The Pareto Frontiers of Magic and Entanglement: The Case of Two Qubits

Abstract

Magic and entanglement are two measures that are widely used to characterize quantum resources. We study the interplay between magic and entanglement in two-qubit systems, focusing on the two extremes: maximal magic and minimal magic for a given level of entanglement. We quantify magic by the R\'{e}nyi entropy of order 2, $M_2$, and entanglement by the concurrence $\Delta$. We find that the Pareto frontier of maximal magic $M_2^{(max)}(\Delta)$ is composed of three separate segments, while the boundary of minimal magic $M_2^{(min)}(\Delta)$ is a single continuous line. We derive simple analytical formulas for all these four cases, and explicitly parametrize all distinct quantum states of maximal or minimal magic at a given level of entanglement.