Communication Cost for Non-Markovianity of Tripartite Quantum States: A Resource Theoretic Approach

To quantify non-Markovianity of tripartite quantum states from an operational viewpoint, we introduce a class Ω° of operations performed by three distant parties. A tripartite quantum state is a free state under Ω° if and only if it is a quantum Markov chain. We introduce a function of tripartite quantum states that we call the non-Markovianity of formation, and prove that it is a faithful measure of non-Markovianity, which is continuous and monotonically nonincreasing under a subclass Ω of Ω°. We consider a task in which the three parties generate a non-Markov state from scratch by operations in Ω, assisted with quantum communication from the third party to the others, which does not belong to Ω. We prove that the minimum cost of quantum communication required therein is asymptotically equal to the regularized non-Markovianity of formation. Based on this result, we provide a direct operational meaning to a measure of bipartite entanglement called the c-squashed entanglement.