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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 R 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 ges 1/2I(C; D|B) and Q + E ges 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 subaddivity and uncover a general organizing principle, in analogy to thermodynamics, that underlies the optimal rates.