and R. Thew, Quantum communication, Nature Pho- tonics1, 165 (2007)

    work page 2007

  3. [3]

    Gottesman, A

    D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in an oscillator, Phys. Rev. A64, 012310 (2001)

    work page 2001

  4. [4]

    Weedbrook, S

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

    work page 2012

  5. [5]

    Serafini,Quantum Continuous Variables: A Primer of The- oretical Methods(CRC Press, 2017)

    A. Serafini,Quantum Continuous Variables: A Primer of The- oretical Methods(CRC Press, 2017)

    work page 2017

  6. [6]

    Walschaers, Non-Gaussian quantum states and where to find them, PRX Quantum2, 030204 (2021)

    M. Walschaers, Non-Gaussian quantum states and where to find them, PRX Quantum2, 030204 (2021)

    work page 2021

  7. [7]

    Eisert, S

    J. Eisert, S. Scheel, and M. B. Plenio, Distilling Gaussian states with Gaussian operations is impossible, Phys. Rev. Lett.89, 137903 (2002)

    work page 2002

  8. [8]

    Giedke and J

    G. Giedke and J. I. Cirac, Characterization of Gaussian op- erations and distillation of Gaussian states, Phys. Rev. A66, 032316 (2002)

    work page 2002

  9. [9]

    Fiur ´aˇsek, Gaussian transformations and distillation of entan- gled Gaussian states, Phys

    J. Fiur ´aˇsek, Gaussian transformations and distillation of entan- gled Gaussian states, Phys. Rev. Lett.89, 137904 (2002)

    work page 2002

  10. [10]

    Banaszek and K

    K. Banaszek and K. W ´odkiewicz, Nonlocality of the Einstein- Podolsky-Rosen state with continuous variables, Phys. Rev. A 58, 4345 (1998)

    work page 1998

  11. [11]

    Lloyd and S

    S. Lloyd and S. L. Braunstein, Quantum computation over con- tinuous variables, Phys. Rev. Lett.82, 1784 (1999)

    work page 1999

  12. [12]

    S. D. Bartlett and B. C. Sanders, Universal continuous-variable quantum computation: Requirement of optical nonlinearity for photon counting, Phys. Rev. A65, 042304 (2002)

    work page 2002

  13. [13]

    Peres, Separability criterion for density matrices, Phys

    A. Peres, Separability criterion for density matrices, Phys. Rev. Lett.77, 1413 (1996)

    work page 1996

  14. [14]

    Horodecki, Separability criterion and inseparable mixed states with positive partial transposition, Phys

    P. Horodecki, Separability criterion and inseparable mixed states with positive partial transposition, Phys. Lett. A232, 333 6 (1997)

    work page 1997

  15. [15]

    Simon, Peres-Horodecki separability criterion for continu- ous variable systems, Phys

    R. Simon, Peres-Horodecki separability criterion for continu- ous variable systems, Phys. Rev. Lett.84, 2726 (2000)

    work page 2000

  16. [16]

    Vidal and R

    G. Vidal and R. F. Werner, Computable measure of entangle- ment, Phys. Rev. A65, 032314 (2002)

    work page 2002

  17. [17]

    Adesso, A

    G. Adesso, A. Serafini, and F. Illuminati, Extremal entangle- ment and mixedness in continuous variable systems, Phys. Rev. A70, 022318 (2004)

    work page 2004

  18. [18]

    Shchukin and W

    E. Shchukin and W. V ogel, Inseparability criteria for continuous bipartite quantum states, Phys. Rev. Lett.95, 230502 (2005)

    work page 2005

  19. [19]

    Hillery and M

    M. Hillery and M. S. Zubairy, Entanglement conditions for two- mode states, Phys. Rev. Lett.96, 050503 (2006)

    work page 2006

  20. [20]

    Jayachandran, L

    P. Jayachandran, L. H. Zaw, and V . Scarani, Dynamics-based entanglement witnesses for non-gaussian states of harmonic os- cillators, Phys. Rev. Lett.130, 160201 (2023)

    work page 2023

  21. [21]

    L. H. Zaw, Certifiable lower bounds of wigner negativity vol- ume and non-gaussian entanglement with conditional displace- ment gates, Phys. Rev. Lett.133, 050201 (2024)

    work page 2024

  22. [22]

    S. Liu, J. Guo, Q. He, and M. Fadel, Quantum entanglement in phase space, Quantum10, 2040 (2026)

    work page 2040

  23. [23]

    S. Liu, J. Guo, M. Fadel, Q. He, M. Huber, and G. Vitagliano, Entanglement dimensionality of continuous variable states from phase-space quasi-probabilities, arXiv:2509.02743

    work page arXiv

  24. [24]

    B. M. Terhal and P. Horodecki, Schmidt number for density matrices, Phys. Rev. A61, 040301 (2000)

    work page 2000

  25. [25]

    Sanpera, D

    A. Sanpera, D. Bruß, and M. Lewenstein, Schmidt-number witnesses and bound entanglement, Phys. Rev. A63, 050301 (2001)

    work page 2001

  26. [26]

    Huber and J

    M. Huber and J. I. de Vicente, Structure of multidimensional entanglement in multipartite systems, Phys. Rev. Lett.110, 030501 (2013)

    work page 2013

  27. [27]

    Friis, G

    N. Friis, G. Vitagliano, M. Malik, and M. Huber, Entangle- ment certification from theory to experiment, Nature Reviews Physics1, 72 (2019)

    work page 2019

  28. [28]

    X. Zhao, P. Liao, F. A. Mele, U. Chabaud, and Q. Zhuang, Com- plexity of quantum tomography from genuine non-Gaussian en- tanglement, Nat. Commun.17, 373 (2025)

    work page 2025

  29. [29]

    Aaronson and A

    S. Aaronson and A. Arkhipov, The computational complexity of linear optics, inProceedings of the 43rd Annual ACM Sym- posium on Theory of Computing (STOC)(2011) p. 333

    work page 2011

  30. [30]

    L. S. Madsen, F. Laudenbach, M. F. Askarani,et al., Quantum computational advantage with a programmable photonic pro- cessor, Nature606, 75 (2022)

    work page 2022

  31. [31]

    M. V . Larsen, J. E. Bourassa, S. Kocsis,et al., Integrated pho- tonic source of Gottesman–Kitaev–Preskill qubits, Nature642, 587 (2025), verify DOI against published record

    work page 2025

  32. [32]

    A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Quantum interferometric optical lithography: Exploiting entanglement to beat the diffraction limit, Phys. Rev. Lett.85, 2733 (2000)

    work page 2000

  33. [33]

    M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, Super- resolving phase measurements with a multiphoton entangled state, Nature429, 161 (2004)

    work page 2004

  34. [34]

    Nagata, R

    T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, Beating the standard quantum limit with four- entangled photons, Science316, 726 (2007)

    work page 2007

  35. [35]

    Certification of genuine non-Gaussian entanglement

    L. Lachman, C. E. Lopetegui-Gonz ´alez, M. Frigerio, and M. Walschaers, Certification of genuine non-gaussian entangle- ment, arXiv:2604.22295

    work page internal anchor Pith review Pith/arXiv arXiv

  36. [36]

    C. K. Hong, Z. Y . Ou, and L. Mandel, Measurement of subpi- cosecond time intervals between two photons by interference, Phys. Rev. Lett.59, 2044 (1987)

    work page 2044

  37. [37]

    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. Fick- ler, T. Giordani, F. Sciarrino, Y . Zheng, J. Wang, M. Huber, A. Tavakoli, R. Uola, N. Brunner, N. Friis, N...

    work page internal anchor Pith review Pith/arXiv arXiv

  38. [38]

    Eltschka and J

    C. Eltschka and J. Siewert, Negativity as an estimator of entan- glement dimension, Phys. Rev. Lett.111, 100503 (2013)

    work page 2013

  39. [39]

    See Supplementary Materials for the properties of non- Gaussian entanglement witnesses, the analysis on the effec- tive Gaussian unitary, more examples of Gaussian-entagnlable states, and the application of non-Gaussian entanglement in en- tanglement distillation

    work page

  40. [40]

    C. W. S. Chang, C. Sab ´ın, P. Forn-D´ıaz, F. Quijandr´ıa, A. M. Vadiraj, I. Nsanzineza, G. Johansson, and C. M. Wilson, Observation of three-photon spontaneous parametric down- conversion in a superconducting parametric cavity, Phys. Rev. X10, 011011 (2020)

    work page 2020

  41. [41]

    G. J. Milburn and C. A. Holmes, Quantum coherence and clas- sical chaos in a pulsed parametric oscillator with a kerr nonlin- earity, Phys. Rev. A44, 4704 (1991)

    work page 1991

  42. [42]

    Wielinga and G

    B. Wielinga and G. J. Milburn, Quantum tunneling in a kerr medium with parametric pumping, Phys. Rev. A48, 2494 (1993)

    work page 1993

  43. [43]

    Miranowicz, R

    A. Miranowicz, R. Tanas, and S. Kielich, Generation of discrete superpositions of coherent states in the anharmonic oscillator model, Quantum Opt.2, 253 (1990)

    work page 1990

  44. [44]

    Imamo ¯glu, H

    A. Imamo ¯glu, H. Schmidt, G. Woods, and M. Deutsch, Strongly interacting photons in a nonlinear cavity, Phys. Rev. Lett.79, 1467 (1997)

    work page 1997

  45. [45]

    J. Guo, Q. He, and M. Fadel, Quantum metrology with a squeezed kerr oscillator, Phys. Rev. A109, 052604 (2024)

    work page 2024

  46. [46]

    Kirchmair, B

    G. Kirchmair, B. Vlastakis, Z. Leghtas, S. E. Nigg, H. Paik, E. Ginossar, M. Mirrahimi, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, Observation of quantum state collapse and revival due to the single-photon kerr effect, Nature495, 205 (2013)

    work page 2013

  47. [47]

    Grimm, N

    A. Grimm, N. E. Frattini, S. Puri, S. O. Mundhada, S. Touzard, M. Mirrahimi, S. M. Girvin, S. Shankar, and M. H. Devoret, Stabilization and operation of a kerr-cat qubit, Nature584, 205 (2020)

    work page 2020

  48. [48]

    Marti, U

    S. Marti, U. V on L ¨upke, O. Joshi, Y . Yang, M. Bild, A. Oma- hen, Y . Chu, and M. Fadel, Quantum squeezing in a nonlinear mechanical oscillator, Nature Physics20, 1448 (2024)

    work page 2024

  49. [49]

    C. Wang, Y . Y . Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. De- voret, and R. J. Schoelkopf, A schr ¨odinger cat living in two boxes, Science352, 1087 (2016)

    work page 2016

  50. [50]

    Y . Y . Gao, B. J. Lester, K. S. Chou, L. Frunzio, M. H. De- voret, L. Jiang, S. M. Girvin, and R. J. Schoelkopf, Entangle- ment of bosonic modes through an engineered exchange inter- action, Nature566, 509 (2019)

    work page 2019

  51. [51]

    von L ¨upke, I

    U. von L ¨upke, I. C. Rodrigues, Y . Yang, M. Fadel, and Y . Chu, Engineering multimode interactions in circuit quantum acous- todynamics, Nat. Phys.20, 564 (2024)

    work page 2024

  52. [52]

    NON-GAUSSIAN ENTANGLEMENT HIERARCHY BASED ON THE SCHMIDT NUMBER

    J. Eisert, S. Scheel, and M. B. Plenio, Distilling gaussian states with gaussian operations is impossible, Phys. Rev. Lett.89, 137903 (2002). 7 SUPPLEMENTAL MA TERIAL FOR “NON-GAUSSIAN ENTANGLEMENT HIERARCHY BASED ON THE SCHMIDT NUMBER” I. NON-GAUSSIAN ENTANGLEMENT WITNESS A. Properties of non-Gaussian entanglement witnessE NG In this section, we prove so...

    work page 2002

  53. [53]

    We chooseˆUG = ˆUBS ˆRA(ϕ), where ˆRA(ϕ)=e −iϕˆa† ˆais a local rotation operator and ˆUBS(π/4)=exp h π/4 ˆa† ˆb−ˆaˆb† i is a balanced beam-splitter

    This yields fG ≥λ 2 max ˆI|ψNOON⟩ = 1 2 .(62) A second lower bound is obtained from a balanced beam splitter together with a local phase rotation. We chooseˆUG = ˆUBS ˆRA(ϕ), where ˆRA(ϕ)=e −iϕˆa† ˆais a local rotation operator and ˆUBS(π/4)=exp h π/4 ˆa† ˆb−ˆaˆb† i is a balanced beam-splitter. This gives ˆUG|ψNOON⟩=2 −(N+1)/2 NX j=0 s N j !h e−iNϕ(−1)N−j...

    work page