The definition of [entanglement entropy](https://en.wikipedia.org/wiki/Quantum_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?

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I am looking for "reasonable" Hilbert spaces to plug into $\mathcal{H}_A, \mathcal{H}_B$ in the above discussion on Entanglement entropy.  Likely it is the [bosonic fock space](https://en.wikipedia.org/wiki/Fock_space).  I would then wonder how to take the "partial trace" over $A$.

An interesting issue was raised as to what a *natural* isomorphism might be for $L^2([0,t]) \otimes L^2([t,1]) \simeq L^2([0,1])$ with $t \in [0,1]$.  In general I am pleased to learn that at the isomorphism $$L^2(A) \otimes L^2(B) \simeq L^2(A \cup B)$.