Entangling Power: A Probe of Symmetry and Integrability in Quantum Many-Body Systems

TL;DR

This paper investigates how the entangling power of unitary operators in quantum many-body systems reflects symmetry and integrability, revealing extremal behaviors at symmetry points and decomposing scattering matrices into quantum gates.

Contribution

It provides a detailed analysis of entangling power in spin chains and scattering matrices, linking entanglement generation to symmetry and integrability in quantum systems.

Findings

Entangling power decreases with increasing symmetry group size in two-site models.

Finite-size XXZ chains show sharp dips in entangling power at symmetry points.

In the thermodynamic limit, the two-magnon S-matrix reduces to quantum gates with zero entangling power at symmetry points.

Abstract

The entangling power of a unitary operator quantifies its ability to generate entanglement from product states and provides a natural probe of quantum many-body dynamics. Entanglement extremization at points of enhanced symmetry has previously been observed in high-energy scattering. In this work we compute the time-averaged entangling power of anisotropic Heisenberg spin chains across two-site models and finite-size systems, as well as the entangling power of the two-magnon $S$-matrix in the thermodynamic limit. For two-site models we establish a monotonic hierarchy: the entangling power decreases as the symmetry group grows, reaching its minimum at the $SU(2)$ XXX point. Finite-size XXZ chains exhibit sharp dips at the $SU(2)$ points $\Delta=\pm 1$ and the free-fermion point $\Delta=0$, with the free-fermion dip decaying much more slowly with system size. In the thermodynamic limit, we decompose the two-magnon $S$-matrix into quantum logic gates -- Identity, SWAP, and $\sigma_z\otimes\sigma_z$ -- and show that the entangling power vanishes for all scattering energies at the $SU(2)$ points, where the $S$-matrix reduces to the Identity gate, while the free-fermion point achieves the maximum -- the opposite of the finite-size many-body behavior. The entangling power can serve as an {\em operator} diagnostic for symmetry and selected aspects of integrability in quantum simulations of spin-chain dynamics.