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

Abstract

To quantify non-Markovianity of tripartite quantum states from an operational viewpoint, we introduce a class ${\Omega }^{\ast }$ of operations performed by three distant parties. A tripartite quantum state is a free state under ${\Omega }^{\ast }$ 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 ${\Omega }^{\ast }$. 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.

19 pages, 6 figures

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• Resource theories of multi-time processes: A window into quantum non-Markovianity
• Advanced Alicki-Fannes-Winter method for energy-constrained quantum systems and its use
• Continuity of characteristics of composite quantum systems
• Asymptotically Consistent Measures of General Quantum Resources: Discord, Non-Markovianity, and Non-Gaussianity
• Uniform continuity bounds for characteristics of multipartite quantum systems
• Squashed quantum non-Markovianity: a measure of genuine quantum non-Markovianity in states