The definition of entanglement entropy in Quantum Field Theory involves decompositing a Hilbert space into a tensor product $\mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B$.

As an example, is it possible to decompose the $L^2$ hilbert space $L^2([0,1], dx)$ into the tensor product of two Hilbert spaces just by splitting the interval?

Is there "mutual information"? To what extent is it possible to construct the value of an $L^2$ function on $[\tfrac{1}{2},1]$ using information from $[0,\tfrac{1}{2}]$?

Or is there no mutual information, and it is possible to have an $L^2$ function on $[0,1]$ gluing together such functions on the left and right halves?