Symmetric extension of two-qubit states

A bipartite state ${\ensuremath{\rho}}_{AB}$ is symmetric extendible if there exists a tripartite state ${\ensuremath{\rho}}_{AB{B}^{\ensuremath{'}}}$ whose $AB$ and $A{B}^{\ensuremath{'}}$ marginal states are both identical to ${\ensuremath{\rho}}_{AB}$. Symmetric extendibility of bipartite states is of vital importance in quantum information because of its central role in separability tests, one-way distillation of Einstein-Podolsky-Rosen pairs, one-way distillation of secure keys, quantum marginal problems, and antidegradable quantum channels. We establish a simple analytic characterization for symmetric extendibility of any two-qubit quantum state ${\ensuremath{\rho}}_{AB}$; specifically, $\mathrm{tr}({\ensuremath{\rho}}_{B}^{2})\ensuremath{\ge}\mathrm{tr}({\ensuremath{\rho}}_{AB}^{2})\ensuremath{-}4\sqrt{det{\ensuremath{\rho}}_{AB}}$. As a special case we solve the bosonic three-representability problem for the two-body reduced density matrix.