Gauge invariance from quantum information principles

TL;DR

The paper investigates whether gauge invariance and diffeomorphism invariance can be derived from quantum-information principles by treating gluon-gluon and graviton-graviton tree-level scattering as two-qubit pure states in the polarization basis. It quantifies entanglement with the concurrence $\Delta$ and non-Clifford resources with the Stabilizer Renyi Entropy $M_2$, exploring how these resources arise when the 4-vertex is modified by a parameter $k$ to break gauge invariance. The key finding is that maximal entanglement alone does not single out the gauge-invariant interaction, but combining MaxEnt with minimal nonzero magic selects the physical solution $k=1$ in both gauge and gravitational cases, implying a dual informational constraint may underlie gauge invariance. This suggests a fundamental principle where nature favors strong quantum correlations while maintaining limited non-Clifford complexity, with potential implications for understanding the emergence of symmetries in fundamental interactions. The work also highlights the limitations of current magic measures for mixed or multipartite scenarios and calls for extending this informational perspective beyond two-particle tree-level processes.

Abstract

Entanglement is a hallmark of quantum theory, yet it alone does not capture the full extent of quantum complexity: some highly entangled states can still be classically simulated. Non-classical behavior also requires magic, the non-Clifford component that enables universal quantum computation. Here, we investigate whether the interplay between entanglement and magic constrains the structure of fundamental interactions. We study gluon-gluon and graviton-graviton scattering at tree level, explicitly breaking gauge and general covariance by modifying the quartic vertices and analyzing the resulting generation of entanglement and magic. We find that imposing maximal entanglement (MaxEnt) alone does not uniquely recover gauge-invariant and diffeomorphism-invariant interactions, but adding the condition of minimal, but nonzero, magic singles it out. Our results indicate that nature favors MaxEnt and low magic: maximal quantum correlations with limited non-Cliffordness, sufficient for universal quantum computing but close to classical simulability. This dual informational principle may underlie the emergence of gauge invariance in fundamental physics.

Figures (3)

• Figure 1: Concurrence as a function of the 4 vertex parameter $k$ for initial polarizations $RL$ and $RR$ and COM angle $\theta_{COM}=\pi/2$. The gauge-invariant solutions correspond to $k=1$, where MaxEnt is achieved for initial $RL$ polarization while $RR$ generate a product state. Left: for QCD, MaxEnt is also obtained for two other unphysical solutions, $k=-3$ and $k=11/3$, for an initial polarization of $|RR\rangle$. Right: in gravity, another non-physical solution is obtained, $k=3$, for an initial polarization of $|RR\rangle$.
• Figure 2: Magic measured with $M_{2}$ as a function of $\theta_{COM}$ for different values of the 4-vertex parameter $k$ and initial $RL$ polarization. Left, for gluons, and right, for gravitons. For initial $RL$ polarization, Magic has a local minima at $\theta_{COM}=\pi/2$. Then, it shows two symmetric maxima. The value of that maximum depends on $k$ and it is minimal at $k=1$, as shown Fig.\ref{['fig:magick']}. For initial $RR$ polarization, other values of $k$ show some magic, while $k=1$ is a product state with no magic. For $k\neq1$, the gluons result depends on the color, in particular to the relations between the structure constants $F_1\equiv f^{aa'c}f^{bb'c}$, $F_2\equiv f^{ab'c}f^{ba'c}$ and $F_{3}=f^{abc}f^{a'b'c}$: there are six possible combinations between these $F_{1}$, $F_{2}$ and $F_{3}$ specified in the plots with a shade between the maximum and minimum values, while the solid line corresponds to the median. For comparison, the maximal amount of $M_{2}$ (MaxMagic) for a pure two-qubit state is $\log(16/7)$.
• Figure 3: Maximum $M_{2}$ achieved as a function of the 4-vertex parameter $k$. The global minimum is obtained for $k=1$, i.e. the gauge-invariant solution, and corresponds to $M_{2}=\log(4/3)$ (MinMagic) for initial $RL$ polarization, and zero por initial $RR$ polarization.