Pith. sign in

REVIEW 4 major objections 5 minor 1 cited by

Measuring non-Gaussianity with Correlation

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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

Pith's one-line read This paper claims that a quantum state is non-Gaussian exactly when two of its copies become correlated at a 50:50 beam splitter, and uses that correlation as a faithful, experimentally accessible measure of non-Gaussianity.

desk verdict Solid operational framework with a fixable but real overclaim in the main theorem: the pure-state Rényi measure doesn't extend monotonically to mixed outputs, and faithfulness needs the 'faithful total correlation' qualifier. read the letter →

arxiv 2508.19890 v1 pith:D6Q7TED6 submitted 2025-08-27 quant-ph

Oliver Hahn , Ryuji Takagi This is my paper · ORCID
classification quant-ph
keywords non-GaussianitycontinuousvariablesbeamsplitterRényientropySWAPtestWignernegativityGaussianchannelscorrelationmeasures
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Non-Gaussian states are the resource behind quantum advantage in continuous-variable computing, but existing quantifiers are hard to measure. This paper proposes to quantify non-Gaussianity by what it does: mixing two copies of a state on a 50:50 beam splitter produces a correlated output exactly when the input is non-Gaussian, and a product output exactly when the input is Gaussian. The authors prove that any correlation measure applied to the beam-splitter output is a faithful non-Gaussianity monotone under Gaussian channels. For pure states the Rényi-2 version equals the purity of one output mode, which a SWAP test can estimate using four copies and O(1/ε²) samples without knowing the state, independent of mode number and energy. They also prove a lower bound showing that estimating Wigner negativity costs at least a cube-root-of-energy number of samples, so their protocol can be substantially cheaper.

What carries the argument

The load-bearing object is the 50:50 beam splitter acting on two identical copies: U_BS=exp(iπ/4 Σ [p_A q_B − q_A p_B]). The paper defines N_C(ρ)=C(U_BS ρ⊗ρ U_BS†) with C an arbitrary measure of correlation; the central identity is the characteristic-function factorization χ_{U_BS ρ⊗ρ U†}(r1,r2)=χρ((r1−r2)/√2)χρ((r1+r2)/√2), which yields a product state iff ρ is Gaussian (via the cited quantum Darmois–Skitovich theorem). For pure states, correlation reduces to entanglement, so the Rényi-α entanglement entropy of one output mode—at α=2, the negative log of the reduced-state purity—serves as the measure; the purity is estimated by a SWAP test (standard or destructive, with PNR detectors), requ

What would settle it

Take a Gaussian thermal state, compute the Rényi-2 purity of one beam-splitter output using Eq. (8); the result is log2(1+2n)>0, nonzero for a Gaussian state. If the paper's monotonicity claim is extended to this measure, a Gaussian channel applied to a pure Gaussian state would move the value from 0 to positive, violating N_C(Φ(ρ)) ≤ N_C(ρ). So the crux is whether the cited Darmois–Skitovich theorem's regularity hypotheses hold for the states in question.

Watch

Extended reading notes

Core claim

At the paper's center is Theorem 1: for any correlation measure C, N_C(ρ)=C(U_BS ρ⊗ρ U_BS†) is a faithful measure of non-Gaussianity—zero if and only if ρ is Gaussian—and it is monotonic under Gaussian channels. The fact that makes this work is the quantum Darmois–Skitovich theorem, invoked to show that two copies of a state become uncorrelated (in fact, product) after a 50:50 beam splitter exactly when the input is Gaussian. For pure states, the generated correlation is entanglement, so the Rényi-α entropy of one output mode is a non-Gaussianity quantifier; the α=2 case is the purity of the reduced state, which is directly accessible by a SWAP test using four copies and constant depth. For

Load-bearing premise

The entire faithfulness statement rests on the cited quantum Darmois–Skitovich theorem, whose technical hypotheses are not stated, and the monotonicity proof presumes the pure-state Rényi measure remains a valid correlation measure when a Gaussian channel outputs a mixed state.

Editorial extensions

If this is right

  • If Theorem 1 is right, any measure of correlation—not just entropy—turns into a faithful non-Gaussianity monotone under Gaussian channels, giving a one-line recipe for new measures.
  • The Rényi-2 instance makes non-Gaussianity experimentally accessible for pure states using only a SWAP test (or a destructive PNR version), with constant sample complexity independent of mode number and energy, in contrast to full state tomography.
  • The framework unifies and generalizes the Hong–Ou–Mandel experiment: the HOM dip becomes a special case of correlation generation for Fock-state inputs, applicable to arbitrary states.
  • For mixed states, Rényi-α mutual information is the appropriate correlation measure; the paper shows these are also monotones and gives bounds amenable to future measurement protocols.
  • Estimating Wigner negativity requires sample complexity at least growing with the cube root of mean photon number for cubic phase states, so the correlation-based measure can be dramatically cheaper.

Reading between the lines

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

  • Not pursued in the paper but a direct corollary of the construction: any observed correlation between the two beam-splitter outputs certifies that the input was outside the Gaussian set, so the setup can serve as a state-agnostic non-Gaussianity witness without tomography.
  • A natural extension the authors leave open is applying the same correlation-generation logic to channels rather than states, e.g., sending a Gaussian probe through an unknown process and measuring the correlation it induces as a quantifier of non-Gaussian operation.
  • The lower-bound result suggests, though the paper does not design it, that direct estimation protocols for Wigner negativity alone could outperform full state reconstruction; finding such a protocol is an open research direction.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper introduces a framework for quantifying non-Gaussianity of a continuous-variable state ρ by the correlation generated when two copies of ρ are mixed on a 50:50 beam splitter, defining N_C(ρ) := C(U_BS ρ⊗ρ U_BS†) for a correlation measure C. The central claim (Theorem 1) is that for any correlation measure C, N_C is monotonic under Gaussian channels and faithful, i.e., N_C(ρ)=0 iff ρ is Gaussian. The authors instantiate the construction with the pure-state Rényi-α entanglement entropy and the mixed-state Rényi-α mutual information, propose a SWAP-test protocol for estimating the Rényi-2 purity with four copies and O(1/ϵ²) samples, and derive a lower bound on the sample complexity of estimating Wigner negativity. Appendix B proves that Gaussian channels can be commuted through the beam splitter so that they act locally on the outputs; Appendix C exhibits non-Gaussian states whose beam-splitter output is separable.

Significance. The idea of characterizing non-Gaussianity through correlation generation at a beam splitter is elegant and provides a useful operational connection, generalizing the Hong–Ou–Mandel effect. For the restricted class of faithful total-correlation measures that are monotone under local operations, the monotonicity argument in Appendix B is sound and the framework is a genuine contribution. The Rényi-2 instance gives a concrete, state-agnostic estimation protocol with constant sample complexity in the ideal setting, which is a practical advantage over full tomography. The analytical examples and the Wigner-negativity sample-complexity comparison are also valuable. However, the theorem as stated is substantially overbroad, and several load-bearing claims need to be tightened before the results can be accepted as stated.

major comments (4)
  1. [Theorem 1 / Eq. (6)] The statement that N_C is faithful for 'C being an arbitrary measure of correlation' is contradicted by the paper's own Appendix C. Equation (C5) constructs a non-Gaussian state ρ = Σ_i p_i D(r_i)ρ_G D(r_i)† whose beam-splitter output is separable. If C is any entanglement-based correlation measure, including the pure-state Rényi entropy E_α of Eq. (7) when used in its natural role, then N_C(ρ)=0 for this non-Gaussian state. Faithfulness therefore holds only for C that is a faithful total-correlation measure (vanishing exactly on product states) and that is monotone under local operations. The theorem should be restated with this class of C; otherwise the central 'iff' claim is false as written.
  2. [Eqs. (7)-(8) and Experimental access] The pure-state measure E_α is not a valid correlation measure for mixed states if Eq. (8) is used as the definition. For a Gaussian thermal state ρ_G, the beam-splitter output is product, ρ_G⊗ρ_G, yet the right-hand side of Eq. (8), namely ∫|χ_ρ(r/√2)|^4/(2π)^m dr, is strictly less than 1, so N_E2(ρ_G)>0. Thus E_2 as defined is nonzero on a Gaussian state and is not a faithful measure. This also breaks the claimed monotonicity under Gaussian channels: a Gaussian noise channel can map a pure input to a thermal state, and the Appendix B proof applies only if C is a local-operation monotone on all states, which E_α is not. The authors should either restrict Theorem 1 to pure states and Gaussian unitaries, or use a genuinely mixed-state total-correlation measure throughout.
  3. [Faithfulness / Ref. [31]] The assertion that the beam-splitter output is product iff the input is Gaussian is the entire basis for faithfulness, but the paper does not state the theorem from Ref. [31] or verify its hypotheses. The title of Ref. [31] ('A stable quantum Darmois-Skitovich theorem') suggests that stability or regularity conditions may be involved. If those conditions are not satisfied by all states considered, the equivalence N_C=0 iff Gaussian can fail. The manuscript should state the precise theorem used and confirm that every state in the claimed domain satisfies its assumptions.
  4. [Appendix H] The lower bound on the sample complexity of estimating Wigner negativity is advertised as a main result, but the derivation contains uncontrolled approximations. The bound ∥W∥1 ≳ x^{1/6} in Eqs. (H20)-(H29) relies on statements such as 'the term ... does not contribute', 'neglect constant factors', and 'approximate with a constant c'. The plotted scaling in Figure 4 additionally uses a numerically observed x^{1/3} growth. To support the claimed third-root scaling of the sample complexity, the Wigner-negativity lower bound should be proved rigorously, or the scaling should be explicitly labeled as numerical/heuristic.
minor comments (5)
  1. [Background] Typo: 'states that cen be represented' should be 'states that can be represented'.
  2. [Theorem 1] Typo: 'faithfull' should be 'faithful'.
  3. [Appendix C] Typo: 'psot beam splitter' should be 'post beam splitter'.
  4. [Discussions] Grammar: 'the number of sample required' should be 'the number of samples required'.
  5. [Experimental access] The claim that the protocol is energy-independent should be qualified by the detector-saturation requirement 1−q_{2M}≤ϵ, which is energy-dependent.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: N_C is defined from correlation measures, monotonicity is proved by commuting Gaussian channels through the beam splitter, and faithfulness is imported from an external Darmois–Skitovich theorem rather than from the paper's own construction.

full rationale

The central objects are definitions (N_C(rho)=C(U_BS rho⊗rho U_BS†)), and the derived statements are theorems, not fitted predictions. Appendix B proves monotonicity by showing a Gaussian channel can be commuted through the beam splitter to act locally on both outputs, so any correlation measure non-increasing under local operations is a Gaussian-channel monotone; this does not presuppose non-Gaussianity. The faithfulness direction ('product iff Gaussian') is taken from Ref. [31], an external, independently published quantum Darmois–Skitovich theorem, and is not re-derived from N_C itself; the paper does not state the theorem's hypotheses, which is a rigor gap but not circularity. The cubic-phase-state formulas used in the Wigner-negativity lower bound come from the authors' own Ref. [27], but they are standard published, externally checkable identities (fidelity and mean photon number), not fitted inputs, so they do not make the argument circular. The main internal tension is that Theorem 1 asserts faithfulness for 'arbitrary measure of correlation,' whereas Appendix C explicitly constructs non-Gaussian states whose beam-splitter output is separable, so an entanglement-only C would fail faithfulness; the text acknowledges this and restricts the mixed-state faithful case to total-correlation measures. That is an overgeneralization/correctness caveat, not a circular reduction: no equation is defined in terms of the conclusion it is used to prove. No self-citation chain forces the result.

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

The central claim rests on one external mathematical fact (the quantum Darmois-Skitovich theorem, cited to [31]), standard CV formalism, and published results for Renyi mutual information, destructive SWAP tests, and cubic phase states. No new physical entities and no fitted parameters are introduced. The main unstated burden is the implicit set of axioms a 'correlation measure' must satisfy for Theorem 1 to hold verbatim: faithfulness and non-increase under local operations, which the Renyi mutual information satisfies but the pure-state Renyi entropy does not on mixed channel outputs.

assumptions (7)
  • domain assumption Quantum Darmois-Skitovich: the output of the 50:50 beam splitter acting on rho (x) rho is a product state if and only if rho is Gaussian.
    Load-bearing for faithfulness (N_C = 0 iff Gaussian, Theorem 1). Cited to Cuesta [31]; the theorem's technical conditions are not stated in the paper.
  • domain assumption C is a faithful correlation measure, non-increasing under local operations and partial trace.
    Theorem 1 asserts monotonicity 'for an arbitrary measure of correlation' without listing these axioms. The Renyi-alpha mutual information satisfies them (data-processing inequality), but the pure-state Renyi entropy satisfies them only on pure states under purity-preserving channels.
  • standard math Gaussian channels have a Stinespring dilation with vacuum ancillas and Gaussian unitaries (Eq. 5).
    Used in Appendix B to reduce general Gaussian channels to n-to-n channels, partial trace, and addition of Gaussian subsystems.
  • standard math Definitions and bounds for Renyi-alpha mutual information from McKinlay-Tomamichel [34] and Tomamichel [35].
    Used for the mixed-state measure and its bounds; not re-derived in this paper.
  • domain assumption Cubic phase state facts: fidelity F(|gamma,r>,|gamma,r'>) = 1/cosh(r-r'), mean photon number formula, and Wigner function expression.
    Appendix H relies on Refs. [27, 60] for these properties to instantiate the Wigner-negativity sample-complexity lower bound.
  • domain assumption PNR detector truncation model: resolution up to 2M photons with estimator Tr[SWAP_2M rho (x) sigma] and the stated systematic error bound (Eq. 11).
    Modeling assumption for the experimental protocol; the bound is stated without proof but is a plausible projection-argument bound.
  • domain assumption The destructive SWAP test measures purity for arbitrary (multi-mode) states.
    Relied on for the state-agnostic protocol; cited to Volkoff-Subasi [46].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Measuring non-Gaussianity with Correlation." pith.science (2026). https://pith.science/paper/D6Q7TED6

@misc{pith2026250819890,
  author       = {Pith},
  title        = {Pith review of: Measuring non-Gaussianity with Correlation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D6Q7TED6}},
  note         = {Machine review of arXiv:2508.19890}
}
abstract

Quantum non-Gaussianity is a key resource for quantum advantage in continuous-variable systems. We introduce a general framework to quantify non-Gaussianity based on correlation generation: two copies of a state become correlated at a $50{:}50$ beam splitter if and only if the state is non-Gaussian, with correlations reducing to entanglement in the pure-state case. This connection enables operational measures of non-Gaussianity, defined through correlation quantifiers such as R\'enyi-$\alpha$ entropy for pure states and R\'enyi-$\alpha$ mutual information for mixed states. We prove that all such measures are monotonic under Gaussian channels. Building on this framework, we propose a sample-efficient experimental protocol to estimate non-Gaussianity using standard optical components, even in the state agnostic setting. Finally, we establish a lower bound on the sample complexity of estimating Wigner negativity, allowing a direct comparison with our protocol. Our results provide both a unifying theoretical framework for non-Gaussianity and a practical route toward its experimental quantification.

Figures

Figures reproduced from arXiv: 2508.19890 by the authors.

Figure 1
Figure 1. FIG. 1. The underlying idea of our framework. (a) The output of two [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. This figures shows the measure of non-Gaussianity [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Circuits for experimentally measuring the purity. (a) Measur [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Lower bound of the sample complexity to estimate the [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Plot showing how the Wigner negativity increases with higher [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. This figure shows the number of samples required over the mean photon number using directly the values for the Wigner negativity of [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Quantum Computational Resources and Conformal Field Theory: Unifying Spins, Bosons, and Fermions

    quant-ph 2026-07 accept novelty 8.0 of 10

    A unified Magic Rényi Entropy measure for spins, bosons, and fermions is shown to have a universal critical contribution determined by the Affleck-Ludwig boundary entropy.

Reference graph

Works this paper leans on

79 extracted references · 63 canonical work pages · cited by 1 Pith paper

  1. [31]

    J. Cuesta, A stable quantum darmois-skitovich theorem, Journal of Mathematical Physics 61, 022201 (2020), https://pubs.aip.org/aip/jmp/article- pdf/doi/10.1063/1.5122955/13296574/022201 1 online.pdf

    work page doi:10.1063/1.5122955/13296574/022201 2020

  2. [1]

    Konno, W

    S. Konno, W. Asavanant, F. Hanamura, H. Nagayoshi, K. Fukui, A. Sakaguchi, R. Ide, F. China, M. Yabuno, S. Miki, H. Terai, K. Takase, M. Endo, P. Marek, R. Filip, P. van Loock, and A. Furusawa, Logical states for fault-tolerant quantum compu- tation with propagating light, Science 383, 289 (2024)

    work page 2024

  3. [2]

    Campagne-Ibarcq, A

    P. Campagne-Ibarcq, A. Eickbusch, S. Touzard, E. Zalys- Geller, N. E. Frattini, V . V . Sivak, P. Reinhold, S. Puri, S. Shankar, R. J. Schoelkopf, et al., Quantum error correction of a qubit encoded in grid states of an oscillator, Nature 584, 368 (2020)

    work page 2020

  4. [3]

    Sivak, A

    V . Sivak, A. Eickbusch, B. Royer, S. Singh, I. Tsioutsios, S. Ganjam, A. Miano, B. Brock, A. Ding, L. Frunzio, et al. , Real-time quantum error correction beyond break-even, Nature 616, 50 (2023)

    work page 2023

  5. [4]

    Fl ¨uhmann, T

    C. Fl ¨uhmann, T. L. Nguyen, M. Marinelli, V . Negnevitsky, K. Mehta, and J. P. Home, Encoding a qubit in a trapped-ion mechanical oscillator, Nature 566, 513–517 (2019)

    work page 2019

  6. [5]

    De Neeve, T.-L

    B. De Neeve, T.-L. Nguyen, T. Behrle, and J. P. Home, Error correction of a logical grid state qubit by dissipative pumping, Nature Physics 18, 296 (2022)

    work page 2022

  7. [6]

    Vaidman, Teleportation of quantum states, Phys

    L. Vaidman, Teleportation of quantum states, Phys. Rev. A 49, 1473 (1994)

    work page 1994

  8. [7]

    S. L. Braunstein and H. J. Kimble, Teleportation of continuous quantum variables, Phys. Rev. Lett. 80, 869 (1998)

    work page 1998

Show all 79 references
  1. [8]

    T. C. Ralph and P. K. Lam, Teleportation with bright squeezed light, Phys. Rev. Lett. 81, 5668 (1998)

    1998

  2. [9]

    Grosshans and P

    F. Grosshans and P. Grangier, Continuous variable quantum cryptography using coherent states, Phys. Rev. Lett.88, 057902 (2002)

    2002

  3. [10]

    Eisert and M

    J. Eisert and M. B. Plenio, Conditions for the local manipulation of gaussian states, Phys. Rev. Lett. 89, 097901 (2002)

    2002

  4. [11]

    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)

    2002

  5. [12]

    Giedke and J

    G. Giedke and J. Ignacio Cirac, Characterization of gaussian operations and distillation of gaussian states, Phys. Rev. A 66, 032316 (2002)

    2002

  6. [13]

    L. Lami, B. Regula, X. Wang, R. Nichols, A. Winter, and G. Adesso, Gaussian quantum resource theories, Phys. Rev. A 98, 022335 (2018)

    2018

  7. [14]

    Niset, J

    J. Niset, J. Fiur ´aˇsek, and N. J. Cerf, No-go theorem for gaussian quantum error correction, Phys. Rev. Lett. 102, 120501 (2009)

    2009

  8. [15]

    R. I. Booth, U. Chabaud, and P.-E. Emeriau, Contextuality and wigner negativity are equivalent for continuous-variable quan- tum measurements, Phys. Rev. Lett. 129, 230401 (2022)

    2022

  9. [16]

    Mari and J

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

    2012

  10. [17]

    M. G. Genoni, M. G. A. Paris, and K. Banaszek, Quantifying the non-gaussian character of a quantum state by quantum rela- tive entropy, Phys. Rev. A78, 060303 (2008)

    2008

  11. [18]

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

    2010

  12. [19]

    Marian and T

    P. Marian and T. A. Marian, Relative entropy is an exact mea- sure of non-gaussianity, Phys. Rev. A88, 012322 (2013)

    2013

  13. [20]

    Regula, L

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

    2021

  14. [21]

    L. Lami, B. Regula, R. Takagi, and G. Ferrari, Framework for resource quantification in infinite-dimensional general proba- bilistic theories, Phys. Rev. A 103, 032424 (2021)

    2021

  15. [22]

    Turner, M

    L. Turner, M. Guta, and G. Adesso, All non-gaussian states are advantageous for channel discrimination: Robustness of non-convex continuous variable quantum resources (2024), arXiv:2412.13011 [quant-ph]

    2024

  16. [23]

    Chabaud, D

    U. Chabaud, D. Markham, and F. Grosshans, Stellar represen- tation of non-gaussian quantum states, Phys. Rev. Lett. 124, 063605 (2020)

    2020

  17. [24]

    O. Hahn, R. Takagi, G. Ferrini, and H. Yamasaki, Classical simulation and quantum resource theory of non-gaussian optics (2024), arXiv:2404.07115 [quant-ph]

    2024

  18. [25]

    Dias and R

    B. Dias and R. K ¨onig, Classical simulation of non-gaussian bosonic circuits, Phys. Rev. A 110, 042402 (2024)

    2024

  19. [26]

    Kenfack and K

    A. Kenfack and K. ˙Zyczkowski, Negativity of the wigner func- tion as an indicator of non-classicality, J. Opt. B 6, 396 (2004)

    2004

  20. [27]

    Takagi and Q

    R. Takagi and Q. Zhuang, Convex resource theory of non- gaussianity, Phys. Rev. A97, 062337 (2018)

    2018

  21. [28]

    Albarelli, M

    F. Albarelli, M. G. Genoni, M. G. A. Paris, and A. Ferraro, Resource theory of quantum non-gaussianity and wigner nega- tivity, Phys. Rev. A98, 052350 (2018)

    2018

  22. [29]

    F. A. Mele, A. A. Mele, L. Bittel, J. Eisert, V . Giovannetti, L. Lami, L. Leone, and S. F. E. Oliviero, Learning quantum states of continuous variable systems (2024), arXiv:2405.01431 [quant-ph]

    2024

  23. [30]

    Henderson and V

    L. Henderson and V . Vedral, Classical, quantum and total cor- relations, Journal of Physics A: Mathematical and General 34, 6899 (2001)

    2001

  24. [32]

    Serafini, Quantum continuous variables: a primer of theo- retical methods (CRC press, 2023)

    A. Serafini, Quantum continuous variables: a primer of theo- retical methods (CRC press, 2023)

    2023

  25. [33]

    Wang, L.-Z

    Y .-X. Wang, L.-Z. Mu, V . Vedral, and H. Fan, Entanglement r´enyi α entropy, Phys. Rev. A93, 022324 (2016)

    2016

  26. [34]

    McKinlay and M

    A. McKinlay and M. Tomamichel, Decomposition rules for quantum r ´enyi mutual information with an application to information exclusion relations, Journal of Mathematical Physics 61, 072202 (2020), https://pubs.aip.org/aip/jmp/article- pdf/doi/10.1063/1.5143862/14042381/072202 ...

    2020 doi

  27. [35]

    Tomamichel, Quantum Information Processing with Finite Resources (Springer International Publishing, 2016)

    M. Tomamichel, Quantum Information Processing with Finite Resources (Springer International Publishing, 2016)

    2016

  28. [36]

    Becker, N

    S. Becker, N. Datta, L. Lami, and C. Rouz ´e, Convergence rates for the quantum central limit theorem, Communications in Mathematical Physics 383, 223–279 (2021)

    2021

  29. [37]

    Coffman, G

    L. Coffman, G. Smith, and X. Gao, Measuring non-gaussian magic in fermions: Convolution, entropy, and the viola- tion of wick’s theorem and the matchgate identity (2025), arXiv:2501.06179 [quant-ph]

    2025 arXiv

  30. [38]

    Lyu and K

    X. Lyu and K. Bu, Fermionic gaussian testing and non-gaussian measures via convolution (2024), arXiv:2409.08180 [quant- ph]

    2024 arXiv

  31. [39]

    K. Bu, W. Gu, and A. Jaffe, Quantum entropy and central limit theorem, Proceedings of the National Academy of Sciences 120, 10.1073/pnas.2304589120 (2023)

    2023 doi

  32. [40]

    K. Bu, W. Gu, and A. Jaffe, Stabilizer testing and magic entropy (2023), arXiv:2306.09292 [quant-ph]

    2023 arXiv

  33. [41]

    K. Bu, W. Gu, and A. Jaffe, Quantum ruzsa divergence to quan- tify magic, IEEE Trans. Inf. Theory 71, 2726 (2025)

    2025

  34. [42]

    A. K. Ekert, C. M. Alves, D. K. L. Oi, M. Horodecki, P. Horodecki, and L. C. Kwek, Direct estimations of linear and nonlinear functionals of a quantum state, Phys. Rev. Lett. 88, 217901 (2002). 7

    2002

  35. [43]

    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, Nature 566, 509–512 (2019)

    2019

  36. [44]

    Pietik ¨ainen, O

    I. Pietik ¨ainen, O. ˇCernot´ık, S. Puri, R. Filip, and S. M. Girvin, Controlled beam splitter gate transparent to dominant ancilla errors, Quantum Science and Technology 7, 035025 (2022)

    2022

  37. [45]

    Filip, Overlap and entanglement-witness measurements, Phys

    R. Filip, Overlap and entanglement-witness measurements, Phys. Rev. A 65, 062320 (2002)

    2002

  38. [46]

    T. J. V olkoff and Y . Subas ¸ı, Ancilla-free continuous-variable SW AP test, Quantum6, 800 (2022)

    2022

  39. [47]

    T. Ono, R. Okamoto, M. Tanida, H. F. Hofmann, and S. Takeuchi, Implementation of a quantum controlled-swap gate with photonic circuits, Scientific reports 7, 45353 (2017)

    2017

  40. [48]

    J. C. Garcia-Escartin and P. Chamorro-Posada, swap test and hong-ou-mandel effect are equivalent, Phys. Rev. A87, 052330 (2013)

    2013

  41. [49]

    Islam, R

    R. Islam, R. Ma, P. M. Preiss, M. Eric Tai, A. Lukin, M. Rispoli, and M. Greiner, Measuring entanglement entropy in a quantum many-body system, Nature 528, 77–83 (2015)

    2015

  42. [50]

    Gily ´en, Y

    A. Gily ´en, Y . Su, G. H. Low, and N. Wiebe, Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics, in Proceedings of the 51st An- nual ACM SIGACT Symposium on Theory of Computing, STOC 2019 (Association for Computing Machin...

    2019

  43. [51]

    Becker, N

    S. Becker, N. Datta, L. Lami, and C. Rouze, Classical shadow tomography for continuous variables quantum systems, IEEE Transactions on Information Theory 70, 3427–3452 (2024)

    2024

  44. [52]

    A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, Quantum state reconstruction of the single- photon fock state, Phys. Rev. Lett. 87, 050402 (2001)

    2001

  45. [53]

    Ourjoumtsev, R

    A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grang- ier, Generating optical schr ¨odinger kittens for quan- tum information processing, Science 312, 83 (2006), https://www.science.org/doi/pdf/10.1126/science.1122858

    2006 doi

  46. [54]

    M. S. Kim, W. Son, V . Buˇzek, and P. L. Knight, Entanglement by a beam splitter: Nonclassicality as a prerequisite for entan- glement, Phys. Rev. A 65, 032323 (2002)

    2002

  47. [55]

    J. K. Asb ´oth, J. Calsamiglia, and H. Ritsch, Computable mea- sure of nonclassicality for light, Phys. Rev. Lett. 94, 173602 (2005)

    2005

  48. [56]

    Bu and B

    K. Bu and B. Li, Efficient measurement of bosonic non- gaussianity (2025), arXiv:2507.10272 [quant-ph]

    2025 arXiv

  49. [57]

    Huang, R

    H.-Y . Huang, R. Kueng, G. Torlai, V . V . Albert, and J. Preskill, Provably efficient machine learning for quantum many-body problems, Science 377, 10.1126/science.abk3333 (2022)

    2022 doi

  50. [58]

    Gottesman, A

    D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in an oscillator, Phys. Rev. A 64, 012310 (2001)

    2001

  51. [59]

    Lloyd and S

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

    1999

  52. [60]

    D. W. Moore and R. Filip, Nonlinear phase gates as airy trans- forms of the wigner function (2025), arXiv:2504.04851 [quant- ph]. 8 SUPPLEMENTAL MA TERIAL A. Background: Continuous variables 8 B. Monotonicity 10

    2025

  53. [61]

    n mode to n mode Gaussian channels 10

  54. [62]

    Tracing out subsystem 10

  55. [63]

    Adding Gaussian subsystem 11

  56. [64]

    Correlation without entanglement 11 D

    Multiplicative output 11 C. Correlation without entanglement 11 D. Examples 12

  57. [65]

    The correlation cannot be detected by covariance matrix 14 F

    Cat states 13 E. The correlation cannot be detected by covariance matrix 14 F. Measuring mutual information 14 G. Classical shadow 15

  58. [66]

    Sampling cost lower bound for estimating Wigner negativity 16

    Non-linear functionals 15 H. Sampling cost lower bound for estimating Wigner negativity 16

  59. [67]

    State discrimination 17

  60. [68]

    Cubic phase state 17 Lower bound for Wigner negativity 19

  61. [69]

    In this work, we will use the canonical operators ˆq, ˆp that fulfill the following commutation relations [ˆq, ˆp] = i

    Sampling lower bound using the cubic phase state 20 Appendix A: Background: Continuous variables We start the supplemental material by introducing the necessary background for continuous-variable quantum systems [32] . In this work, we will use the canonical operators ˆq, ˆp t...

  62. [70]

    In later sections we will show monotonicity under partial trace and Gaussian state preparation, which then directly implies monotonicity under Gaussian channels

    n mode to n mode Gaussian channels In this section we show monotonicity of the measures using our construction under n mode to n mode Gaussian channels. In later sections we will show monotonicity under partial trace and Gaussian state preparation, which then directly implies ...

  63. [71]

    We start by considering as the input a state out of which a sub system C is traced out

    Tracing out subsystem In this section we will show monotonicity under partial trace. We start by considering as the input a state out of which a sub system C is traced out. The input state is then given as σ = TrC [ρ], while the output of the beam splitter is χ ˆUBS (σ⊗σ) ˆU †...

  64. [72]

    As Gaussian states are product states after the beam splitters, we can add Gaussian subsystems without increasing the correlation of the output state

    Adding Gaussian subsystem We note that the output of tensor product states in our setup is a tensor product of the separate subsystems. As Gaussian states are product states after the beam splitters, we can add Gaussian subsystems without increasing the correlation of the output state

  65. [73]

    We take two states of the form ρ ⊗ σ as the input of the beam splitter

    Multiplicative output In this section we show that if the input states are product states then the output will be multiplicative along the same axis. We take two states of the form ρ ⊗ σ as the input of the beam splitter. The characteristic function before the beam splitter is...

  66. [74]

    Fock states Here, we derive analytical expressions quantifying the non-Gaussianity of Fock states. The output state for two identical Fock states after the beam splitter is [54] |Ψ⟩ = ˆUBS |n, n⟩ = nX m,k=0 (−1)n−k n k n 2m − k p 2m!(2n − 2m)! n! |2m, 2n − 2m⟩ (D8) = nX m=0 cn...

  67. [75]

    See Appendix A for the precise definitions

    Cat states In this section we are providing analytical expression for squeezed cat states of the form|ψ⟩ = P1 i=0 ci ˆS(s) |αi⟩, where ˆS(s) is the squeezing operator and a coherent state |αi⟩ with real αi. See Appendix A for the precise definitions. The output after the beam ...

  68. [76]

    The examples we are interested are purity of a subsystem P (ρA) = Tr ρ2 A and the von Neumann entropy of a subsystems S(ρA) = − Tr [ρA log ρA]

    Non-linear functionals Our main interest is to use classical shadows to obtain non-linear functionals of the state ρ. The examples we are interested are purity of a subsystem P (ρA) = Tr ρ2 A and the von Neumann entropy of a subsystems S(ρA) = − Tr [ρA log ρA]. We consider onl...

  69. [77]

    The optimal probability to succeed in discriminating between the states ρ and σ using n copies is given by 1 2 1 2 ρ⊗n − σ⊗n 1 + 1 = psucc,n

    State discrimination We prove the lower bound on the number of samples required to obtain the Wigner negativity of a quantum state by connecting the task of estimating the Wigner negativity to a state discrimination task. The optimal probability to succeed in discriminating be...

  70. [78]

    In what follows, we focus on the case of cubic phase state

    Cubic phase state The general bound derived above can now be made explicit by considering concrete examples. In what follows, we focus on the case of cubic phase state. This section provides a summary of the most important properties of the cubic phase state as well as a lower...

  71. [79]

    Sampling lower bound using the cubic phase state Having established a general sample lower bound for estimating Wigner negativity, we now employ the cubic phase state to obtain concrete bounds. Assume now we have for the Wigner negativity of both cubic phase states that W(e3∆r...

    2000

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.