A bipartite state ${\rho }_{AB}$ is symmetric extendible if there exists a tripartite state ${\rho }_{AB{B}^{\prime }}$ whose $AB$ and $A{B}^{\prime }$ marginal states are both identical to ${\rho }_{AB}$. 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 ${\rho }_{AB}$; specifically, $\mathrm{tr}\left({\rho }_B^2\right)\ge \mathrm{tr}\left({\rho }_{AB}^2\right)-4\sqrt{det{\rho }_{AB}}$. As a special case we solve the bosonic three-representability problem for the two-body reduced density matrix.

• Received 3 January 2014

DOI:https://doi.org/10.1103/PhysRevA.90.032318