Spectrum conditions for symmetric extendible states

We analyze bipartite quantum states that admit a symmetric extension. Any such state can be decomposed into a convex combination of states that allow a pure symmetric extension. A necessary condition for a state to admit a pure symmetric extension is that the spectra of the local and global density matrices are equal. This condition is also sufficient for two qubits but not for any larger systems. Using this condition, we present a conjectured necessary and sufficient condition for a two-qubit state to admit symmetric extension, which we prove in some special cases. The results from symmetric extension carry over to degradable and antidegradable channels and we use this to prove that all degradable channels with qubit output have a qubit environment.