Quantum key distribution uses public discussion protocols to establish shared

secret keys. In the exploration of ultimate limits to such protocols, the

property of symmetric extendibility of underlying bipartite states $\rho_{AB}$

plays an important role. A bipartite state $\rho_{AB}$ is symmetric extendible

if there exits a tripartite state $\rho_{ABB'}$, such that the $AB$ marginal

state is identical to the $AB'$ marginal state, i.e. $\rho_{AB'}=\rho_{AB}$.

For a symmetric extendible state $\rho_{AB}$, the first task of the public

discussion protocol is to break this symmetric extendibility. Therefore to

characterize all bi-partite quantum states that possess symmetric extensions is

of vital importance. We prove a simple analytical formula that a two-qubit

state $\rho_{AB}$ admits a symmetric extension if and only if

$\tr(\rho_B^2)\geq \tr(\rho_{AB}^2)-4\sqrt{\det{\rho_{AB}}}$. Given the

intimate relationship between the symmetric extension problem and the quantum

marginal problem, our result also provides the first analytical necessary and

sufficient condition for the quantum marginal problem with overlapping

marginals.