Bra-ket entanglement, an indicator bridging entanglement, magic, and coherence

Abstract

Understanding the intricate interplay between distinct quantum resources is a fundamental prerequisite for rigorously characterizing the boundary between classical and quantum technologies. Among the vast landscape of quantum resources, entanglement, magic, and coherence have arguably attracted the most intense investigation. However, while universally recognized as the core drivers of quantum advantage, our understanding of their structural interplay remains fragmented and compartmentalized. In this work, we introduce an indicator called {\em bra-ket entanglement} (BKE) defined in the operator vectorization space to bridge all three quantum resources. Specifically, we show that BKE governs a resource dependence transition in the generation of entanglement: in the low-BKE regime, the growth of entanglement is dominated by coherence, largely independent of magic. However, as BKE increases, the dependence on coherence will gradually be replaced by a dependence on magic. Consequently, in the high-BKE regime, entanglement generation becomes dominated by magic, largely independent of coherence. These results are built on a series of new entropy-theoretic relations and are verified through numerical experiments. We also discuss implications of our results for the resource transitions in classical simulations of mixed states and marginal probabilities and for relating different classical simulation methods.