The Standard Model Symmetry and Qubit Entanglement

TL;DR

This work proposes a quantum-first paradigm in which spacetime geometry emerges from the area-law component of entanglement, while gauge and matter fields arise from area-law-violating, encoded entanglement. By employing Hopf fibrations over complex, quaternionic, and octonionic state spaces, the authors map few-qubit systems to higher-dimensional spacetimes and extract residual symmetry groups that coincide with the Standard Model gauge group after dimensional reduction. The three-qubit (octonionic) construction yields the exact SM gauge content for a single fermion generation, with a proposed path to include masses, chirality, and potentially all generations via four-qubit extensions. The framework also suggests practical quantum-simulation avenues for gauge theories, linking quantum information, division algebras, and high-energy physics in a unified picture. However, foundational issues such as tensor-product decomposition, Higgs mechanisms, and empirical validation remain open.

Abstract

Research at the intersection of quantum gravity and quantum information theory has seen significant success in describing the emergence of spacetime and gravity from quantum states whose entanglement entropy approximately obeys an area law. In a different direction, the Kaluza-Klein proposal aims to recover gauge symmetries by means of dimensional reduction of higher-dimensional gravitational theories. Integrating both, gravitational and gauge degrees of freedom in $3+1$ dimensions may be obtained upon dimensional reduction of higher-dimensional emergent gravity. To this end, we show that entangled systems of two and three qubits can be associated with $5+1$ and $9+1$ dimensional spacetimes respectively, which are reduced to $3+1$ dimensions upon singling out a preferred complex direction. In the latter case, this reduction is invariant under a residual $SU(3) \times SU(2) \times U(1) /\mathbb{Z}_6$ symmetry, the Standard Model gauge group. This motivates a picture in which spacetime emerges from the area law-contribution to the entanglement entropy, while gauge and matter degrees of freedom are due to area law-violating terms. We remark on a possible natural origin of the chirality of the weak force in the given construction. Furthermore, we highlight the possibility of using this construction in quantum simulations of Standard Model fields.

Figures (4)

• Figure 1: The entanglement entropy of a region $R$ across its boundary $\partial R$ is computed by summing up the contributions of entanglement 'links' cut by the boundary. For a state with 'short-range' entanglement, this yields an area law.
• Figure 2: A state with additional long-range entanglement, or entanglement due to a state 'encoded' into it in the sense of quantum error correction, receives subleading corrections to the entanglement entropy from these additional terms.
• Figure 3: The iterated Hopf fibration, starting out with $S^{15}$ containing the 3-qubit state as an $S^{7}$ fibration over $S^{8}$, with the $S^7$ fiber itself being an $S^3$-fiber over $S^4$, and similarly for the $S^3$-fiber again (not shown explicitly). The colors highlight the coordinates permuted by the various symmetry groups.
• Figure 4: The 3-qubit state with the association between qubit- and entanglement degrees of freedom and the respective action of the symmetry groups highlighted.