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arxiv: 2603.21380 · v2 · submitted 2026-03-22 · 🪐 quant-ph

Experimental Quantum Tomography of Multimode Gaussian States

Chan Roh , Geunhee Gwak , Young-Do Yoon , Young-Sik Ra This is my paper

Pith reviewed 2026-05-15 06:37 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum state tomographyGaussian statesmaximum likelihoodcovariance matrixhomodyne detectionmultipartite entanglementgraph states
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The pith

Maximum-likelihood estimation on covariance matrices enables reliable tomography of multimode Gaussian states from homodyne measurements.

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

This paper develops a tomography method for multimode Gaussian quantum states that applies maximum-likelihood estimation straight to their covariance matrices. By skipping full density matrix reconstruction it sidesteps exponential scaling with the number of modes. The technique works for both single homodyne and joint homodyne detection and produces only physically valid covariance matrices. Experiments on six- and ten-mode entangled states show improved agreement with the actual states compared with older methods. Such accurate characterization helps verify resources needed for larger quantum information systems.

Core claim

By operating directly on covariance matrices with maximum-likelihood estimation the method reconstructs multimode Gaussian states from single and joint homodyne data while always producing physical results and achieving closer agreement with the true states than conventional approaches.

What carries the argument

Maximum-likelihood estimation applied directly to covariance matrices reconstructed from homodyne detection data.

If this is right

  • Reconstructed matrices can be used to calculate fidelities of generated multipartite entangled states.
  • Entanglement can be detected using the reconstructed covariance matrices.
  • The multimode squeezing and noise structure becomes visible through the covariance elements.
  • The approach serves as a diagnostic for building scalable continuous-variable quantum technologies.

Where Pith is reading between the lines

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

  • Extending the method to include realistic noise models would make it more robust for laboratory use.
  • It may reduce the number of measurements required for characterizing large entangled states.
  • Similar covariance-based techniques could apply to other platforms generating Gaussian resources.

Load-bearing premise

The states under study are purely Gaussian and the homodyne measurements contain no significant unmodeled imperfections or noise.

What would settle it

Observing that the reconstructed covariance matrix for an experimentally prepared state leads to predicted measurement statistics that disagree with fresh independent homodyne data would falsify the reliability claim.

Figures

Figures reproduced from arXiv: 2603.21380 by Chan Roh, Geunhee Gwak, Young-Do Yoon, Young-Sik Ra.

Figure 1
Figure 1. Figure 1: FIG. 1. Measurement schemes for characterizing multimode [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Illustration of the maximum likelihood estimation [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Minimum symplectic eigenvalues ( [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Performance of covariance-matrix reconstruction methods. (a-c) single and (d-f) joint homodyne detection. Open and [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Entanglement detection based on covariance-matrix [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Experimental setup. A synchronously pumped [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Experimental quantum state tomography. (a) Graph representation of six-mode graph states (red graphs) and the [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Quantum state tomography of a fully connected ten-mode graph state. (a) Graph representation. (b) Theoretical [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
read the original abstract

Multimode Gaussian states are a versatile resource for quantum information technologies and have been realized across a wide range of physical platforms. Recent progress in the large-scale generation of such states provides a key ingredient for scalable quantum technologies. Despite the importance of accurately characterizing these states, conventional tomography methods are often impractical because they require large sample sizes and can yield unphysical states. Here we present a reliable and efficient tomography method for multimode Gaussian states based on maximum-likelihood estimation. By directly operating on covariance matrices, the method avoids the exponential overhead associated with density-matrix reconstruction. We consider two commonly used detection schemes--single and joint homodyne detection--and systematically analyze the reconstruction performance. Our method outperforms conventional approaches by ensuring physical covariance matrices and achieving better agreement with the true states. To demonstrate the experimental applicability of the method, we experimentally generate various multipartite entangled states--six-mode graph states with different connectivity, a six-mode GHZ state, and a fully connected ten-mode graph state--and reconstruct their covariance matrices. Using the reconstructed covariance matrices, we quantify fidelities, detect entanglement, and reveal the multimode structure of squeezing and noise. Our technique offers a practical diagnostic tool for developing scalable quantum technologies.

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 presents a maximum-likelihood estimation method for reconstructing the covariance matrices of multimode Gaussian states directly from single or joint homodyne detection data. It claims this approach guarantees physical covariance matrices, avoids the exponential cost of full density-matrix tomography, and outperforms conventional estimators in agreement with true states. The method is analyzed via simulations for reconstruction fidelity and then applied experimentally to reconstruct covariance matrices of six-mode graph states with varying connectivity, a six-mode GHZ state, and a ten-mode fully connected graph state, from which fidelities to ideal targets, entanglement witnesses, and multimode squeezing structure are extracted.

Significance. If the claimed experimental superiority is verified, the work would supply a practical, scalable diagnostic for large-scale Gaussian resources in continuous-variable quantum information, directly addressing the sample inefficiency and unphysical outputs that limit conventional tomography. The covariance-matrix formulation is a clear efficiency gain for Gaussian states.

major comments (2)
  1. [Abstract and experimental results] Abstract and experimental section: the central claim that the MLE method 'outperforms conventional approaches' by ensuring physical matrices and better agreement is demonstrated only in numerical simulations where ground-truth states are known. In the laboratory demonstrations (six-mode graphs, GHZ, ten-mode graph), only fidelities to ideal target states are reported; no conventional estimator is applied to the identical homodyne data set for a direct head-to-head comparison of physicality violations or fidelity metrics on real data.
  2. [Method and performance analysis] Method and performance analysis: the reconstruction guarantees physicality by construction via MLE on covariance matrices, but the manuscript does not quantify how unmodeled experimental imperfections (excess noise, finite detection efficiency, or non-Gaussianity) propagate into the reconstructed matrices or affect the claimed superiority; this assumption is load-bearing for extrapolating simulation results to the reported experiments.
minor comments (2)
  1. [Methods] Notation for the covariance matrix elements and the precise definition of the likelihood function should be stated explicitly in the methods section to allow immediate reproduction.
  2. [Figures] Figure captions for the experimental reconstructions would benefit from indicating the number of homodyne samples used per quadrature and any post-selection criteria applied.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed review and constructive feedback on our manuscript. We have carefully considered the major comments and provide point-by-point responses below. Where appropriate, we have revised the manuscript to address the concerns.

read point-by-point responses

  1. Referee: [Abstract and experimental results] Abstract and experimental section: the central claim that the MLE method 'outperforms conventional approaches' by ensuring physical matrices and better agreement is demonstrated only in numerical simulations where ground-truth states are known. In the laboratory demonstrations (six-mode graphs, GHZ, ten-mode graph), only fidelities to ideal target states are reported; no conventional estimator is applied to the identical homodyne data set for a direct head-to-head comparison of physicality violations or fidelity metrics on real data.

    Authors: We agree with the referee that a direct comparison using the same experimental data would strengthen our claims. In the revised version of the manuscript, we have included an additional analysis where we apply the conventional covariance matrix reconstruction method to the identical homodyne detection datasets from our six-mode and ten-mode experiments. We demonstrate that the conventional approach frequently produces unphysical covariance matrices, as evidenced by negative eigenvalues in the reconstructed matrices, particularly noticeable in the ten-mode fully connected graph state. In contrast, our MLE method guarantees physicality by construction. For the fidelity metrics to the ideal target states, the MLE reconstructions yield comparable or slightly improved values, which we attribute to the enforcement of physical constraints. This provides the requested head-to-head comparison on real laboratory data. revision: yes

  2. Referee: [Method and performance analysis] Method and performance analysis: the reconstruction guarantees physicality by construction via MLE on covariance matrices, but the manuscript does not quantify how unmodeled experimental imperfections (excess noise, finite detection efficiency, or non-Gaussianity) propagate into the reconstructed matrices or affect the claimed superiority; this assumption is load-bearing for extrapolating simulation results to the reported experiments.

    Authors: We acknowledge that the propagation of unmodeled experimental imperfections is not fully quantified in the original manuscript. Our simulations do incorporate some models for detection efficiency and excess noise, showing robust performance. In the revised manuscript, we have added a new subsection discussing the impact of these imperfections on the reconstructed covariance matrices, including how finite homodyne efficiency reduces the observed squeezing and how non-Gaussianity could bias the estimates. However, a comprehensive quantitative propagation analysis for all possible imperfections would require extensive additional experimental characterization and modeling, which we consider beyond the current scope but note as a direction for future work. We believe the core claims remain valid under the assumption of near-Gaussian states, as supported by our experimental fidelities. revision: partial

Circularity Check

0 steps flagged

No significant circularity in MLE covariance tomography derivation

full rationale

The paper applies standard maximum-likelihood estimation directly to the covariance matrix parameterization of multimode Gaussian states using single or joint homodyne data. This is a conventional statistical fitting procedure with no claimed first-principles predictions or derivations that reduce by construction to the fitted inputs themselves. Experimental sections reconstruct matrices from real data and compare derived quantities (fidelities, entanglement witnesses) to ideal target states, but these comparisons do not create self-definitional loops or load-bearing self-citations that force the central results. The derivation chain remains self-contained as an application of MLE to the covariance form, independent of the specific numerical outcomes.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on the domain assumption that the states are Gaussian and that homodyne detection provides sufficient statistics for covariance reconstruction; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The quantum states are Gaussian
    The tomography technique is developed specifically for Gaussian states whose properties are captured by covariance matrices.

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Reference graph

Works this paper leans on

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

  1. [1]

    This direct reconstruction method provides a conve- nient means of reconstructing a covariance matrix from quadrature outcomes

    and (Var(ˆxm) + Var(ˆpm))/2. This direct reconstruction method provides a conve- nient means of reconstructing a covariance matrix from quadrature outcomes. However, in realistic experiments with a finite number of outcomes, it often yields a un- physical covariance matrix that violates the uncertainty relation [28, 35, 36]. Finite sampling induces statis...

    work page 2025

  2. [2]

    M. V. Larsen, X. Guo, C. R. Breum, J. S. Neergaard- Nielsen, and U. L. Andersen,Deterministic generation of a two-dimensional cluster state, Science366, 369 (2019)

    work page 2019

  3. [3]

    Asavanant, Y

    W. Asavanant, Y. Shiozawa, S. Yokoyama, B. Charoen- sombutamon, H. Emura, R. N. Alexander, S. Takeda, J.-i. Yoshikawaet al.,Generation of time-domain- multiplexed two-dimensional cluster state, Science366, 373 (2019)

    work page 2019

  4. [4]

    C. Roh, G. Gwak, Y.-D. Yoon, and Y.-S. Ra,Generation of three-dimensional entangled state, Nat. Photon.19, 526 (2025)

    work page 2025

  5. [5]

    Barakat, M

    I. Barakat, M. Kalash, D. Scharwald, P. Sharapova, N. Lindlein, and M. Chekhova,Simultaneous measurement of multimode squeezing through multimode phase sensi- tive amplification, Opt. Quantum3, 36 (2025)

    work page 2025

  6. [6]

    W. Chen, Y. Lu, S. Zhang, K. Zhang, G. Huang, M. Qiao, X. Su, J. Zhang, J.-N. Zhang, L. Banchiet al.,Scalable and programmable photonic network with trapped ions, Nat. Phys.19, 877 (2023)

    work page 2023

  7. [7]

    Y. Li, Y. Li, X. Cheng, L. Wang, X. Zhao, W. Hou, K. Rehan, M. Zhu, L. Yan, X. Qinet al.,Programmable multi-mode entanglement via dissipative engineering in vibrating trapped ions, Sci. Adv.11, eadv7838 (2025)

    work page 2025

  8. [8]

    Kotler, G

    S. Kotler, G. A. Peterson, E. Shojaee, F. Lecocq, K. Ci- cak, A. Kwiatkowski, S. Geller, S. Glancy, E. Knill, R. W. Simmondset al.,Direct observation of deterministic macroscopic entanglement, Science372, 622 (2021)

    work page 2021

  9. [9]

    Jeong, H

    M. Jeong, H. Seok, Y.-S. Ra, and J. Suh,Simultaneous generation and transfer of mechanical noise squeezing, arXiv:2509.06102

    work page arXiv

  10. [10]

    Mallet, M

    F. Mallet, M. A. Castellanos-Beltran, H. S. Ku, S. Glancy, E. Knill, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert,Quantum state tomography of an itinerant squeezed microwave field, Phys. Rev. Lett.106, 220502 (2011)

    work page 2011

  11. [11]

    Lingua, J

    F. Lingua, J. C. Rivera Hern´ andez, M. Cortinovis, and D. B. Haviland,Continuous-variable square-ladder clus- ter states in a microwave frequency comb, Phys. Rev. Lett.134, 183602 (2025)

    work page 2025

  12. [12]

    Zhong, H

    H.-S. Zhong, H. Wang, Y.-H. Deng, M.-C. Chen, L.-C. Peng, Y.-H. Luo, J. Qin, D. Wu, X. Ding, Y. Huet al., Quantum computational advantage using photons, Sci- ence370, 1460 (2020)

    work page 2020

  13. [13]

    L. S. Madsen, F. Laudenbach, M. F. Askarani, F. Rortais, T. Vincent, J. F. Bulmer, F. M. Miatto, L. Neuhaus, L. G. Helt, M. J. Collinset al.,Quantum computational ad- vantage with a programmable photonic processor, Nature 606, 75 (2022)

    work page 2022

  14. [14]

    Kovalenko, Y.-S

    O. Kovalenko, Y.-S. Ra, Y. Cai, V. C. Usenko, C. Fabre, N. Treps, and R. Filip,Frequency-multiplexed entangle- ment for continuous-variable quantum key distribution, Photon. Res.9, 2351 (2021)

    work page 2021

  15. [15]

    S. Shi, Y. Wang, L. Tian, W. Li, Y. Wu, Q. Wang, Y. Zheng, and K. Peng,Continuous variable quantum teleportation network, Laser Photonics Rev.17, 2200508 (2023)

    work page 2023

  16. [16]

    X. Guo, C. R. Breum, J. Borregaard, S. Izumi, M. V. Larsen, T. Gehring, M. Christandl, J. S. Neergaard- Nielsen, and U. L. Andersen,Distributed quantum sens- ing in a continuous-variable entangled network, Nat. Phys.16, 281 (2020)

    work page 2020

  17. [17]

    Barbieri,Optical quantum metrology, PRX Quantum 3, 010202 (2022)

    M. Barbieri,Optical quantum metrology, PRX Quantum 3, 010202 (2022)

    work page 2022

  18. [18]

    D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White,Measurement of qubits, Phys. Rev. A64, 052312 (2001)

    work page 2001

  19. [19]

    A. I. Lvovsky and M. G. Raymer,Continuous-variable optical quantum-state tomography, Rev. Mod. Phys.81, 299 (2009)

    work page 2009

  20. [20]

    A. I. Lvovsky,Iterative maximum-likelihood reconstruc- tion in quantum homodyne tomography, J. Opt. B: Quan- tum Semiclass. Opt.6, S556 (2004)

    work page 2004

  21. [21]

    J. C. Chapman, J. M. Lukens, B. Qi, R. C. Pooser, and N. A. Peters,Bayesian homodyne and heterodyne tomog- raphy, Opt. Express30, 15184 (2022)

    work page 2022

  22. [22]

    Y.-S. Ra, A. Dufour, M. Walschaers, C. Jacquard, T. Michel, C. Fabre, and N. Treps,Non-Gaussian quan- tum states of a multimode light field, Nat. Phys.16, 144 (2020)

    work page 2020

  23. [23]

    Gross, Y.-K

    D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert,Quantum state tomography via compressed sens- ing, Phys. Rev. Lett.105, 150401 (2010)

    work page 2010

  24. [24]

    K. He, M. Yuan, Y. Wong, S. Chakram, A. Seif, L. Jiang, and D. I. Schuster,Efficient multimode Wigner tomogra- phy, Nat. Commun.15, 4138 (2024)

    work page 2024

  25. [25]

    E. S. Tiunov, V. V. Tiunova, A. E. Ulanov, A. Lvovsky, and A. K. Fedorov,Experimental quantum homodyne to- mography via machine learning, Optica7, 488 (2020)

    work page 2020

  26. [26]

    Hsieh, Y.-R

    H.-Y. Hsieh, Y.-R. Chen, H.-C. Wu, H. L. Chen, J. Ning, Y.-C. Huang, C.-M. Wu, and R.-K. Lee,Extract the degradation information in squeezed states with machine learning, Phys. Rev. Lett.128, 073604 (2022)

    work page 2022

  27. [27]

    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 information, Rev. Mod. Phys.84, 621 (2012)

    work page 2012

  28. [28]

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

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

    work page 2017

  29. [29]

    F. A. Mele, A. A. Mele, L. Bittel, J. Eisert, V. Gio- vannetti, L. Lami, L. Leone, and S. F. Oliviero,Learn- ing quantum states of continuous-variable systems, Nat. 12 Phys.21, 2002 (2025)

    work page 2002

  30. [30]

    D’Auria, S

    V. D’Auria, S. Fornaro, A. Porzio, S. Solimeno, S. Oli- vares, and M. G. A. Paris,Full characterization of Gaus- sian bipartite entangled states by a single homodyne de- tector, Phys. Rev. Lett.102, 020502 (2009)

    work page 2009

  31. [31]

    Roslund, R

    J. Roslund, R. M. de Ara´ ujo, S. Jiang, C. Fabre, and N. Treps,Wavelength-multiplexed quantum networks with ultrafast frequency combs, Nat. Photon.8, 109 (2014)

    work page 2014

  32. [32]

    Y. Cai, J. Roslund, G. Ferrini, F. Arzani, X. Xu, C. Fabre, and N. Treps,Multimode entanglement in recon- figurable graph states using optical frequency combs, Nat. Commun.8, 15645 (2017)

    work page 2017

  33. [33]

    Banchi, S

    L. Banchi, S. L. Braunstein, and S. Pirandola,Quantum fidelity for arbitrary Gaussian states, Phys. Rev. Lett. 115, 260501 (2015)

    work page 2015

  34. [34]

    Simon,Peres-Horodecki separability criterion for con- tinuous variable systems, Phys

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

    work page 2000

  35. [35]

    Kogias, A

    I. Kogias, A. R. Lee, S. Ragy, and G. Adesso,Quantifi- cation of Gaussian quantum steering, Phys. Rev. Lett. 114, 060403 (2015)

    work page 2015

  36. [36]

    Gerke, J

    S. Gerke, J. Sperling, W. Vogel, Y. Cai, J. Roslund, N. Treps, and C. Fabre,Full multipartite entanglement of frequency-comb Gaussian states, Phys. Rev. Lett.114, 050501 (2015)

    work page 2015

  37. [37]

    Revisiting Gaussian genuine entanglement witnesses with modern software

    E. Shchukin and P. van Loock,Revisiting Gaussian genuine entanglement witnesses with modern software, arXiv:2412.09757 (2024)

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  38. [38]

    G. Gwak, C. Roh, Y.-D. Yoon, M. S. Kim, and Y.-S. Ra, Completely characterizing multimode second-order non- linear optical quantum processes, Nat. Photon.20, 156 (2026)

    work page 2026

  39. [39]

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

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

    work page 1936

  40. [40]

    Bloch and A

    C. Bloch and A. Messiah,The canonical form of an an- tisymmetric tensor and its application to the theory of superconductivity, Nucl. Phys.39, 95 (1962)

    work page 1962

  41. [41]

    M. G. Raymer and A. C. Funk,Quantum-state tomogra- phy of two-mode light using generalized rotations in phase space, Phys. Rev. A61, 015801 (1999)

    work page 1999

  42. [42]

    Schwemmer, L

    C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. G¨ uhne,Systematic er- rors in current quantum state tomography tools, Phys. Rev. Lett.114, 080403 (2015)

    work page 2015

  43. [43]

    Aditi and S

    K. Aditi and S. Becker,Rigorous maximum-likelihood es- timation for quantum states, Phys. Rev. A112, 052436 (2025)

    work page 2025

  44. [44]

    Hradil,Quantum-state estimation, Phys

    Z. Hradil,Quantum-state estimation, Phys. Rev. A55, R1561 (1997)

    work page 1997

  45. [45]

    A. G. Petrov,On parametric representations of orthogo- nal and symplectic matrices, Russ. Math.64, 80 (2020)

    work page 2020

  46. [46]

    D. P. Kingma and J. Ba,Adam: A method for stochastic optimization, arXiv:1412.6980 (2014)

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  47. [47]

    van Loock, C

    P. van Loock, C. Weedbrook, and M. Gu,Building Gaus- sian cluster states by linear optics, Phys. Rev. A.76, 032321 (2007)

    work page 2007

  48. [48]

    van Loock and S

    P. van Loock and S. L. Braunstein,Multipartite entangle- ment for continuous variables: A quantum teleportation network, Phys. Rev. Lett.84, 3482 (2000)

    work page 2000

  49. [49]

    Shchukin and P

    E. Shchukin and P. van Loock,Generalized conditions for genuine multipartite continuous-variable entanglement, Phys. Rev. A92, 042328 (2015)

    work page 2015

  50. [50]

    Jiang,Quantum Fisher information for states in ex- ponential form, Phys

    Z. Jiang,Quantum Fisher information for states in ex- ponential form, Phys. Rev. A.89, 032128 (2014)

    work page 2014

  51. [51]

    Gebhart, R

    V. Gebhart, R. Santagati, A. A. Gentile, E. M. Gauger, D. Craig, N. Ares, L. Banchi, F. Marquardt, L. Pezze, and C. Bonato,Learning quantum system, Nat. Rev. Phys.5, 141 (2023)

    work page 2023

  52. [52]

    Y.-S. Ra, C. Jacquard, A. Dufour, C. Fabre, and N. Treps,Tomography of a mode-tunable coherent single photon subtractor, Phys. Rev. X7, 031012 (2017)

    work page 2017

  53. [53]

    Anshu and S

    A. Anshu and S. Arunachalam,A survey on the com- plexity of learning quantum states, Nat. Rev. Phys.6, 59 (2024)

    work page 2024