It is well known that the Minkowski vacuum is expressed as an entangled state between the states constructed on the left and right Rindler space-times. We extend this expression of the Minkowski vacuum with the ordinary left and right Rindler space-times to the entire Minkowski space-time, including the Kasner expanding universe and Kasner shrinking universe. As a result of this research, the structure of the quantum entanglement of Minkowski vacuum state of two-dimensional and four dimensional massless scalar fields is clarified by connecting the states on the Rindler spacetimes and those on the Kasner spacetimes. This result is useful to understand the origin of the quantum radiation associated with the Unruh effect.