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  • Open Access

Symplectic Rank of Non-Gaussian Quantum States

Francesco A. Mele1,*, Salvatore F.E. Oliviero1,†, Varun Upreti2,‡, and Ulysse Chabaud2,§

  • *Contact author: francesco.mele@sns.it
  • †Contact author: salvatore.oliviero@sns.it
  • ‡Contact author: varun.upreti@inria.fr
  • §Contact author: ulysse.chabaud@inria.fr
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PRX Quantum 7, 020366 – Published 22 June, 2026

DOI: https://doi.org/10.1103/1rtk-1jsn

Abstract

Non-Gaussianity is a key resource for achieving quantum advantages in bosonic platforms. Here, we investigate the symplectic rank: a non-Gaussianity monotone that satisfies remarkable operational and resource-theoretic properties. Mathematically, the symplectic rank of a pure state is the number of symplectic eigenvalues of the covariance matrix that are strictly larger than those of the vacuum. Operationally, it (i) is easy to compute, (ii) emerges as the smallest number of modes onto which all the non-Gaussianity can be compressed via Gaussian unitaries, (iii) lower bounds the non-Gaussian gate complexity of state preparation independently of the gate set, (iv) governs the sample complexity of quantum tomography, and (v) bounds the computational complexity of bosonic circuits. Crucially, the symplectic rank is nonincreasing under postselected Gaussian operations, leading to new no-go theorems for Gaussian conversion. Remarkably, this allows us to show that the resource theory of non-Gaussianity is irreversible under exact Gaussian operations. Finally, we show that the symplectic rank is a robust non-Gaussian measure, explaining how to witness it in experiments and how to exploit it to meaningfully benchmark different bosonic platforms. In doing so, we derive lower bounds on the trace distance (resp. total variation distance) between arbitrary states (resp. classical probability distributions) in terms of the norm distance between their covariance matrices, which may be of independent interest.

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Popular Summary

We introduce the symplectic rank, a rigorous, robust, and experimentally accessible measure of non-Gaussianity in bosonic quantum states. Defined via covariance matrices, it quantifies non-Gaussian resources with clear operational meaning, characterizing compression under Gaussian unitaries, circuit complexity, tomography, and classical simulation. We prove that symplectic rank is nonincreasing under Gaussian operations, establishing stronger no-go theorems and revealing the inherent irreversibility of non-Gaussian resource theory. Robust to noise and directly measurable, it provides a unifying benchmark for bosonic platforms. Our results bridge foundational theory with practical applications, offering a powerful tool for advancing quantum technologies.

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PRX Quantum 7, 020366– Published 22 June, 2026
  • Received 15 September 2025
  • Revised 18 February 2026
  • Accepted 6 April 2026
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