Extendibility of Fermionic Gaussian States

TL;DR

This paper characterizes fermionic Gaussian states' extendibility, provides a covariance-matrix criterion, and establishes a finite de Finetti theorem with improved bounds, impacting quantum correlations and channel properties.

Contribution

It offers a complete covariance-matrix characterization of fermionic Gaussian state extendibility and derives a finite de Finetti theorem with linear scaling bounds, advancing understanding of quantum correlations.

Findings

Characterization of extendibility via covariance matrices

Finite de Finetti theorem with linear bounds in modes

SDP criteria for fermionic Gaussian channel properties

Abstract

We investigate $(k_1,k_2)$-extendibility of fermionic Gaussian states, a property central to quantum correlations and approximations of separability. We show that these states are $(k_1,k_2)$-extendible if and only if they admit a fermionic Gaussian extension, yielding a complete covariance-matrix characterization and a simple semidefinite program (SDP) whose size scales linearly with the number of modes. This provides necessary conditions for arbitrary fermionic states and is sufficient within the Gaussian setting. Our main result is a finite de Finetti--type theorem: we derive trace-norm bounds between $(k_1,k_2)$-extendible fermionic Gaussian states and separable states, improving previous exponential scaling to linear in the number of modes, with complementary relative entropy and squashed entanglement bounds. For two modes, upper and lower bounds match at order $1/\sqrt{k_1 k_2}$. Extendibility also provides operational support for one of the different notions of separability in fermionic systems. Finally, for fermionic Gaussian channels, we provide an SDP criterion for anti-degradability and show that entanglement-breaking channels coincide with replacement channels, implying no nontrivial entanglement-breaking fermionic Gaussian channels exist.