Non-Gaussian Entanglement Hierarchy Based on the Schmidt Number

TL;DR

This paper introduces a quantitative hierarchy for non-Gaussian entanglement in bipartite bosonic systems, providing a new witness and benchmarks for various non-Gaussian states, with practical measurement schemes.

Contribution

It defines a new entanglement witness $E_{NG}$ and a hierarchy based on the Schmidt number, offering a practical and experimentally accessible framework for non-Gaussian entanglement detection.

Findings

The witness $E_{NG}$ equals 1 for Gaussian-entanglable states and exceeds 1 for non-Gaussian entanglement.

The hierarchy provides a lower bound on the Schmidt number, reflecting state complexity.

An economical NOON-type witness requiring only four measurements is constructed.

Abstract

Non-Gaussian entanglement is a promising resource in various quantum tasks. A recently defined class identifies entanglement that cannot be generated by applying Gaussian operations to separable inputs. To further explore the entanglement in this context, we introduce a quantitative witness $E_{\rm NG}$ in bipartite bosonic systems, which satisfies $E_{\rm NG}=1$ for all Gaussian-entanglable states, while $E_{\rm NG}>1$ certifies non-Gaussian entanglement. Its ceiling $d=\lceil E_{\rm NG}\rceil$ provides a lower bound on the Schmidt number irreducible by Gaussian transformations, thereby defining a natural hierarchy of non-Gaussian entanglement. For pure states, the condition is sharp and the hierarchy reflects the complexity of state learning. We benchmark the framework with some paradigmatic non-Gaussian states, such as NOON states and squeezed Kerr states, and analyze its robustness against loss. Moreover, we construct an experimentally economical NOON-type witness requiring only four projective measurements, where an analytical Gaussian-entanglable threshold is derived. These results establish an operationally meaningful and experimentally accessible framework for identifying non-Gaussian entanglement resources in continuous-variable quantum platforms.