Assessing non-Gaussian quantum state conversion with the stellar rank
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AbstractState conversion is a fundamental task in quantum information processing. Quantum resource theories allow for analyzing and bounding conversions that use restricted sets of operations. In the context of continuous-variable systems, state conversions restricted to Gaussian operations are crucial for both fundamental and practical reasons, particularly in state preparation and quantum computing with bosonic codes. However, previous analysis did not consider the relevant case of approximate state conversion. In this work, we introduce a framework for assessing approximate Gaussian state conversion by extending the stellar rank to the approximate stellar rank, which serves as an operational measure of non-Gaussianity. We derive bounds for Gaussian state conversion and distillation under approximate and probabilistic conditions, yielding new no-go results for non-Gaussian state preparation and enabling a reliable assessment of the performance of Gaussian conversion protocols. We also provide an open-source Python library to compute stellar-rank-related quantities and to assess Gaussian conversion.Popular summaryIn quantum optics, we typically distinguish between "easy" Gaussian states and operations and "difficult" non-Gaussian ones. Non-Gaussian states and operations are necessary to build quantum computers capable of outperforming their classical counterparts. Non-Gaussian states and operations are thus conceivable as resources for quantum information processing. Various metrics have been introduced to quantify these non-Gaussian resources, such as the stellar rank. However, these metrics only worked in perfect, idealized settings. In the lab, we deal with noise and other imperfections, while we only need to get "close enough" to a target non-Gaussian state. In this article, we introduce the approximate stellar rank. This is a more realistic non-Gaussian measure that can account for the small errors inherent in experimental physics. We use this new measure to establish the first no-go theorems for approximate non-Gaussian state conversion using Gaussian operations. Our results provide a clear boundary for what can and cannot be achieved in the lab, helping us better manage the non-Gaussian resources required for useful quantum technologies.► BibTeX data@article{Hahn2026assessingnon, doi = {10.22331/q-2026-05-05-2095}, url = {https://doi.org/10.22331/q-2026-05-05-2095}, title = {Assessing non-{G}aussian quantum state conversion with the stellar rank}, author = {Hahn, Oliver and Garnier, Maxime and Ferrini, Giulia and Ferraro, Alessandro and Chabaud, Ulysse}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2095}, month = may, year = {2026} }► References [1] E. Chitambar and G. Gour, ``Quantum resource theories,'' Rev. Mod. Phys. 91, 025001 (2019). https://doi.org/10.1103/RevModPhys.91.025001 [2] S. Lloyd and S. L. Braunstein, ``Quantum computation over continuous variables,'' in Quantum Information with Continuous Variables, pp. 9–17. Springer, 1999. https://doi.org/10.1007/978-94-015-1258-9_2 [3] N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, ``Universal Quantum Computation with Continuous-Variable Cluster States,'' Phys. Rev. Lett. 97, 110501 (2006). https://doi.org/10.1103/PhysRevLett.97.110501 [4] C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, ``Gaussian quantum information,'' Rev. Mod. Phys. 84, 621–669 (2012). https://doi.org/10.1103/RevModPhys.84.621 [5] O. Pfister, ``Continuous-variable quantum computing in the quantum optical frequency comb,'' J. Phys. B 53, 012001 (2019). https://doi.org/10.1088/1361-6455/ab526f [6] K. Fukui and S. Takeda, ``Building a large-scale quantum computer with continuous-variable optical technologies,'' J. Phys. B. 55, 012001 (2022). https://doi.org/10.1088/1361-6455/ac489c [7] B. M. Terhal, J. Conrad, and C. Vuillot, ``Towards scalable bosonic quantum error correction,'' Quantum Sci. Technol. 5, 043001 (2020). https://doi.org/10.1088/2058-9565/ab98a5 [8] A. L. Grimsmo and S. Puri, ``Quantum Error Correction with the Gottesman-Kitaev-Preskill Code,'' PRX Quantum 2, 020101 (2021). https://doi.org/10.1103/PRXQuantum.2.020101 [9] A. Joshi, K. Noh, and Y. Y. Gao, ``Quantum information processing with bosonic qubits in circuit QED,'' Quantum Sci. Technol. 6, 033001 (2021). https://doi.org/10.1088/2058-9565/abe989 [10] W. Cai, Y. Ma, W. Wang, C.-L. Zou, and L. Sun, ``Bosonic quantum error correction codes in superconducting quantum circuits,'' Fundam. Res. 1, 50–67 (2021). https://doi.org/10.1016/j.fmre.2020.12.006 [11] V. V. Albert, ``Bosonic coding: introduction and use cases,'', vol. 209 of International School of Physics “Enrico Fermi”. IOS Press, Jan., 2025. https://doi.org/10.3254/ENFI250007 [12] A. J. Brady, A. Eickbusch, S. Singh, J. Wu, and Q. Zhuang, ``Advances in bosonic quantum error correction with Gottesman–Kitaev–Preskill Codes: Theory, engineering and applications,'' Prog. Quantum Electron. 93, 100496 (2024). https://doi.org/10.1016/j.pquantelec.2023.100496 [13] A. Ferraro, S. Olivares, and M. G. A. Paris, ``Gaussian States in Continuous Variable Quantum Information,''. Bibliopolis, Napoli, 2005, ISBN 88-7088-483-X. https://doi.org/10.48550/arXiv.quant-ph/0503237 arXiv:quant-ph/0503237 [14] G. Adesso, S. Ragy, and A. R. Lee, ``Continuous Variable Quantum Information: Gaussian States and Beyond,'' Open Syst. Inf. Dyn. 21, 1440001 (2014). https://doi.org/10.1142/s1230161214400010 [15] M. Walschaers, ``Non-Gaussian quantum states and where to find them,'' PRX Quantum 2, 030204 (2021). https://doi.org/10.1103/PRXQuantum.2.030204 [16] S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, ``Efficient Classical Simulation of Continuous Variable Quantum Information Processes,'' Phys. Rev. Lett. 88, 097904 (2002). https://doi.org/10.1103/PhysRevLett.88.097904 [17] A. Mari and J. Eisert, ``Positive Wigner Functions Render Classical Simulation of Quantum Computation Efficient,'' Phys. Rev. Lett. 109, 230503 (2012). https://doi.org/10.1103/PhysRevLett.109.230503 [18] V. Veitch, N. Wiebe, C. Ferrie, and J. Emerson, ``Efficient simulation scheme for a class of quantum optics experiments with non-negative Wigner representation,'' New J. Phys. 15, 013037 (2013). https://doi.org/10.1088/1367-2630/15/1/013037 [19] S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, ``Efficient Classical Simulation of Continuous Variable Quantum Information Processes,'' Phys. Rev. Lett. 88, 097904 (2002). https://doi.org/10.1103/PhysRevLett.88.097904 [20] S. Rahimi-Keshari, T. C. Ralph, and C. M. Caves, ``Sufficient Conditions for Efficient Classical Simulation of Quantum Optics,'' Phys. Rev. X 6, 021039 (2016). https://doi.org/10.1103/PhysRevX.6.021039 [21] A. Kenfack and K. Życzkowski, ``Negativity of the Wigner function as an indicator of non-classicality,'' J. Opt. B: Quantum Semiclassical Opt.. 6, 396 (2004). https://doi.org/10.1088/1464-4266/6/10/003 [22] F. Albarelli, M. G. Genoni, M. G. Paris, and A. Ferraro, ``Resource theory of quantum non-Gaussianity and Wigner negativity,'' Phys. Rev. A 98, 052350 (2018). https://doi.org/10.1103/PhysRevA.98.052350 [23] R. Takagi and Q. Zhuang, ``Convex resource theory of non-Gaussianity,'' Phys. Rev. A 97, 062337 (2018). https://doi.org/10.1103/PhysRevA.97.062337 [24] U. Chabaud, D. Markham, and F. Grosshans, ``Stellar representation of non-Gaussian quantum states,'' Phys. Rev. Lett. 124, 063605 (2020). https://doi.org/10.1103/PhysRevLett.124.063605 [25] U. Chabaud and S. Mehraban, ``Holomorphic representation of quantum computations,'' Quantum 6, 831 (2022). https://doi.org/10.22331/q-2022-10-06-831 [26] L. c. v. Lachman, I. Straka, J. Hloušek, M. Ježek, and R. Filip, ``Faithful Hierarchy of Genuine $n$-Photon Quantum Non-Gaussian Light,'' Phys. Rev. Lett. 123, 043601 (2019). https://doi.org/10.1103/PhysRevLett.123.043601 [27] O. Hahn, R. Takagi, G. Ferrini, and H. Yamasaki, ``Classical simulation and quantum resource theory of non-Gaussian optics,'' Quantum 9, 1881 (2025). https://doi.org/10.22331/q-2025-10-13-1881 [28] B. Dias and R. König, ``Classical simulation of non-Gaussian bosonic circuits,'' Phys. Rev. A 110, 042402 (2024). https://doi.org/10.1103/PhysRevA.110.042402 [29] M. Garnier, T. Martinez, and U. Chabaud, ``stellar-rank-numerics.'' https://github.com/qat-inria/stellar-numerics, 2026. https://github.com/qat-inria/stellar-numerics [30] U. Chabaud, G. Roeland, M. Walschaers, F. Grosshans, V. Parigi, D. Markham, and N. Treps, ``Certification of Non-Gaussian States with Operational Measurements,'' PRX Quantum 2, 020333 (2021). https://doi.org/10.1103/PRXQuantum.2.020333 [31] J. Eisert and M. B. Plenio, ``Conditions for the Local Manipulation of Gaussian States,'' Phys. Rev. Lett. 89, 097901 (2002). https://doi.org/10.1103/PhysRevLett.89.097901 [32] J. Fiurášek, ``Gaussian Transformations and Distillation of Entangled Gaussian States,'' Phys. Rev. Lett. 89, 137904 (2002). https://doi.org/10.1103/PhysRevLett.89.137904 [33] G. Giedke and J. Ignacio Cirac, ``Characterization of Gaussian operations and distillation of Gaussian states,'' Phys. Rev. A 66, 032316 (2002). https://doi.org/10.1103/PhysRevA.66.032316 [34] J. Niset, J. Fiurášek, and N. J. Cerf, ``No-Go Theorem for Gaussian Quantum Error Correction,'' Phys. Rev. Lett. 102, 120501 (2009). https://doi.org/10.1103/PhysRevLett.102.120501 [35] R. I. Booth, U. Chabaud, and P.-E. Emeriau, ``Contextuality and Wigner Negativity Are Equivalent for Continuous-Variable Quantum Measurements,'' Phys. Rev. Lett. 129, 230401 (2022). https://doi.org/10.1103/PhysRevLett.129.230401 [36] U. Chabaud, G. Ferrini, F. Grosshans, and D. Markham, ``Classical simulation of Gaussian quantum circuits with non-Gaussian input states,'' Phys. Rev. Res. 3, 033018 (2021). https://doi.org/10.1103/PhysRevResearch.3.033018 [37] Y. Yao, F. Miatto, and N. Quesada, ``Riemannian optimization of photonic quantum circuits in phase and Fock space,'' SciPost Phys. 17, 082 (2024). https://doi.org/10.21468/SciPostPhys.17.3.082 [38] I. Segal, ``Mathematical Problems of Relativistic Physics: With an Appendix on Group Representations in Hilbert Space,''. Lectures in applied mathematics.
American Mathematical Society, 1967. [39] V. Bargmann, ``On a Hilbert space of analytic functions and an associated integral transform part I,'' Commun. Pure Appl. Math. 14, 187–214 (1961). https://doi.org/10.1002/cpa.3160140303 [40] R. Filip and L. Mišta Jr, ``Detecting quantum states with a positive Wigner function beyond mixtures of Gaussian states,'' Phys. Rev. Lett. 106, 200401 (2011). https://doi.org/10.1103/PhysRevLett.106.200401 [41] J. Fiurášek, ``Efficient construction of witnesses of the stellar rank of nonclassical states of light,'' Opt. Express 30, 30630–30639 (2022). https://doi.org/10.1364/OE.466175 [42] U. Chabaud and M. Walschaers, ``Resources for bosonic quantum computational advantage,'' Physical Review Letters 130, 090602 (2023). https://doi.org/10.1103/PhysRevLett.130.090602 [43] U. Chabaud, G. Roeland, M. Walschaers, F. Grosshans, V. Parigi, D. Markham, and N. Treps, ``Erratum: Certification of non-Gaussian states with operational measurements,'' PRX Quantum 6, 010902 (2025). https://doi.org/10.1103/PRXQuantum.6.010902 [44] L. Lami, B. Regula, R. Takagi, and G. Ferrari, ``Framework for resource quantification in infinite-dimensional general probabilistic theories,'' Phys, Rev, A 103, 032424 (2021). https://doi.org/10.1103/PhysRevA.103.032424 [45] A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, ``Generation of optical ‘Schrödinger cats’ from photon number states,'' Nature 448, 784 (2007). https://doi.org/10.1038/nature06054 [46] J. Etesse, R. Blandino, B. Kanseri, and R. Tualle-Brouri, ``Proposal for a loophole-free violation of Bell's inequalities with a set of single photons and homodyne measurements,'' New J. Phys. 16, 053001 (2014). https://doi.org/10.1088/1367-2630/16/5/053001 [47] D. J. Weigand and B. M. Terhal, ``Generating grid states from Schrödinger-cat states without postselection,'' Phys. Rev. A 97, 022341 (2018). https://doi.org/10.1103/PhysRevA.97.022341 [48] K. Takase, F. Hanamura, H. Nagayoshi, J. E. Bourassa, R. N. Alexander, A. Kawasaki, W. Asavanant, M. Endo, and A. Furusawa, ``Generation of flying logical qubits using generalized photon subtraction with adaptive Gaussian operations,'' Phys. Rev. A 110, 012436 (2024). https://doi.org/10.1103/PhysRevA.110.012436 [49] Y. Zheng, O. Hahn, P. Stadler, P. Holmvall, F. Quijandría, A. Ferraro, and G. Ferrini, ``Gaussian Conversion Protocols for Cubic Phase State Generation,'' PRX Quantum 2, 010327 (2021). https://doi.org/10.1103/PRXQuantum.2.010327 [50] O. Hahn, P. Holmvall, P. Stadler, G. Ferrini, and A. Ferraro, ``Deterministic Gaussian conversion protocols for non-Gaussian single-mode resources,'' Phys. Rev. A 105, 062446 (2022). https://doi.org/10.1103/PhysRevA.105.062446 [51] Y. Zheng, A. Ferraro, A. F. Kockum, and G. Ferrini, ``Gaussian conversion protocol for heralded generation of generalized Gottesman-Kitaev-Preskill states,'' Phys. Rev. A 108, 012603 (2023). https://doi.org/10.1103/PhysRevA.108.012603 [52] J. Etesse, B. Kanseri, and R. Tualle-Brouri, ``Iterative tailoring of optical quantum states with homodyne measurements,'' Opt. Express 22, 30357–30367 (2014). https://doi.org/10.1364/OE.22.030357 [53] F. Arzani, N. Treps, and G. Ferrini, ``Polynomial approximation of non-Gaussian unitaries by counting one photon at a time,'' Phys. Rev. A 95, 052352 (2017). https://doi.org/10.1103/PhysRevA.95.052352 [54] P. V. Parellada, V. G. i Garcia, J. J. Moyano-Fernández, and J. C. Garcia-Escartin, ``No-go theorems for photon state transformations in quantum linear optics,'' Results Phys. 54, 107108 (2023). https://doi.org/10.1016/j.rinp.2023.107108 [55] U. Chabaud, ``Continuous Variable Quantum Advantages and Applications in Quantum Optics.'' Phd thesis arXiv:2102.05227 [quant-ph], Sorbonne University, 2021. arXiv:2102.05227 [56] A. J. Pizzimenti and D. Soh, ``Optical Gottesman-Kitaev-Preskill qubit generation via approximate squeezed coherent state superposition breeding,'' Phys. Rev. A 110, 062619 (2024). https://doi.org/10.1103/PhysRevA.110.062619 [57] T. Matsuura, H. Yamasaki, and M. Koashi, ``Equivalence of approximate Gottesman-Kitaev-Preskill codes,'' Phys. Rev. A 102, 032408 (2020). https://doi.org/10.1103/PhysRevA.102.032408 [58] D. Gottesman, A. Kitaev, and J. Preskill, ``Encoding a qubit in an oscillator,'' Phys. Rev. A 64, 012310 (2001). https://doi.org/10.1103/PhysRevA.64.012310 [59] S. Aaronson and A. Arkhipov, ``The computational Complexity of Linear Optics,'' Theory Comput. 9, 143 (2013). https://doi.org/10.1145/1993636.1993682 [60] E. Knill, R. Laflamme, and G. J. Milburn, ``A scheme for efficient quantum computation with linear optics,'' Nature 409, 46–52 (2001). https://doi.org/10.1038/35051009 [61] M. M. Wilde, ``Quantum Information Theory,''.
Cambridge University Press, Nov., 2016. https://doi.org/10.1017/9781316809976 [62] D. S. Mitrinović and P. M. Vasić, ``Analytic Inequalities,'', vol. 165 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin, Heidelberg, 1970. https://doi.org/10.1007/978-3-642-99970-3 [63] A. Rastegin, ``A lower bound on the relative error of mixed-state cloning and related operations,'' J. Opt. B Quantum Semiclassical Opt. 5, S647 (2003). https://doi.org/10.1088/1464-4266/5/6/017 [64] U. Chabaud, T. Douce, F. Grosshans, E. Kashefi, and D. Markham, ``Building Trust for Continuous Variable Quantum States,'' LIPIcs, TQC 2020 158, 3:1–3:15 (2020). https://doi.org/10.4230/LIPICS.TQC.2020.3 [65] F. M. Miatto and N. Quesada, ``Fast optimization of parametrized quantum optical circuits,'' Quantum 4, 366 (2020). https://doi.org/10.22331/q-2020-11-30-366 [66] XanaduAI, ``Mr Mustard.'' https://github.com/XanaduAI/MrMustard, 2021. https://github.com/XanaduAI/MrMustardCited byCould not fetch Crossref cited-by data during last attempt 2026-05-05 11:19:58: Could not fetch cited-by data for 10.22331/q-2026-05-05-2095 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-05-05 11:19:59: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractState conversion is a fundamental task in quantum information processing. Quantum resource theories allow for analyzing and bounding conversions that use restricted sets of operations. In the context of continuous-variable systems, state conversions restricted to Gaussian operations are crucial for both fundamental and practical reasons, particularly in state preparation and quantum computing with bosonic codes. However, previous analysis did not consider the relevant case of approximate state conversion. In this work, we introduce a framework for assessing approximate Gaussian state conversion by extending the stellar rank to the approximate stellar rank, which serves as an operational measure of non-Gaussianity. We derive bounds for Gaussian state conversion and distillation under approximate and probabilistic conditions, yielding new no-go results for non-Gaussian state preparation and enabling a reliable assessment of the performance of Gaussian conversion protocols. We also provide an open-source Python library to compute stellar-rank-related quantities and to assess Gaussian conversion.Popular summaryIn quantum optics, we typically distinguish between "easy" Gaussian states and operations and "difficult" non-Gaussian ones. Non-Gaussian states and operations are necessary to build quantum computers capable of outperforming their classical counterparts. Non-Gaussian states and operations are thus conceivable as resources for quantum information processing. Various metrics have been introduced to quantify these non-Gaussian resources, such as the stellar rank. However, these metrics only worked in perfect, idealized settings. In the lab, we deal with noise and other imperfections, while we only need to get "close enough" to a target non-Gaussian state. In this article, we introduce the approximate stellar rank. This is a more realistic non-Gaussian measure that can account for the small errors inherent in experimental physics. We use this new measure to establish the first no-go theorems for approximate non-Gaussian state conversion using Gaussian operations. Our results provide a clear boundary for what can and cannot be achieved in the lab, helping us better manage the non-Gaussian resources required for useful quantum technologies.► BibTeX data@article{Hahn2026assessingnon, doi = {10.22331/q-2026-05-05-2095}, url = {https://doi.org/10.22331/q-2026-05-05-2095}, title = {Assessing non-{G}aussian quantum state conversion with the stellar rank}, author = {Hahn, Oliver and Garnier, Maxime and Ferrini, Giulia and Ferraro, Alessandro and Chabaud, Ulysse}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2095}, month = may, year = {2026} }► References [1] E. Chitambar and G. Gour, ``Quantum resource theories,'' Rev. Mod. Phys. 91, 025001 (2019). https://doi.org/10.1103/RevModPhys.91.025001 [2] S. Lloyd and S. L. Braunstein, ``Quantum computation over continuous variables,'' in Quantum Information with Continuous Variables, pp. 9–17. Springer, 1999. https://doi.org/10.1007/978-94-015-1258-9_2 [3] N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, ``Universal Quantum Computation with Continuous-Variable Cluster States,'' Phys. Rev. Lett. 97, 110501 (2006). https://doi.org/10.1103/PhysRevLett.97.110501 [4] C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, ``Gaussian quantum information,'' Rev. Mod. Phys. 84, 621–669 (2012). https://doi.org/10.1103/RevModPhys.84.621 [5] O. Pfister, ``Continuous-variable quantum computing in the quantum optical frequency comb,'' J. Phys. B 53, 012001 (2019). https://doi.org/10.1088/1361-6455/ab526f [6] K. Fukui and S. Takeda, ``Building a large-scale quantum computer with continuous-variable optical technologies,'' J. Phys. B. 55, 012001 (2022). https://doi.org/10.1088/1361-6455/ac489c [7] B. M. Terhal, J. Conrad, and C. Vuillot, ``Towards scalable bosonic quantum error correction,'' Quantum Sci. Technol. 5, 043001 (2020). https://doi.org/10.1088/2058-9565/ab98a5 [8] A. L. Grimsmo and S. Puri, ``Quantum Error Correction with the Gottesman-Kitaev-Preskill Code,'' PRX Quantum 2, 020101 (2021). https://doi.org/10.1103/PRXQuantum.2.020101 [9] A. Joshi, K. Noh, and Y. Y. Gao, ``Quantum information processing with bosonic qubits in circuit QED,'' Quantum Sci. Technol. 6, 033001 (2021). https://doi.org/10.1088/2058-9565/abe989 [10] W. Cai, Y. Ma, W. Wang, C.-L. Zou, and L. Sun, ``Bosonic quantum error correction codes in superconducting quantum circuits,'' Fundam. Res. 1, 50–67 (2021). https://doi.org/10.1016/j.fmre.2020.12.006 [11] V. V. Albert, ``Bosonic coding: introduction and use cases,'', vol. 209 of International School of Physics “Enrico Fermi”. IOS Press, Jan., 2025. https://doi.org/10.3254/ENFI250007 [12] A. J. Brady, A. Eickbusch, S. Singh, J. Wu, and Q. Zhuang, ``Advances in bosonic quantum error correction with Gottesman–Kitaev–Preskill Codes: Theory, engineering and applications,'' Prog. Quantum Electron. 93, 100496 (2024). https://doi.org/10.1016/j.pquantelec.2023.100496 [13] A. Ferraro, S. Olivares, and M. G. A. Paris, ``Gaussian States in Continuous Variable Quantum Information,''. Bibliopolis, Napoli, 2005, ISBN 88-7088-483-X. https://doi.org/10.48550/arXiv.quant-ph/0503237 arXiv:quant-ph/0503237 [14] G. Adesso, S. Ragy, and A. R. Lee, ``Continuous Variable Quantum Information: Gaussian States and Beyond,'' Open Syst. Inf. Dyn. 21, 1440001 (2014). https://doi.org/10.1142/s1230161214400010 [15] M. Walschaers, ``Non-Gaussian quantum states and where to find them,'' PRX Quantum 2, 030204 (2021). https://doi.org/10.1103/PRXQuantum.2.030204 [16] S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, ``Efficient Classical Simulation of Continuous Variable Quantum Information Processes,'' Phys. Rev. Lett. 88, 097904 (2002). https://doi.org/10.1103/PhysRevLett.88.097904 [17] A. Mari and J. Eisert, ``Positive Wigner Functions Render Classical Simulation of Quantum Computation Efficient,'' Phys. Rev. Lett. 109, 230503 (2012). https://doi.org/10.1103/PhysRevLett.109.230503 [18] V. Veitch, N. Wiebe, C. Ferrie, and J. Emerson, ``Efficient simulation scheme for a class of quantum optics experiments with non-negative Wigner representation,'' New J. Phys. 15, 013037 (2013). https://doi.org/10.1088/1367-2630/15/1/013037 [19] S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, ``Efficient Classical Simulation of Continuous Variable Quantum Information Processes,'' Phys. Rev. Lett. 88, 097904 (2002). https://doi.org/10.1103/PhysRevLett.88.097904 [20] S. Rahimi-Keshari, T. C. Ralph, and C. M. Caves, ``Sufficient Conditions for Efficient Classical Simulation of Quantum Optics,'' Phys. Rev. X 6, 021039 (2016). https://doi.org/10.1103/PhysRevX.6.021039 [21] A. Kenfack and K. Życzkowski, ``Negativity of the Wigner function as an indicator of non-classicality,'' J. Opt. B: Quantum Semiclassical Opt.. 6, 396 (2004). https://doi.org/10.1088/1464-4266/6/10/003 [22] F. Albarelli, M. G. Genoni, M. G. Paris, and A. Ferraro, ``Resource theory of quantum non-Gaussianity and Wigner negativity,'' Phys. Rev. A 98, 052350 (2018). https://doi.org/10.1103/PhysRevA.98.052350 [23] R. Takagi and Q. Zhuang, ``Convex resource theory of non-Gaussianity,'' Phys. Rev. A 97, 062337 (2018). https://doi.org/10.1103/PhysRevA.97.062337 [24] U. Chabaud, D. Markham, and F. Grosshans, ``Stellar representation of non-Gaussian quantum states,'' Phys. Rev. Lett. 124, 063605 (2020). https://doi.org/10.1103/PhysRevLett.124.063605 [25] U. Chabaud and S. Mehraban, ``Holomorphic representation of quantum computations,'' Quantum 6, 831 (2022). https://doi.org/10.22331/q-2022-10-06-831 [26] L. c. v. Lachman, I. Straka, J. Hloušek, M. Ježek, and R. Filip, ``Faithful Hierarchy of Genuine $n$-Photon Quantum Non-Gaussian Light,'' Phys. Rev. Lett. 123, 043601 (2019). https://doi.org/10.1103/PhysRevLett.123.043601 [27] O. Hahn, R. Takagi, G. Ferrini, and H. Yamasaki, ``Classical simulation and quantum resource theory of non-Gaussian optics,'' Quantum 9, 1881 (2025). https://doi.org/10.22331/q-2025-10-13-1881 [28] B. Dias and R. König, ``Classical simulation of non-Gaussian bosonic circuits,'' Phys. Rev. A 110, 042402 (2024). https://doi.org/10.1103/PhysRevA.110.042402 [29] M. Garnier, T. Martinez, and U. Chabaud, ``stellar-rank-numerics.'' https://github.com/qat-inria/stellar-numerics, 2026. https://github.com/qat-inria/stellar-numerics [30] U. Chabaud, G. Roeland, M. Walschaers, F. Grosshans, V. Parigi, D. Markham, and N. Treps, ``Certification of Non-Gaussian States with Operational Measurements,'' PRX Quantum 2, 020333 (2021). https://doi.org/10.1103/PRXQuantum.2.020333 [31] J. Eisert and M. B. Plenio, ``Conditions for the Local Manipulation of Gaussian States,'' Phys. Rev. Lett. 89, 097901 (2002). https://doi.org/10.1103/PhysRevLett.89.097901 [32] J. Fiurášek, ``Gaussian Transformations and Distillation of Entangled Gaussian States,'' Phys. Rev. Lett. 89, 137904 (2002). https://doi.org/10.1103/PhysRevLett.89.137904 [33] G. Giedke and J. Ignacio Cirac, ``Characterization of Gaussian operations and distillation of Gaussian states,'' Phys. Rev. A 66, 032316 (2002). https://doi.org/10.1103/PhysRevA.66.032316 [34] J. Niset, J. Fiurášek, and N. J. Cerf, ``No-Go Theorem for Gaussian Quantum Error Correction,'' Phys. Rev. Lett. 102, 120501 (2009). https://doi.org/10.1103/PhysRevLett.102.120501 [35] R. I. Booth, U. Chabaud, and P.-E. Emeriau, ``Contextuality and Wigner Negativity Are Equivalent for Continuous-Variable Quantum Measurements,'' Phys. Rev. Lett. 129, 230401 (2022). https://doi.org/10.1103/PhysRevLett.129.230401 [36] U. Chabaud, G. Ferrini, F. Grosshans, and D. Markham, ``Classical simulation of Gaussian quantum circuits with non-Gaussian input states,'' Phys. Rev. Res. 3, 033018 (2021). https://doi.org/10.1103/PhysRevResearch.3.033018 [37] Y. Yao, F. Miatto, and N. Quesada, ``Riemannian optimization of photonic quantum circuits in phase and Fock space,'' SciPost Phys. 17, 082 (2024). https://doi.org/10.21468/SciPostPhys.17.3.082 [38] I. Segal, ``Mathematical Problems of Relativistic Physics: With an Appendix on Group Representations in Hilbert Space,''. Lectures in applied mathematics.
American Mathematical Society, 1967. [39] V. Bargmann, ``On a Hilbert space of analytic functions and an associated integral transform part I,'' Commun. Pure Appl. Math. 14, 187–214 (1961). https://doi.org/10.1002/cpa.3160140303 [40] R. Filip and L. Mišta Jr, ``Detecting quantum states with a positive Wigner function beyond mixtures of Gaussian states,'' Phys. Rev. Lett. 106, 200401 (2011). https://doi.org/10.1103/PhysRevLett.106.200401 [41] J. Fiurášek, ``Efficient construction of witnesses of the stellar rank of nonclassical states of light,'' Opt. Express 30, 30630–30639 (2022). https://doi.org/10.1364/OE.466175 [42] U. Chabaud and M. Walschaers, ``Resources for bosonic quantum computational advantage,'' Physical Review Letters 130, 090602 (2023). https://doi.org/10.1103/PhysRevLett.130.090602 [43] U. Chabaud, G. Roeland, M. Walschaers, F. Grosshans, V. Parigi, D. Markham, and N. Treps, ``Erratum: Certification of non-Gaussian states with operational measurements,'' PRX Quantum 6, 010902 (2025). https://doi.org/10.1103/PRXQuantum.6.010902 [44] L. Lami, B. Regula, R. Takagi, and G. Ferrari, ``Framework for resource quantification in infinite-dimensional general probabilistic theories,'' Phys, Rev, A 103, 032424 (2021). https://doi.org/10.1103/PhysRevA.103.032424 [45] A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, ``Generation of optical ‘Schrödinger cats’ from photon number states,'' Nature 448, 784 (2007). https://doi.org/10.1038/nature06054 [46] J. Etesse, R. Blandino, B. Kanseri, and R. Tualle-Brouri, ``Proposal for a loophole-free violation of Bell's inequalities with a set of single photons and homodyne measurements,'' New J. Phys. 16, 053001 (2014). https://doi.org/10.1088/1367-2630/16/5/053001 [47] D. J. Weigand and B. M. Terhal, ``Generating grid states from Schrödinger-cat states without postselection,'' Phys. Rev. A 97, 022341 (2018). https://doi.org/10.1103/PhysRevA.97.022341 [48] K. Takase, F. Hanamura, H. Nagayoshi, J. E. Bourassa, R. N. Alexander, A. Kawasaki, W. Asavanant, M. Endo, and A. Furusawa, ``Generation of flying logical qubits using generalized photon subtraction with adaptive Gaussian operations,'' Phys. Rev. A 110, 012436 (2024). https://doi.org/10.1103/PhysRevA.110.012436 [49] Y. Zheng, O. Hahn, P. Stadler, P. Holmvall, F. Quijandría, A. Ferraro, and G. Ferrini, ``Gaussian Conversion Protocols for Cubic Phase State Generation,'' PRX Quantum 2, 010327 (2021). https://doi.org/10.1103/PRXQuantum.2.010327 [50] O. Hahn, P. Holmvall, P. Stadler, G. Ferrini, and A. Ferraro, ``Deterministic Gaussian conversion protocols for non-Gaussian single-mode resources,'' Phys. Rev. A 105, 062446 (2022). https://doi.org/10.1103/PhysRevA.105.062446 [51] Y. Zheng, A. Ferraro, A. F. Kockum, and G. Ferrini, ``Gaussian conversion protocol for heralded generation of generalized Gottesman-Kitaev-Preskill states,'' Phys. Rev. A 108, 012603 (2023). https://doi.org/10.1103/PhysRevA.108.012603 [52] J. Etesse, B. Kanseri, and R. Tualle-Brouri, ``Iterative tailoring of optical quantum states with homodyne measurements,'' Opt. Express 22, 30357–30367 (2014). https://doi.org/10.1364/OE.22.030357 [53] F. Arzani, N. Treps, and G. Ferrini, ``Polynomial approximation of non-Gaussian unitaries by counting one photon at a time,'' Phys. Rev. A 95, 052352 (2017). https://doi.org/10.1103/PhysRevA.95.052352 [54] P. V. Parellada, V. G. i Garcia, J. J. Moyano-Fernández, and J. C. Garcia-Escartin, ``No-go theorems for photon state transformations in quantum linear optics,'' Results Phys. 54, 107108 (2023). https://doi.org/10.1016/j.rinp.2023.107108 [55] U. Chabaud, ``Continuous Variable Quantum Advantages and Applications in Quantum Optics.'' Phd thesis arXiv:2102.05227 [quant-ph], Sorbonne University, 2021. arXiv:2102.05227 [56] A. J. Pizzimenti and D. Soh, ``Optical Gottesman-Kitaev-Preskill qubit generation via approximate squeezed coherent state superposition breeding,'' Phys. Rev. A 110, 062619 (2024). https://doi.org/10.1103/PhysRevA.110.062619 [57] T. Matsuura, H. Yamasaki, and M. Koashi, ``Equivalence of approximate Gottesman-Kitaev-Preskill codes,'' Phys. Rev. A 102, 032408 (2020). https://doi.org/10.1103/PhysRevA.102.032408 [58] D. Gottesman, A. Kitaev, and J. Preskill, ``Encoding a qubit in an oscillator,'' Phys. Rev. A 64, 012310 (2001). https://doi.org/10.1103/PhysRevA.64.012310 [59] S. Aaronson and A. Arkhipov, ``The computational Complexity of Linear Optics,'' Theory Comput. 9, 143 (2013). https://doi.org/10.1145/1993636.1993682 [60] E. Knill, R. Laflamme, and G. J. Milburn, ``A scheme for efficient quantum computation with linear optics,'' Nature 409, 46–52 (2001). https://doi.org/10.1038/35051009 [61] M. M. Wilde, ``Quantum Information Theory,''.
Cambridge University Press, Nov., 2016. https://doi.org/10.1017/9781316809976 [62] D. S. Mitrinović and P. M. Vasić, ``Analytic Inequalities,'', vol. 165 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin, Heidelberg, 1970. https://doi.org/10.1007/978-3-642-99970-3 [63] A. Rastegin, ``A lower bound on the relative error of mixed-state cloning and related operations,'' J. Opt. B Quantum Semiclassical Opt. 5, S647 (2003). https://doi.org/10.1088/1464-4266/5/6/017 [64] U. Chabaud, T. Douce, F. Grosshans, E. Kashefi, and D. Markham, ``Building Trust for Continuous Variable Quantum States,'' LIPIcs, TQC 2020 158, 3:1–3:15 (2020). https://doi.org/10.4230/LIPICS.TQC.2020.3 [65] F. M. Miatto and N. Quesada, ``Fast optimization of parametrized quantum optical circuits,'' Quantum 4, 366 (2020). https://doi.org/10.22331/q-2020-11-30-366 [66] XanaduAI, ``Mr Mustard.'' https://github.com/XanaduAI/MrMustard, 2021. https://github.com/XanaduAI/MrMustardCited byCould not fetch Crossref cited-by data during last attempt 2026-05-05 11:19:58: Could not fetch cited-by data for 10.22331/q-2026-05-05-2095 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-05-05 11:19:59: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.