Quantum Singleton bound ($n-k\ge 2(d-1)$)

For a code carrying $k>0$ logical qubits, two disjoint subsets $A,B$ of $d-1$ physical qubits are each correctable erasures. Purify the maximally mixed logical input using a reference $R$, and call the remaining physical subsystem $C$. Erasure decoupling gives $S(RA)=S(R)+S(A)$ and $S(RB)=S(R)+S(B)$. Purity and Subadditivity of Von Neumann entropy imply $2S(R)\le 2S(C)$, hence $k\le n-2(d-1)$. If two such disjoint subsets would not fit, partition the entire physical system into two correctable subsets; purity and decoupling then force $S(R)=0$, a contradiction. Thus the partition used in the proof is justified independently of the bound.