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

TL;DR

This paper explores the relationship between magic and entanglement in two-qubit systems, deriving analytical formulas for their Pareto frontiers at extremal levels of magic for given entanglement.

Contribution

It provides the first analytical characterization of the Pareto frontiers of magic and entanglement in two-qubit states, including explicit parametrizations.

Findings

The Pareto frontier of maximal magic has three segments.

The boundary of minimal magic is a single continuous line.

Explicit formulas for extremal magic states at given entanglement levels.

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.