Modular Theory for Operator Algebra in Bounded Region of Space-Time and Quantum Entanglement

TL;DR

This paper explores the quantum states in a bounded space-time region, revealing their entangled nature and calculating thermodynamic quantities, thereby linking quantum entanglement with thermodynamics and information theory.

Contribution

It identifies the vacuum state in a diamond-shaped region as an entangled state for a 2D free scalar field, connecting operator algebra, thermodynamics, and quantum information.

Findings

The vacuum state is equivalent to an entangled quantum state.

Thermodynamic quantities like Casimir energy and entropy are computed.

The entropy-to-energy ratio saturates the Bekenstein bound.

Abstract

We consider the quantum state seen by an observer in the diamond-shaped region, which is a globally hyperbolic open submanifold of the Minkowski space-time. It is known from the operator-algebraic argument that the vacuum state of the quantum field transforming covariantly under the conformal group looks like a thermal state on the von Neumann algebra generated by the field operators on the diamond-shaped region of the Minkowski space-time. Here, we find, in the case of the free massless Hermitian scalar field in the 2-dimensional Minkowski space-time, that such a state can in fact be identified with a certain entangled quantum state. By doing this, we obtain the thermodynamic quantities such as the Casimir energy and the von Neumann entropy of the thermal state in the diamond-shaped region, and show that the Bekenstein bound for the entropy-to-energy ratio is saturated. We further speculate on a possible information-theoretic interpretation of the entropy in terms of the probability density functions naturally determined from the Tomita-Takesaki modular flow in the diamond-shaped region.