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Operator Entanglement from Non-Commutative Symmetries

Michele Arzano, Goffredo Chirco

TL;DR

This work shows that Hopf-algebra deformations of symmetry, exemplified by $U_q(\mathfrak{su}(2))$, enforce a nontrivial operator entanglement at the level of composite generators through the non-cocommutative coproduct. In the spin-1/2 two-qubit realization, the single-qubit Hamiltonian is indistinguishable from the undeformed case, but the coproduct induces a nonlocal two-qubit Hamiltonian whose unitary evolution exhibits oscillatory operator entanglement $E(U(t))$ for $q\neq1$, with a closed-form expression. Crucially, the entangling power under Haar-uniform product inputs is slaved to $E(U(t))$, i.e., $e_{p_0}(U)=\frac{4}{9}E(U)$, making the dynamical entanglement content fully determined by the coproduct structure. The dynamics is integrable with operator-space closure, showing revivals rather than chaotic growth, and this framework provides a minimal, exact setting to study how noncommutative symmetries influence information dynamics and potential scrambling in quantum-spacetime contexts. The results motivate extending to higher representations and many-body systems to explore richer operator growth and its interplay with quantum-group symmetries.

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.

Operator Entanglement from Non-Commutative Symmetries

TL;DR

This work shows that Hopf-algebra deformations of symmetry, exemplified by , enforce a nontrivial operator entanglement at the level of composite generators through the non-cocommutative coproduct. In the spin-1/2 two-qubit realization, the single-qubit Hamiltonian is indistinguishable from the undeformed case, but the coproduct induces a nonlocal two-qubit Hamiltonian whose unitary evolution exhibits oscillatory operator entanglement for , with a closed-form expression. Crucially, the entangling power under Haar-uniform product inputs is slaved to , i.e., , making the dynamical entanglement content fully determined by the coproduct structure. The dynamics is integrable with operator-space closure, showing revivals rather than chaotic growth, and this framework provides a minimal, exact setting to study how noncommutative symmetries influence information dynamics and potential scrambling in quantum-spacetime contexts. The results motivate extending to higher representations and many-body systems to explore richer operator growth and its interplay with quantum-group symmetries.

Abstract

We argue that Hopf-algebra deformations of symmetries -- as encountered in non-commutative models of quantum spacetime -- carry an intrinsic content of that is enforced by the coproduct-defined notion of composite generators. As a minimal and exactly solvable example, we analyze the quantum group and a two-qubit realization obtained from the coproduct of a -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.
Paper Structure (14 sections, 45 equations, 2 figures)

This paper contains 14 sections, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Operator entanglement
  3. $q$-deformed $\mathfrak{su}(2)$ and Hopf structure
  4. Single-spin Hamiltonian and the "accidental" qubit
  5. Two-qubit Hamiltonian from the coproduct
  6. Operator entanglement
  7. Entangling power
  8. Coproduct-induced, irreducible entanglement
  9. Conclusions and outlook
  10. Single-spin representation and $q$-exponentials
  11. Coproduct-derived two-qubit Hamiltonian
  12. Exponential map $U(t)=e^{-itH_{AB}(q)}$
  13. Operator entanglement from the coproduct unitary
  14. The mixed invariant $\widetilde{E}(U)$ and entangling power

Figures (2)

  • Figure 1: Operator entanglement $E(U(t))$ for several values of $q$. At $q=1$, $E(U(t))=0$. For $q\neq1$, $E(U(t))$ oscillates with frequency $\alpha=q+q^{-1}$.
  • Figure 2: Numerical maximization of $E(U(t))$ over $t$. The asymptotic limit $E_{\max}\to \tfrac{1}{2}$ as $q\to\infty$ is shown; the maximum is effectively saturated already for $q\gtrsim2$.