Modular theory for operator algebra in a bounded region of space-time and quantum entanglement

• Published in: Progress of theoretical and experimental physics Vol. 2013; no. 8
• Main Authors: Ida, Daisuke, Okamoto, Takahiro, Saito, Miyuki
• Format: Journal Article
• Language: English
• Published: Oxford University Press 01.08.2013
• Subjects: E01; A60; B30
• ISSN: 2050-3911; 2050-3911
• DOI: 10.1093/ptep/ptt061

We consider the quantum state seen by an observer in a diamond-shaped region, which is a globally hyperbolic open submanifold of 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 Minkowski space-time. Here, we find, in the case of the free massless Hermitian scalar field in 2D Minkowski space-time, that such a state can in fact be identified with a certain entangled quantum state. By doing this, we obtain 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.