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  • Open Access

Gravitational Backreaction is Magical

ChunJun Cao1,*, Gong Cheng1, Alioscia Hamma2,3,4, Lorenzo Leone5, William Munizzi6, and Savatore F.E. Oliviero7

  • *Contact author: cjcao@vt.edu
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PRX Quantum 6, 040375 – Published 31 December, 2025

DOI: https://doi.org/10.1103/z3vr-w5c5

Abstract

We study the interplay between magic and entanglement in quantum many-body systems. We show that nonlocal magic, which is supported by the quantum correlations is lower bounded by the nonflatness of entanglement spectrum and upper bounded by the amount of entanglement in the system. We then argue that a smoothed version of nonlocal magic bounds the hardness of classical simulations for incompressible states. In conformal field theories (CFTs), we conjecture that the nonlocal magic should scale linearly with entanglement entropy but sublinearly when an approximation of the state is allowed. We support the conjectures using both analytical arguments based on unitary distillation and numerical data from an Ising CFT. If the CFT has a holographic dual, then we prove that the nonlocal magic vanishes if and only if there is no gravitational backreaction. Furthermore, we show that nonlocal magic is approximately equal to the rate of change of the minimal surface area in response to the change of cosmic brane tension in the bulk.

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Physics Subject Headings (PhySH)

Popular Summary

The concept of “magic” in physics describes the quantumness, or nonclassical behavior, in interacting many-body systems. Accordingly, magic plays a critical role in understanding the complexity of simulating quantum systems using classical computers. Despite its importance, little is understood about quantum magic, and the precise connection between magic and computational complexity has remained elusive.

In conformal field theories (CFTs), quantum systems with a high level of symmetry, magic plays a significant role in producing the correct entanglement structure, yielding power-law correlations. Furthermore, for holographic CFTs, those with a dual gravitational interpretation, magic enables the construction of area operators that are essential to the emergence of gravitational phenomena in quantum gravity. However, the precise mechanism by which magic generates gravitational phenomena remains poorly understood.

Our research introduces the concept of “nonlocal magic,” which captures the interplay between entanglement and magic. We provide strong evidence linking nonlocal magic to the difficulty of classically simulating incompressible quantum states. Furthermore, we derive rigorous bounds and computable estimates for nonlocal magic measures, demonstrating the dependence of nonlocal magic on a state’s entanglement spectrum, as well as various other entropy measures. Applying our protocol to CFTs, we provide a direct relationship between the quantum properties of magic, entropy, and spectrum antiflatness. For CFTs with a holographic dual we demonstrate how nonlocal magic determines a gravitational response to stress energy and thereby plays a crucial role in describing gravitational emergence in quantum gravity.

Our findings advance the current theoretical understanding of magic, entanglement, and gravity, while also providing practical tools for estimating key quantum resources. The relationship between magic and entanglement motivates further investigation into nonlocal magic as a distinct indicator for quantum behavior. The role of nonlocal magic in quantum phase transitions, symmetry breaking, and nonequilibrium systems is intriguing and certainly warrants further exploration. In the context of quantum gravity, we augment the current gauge or gravity duality dictionary and take a significant first step toward addressing the open question: What is the holographic dual of magic? Finally, studying holographic magic using dynamics, such as quantum chaos and classical complexity, can also offer unique future research perspectives.

Article Text

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PRX Quantum 6, 040375– Published 31 December, 2025
  • Received 17 June 2024
  • Revised 23 July 2025
  • Accepted 10 November 2025
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