# Extendibility of bosonic Gaussian states

## TL;DR

This paper characterizes the extendibility of bosonic Gaussian states, providing efficient criteria, bounds on their distance to separable states, and implications for quantum steerability and entanglement.

## Contribution

It introduces a semidefinite program for deciding $k$-extendibility of Gaussian states and derives Gaussian de Finetti theorems with universal scaling.

## Key findings

- Efficient semidefinite program for $k$-extendibility
- Universal bounds on distance to separable states
- Upper bounds on entanglement of formation for Gaussian states

## Abstract

Extendibility of bosonic Gaussian states is a key issue in continuous-variable quantum information. We show that a bosonic Gaussian state is $k$-extendible if and only if it has a Gaussian $k$-extension, and we derive a simple semidefinite program, whose size scales linearly with the number of local modes, to efficiently decide $k$-extendibility of any given bosonic Gaussian state. When the system to be extended comprises one mode only, we provide a closed-form solution. Implications of these results for the steerability of quantum states and for the extendibility of bosonic Gaussian channels are discussed. We then derive upper bounds on the distance of a $k$-extendible bosonic Gaussian state to the set of all separable states, in terms of trace norm and R\'enyi relative entropies. These bounds, which can be seen as "Gaussian de Finetti theorems," exhibit a universal scaling in the total number of modes, independently of the mean energy of the state. Finally, we establish an upper bound on the entanglement of formation of Gaussian $k$-extendible states, which has no analogue in the finite-dimensional setting.