Operator Entanglement from Non-Commutative Symmetries

TL;DR

This paper demonstrates that non-commutative symmetries, modeled by Hopf algebras like quantum groups, inherently generate operator entanglement in multi-qubit systems, affecting quantum information dynamics.

Contribution

It introduces a solvable model linking non-commutative symmetry deformations to intrinsic operator entanglement, revealing their role in quantum information processing.

Findings

Operator entanglement is analytically computed for deformed quantum groups.

Non-commutative coproducts produce nonlocal unitaries with entangling power.

Non-commutative symmetries set a baseline for entanglement in quantum systems.

Abstract

We argue that Hopf-algebra deformations of symmetries -- as encountered in non-commutative models of quantum spacetime -- carry an intrinsic content of $operator$ $entanglement$ that is enforced by the coproduct-defined notion of composite generators. As a minimal and exactly solvable example, we analyze the $U_q(\mathfrak{su}(2))$ quantum group and a two-qubit realization obtained from the coproduct of a $q$-deformed single-spin Hamiltonian. Although the deformation is invisible on a single qubit, it resurfaces in the two-qubit sector through the non-cocommutative coproduct, yielding a family of intrinsically nonlocal unitaries. We compute their operator entanglement in closed form and show that, for Haar-uniform product inputs, their entangling power is fully determined by the latter. This provides a concrete mechanism by which non-commutative symmetries enforce a baseline of entanglement at the algebraic level, with implications for information dynamics in quantum-spacetime settings and quantum information processing.