 Solution

For a bipartite density operator ${\rho }_{AB}$, the Subadditivity of Von Neumann entropy isHere $S(A)=S({\mathrm{Tr}}_B{\rho }_{AB})$ and likewise for $B$, using the partial trace. For a tripartite density operator, the Strong subadditivity of Von Neumann entropy isEquivalently, quantum conditional entropy cannot increase upon adding a conditioning system: $S(A\mid BC)\le S(A\mid B)$.