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arxiv: 2606.31150 · v1 · pith:AQRVS7UPnew · submitted 2026-06-30 · 🪐 quant-ph

Energetics of non-Gaussianity in single mode cavities

Sahil Sardar Jafar , Kian Hwee Lim , Kiarn T. Laverick , Samyak P. Prasad , Maria Maffei , Alexia Auff\`eves This is my paper

Pith reviewed 2026-07-01 05:42 UTC · model grok-4.3

classification 🪐 quant-ph
keywords non-Gaussianitybosonic statesenergy decompositionrelative entropyWigner negativityquantum resource theorysingle-mode cavity
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The pith

The non-Gaussian part of a bosonic state's energy forms a valid measure of non-Gaussianity for pure states and a witness for mixed states.

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

The paper decomposes the total energy of a single-mode bosonic state into Gaussian and non-Gaussian parts and shows that the non-Gaussian share behaves as a resource quantifier. For pure states this quantity satisfies the axioms of a non-Gaussianity measure and is monotonically related to the relative entropy of non-Gaussianity. For mixed states the same quantity remains strictly positive precisely when the state is non-Gaussian. The authors illustrate the measure by comparing it with Wigner-function negativity and find that both peak in nearby regions of parameter space. The framework therefore supplies an energy-based handle on non-Gaussianity that is directly tied to the physical resources needed to produce such states.

Core claim

The authors partition the expectation value of the Hamiltonian of a single-mode cavity into a Gaussian contribution (fixed by the first and second moments) and a remainder that isolates the non-Gaussian content. For any pure state the non-Gaussian remainder is non-negative, vanishes if and only if the state is Gaussian, and is monotonically related to the relative entropy of non-Gaussianity. For mixed states the same remainder is strictly positive whenever the state is non-Gaussian, thereby serving as a faithful witness.

What carries the argument

The energetic decomposition of the total energy into Gaussian and non-Gaussian contributions, obtained by subtracting the energy fixed by the covariance matrix from the full energy expectation value.

If this is right

  • Any protocol that lowers the non-Gaussian energy component must also reduce the relative entropy of non-Gaussianity for pure states.
  • The non-Gaussian energy witness can be evaluated from a single energy measurement once the Gaussian part has been subtracted via quadrature measurements.
  • States that maximize Wigner negativity are expected to lie near the states that maximize the non-Gaussian energy component.
  • The decomposition supplies a concrete energy cost for generating non-Gaussianity that is independent of the particular form of the driving Hamiltonian.

Where Pith is reading between the lines

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

  • If the non-Gaussian energy can be extracted by a Gaussian operation, it would furnish a thermodynamic resource theory of non-Gaussianity.
  • The same partition may extend to multimode systems and thereby link non-Gaussianity to multipartite entanglement measures.
  • Experimental platforms that already track total energy (optomechanical cavities, superconducting resonators) could monitor the witness in real time without full state tomography.

Load-bearing premise

The total energy of the state can be cleanly split into a part determined solely by its first two moments and a remainder that captures everything else.

What would settle it

Prepare a pure single-mode state whose covariance matrix is fixed but whose higher moments are varied; measure whether the excess energy above the Gaussian value remains constant or changes with those higher moments.

Figures

Figures reproduced from arXiv: 2606.31150 by Alexia Auff\`eves, Kian Hwee Lim, Kiarn T. Laverick, Maria Maffei, Sahil Sardar Jafar, Samyak P. Prasad.

Figure 1
Figure 1. Figure 1: FIG. 1. A diagram of our energetic approach to quantify [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Wigner negativity (solid lines) and non-Gaussian [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. A diagrammatic representation of our energetic wit [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. A phase space representation of the relative entropy of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The change in [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
read the original abstract

Non-Gaussian states play a central role in quantum technologies, making the ability to quantify non-Gaussianity essential. We introduce an energetic framework to characterize non-Gaussianity in single-mode bosonic states by decomposing the total energy into Gaussian and non-Gaussian contributions. For pure states, we show that the non-Gaussian component defines a valid measure of non-Gaussianity and establish its connection to the relative entropy of non-Gaussianity. As an illustration, we compare this measure with Wigner negativity and find that both are maximized in closely related parameter regimes. For mixed states, we demonstrate that the non-Gaussian contribution acts as a faithful witness of non-Gaussianity. Our results reveal an energetic fine structure underlying non-Gaussianity and may provide practical insights for the efficient generation of non-Gaussian states.

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 / 1 minor

Summary. The paper introduces an energetic framework for characterizing non-Gaussianity in single-mode bosonic states by decomposing the total energy into Gaussian (E_G) and non-Gaussian (E_NG) contributions. For pure states, it claims that E_NG defines a valid measure of non-Gaussianity and is connected to the relative entropy of non-Gaussianity. For mixed states, E_NG acts as a faithful witness. The approach is illustrated by comparing the measure to Wigner negativity, with both maximized in related parameter regimes.

Significance. If the energy decomposition were rigorously valid and consistent with standard bosonic Hamiltonians, the framework would provide a novel energetic interpretation of non-Gaussianity with potential practical value for state generation in quantum technologies. However, the central construction appears internally inconsistent with the properties of quadratic Hamiltonians, limiting any potential significance.

major comments (2)
  1. [Abstract] Abstract, paragraph 2: The claim that the total energy can be cleanly partitioned into Gaussian and non-Gaussian contributions without additional assumptions about the state preparation or Hamiltonian form is contradicted by the standard single-mode cavity Hamiltonian H = ω(a†a + 1/2). For this H, <H> is a function of the covariance matrix alone, so any Gaussian reference state with matching first and second moments yields identical <H>, forcing E_NG ≡ 0 for every state. This directly undermines the assertion that E_NG is positive for non-Gaussian states and connected to the relative entropy of non-Gaussianity.
  2. [Abstract] Abstract: The abstract asserts that for pure states the non-Gaussian component defines a valid measure and establishes a connection to the relative entropy of non-Gaussianity, yet supplies no derivation steps, explicit conditions on the decomposition, or error analysis. The central claim that E_NG serves as a measure or witness cannot be checked or verified from the available text.
minor comments (1)
  1. The abstract refers to an illustration comparing the energetic measure with Wigner negativity but provides no details on the specific states, parameter regimes, or quantitative comparison results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comments on our manuscript. We address each major comment below and indicate where revisions will be made to improve clarity and rigor.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph 2: The claim that the total energy can be cleanly partitioned into Gaussian and non-Gaussian contributions without additional assumptions about the state preparation or Hamiltonian form is contradicted by the standard single-mode cavity Hamiltonian H = ω(a†a + 1/2). For this H, <H> is a function of the covariance matrix alone, so any Gaussian reference state with matching first and second moments yields identical <H>, forcing E_NG ≡ 0 for every state. This directly undermines the assertion that E_NG is positive for non-Gaussian states and connected to the relative entropy of non-Gaussianity.

    Authors: We agree that the referee's observation is correct for the standard harmonic-oscillator Hamiltonian H = ω(a†a + 1/2): the energy expectation value is fully determined by the second moments, so the proposed decomposition would yield E_NG = 0 for all states. Our framework is formulated under the assumption of a Hamiltonian containing non-quadratic terms (or an effective energetic functional that isolates higher-order contributions), which permits a non-trivial partition. We will revise the abstract and add an explicit statement of the Hamiltonian assumptions together with the derivation of the decomposition in the main text. revision: yes

  2. Referee: [Abstract] Abstract: The abstract asserts that for pure states the non-Gaussian component defines a valid measure and establishes a connection to the relative entropy of non-Gaussianity, yet supplies no derivation steps, explicit conditions on the decomposition, or error analysis. The central claim that E_NG serves as a measure or witness cannot be checked or verified from the available text.

    Authors: Abstracts are necessarily concise and cannot contain full derivations. The manuscript body contains the proofs that E_NG is a valid measure for pure states, its connection to the relative entropy of non-Gaussianity, the explicit conditions on the decomposition, and the witness property for mixed states. We will revise the abstract to better signpost these results and to qualify the claims by reference to the main text. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation defines new decomposition independently

full rationale

Abstract and provided text introduce an energetic decomposition of total energy into Gaussian and non-Gaussian parts as a new framework. No equations, self-citations, or fitted parameters are visible that would reduce the non-Gaussian component to a tautological input or prior self-result. The connection to relative entropy of non-Gaussianity is asserted as a shown property rather than presupposed by definition. This is the common case of a self-contained proposal with no load-bearing circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; ledger left empty pending full text.

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Reference graph

Works this paper leans on

71 extracted references · 6 canonical work pages · 3 internal anchors

  1. [1]

    It is a well-defined, non-negative quantity,i.e., EN G ≥0

  2. [2]

    (Faithfulness)E N G = 0, iff|ψ⟩is pure Gaussian state

  3. [3]

    maximally non- Gaussian

    (Invariant under Gaussian unitaries)E N G[|ψ⟩] = EN G[ ˆUG |ψ⟩], where ˆUG is a Gaussian unitary,i.e., ˆUG =e i ˆG where ˆGis a Hermitian operator at most quadratic in annihilation and creation operators. Properties 1 and 2 follow directly from the Robertson- Schr¨ odinger uncertainty relation [13, 44], which states that, detV≥ 1 4 .(7) The equality is sa...

  4. [4]

    This means that, under a fixed total energy budget, the class of pure states with zero first and second moment uniquely define the most non-Gaussian states. In the same spirit, one may ask: given a class of non- Gaussian unitaries, which Gaussian input state yields an output state with maximum non-Gaussianity? As an ex- ample, consider the Selective Numbe...

  5. [5]

    It is a well defined non-negative quantity,i.e., WN G ≥0

  6. [6]

    (Faithfulness)W N G = 0, iffρis Gaussian

  7. [7]

    This follows from the property (3) of non-Gaussian energy for pure states and the fact thatn th, which depends on the von Neumann entropy, is invariant under unitaries

    (Invariant under Gaussian unitaries)W N G[ρ] = WN G[ ˆUGρ ˆU † G], where ˆUG is a Gaussian unitary. This follows from the property (3) of non-Gaussian energy for pure states and the fact thatn th, which depends on the von Neumann entropy, is invariant under unitaries. These properties makeW N G a faithful witness of non- Gaussianity. Physically, it quanti...

    2000

  8. [8]

    However, they exhibit qualitatively similar behavior, in the sense that both the quantities increase in the regime of large √ detVand smalln th + 1 2

    As expected, the constant lines of these quantities are not parallel. However, they exhibit qualitatively similar behavior, in the sense that both the quantities increase in the regime of large √ detVand smalln th + 1 2. AlthoughW N G and ΣN G[ρ] are not equivalent quanti- fiers for mixed states, in Appendix A, we derive a neces- sary and sufficient condi...

    2030

  9. [9]

    Ob- serving the progressive decoherence of the “meter

    M. Brune, E. Hagley, J. Dreyer, X. Maˆ ıtre, A. Maali, C. Wunderlich, J. M. Raimond, and S. Haroche, “Ob- serving the progressive decoherence of the “meter” in a quantum measurement,” Phys. Rev. Lett.77, 4887–4890 (1996)

    1996

  10. [10]

    Haroche and J.-M

    S. Haroche and J.-M. Raimond,Exploring the quantum: atoms, cavities, and photons(Oxford University Press, 2006)

    2006

  11. [11]

    Fast generation of Schr¨ odinger cat states using a Kerr-tunable superconducting resonator,

    X. L. He, Y. Lu, D. Q. Bao, H. Xue, W. B. Jiang, Z. Wang, A. F. Roudsari, P. Delsing, J. S. Tsai, and Z. R. Lin, “Fast generation of Schr¨ odinger cat states using a Kerr-tunable superconducting resonator,” Nat. Commun.14, 6358 (2023)

    2023

  12. [12]

    H. M. Wiseman and G. J. Milburn,Quantum Measure- ment and Control(Cambridge University Press, Cam- bridge, 2010)

    2010

  13. [13]

    Schlosshauer,Decoherence and the Quantum-To- Classical Transition(Springer, 2007)

    M. Schlosshauer,Decoherence and the Quantum-To- Classical Transition(Springer, 2007)

    2007

  14. [14]

    Decoherence, einselection, and the quan- tum origins of the classical,

    W. H. Zurek, “Decoherence, einselection, and the quan- tum origins of the classical,” Rev. Mod. Phys.75, 715– 775 (2003)

    2003

  15. [15]

    Quantum computation over continuous variables,

    S. Lloyd and S. L. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. Lett.82, 1784– 1787 (1999)

    1999

  16. [16]

    Resources for bosonic quantum computational advantage,

    U. Chabaud and M. Walschaers, “Resources for bosonic quantum computational advantage,” Phys. Rev. Lett. 130, 090602 (2023)

    2023

  17. [17]

    Bounding the computational power of bosonic systems,

    V. Upreti, D. Rudolph, and U. Chabaud, “Bounding the computational power of bosonic systems,” npj Quantum Inf. (2026), 10.1038/s41534-026-01255-6

    work page doi:10.1038/s41534-026-01255-6 2026

  18. [18]

    Quantum informa- tion with continuous variables,

    S. L. Braunstein and P. van Loock, “Quantum informa- tion with continuous variables,” Rev. Mod. Phys.77, 513–577 (2005)

    2005

  19. [19]

    Continuous- variable quantum communication,

    V. C. Usenko, A. Ac´ ın, R. All´ eaume, U. L. Andersen, E. Diamanti, T. Gehring, A. A. E. Hajomer, F. Kan- itschar, C. Pacher, S. Pirandola,et al., “Continuous- variable quantum communication,” Rev. Mod. Phys.98, 015003 (2026)

    2026

  20. [20]

    Quantum information processing with bosonic qubits in circuit QED,

    A. Joshi, K. Noh, and Y. Y. Gao, “Quantum information processing with bosonic qubits in circuit QED,” Quan- tum Sci. Technol.6, 033001 (2021)

    2021

  21. [21]

    Gaus- sian quantum information,

    C. Weedbrook, S. Pirandola, R. Garc´ ıa-Patr´ on, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaus- sian quantum information,” Rev. Mod. Phys.84, 621– 669 (2012)

    2012

  22. [22]

    Advances in photonic quantum sensing,

    S. Pirandola, B. R. Bardhan, T. Gehring, C. Weedbrook, and S. Lloyd, “Advances in photonic quantum sensing,” Nat. Photonics12, 724–733 (2018)

    2018

  23. [23]

    Distilling Gaussian states with Gaussian operations is impossible,

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

    2002

  24. [24]

    No-go theorem for Gaussian quantum error correction,

    J. Niset, J. Fiur´ aˇ sek, and N. J. Cerf, “No-go theorem for Gaussian quantum error correction,” Phys. Rev. Lett. 102, 120501 (2009)

    2009

  25. [25]

    No-Go Theorem for Gaussian Quantum Repeaters from Fractional Extendibility

    R. G. Ahmed and G. Smith, “No-go theorem for Gaus- sian quantum repeaters from fractional extendibility,” (2026), arXiv:2606.05097 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  26. [26]

    Positive Wigner functions render classical simulation of quantum computation efficient,

    A. Mari and J. Eisert, “Positive Wigner functions render classical simulation of quantum computation efficient,” Phys. Rev. Lett.109, 230503 (2012)

    2012

  27. [27]

    Nega- tive quasi-probability as a resource for quantum compu- tation,

    V. Veitch, C. Ferrie, D. Gross, and J. Emerson, “Nega- tive quasi-probability as a resource for quantum compu- tation,” New J. Phys.14, 113011 (2012)

    2012

  28. [28]

    Nonlocality of the Einstein-Podolsky-Rosen state in the Wigner represen- tation,

    K. Banaszek and K. W´ odkiewicz, “Nonlocality of the Einstein-Podolsky-Rosen state in the Wigner represen- tation,” Phys. Rev. A58, 4345–4347 (1998)

    1998

  29. [29]

    Con- textuality and Wigner negativity are equivalent for continuous-variable quantum measurements,

    R. I. Booth, U. Chabaud, and P.-E. Emeriau, “Con- textuality and Wigner negativity are equivalent for continuous-variable quantum measurements,” Phys. Rev. Lett.129, 230401 (2022)

    2022

  30. [30]

    Quan- tifying the non-Gaussian character of a quantum state by quantum relative entropy,

    M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quan- tifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A78, 060303 (2008)

    2008

  31. [31]

    Quantifying non- Gaussianity for quantum information,

    M. G. Genoni and M. G. A. Paris, “Quantifying non- Gaussianity for quantum information,” Phys. Rev. A82, 052341 (2010)

    2010

  32. [32]

    Relative entropy is an exact measure of non-Gaussianity,

    P. Marian and T. A. Marian, “Relative entropy is an exact measure of non-Gaussianity,” Phys. Rev. A88, 012322 (2013)

    2013

  33. [33]

    Faithful measure of quantum non-Gaussianity via quantum rela- tive entropy,

    J. Park, J. Lee, K. Baek, S.-W. Ji, and H. Nha, “Faithful measure of quantum non-Gaussianity via quantum rela- tive entropy,” Phys. Rev. A100, 012333 (2019)

    2019

  34. [34]

    Mea- sure of the non-Gaussian character of a quantum state,

    M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Mea- sure of the non-Gaussian character of a quantum state,” Phys. Rev. A76, 042327 (2007). 9

    2007

  35. [35]

    Negativity of the Wigner function as an indicator of non-classicality,

    A. Kenfack and K. ˙Zyczkowski, “Negativity of the Wigner function as an indicator of non-classicality,” J. Opt. B: Quantum and Semiclass. Opt.6, 396 (2004)

    2004

  36. [36]

    Resource theory of quantum non-Gaussianity and Wigner negativity,

    F. Albarelli, M. G. Genoni, M. G. A. Paris, and A. Fer- raro, “Resource theory of quantum non-Gaussianity and Wigner negativity,” Phys. Rev. A98, 052350 (2018)

    2018

  37. [37]

    Quantifying non- Gaussianity of a quantum state by the negative entropy of quadrature distributions,

    J. Park, J. Lee, K. Baek, and H. Nha, “Quantifying non- Gaussianity of a quantum state by the negative entropy of quadrature distributions,” Phys. Rev. A104, 032415 (2021)

    2021

  38. [38]

    Op- erational quantification of continuous-variable quantum resources,

    B. Regula, L. Lami, G. Ferrari, and R. Takagi, “Op- erational quantification of continuous-variable quantum resources,” Phys. Rev. Lett.126, 110403 (2021)

    2021

  39. [39]

    Frame- work for resource quantification in infinite-dimensional general probabilistic theories,

    L. Lami, B. Regula, R. Takagi, and G. Ferrari, “Frame- work for resource quantification in infinite-dimensional general probabilistic theories,” Phys. Rev. A103, 032424 (2021)

    2021

  40. [40]

    Stellar representation of non-Gaussian quantum states,

    U. Chabaud, D. Markham, and F. Grosshans, “Stellar representation of non-Gaussian quantum states,” Phys. Rev. Lett.124, 063605 (2020)

    2020

  41. [41]

    Beyond stellar rank: Control parameters for scalable op- tical non-Gaussian state generation,

    F. Hanamura, K. Takase, H. Nagayoshi, R. Ide, W. Asa- vanant, K. Fukui, P. Marek, R. Filip, and A. Furusawa, “Beyond stellar rank: Control parameters for scalable op- tical non-Gaussian state generation,” Phys. Rev. X16, 021034 (2026)

    2026

  42. [42]

    Ergotropic characterization of continuous- variable entanglement,

    B. Polo-Rodr´ ıguez, F. Centrone, G. Adesso, and M. Al- imuddin, “Ergotropic characterization of continuous- variable entanglement,” Phys. Rev. Lett.136, 050201 (2026)

    2026

  43. [43]

    Energetics of correlations in interacting systems,

    N. Friis, M. Huber, and M. Perarnau-Llobet, “Energetics of correlations in interacting systems,” Phys. Rev. E93, 042135 (2016)

    2016

  44. [44]

    Detecting entanglement by symplectic un- certainty relations,

    A. Serafini, “Detecting entanglement by symplectic un- certainty relations,” J. Opt. Soc. Am. B24, 347–354 (2007)

    2007

  45. [45]

    Optimal entanglement witnesses for continuous-variable systems,

    P. Hyllus and J. Eisert, “Optimal entanglement witnesses for continuous-variable systems,” New J. Phys.8, 51 (2006)

    2006

  46. [46]

    An Energetic Constraint for Qubit-Qubit Entanglement

    K. T. Laverick, S. P. Prasad, P. Senellart, M. Maffei, and A. Auff` eves, “An energetic constraint for qubit-qubit entanglement,” (2026), arXiv:2603.16225 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  47. [47]

    Characterizing bipartite entanglement via the er- gotropic gap,

    X. Yang, M. Alimuddin, Y.-H. Yang, and M.-X. Luo, “Characterizing bipartite entanglement via the er- gotropic gap,” Quantum Inf. Process.23, 382 (2024)

    2024

  48. [48]

    Indepen- dence of work and entropy for equal-energetic finite quan- tum systems: Passive-state energy as an entanglement quantifier,

    M. Alimuddin, T. Guha, and P. Parashar, “Indepen- dence of work and entropy for equal-energetic finite quan- tum systems: Passive-state energy as an entanglement quantifier,” Phys. Rev. E102, 012145 (2020)

    2020

  49. [49]

    Extractable work from correlations,

    M. Perarnau-Llobet, K. V. Hovhannisyan, M. Huber, P. Skrzypczyk, N. Brunner, and A. Ac´ ın, “Extractable work from correlations,” Phys. Rev. X5, 041011 (2015)

    2015

  50. [50]

    Optimizing Wigner Negativity in Scattering Processes Using Energetic Cost Functions

    K. H. Lim, K. T. Laverick, S. S. Jafar, S. P. Prasad, M. Maffei, and A. Auff` eves, “Optimizing Wigner neg- ativity in scattering processes using energetic cost func- tions,” (2026), arXiv:2606.15101 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  51. [51]

    Cavity state manipulation using photon-number selec- tive phase gates,

    R. W. Heeres, B. Vlastakis, E. Holland, S. Krastanov, V. V. Albert, L. Frunzio, L. Jiang, and R. J. Schoelkopf, “Cavity state manipulation using photon-number selec- tive phase gates,” Phys. Rev. Lett.115, 137002 (2015)

    2015

  52. [52]

    Serafini,Quantum Continuous Variables: A Primer of Theoretical Methods(CRC Press, 2017)

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

    2017

  53. [53]

    Thermodynamics of autonomous optical Bloch equations,

    S. P. Prasad, M. Maffei, P. A. Camati, C. Elouard, and A. Auff` eves, “Thermodynamics of autonomous optical Bloch equations,” SciPost Phys.20, 112 (2026)

    2026

  54. [54]

    Passivity and practical work extraction using Gaussian operations,

    E. G. Brown, N. Friis, and M. Huber, “Passivity and practical work extraction using Gaussian operations,” New J. Phys.18, 113028 (2016)

    2016

  55. [55]

    Extract- ing energy via bosonic Gaussian operations,

    F. E. Q Rodriguez, F. A. Mele, S. F. E Oliviero, V. Giovannetti, L. Lami, and V. Cavina, “Extract- ing energy via bosonic Gaussian operations,” (2025), arXiv:2503.21748 [quant-ph]

    work page arXiv 2025

  56. [56]

    On the algebraic problem concerning the normal forms of linear dynamical systems,

    J. Williamson, “On the algebraic problem concerning the normal forms of linear dynamical systems,” Am. J. Math. 58, 141–163 (1936)

    1936

  57. [57]

    Pedagogical introduction to the entropy of entanglement for Gaussian states,

    T. F. Demarie, “Pedagogical introduction to the entropy of entanglement for Gaussian states,” Eur. J. Phys.39, 035302 (2018)

    2018

  58. [58]

    Universal control of an oscillator with dispersive cou- pling to a qubit,

    S. Krastanov, V. V. Albert, C. Shen, C.-L. Zou, R. W. Heeres, B. Vlastakis, R. J. Schoelkopf, and L. Jiang, “Universal control of an oscillator with dispersive cou- pling to a qubit,” Phys. Rev. A92, 040303 (2015)

    2015

  59. [59]

    Robust preparation of Wigner-negative states with op- timized SNAP-displacement sequences,

    M. Kudra, M. Kervinen, I. Strandberg, S. Ahmed, M. Scigliuzzo, A. Osman, D. P Lozano, M. O. Thol´ en, R. Borgani, D. B. Haviland, G. Ferrini, J. Bylander, A. F. Kockum, F. Quijandr´ ıa, P. Delsing, and S. Gasparinetti, “Robust preparation of Wigner-negative states with op- timized SNAP-displacement sequences,” PRX Quantum 3, 030301 (2022)

    2022

  60. [60]

    On the quantum correction for thermody- namic equilibrium,

    E. Wigner, “On the quantum correction for thermody- namic equilibrium,” Phys. Rev.40, 749–759 (1932)

    1932

  61. [61]

    Wigner function as the expectation value of a parity operator,

    A. Royer, “Wigner function as the expectation value of a parity operator,” Phys. Rev. A15, 449–450 (1977)

    1977

  62. [62]

    When is the Wigner quasi-probability density non-negative?

    R. L. Hudson, “When is the Wigner quasi-probability density non-negative?” Rep. Math. Phys.6, 249 (1974)

    1974

  63. [63]

    Ordering states with en- tanglement measures,

    S. Virmani and M. B. Plenio, “Ordering states with en- tanglement measures,” Phys. Lett. A268, 31–34 (2000)

    2000

  64. [64]

    Thermodynamical proof of the Gibbs for- mula for elementary quantum systems,

    A. Lenard, “Thermodynamical proof of the Gibbs for- mula for elementary quantum systems,” J. Stat. Phys. 19, 575–586 (1978)

    1978

  65. [65]

    Passive states and KMS states for general quantum systems,

    W. Pusz and S. L. Woronowicz, “Passive states and KMS states for general quantum systems,” Commun. Math. Phys.58, 273–290 (1978)

    1978

  66. [66]

    Entanglement boost for ex- tractable work from ensembles of quantum batteries,

    R. Alicki and M. Fannes, “Entanglement boost for ex- tractable work from ensembles of quantum batteries,” Phys. Rev. E87, 042123 (2013)

    2013

  67. [67]

    Extremality of Gaussian quantum states,

    M. M. Wolf, G. Giedke, and J. I. Cirac, “Extremality of Gaussian quantum states,” Phys. Rev. Lett.96, 080502 (2006)

    2006

  68. [68]

    Sub-bath cooling in bosonic systems: Gaussian constraints and non-Gaussian enhancements,

    W.-H. Png, X. Hu, and V. Scarani, “Sub-bath cooling in bosonic systems: Gaussian constraints and non-Gaussian enhancements,” PRX Quantum7, 020316 (2026)

    2026

  69. [69]

    Induced measures in the space of mixed quantum states,

    K. ˙Zyczkowski and H.-J. Sommers, “Induced measures in the space of mixed quantum states,” J. Phys. A: Math. Gen.34, 7111 (2001)

    2001

  70. [70]

    The code used to numerically show thatW N G does not increase under Gaussian channels can be found athttps: //zenodo.org/records/20794754

    work page arXiv

  71. [71]

    Gaus- sian states minimize the output entropy of one-mode quantum Gaussian channels,

    G. De Palma, D. Trevisan, and V. Giovannetti, “Gaus- sian states minimize the output entropy of one-mode quantum Gaussian channels,” Phys. Rev. Lett.118, 160503 (2017). Appendix A: Proof of inequality(32) Let us consider an infinitesimal operation on the state ρwhich changes the state fromρ→ρ+ dρ. Beginning 10 from Eq. (29), and simplifying the notation s...

    2017