Unified quantification of entanglement and magic in information scrambling and their trade-off relation

Abstract

Entanglement and magic are among the most fundamental properties unique to quantum systems. Each quantity captures a different aspect of non-classical behavior, and each can be regarded as a resource within its own operational setting. However, the interrelation between them has not yet been fully clarified, and whether a more fundamental measure exists remains an open question. Addressing these issues is essential for deepening our understanding of quantumness. In this study, we establish a unified resource theory of information scrambling, consisting of two types: entanglement scrambling and magic scrambling. We introduce a measure that jointly characterizes both types of scrambling. This unified approach reveals a rigorous trade-off relation between entanglement and magic scrambling, as the exact maximum value of the proposed measure can be derived analytically. Furthermore, we quantify the scrambling capability of unitary transformations in terms of their ability to amplify this measure. Our work provides insights into the connection between entanglement and magic that extend beyond the context of information scrambling.

Figures (2)

• Figure 1: Schematic diagrams illustrating two types of information scrambling. (a) Entanglement scrambling corresponds to transformations that increase the number of non-identity operators in Pauli strings. (b) Magic scrambling corresponds to transformations that increase the number of terms required in the Pauli basis expansion.
• Figure 2: Overview of the $W$-$S$ plane for arbitrary linear operators. A key observation is the trade-off relation between the two types of scrambling, illustrated by the solid red curve. When $W(O)$ reaches its maximum, $S_a(O)$ cannot, and vice versa: when $S_a(O)$ is maximized, $W(O)$ takes a specific, non-maximal value. The dashed curve provides a rough sketch of the general trend (possibly imprecise).