The Entropy Characterization of Quantum MDS Codes

An $[[n,k,d]]$ quantum maximum-distance-separable code maps k source qudits to n coded qudits such that any $n-(d-1)$ coded qudits may recover all source qudits and $n = k + 2 (d-1)$. The entropy of the joint state of the reference system of k qudits and the n coded qudits is fully characterized - the joint state must be pure, i.e., has entropy zero; and any sub-system whose number of qudits is at most half of $k+n$, the total number of qudits in the joint state must be maximally mixed, i.e., has entropy equal to its size.