Geometric characterization of non-Gaussian entanglement for finite stellar rank states

TL;DR

The paper develops a comprehensive framework to characterize non-Gaussian entanglement in pure bosonic states of finite stellar rank by leveraging the stellar representation. It introduces the atomic decomposition of stellar polynomials and the notion of essential variables to expose the full entanglement structure via a structural graph, linking it to passive separability and mode partitions. For two-mode and rank-2 cores, the authors derive complete separability criteria, expressed through zero-set hyperplane decompositions and quadratic forms, and illustrate how the decomposition informs state preparation complexity. The approach isolates genuinely non-Gaussian resources and clarifies how non-Gaussian entanglement can be generated, including via non-orthogonal photon additions and algebraic entanglement, with potential applicability to scalable quantum information processing in continuous-variable platforms.

Abstract

We introduce a general framework for the analysis of non-Gaussian entanglement in bosonic states of finite stellar rank. The central result is the full characterization of their entanglement structure through the atomic decomposition of their stellar polynomial and its associated structural graph, whose connected components determine the mode-intrinsic entanglement content of the state and all partitions compatible with passive separability. An essential ingredient in this construction is the concept of essential variables, which identify the minimal number of effective modes involved in a core state, in direct correspondence with the symplectic rank. This reduction provides the foundation for decomposing stellar polynomials into atomic factors and for revealing the underlying entanglement structure. Building on this, we derive complete separability criteria for two-mode states, expressed through hyperplane decompositions of zero sets, and for stellar-rank-2 states across arbitrary number of modes. Applications to several example states illustrate how the method isolates genuinely non-Gaussian resources and quantifies preparation complexity.

Figures (10)

• Figure 1: A state $|\psi\rangle$ is passive separable with respect to some integer partition of the modes $\mathcal{I}_K$, if there exists a passive unitary $\hat{U}_O$, that disentangles it with respect to such a partition. An equivalent representation implies that the state can be generated from such a separable state by the implementation of the inverse of $\hat{U}_O$.
• Figure 2: A state $|\psi\rangle$ is Gaussian separable with respect to some integer partition of the modes $\mathcal{I}_K$, if there exists a Gaussian unitary $\hat{U}_S$, that disentangles it with respect to such a partition.
• Figure 3: Every core state can be obtained from the vacuum state by the implementation of the corresponding stellar polynomial of the creation operators over all the populated modes.
• Figure 4: Finding the essential variables of the stellar polynomial $p_C(\mathbf z)$, effectively decreases both the complexity of the state preparation and simplifies the analysis of the mode-intrinsic entanglement properties of the state. The modes on the vacuum (those that are not affected by the polynomial of creation and annihilation operators $p_C^E(\mathbf{\hat{a}_E^{\dagger}})$), are irrelevant for the analysis of the passive separability.
• Figure 5: The irreducible factorization of the stellar polynomial of a core state $|C\rangle$, breaks down the state preparation into a sequential application of all the irreducible factors. Moreover it simplifies the analysis of the mode-intrinsic entanglement to the analysis of the geometrical relations among them.

Theorems & Definitions (31)

• Definition 1
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• Definition 5