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arXiv · 1211.3525

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

Daisuke Ida·Takahiro Okamoto·Miyuki Saito
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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.

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Daisuke Ida, Takahiro Okamoto, Miyuki Saito. 2012-11-15. Modular Theory for Operator Algebra in Bounded Region of Space-Time and Quantum Entanglement. https://doi.org/10.1093/ptep%2Fptt061

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