Excluding one of the parties and Markovianizing of tripartite quantum states

We consider a distributed decoupling task for a tripartite system in which Bob and Charlie “exclude” Alice by destroying correlation between BC and A in a tripartite state ρABC shared by the parties using only local random operations. We refer to this task as excluding. In an asymptotic limit of infinite copies of ρABC , we investigate the excluding cost, namely, the cost of randomness required to accomplish excluding. In particular, we consider a case where the initial state is one obtained from a bipartite unitary U as Ψ(U)ABC = (U⊗I)(I/d⊗|Φd〉〈Φd|)(U †AB⊗IC), for which excluding can be accomplished by only Charlie’s local random operation. We prove that the excluding cost in this case is equal to the cost of randomness required for turning a tripartite pure state sufficiently close to a quantum Markov chain by a random operation, in the asymptotic limit of infinite copies. We derive the single-letter formula of this “Markovianizing cost” for arbitrary tripartite pure state by using the data compression theorem for mixed-state quantum information sources. We show that the excluding cost derived here gives an upper bound on the cost of resources required in a model of distributed quantum computation, and that it gives the optimal quantum communication rate required in distributed compression of tripartite quantum state in a particular setting.