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arXiv:2504.10455 · Quantum

The stellar decomposition of Gaussian quantum states

Arsalan Motamedi, Yuan Yao, Kasper Nielsen, Ulysse Chabaud, J. Eli Bourassa, Rafael N. Alexander, Filippo M. Miatto

Published Jan 19, 2026• 3 citations
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Abstract

We introduce the 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 c o r e 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 c o r e 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 c o r e 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 c o r e , T ) always exists with G c o r e pure and T unitary. For mixed states, we establish necessary and sufficient conditions for ( G c o r e , 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.

DOI
10.22331/q-2026-01-19-1971

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We introduce the 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 c o r e 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 c o r e 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 c o r e 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 c o r e , T ) always exists with G c o r e pure and T unitary. For mixed states, we establish necessary and sufficient conditions for ( G c o r e , 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.
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The stellar hierarchy of quantum states of light classifies the states according to the Fock-state resources that are required for their generation together with unitary Gaussian operations. States with stellar rank n can be also equivalently referred to as genuinely n-photon quantum non-Gaussian states. Here we present an efficient method for construction of general witnesses of the stellar rank. The number of parameters that need to be optimized in order to determine the witness does not depend on the stellar rank and it scales quadratically with the number of modes. We illustrate the procedure by constructing stellar rank witnesses based on pairs of Fock state probabilities and also based on pairs of fidelities with superpositions of coherent states.
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The so-called stellar formalism allows us to represent the non-Gaussian properties of single-mode quantum states by the distribution of the zeros of their Husimi Q function in phase space. We use this representation in order to derive an infinite hierarchy of single-mode states based on the number of zeros of the Husimi Q function: the stellar hierarchy. We give an operational characterization of the states in this hierarchy with the minimal number of single-photon additions needed to engineer them, and derive equivalence classes under Gaussian unitary operations. We study in detail the topological properties of this hierarchy with respect to the trace norm, and discuss implications for non-Gaussian state engineering, and continuous variable quantum computing.
We study how many Gaussian components a continuous-variable quantum state needs, adopting the approximate Gaussian rank framework of Hahn–Takagi–Ferrini–Yamasaki and the fidelity-ball relaxation of the stellar-rank literature; no novelty is claimed for the definitions. For mixed states we formulate the measure-theoretic ess-sup rank roof over generalized ensembles and prove its attainment and Gaussian-channel monotonicity — a partial answer, on the rank side, to the discrete-versus-continuous decomposition question the cited work leaves open. Our main contributions are three. First, for single-mode dictionaries of bounded common-squeezing Gaussian atoms, we prove zero-counting lower bounds: robust zeros of a target's Bargmann function survive fidelity-ball perturbation and are counted against an exponential-sum zero budget. The dictionary restriction is provably forced — counterexample families show no bound on the unrestricted rank can follow from raw disk zero counts alone. Second, an inequivalence theorem: the thermal lossy cat of our earlier preprint has Gaussian rank at most 2 yet no finite-rank pre-loss operator representation at any efficiency, separating two natural compressibility notions in one direction. Third, measured curves: family-constrained approximation residuals and a real-data rank proxy on public GKP homodyne data, and a robust-zero census for finite-energy GKP states in which all three computed configurations support the lower-bound premises for the untruncated comb. The census is numerically supported, not computer-certified (float64 without outward rounding; numerical root-finding); a machine-checked claim register binds every quoted number to committed artifacts. Multimode extension and a general-squeezing plateau conjecture remain open.