Geometric additivity of modular commutator for multipartite entanglement

TL;DR

This paper reveals that the modular commutator in two-dimensional gapped quantum many-body systems exhibits geometric additivity, which has implications for understanding multipartite entanglement and topological invariants.

Contribution

It introduces the geometric additivity property of the modular commutator and derives related identities, advancing the understanding of many-body entanglement in quantum systems.

Findings

Modular commutator is geometrically additive in multipartite systems.

Derived a new identity for modular commutators in certain conformal field theories.

Numerical validation using Haldane and π-flux models.

Abstract

A recent surge of research in many-body quantum entanglement has uncovered intriguing properties of quantum many-body systems. A prime example is the modular commutator, which can extract a topological invariant from a single wave function. Here, we unveil novel geometric properties of many-body entanglement via a modular commutator of two-dimensional gapped quantum many-body systems. We obtain the geometric additivity of a modular commutator, indicating that modular commutator for a multipartite system may be an integer multiple of the one for tripartite systems. Using our additivity formula, we also derive a curious identity for the modular commutators involving disconnected intervals in a certain class of conformal field theories. We further illustrate this geometric additivity for both bulk and edge subsystems using numerical calculations of the Haldane and $\pi$-flux models.