Markovianization of tripartite quantum states and its application to distributed quantum computation

We introduce a task that we call Markovianization, in which a tripartite quantum state is transformed to an approximate quantum Markov chain by a random unitary operation on one of the three subsystems. We consider an asymptotic limit of infinite copies and vanishingly small error, and define the Markovianizing cost as the minimum cost of randomness per copy required for the task. We derive a single-letter formula for the Markovianizing cost of pure states based on the Koashi-Imoto decomposition, and prove that it can be computed by a finite-step algorithm.

 

The obtained results are then applied to an analysis of entanglement-assisted LOCC implementations of bipartite unitaries. We consider an information theoretical scenario in which the same bipartite unitary is applied on infinite pairs of input states generated by a completely random i.i.d. quantum information source, and analyze the minimum costs of entanglement and classical communication per input. For the case of two-round LOCC protocols, we prove that the minimum costs are given by the Markovianizing cost of the Jamiolkowski state of the unitary. The result implies a trade-off relation between the entanglement cost and the number of rounds for a bidirectional LOCC task.