Magic-induced computational separation in entanglement theory

Entanglement serves as a foundational pillar in quantum information theory, delineating the boundary between what is classical and what is quantum. The common assumption is that higher entanglement corresponds to a greater degree of 'quantumness'. However, this folk belief is challenged by the fact that classically simulable operations, such as Clifford circuits, can create highly entangled states. The simulability of these states raises a question: what are the differences between 'low-magic' entanglement, and 'high-magic' entanglement? We answer this question in this work with a rigorous investigation into the role of magic in entanglement theory. We take an operational approach to understanding this relationship by studying tasks such as entanglement estimation, distillation and dilution. This approach reveals that magic has surprisingly strong implications for entanglement. Specifically, we find a sharp operational separation that splits Hilbert space into two distinct phases: the entanglement-dominated (ED) phase and magic-dominated (MD) phase. Roughly speaking, ED states have entanglement that significantly surpasses their magic, while MD states have magic that dominates their entanglement. The competition between the two resources in these two phases induces a computational phase separation between them: there are sample- and time-efficient quantum algorithms for almost any entanglement task on ED states, while these tasks are provably computationally intractable in the MD phase. To demonstrate the power of our results beyond entanglement theory, we highlight the relevance of our findings in many-body physics and topological error correction. Additionally, we offer simple theoretical explanations for phenomenological observations made in previous numerical studies using ED-MD phases.