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The stellar decomposition of Gaussian quantum states

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The stellar decomposition of Gaussian quantum states

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AbstractWe introduce the $\textit{stellar decomposition}$, a novel method for characterizing non-Gaussian states produced by photon-counting measurements on Gaussian states. Given an $(m+n)$-mode Gaussian state $G$, we express it as an $(m+n)$-mode "Gaussian core state" $G_{\mathrm{core}}$ followed by an $m$-mode Gaussian transformation $T$ that only acts on the first $m$ modes. The defining property of the Gaussian core state $G_{\mathrm{core}}$ is that measuring the last $n$ of its modes in the photon-number basis leaves the first $m$ modes on a finite Fock support, i.e. a core state. Since $T$ is measurement-independent and $G_{\mathrm{core}}$ has an exact and finite Fock representation, this decomposition exactly describes all non-Gaussian states obtainable by projecting $n$ modes of $G$ onto the Fock basis. For pure states we prove that a physical pair $(G_{\mathrm{core}}, T)$ always exists with $G_{\mathrm{core}}$ pure and $T$ unitary. For mixed states, we establish necessary and sufficient conditions for $(G_{\mathrm{core}}, T)$ to be a Gaussian mixed state and a Gaussian channel. We also develop a semidefinite program to extract the "largest" possible Gaussian channel when these conditions fail. Finally, we present a formal stellar decomposition for generic operators, which is useful in simulations where the only requirement is that the two parts contract back to the original operator. The stellar decomposition leads to practical bounds on achievable state quality in photonic circuits and for GKP state generation in particular. Our results are based on a new characterization of Gaussian completely positive maps in the Bargmann picture, which may be of independent interest.Popular summaryIn quantum photonics, non-Gaussian states are required to achieve universal quantum advantage. Typically, non-Gaussian states are engineered by measuring photons on larger Gaussian states, but simulating these processes is imprecise and computationally expensive due to the infinite nature of the photon number basis. This work introduces the "stellar decomposition," a novel mathematical formalism that factorizes any multimode Gaussian state into a "Gaussian core state" followed by a Gaussian transformation. Crucially, the Gaussian core state possesses a finite photon number support that matches the number of measured photons. Thanks to this, one can replace truncated numerical approximations with exact descriptions of non-Gaussian states. Beyond simulation, this framework leads to rigorous upper bounds on the quality of Gottesman-Kitaev-Preskill (GKP) states producible by Gaussian Boson Sampling devices, establishing fundamental limits imposed by optical loss. This toolkit significantly streamlines the design and benchmarking of fault-tolerant photonic architectures.► BibTeX data@article{Motamedi2026stellar, doi = {10.22331/q-2026-01-19-1971}, url = {https://doi.org/10.22331/q-2026-01-19-1971}, title = {The stellar decomposition of {G}aussian quantum states}, author = {Motamedi, Arsalan and Yao, Yuan and Nielsen, Kasper and Chabaud, Ulysse and Bourassa, J. Eli and Alexander, Rafael N. and Miatto, Filippo M.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1971}, month = jan, year = {2026} }► References [1] C. H. Bennett and S. J. Wiesner. Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett., 69:2881–2884, 1992. https://doi.org/10.1103/PhysRevLett.69.2881. https://doi.org/10.1103/PhysRevLett.69.2881 [2] P. W. Shor. Algorithms for quantum computation: discrete logarithms and factoring. In Proc. 35th Ann. IEEE Symp. on Foundations of Computer Science, pages 124–134, 1994. https://doi.org/10.1109/SFCS.1994.365700. https://doi.org/10.1109/SFCS.1994.365700 [3] Michael A Nielsen and Isaac L Chuang. Quantum computation and quantum information.

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Could not fetch ADS cited-by data during last attempt 2026-01-19 16:31:08: 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. AbstractWe introduce the $\textit{stellar decomposition}$, a novel method for characterizing non-Gaussian states produced by photon-counting measurements on Gaussian states. Given an $(m+n)$-mode Gaussian state $G$, we express it as an $(m+n)$-mode "Gaussian core state" $G_{\mathrm{core}}$ followed by an $m$-mode Gaussian transformation $T$ that only acts on the first $m$ modes. The defining property of the Gaussian core state $G_{\mathrm{core}}$ is that measuring the last $n$ of its modes in the photon-number basis leaves the first $m$ modes on a finite Fock support, i.e. a core state. Since $T$ is measurement-independent and $G_{\mathrm{core}}$ has an exact and finite Fock representation, this decomposition exactly describes all non-Gaussian states obtainable by projecting $n$ modes of $G$ onto the Fock basis. For pure states we prove that a physical pair $(G_{\mathrm{core}}, T)$ always exists with $G_{\mathrm{core}}$ pure and $T$ unitary. For mixed states, we establish necessary and sufficient conditions for $(G_{\mathrm{core}}, T)$ to be a Gaussian mixed state and a Gaussian channel. We also develop a semidefinite program to extract the "largest" possible Gaussian channel when these conditions fail. Finally, we present a formal stellar decomposition for generic operators, which is useful in simulations where the only requirement is that the two parts contract back to the original operator. The stellar decomposition leads to practical bounds on achievable state quality in photonic circuits and for GKP state generation in particular. Our results are based on a new characterization of Gaussian completely positive maps in the Bargmann picture, which may be of independent interest.Popular summaryIn quantum photonics, non-Gaussian states are required to achieve universal quantum advantage. Typically, non-Gaussian states are engineered by measuring photons on larger Gaussian states, but simulating these processes is imprecise and computationally expensive due to the infinite nature of the photon number basis. This work introduces the "stellar decomposition," a novel mathematical formalism that factorizes any multimode Gaussian state into a "Gaussian core state" followed by a Gaussian transformation. Crucially, the Gaussian core state possesses a finite photon number support that matches the number of measured photons. Thanks to this, one can replace truncated numerical approximations with exact descriptions of non-Gaussian states. Beyond simulation, this framework leads to rigorous upper bounds on the quality of Gottesman-Kitaev-Preskill (GKP) states producible by Gaussian Boson Sampling devices, establishing fundamental limits imposed by optical loss. This toolkit significantly streamlines the design and benchmarking of fault-tolerant photonic architectures.► BibTeX data@article{Motamedi2026stellar, doi = {10.22331/q-2026-01-19-1971}, url = {https://doi.org/10.22331/q-2026-01-19-1971}, title = {The stellar decomposition of {G}aussian quantum states}, author = {Motamedi, Arsalan and Yao, Yuan and Nielsen, Kasper and Chabaud, Ulysse and Bourassa, J. Eli and Alexander, Rafael N. and Miatto, Filippo M.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1971}, month = jan, year = {2026} }► References [1] C. H. Bennett and S. J. Wiesner. Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett., 69:2881–2884, 1992. https://doi.org/10.1103/PhysRevLett.69.2881. https://doi.org/10.1103/PhysRevLett.69.2881 [2] P. W. Shor. Algorithms for quantum computation: discrete logarithms and factoring. In Proc. 35th Ann. IEEE Symp. on Foundations of Computer Science, pages 124–134, 1994. https://doi.org/10.1109/SFCS.1994.365700. https://doi.org/10.1109/SFCS.1994.365700 [3] Michael A Nielsen and Isaac L Chuang. Quantum computation and quantum information.

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The stellar decomposition of Gaussian quantum states

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