The Deimginarity Cost of Quantum States

Here we address a task denoted as deimaginarity, which is to transform a state into a real state with the aid of random covariant-free unitary operations. We consider the minimum cost of randomness required for deimaginarity in the scenario of infinite copies and suiciently small error, and we prove that the deimaginarity cost of a state $\rho$ is equal to its regularized relative entropy of imaginarity, which can be seen as an operational interpretation of the relative entropy of imaginarity.