Magic-induced computational separation in entanglement theory

TL;DR

This paper investigates how 'magic' and entanglement interact in quantum systems, revealing a phase separation that impacts computational complexity and has broad implications across quantum physics.

Contribution

It introduces an operational framework distinguishing entanglement-dominated and magic-dominated phases, elucidating their roles in quantum computational tasks.

Findings

Identifies a phase separation between ED and MD regimes in Hilbert space.

Shows efficient quantum algorithms exist in ED states but not in MD states.

Provides explanations for previous numerical observations in quantum physics.

Abstract

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 notable implications for entanglement. Specifically, we find an operational separation that divides Hilbert space into two distinct regimes: 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. Our results find applications in diverse areas such as quantum error correction, many-body physics, and the study of quantum chaos, providing a unifying framework for understanding the behavior of quantum systems. We also offer theoretical explanations for previous numerical observations, highlighting the broad implications of the ED-MD distinction across various subfields of physics.