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 $\Omega ^{*}$ 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 .