Magic-induced computational separation in entanglement theory
arxiv(2024)
摘要
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.
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