# Symmetric Extension of Two-Qubit States

Presenter: Jianxin Chen (University of Guelph)

Co-authors: Zhengfeng Ji (University of Waterloo), David Kribs (University of Guelph), Norbert Lutkenhaus (University of Waterloo), Bei Zeng (University of Guelph)

### Abstract

A bipartite state $\rho_{AB}$ is symmetric extendable if there exits a tripartite state $\rho_{ABB'}$, such that the $AB$ marginal state is identical to the $AB'$ marginal state, i.e. $\rho_{AB'}=\rho_{AB}$. Understanding the conditions for symmetric extendability is of vital importance in analyzing protocols that distill secret key from quantum correlations. We prove a simple analytical formula that a two-qubit state $\rho_{AB}$ admits a symmetric extension if and only if $\tr(\rho_B^2)\geq \tr(\rho_{AB}^2)-4\sqrt{\det{\rho_{AB}}}$. Our result provides the first analytical necessary and sufficient condition for the quantum marginal problem with overlapping marginals.