The stellar decomposition of Gaussian quantum states

TL;DR

The paper addresses the challenge of characterizing non-Gaussian states produced by heralding photons from Gaussian states in continuous-variable photonics. It introduces the stellar decomposition, expressing any $(m+n)$-mode Gaussian state as a Gaussian core state $G_{ m core}$ followed by an $m$-mode Gaussian transformation $T$, with the core enabling finite Fock representations after heralding. The authors establish exact results for pure states, necessary and sufficient conditions for mixed-state decompositions, and a formal decomposition for general Gaussian operators, enabling efficient heralded-state simulations and semidefinite-program-based bounds on the quality of GKP states. These results provide practical bounds on state preparation fidelity in photonic circuits and inform the design of staircase GBS architectures for high-quality GKP generation. Leveraging the Bargmann representation of Gaussian CPTP maps, the work bridges foundational theory with actionable tools for simulation and device benchmarking in quantum optics.

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" $G_{\mathrm{core}}$ 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.

Figures (15)

• Figure 1: Stellar decomposition for pure Gaussian states (see \ref{['prop:pure-state-case']}). The triangle shape indicates a Hilbert space vector (a ket). Every pure Gaussian state $|\psi\rangle$ can be decomposed as a Gaussian core state $|\psi_\mathrm{core}\rangle$ followed by an $m$-mode Gaussian unitary $U$ acting only on $M$. Any $(W|\psi_\mathrm{core}\rangle, UW^\dagger)$ pair for $W$ the unitary of an $m$-mode interferometer is a unitarily equivalent stellar decomposition.
• Figure 2: A visual demonstration that the unitary of the pure stellar decomposition is one that maps the $m$-mode vacuum $\ket{0^m}$ to the heralded state $(\mathbb 1 \otimes\langle0^n|)|\psi\rangle$ corresponding to measuring vacuum on the last $n$ modes of $|\psi\rangle$.
• Figure 3: Measuring a Fock pattern $k\in\mathbb{N}^n$ on subsystem $N$ leaves the state on subsystem $M$ with Fock support on the set of Fock states $J_{\norm{k}_1}^{m}$ (see \ref{['eq:J-notation']} for the definition of such sets). This means the stellar rank is upper bounded by $\norm{k}_1$ i.e. the total number of measured photons.
• Figure 4: Stellar decomposition for mixed Gaussian states (see \ref{['prop:mixed-state-case']}). Not every mixed Gaussian state $\rho$ admits a stellar decomposition in terms of physical parts (i.e. a Gaussian mixed state and a Gaussian channel). The rounded shape indicates density matrices.
• Figure 5: Stellar decomposition for Gaussian operators (see \ref{['prop:formal-stellar-decomp']}). If no physicality requirement is necessary, any multimode Gaussian object $G$ can be decomposed into two Gaussian parts such that $S$ has the core property of needing a finite Fock cutoff on the subsystem $M$, when we project the subsystem $N$ onto the Fock basis.

Theorems & Definitions (41)

• Lemma 1
• Definition 1
• Proposition 1
• Theorem 1
• Theorem 2
• Proposition 2
• Definition 2
• Proposition 3
• Proposition 4
• Proposition 5