the current literature is mostly on the entanglement based on squeeze along the normal coordinates, the shear transformation is an interesting future possibility. The mathematical issues on this problem are clarified.'
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[**Entangled Harmonic Oscillators and Space-time Entanglement**]{}

Sibel Ba[ş]{}kal [^1]\
Department of Physics, Middle East Technical University, 06531 Ankara, Turkey

Young S. Kim[^2]\
Center for Fundamental Physics, University of Maryland,\
College Park, Maryland 20742, U.S.A.

Marilyn E. Noz[^3]\
Department of Radiology,\
New York University New York, New York 10016, U.S.A.

Introduction
============

Entanglement problems deal with fundamental issues in physics. Among them, the Gaussian entanglement is of current interest not only in quantum optics [@gied03; @braun05; @kn05job; @ge15] but also in other dynamical systems [@kn05job; @ging02; @dodd04; @ferra05; @adesso07]. The underlying mathematical language for this form of entanglement is that of harmonic oscillators. In this paper, we present first the mathematical tools which are and may be useful in this branch of physics.

The entangled Gaussian state is based on the formula $$\label{seri00}
\frac{1}{\cosh\eta}\sum_{k} (\tanh{\eta})^k \chi_{k}(x) \chi_{k}(y) ,$$ where $\chi_{n}(x)$ is the $n^{th}$ excited-state oscillator wave function.

In Chapter 16 of their book [@walls08], Walls and Milburn discussed in detail the role of this formula in the theory of 