To quantify non-Markovianity of tripartite quantum states from an operational viewpoint, we introduce a class $Omega^*$ of operations performed by three distant parties. A tripartite quantum state is a free state under $Omega^*$ 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 $Omega$ of $Omega^*$. We consider a task in which the three parties generate a non-Markov state from scratch by operations in $Omega$, assisted with quantum communication from the third party to the others, which does not belong to $Omega$. 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.
Download