Markovianizing Cost of Tripartite Quantum States

TL;DR

This paper introduces the concept of Markovianization for tripartite quantum states, analyzing the minimal randomness cost needed to transform states into quantum Markov chains, with implications for quantum resource simulation.

Contribution

It provides the first single-letter formula for Markovianizing costs of tripartite pure states and reveals non-continuous behavior of these costs.

Findings

Markovianizing cost can be arbitrarily large near quantum Markov states.

Derived a single-letter formula for pure states.

Application to resource costs in bipartite unitary gate simulation.

Abstract

We introduce and analyze a task that we call Markovianization, in which a tripartite quantum state is transformed to a quantum Markov chain by a randomizing operation on one of the three subsystems. We consider cases where the initial state is the tensor product of $n$ copies of a tripartite state $\rho^{ABC}$, and is transformed to a quantum Markov chain conditioned by $B^n$ with a small error, using a random unitary operation on $A^n$. In an asymptotic limit of infinite copies and vanishingly small error, we analyze the Markovianizing cost, that is, the minimum cost of randomness per copy required for Markovianization. For tripartite pure states, we derive a single-letter formula for the Markovianizing costs. Counterintuitively, the Markovianizing cost is not a continuous function of states, and can be arbitrarily large even if the state is close to a quantum Markov chain. Our results have an application in analyzing the cost of resources for simulating a bipartite unitary gate by local operations and classical communication.