Entanglement harvesting and curvature of entanglement: A modular operator approach

TL;DR

The paper develops an operator-algebraic framework based on Tomita–Takesaki modular theory to study quantum entanglement, particularly entanglement harvesting and the curvature of entanglement. By introducing modular reflection $\rho_J=J\rho J$ and entanglement functionals $f(\rho, J\rho J)$, it defines a modular curvature $\mathcal{K}_E(g)$ as the second derivative with respect to a coupling $g$, which reduces to the quantum Fisher information $F(g)$ at points of modular self-duality $\rho=J\rho J$. The formalism is applied to Unruh–DeWitt detectors in a scalar field and to a dissipative two-qubit open system, where the coherence term $2|X|$ from the modular overlap reproduces the entanglement harvesting signal and the curvature–information link $\mathcal{K}_E(g)=-F(g)$ at symmetry points. The work also extends concurrence to mixed states via the modular framework, connects curvature to relative entropy via Araki’s modular operators, and outlines future directions, including extensions to type III algebras and potential applications in quantum metrology and holography.

Abstract

An operator-algebraic framework based on Tomita-Takesaki modular theory is used to study aspects of quantum entanglement via the application of the modular conjugation operator $J$. The entanglement structure of quantum fields is studied through the protocol of entanglement harvesting whereby quantum correlations evolve through the time evolution of qubit detectors coupled to a Bosonic field. Modular conjugation operators are constructed for Unruh-Dewitt type qubits interacting with a scalar field such that initially unentangled qubits become entangled. The entanglement harvested in this process is directly quantified by an expectation value involving $J$ offering a physical application of this operator. The modular operator formalism is then extended to the Markovian open system dynamics of coupled qubits by expressing entanglement monotones as functionals of a state $ρ$ and its modular reflection $JρJ$. The second derivative of such functionals with respect to an external coupling parameter, termed the curvature of entanglement, provides a natural measure of entanglement sensitivity. At points of modular self-duality, the curvature of entanglement coincides with the quantum Fisher information measure. These results demonstrate that the modular conjugation operator $J$ captures both the harvesting of entanglement from quantum fields and the curvature of entanglement in coupled qubit dynamics providing parallel modular structures that connect these systems.