Tomita-Takesaki theory and quantum concurrence

TL;DR

This paper links the Tomita-Takesaki modular operator framework to quantum concurrence, providing a new way to compute and interpret entanglement directly from the algebraic structure of quantum systems.

Contribution

It introduces a novel connection between the Tomita-Takesaki theory and quantum entanglement measures, specifically relating the modular conjugation operator to concurrence.

Findings

Concurrence can be directly calculated from the modular conjugation operator.

The modular conjugation operator $J$ serves as both a symmetry and a measure of entanglement.

Results are consistent with known Bell-CHSH inequality outcomes for entangled systems.

Abstract

The quantum entanglement measure of concurrence is shown to be directly calculable from a Tomita- Takesaki modular operator framework constructed from the local von Neumann algebras of observables for two quantum systems. Specifically, the Tomita-Takesaki modular conjugation operator $J$ that links two separate systems with respect to their von Neumann algebras is related to the quantum concurrence $C$ of a pure bi-variate entangled state composed from these systems. This concurrence relation provides a direct physical meaning to $J$ as both a symmetry operator and a quantitative measure of entanglement. This procedure is then demonstrated for a supersymmetric quantum mechanical system and a real scalar field interacting with two entangled spin-$\frac{1}{2}$ Unruh-DeWitt qubit detectors. For the latter system, the concurrence result is shown to be consistent with some known results on the Bell-CHSH inequality for such a system.