  Consider many instances of an arbitrary quadripartite pure state of four
quantum systems ABCD. Alice holds the AC part of each state, Bob holds B, while
D represents all other parties correlated with ABC. Alice is required to
redistribute the C systems to Bob while asymptotically preserving the overall
purity. We prove that this is possible using Q qubits of communication and E
ebits of shared entanglement between Alice and Bob, provided that Q geq
I(C;D|B)/2 and Q+E geq H(C|B), proving the optimality of the Luo-Devetak outer
bound. The optimal qubit rate provides the first known operational
interpretation of quantum conditional mutual information. We also show how our
protocol leads to a fully operational proof of strong subadditivity and uncover
a general organizing principle, in analogy to thermodynamics, that underlies
the optimal rates.
