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

## TL;DR

This paper introduces a resource-theoretic framework to quantify non-Markovianity in tripartite quantum states, linking it to quantum communication costs and providing operational meaning to c-squashed entanglement.

## Contribution

It defines a new measure called non-Markovianity of formation and establishes its operational significance in quantum communication tasks.

## Key findings

- Minimum quantum communication cost equals the regularized non-Markovianity of formation.
- Non-Markovianity of formation is a faithful, continuous, and monotonic measure.
- Provides operational interpretation for c-squashed entanglement.

## Abstract

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.