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I have a composite system AB, initially the state of the system is $|\psi\rangle_A \otimes |\phi\rangle_B$. $U$ is an operator acting on the composite system. If even after application of the operator the composite system is in a separable state that is in some state $|\psi ^{'}\rangle_A \otimes |\phi ^{'} \rangle_B$, then is it correct to say that unitary is of the type $U=U_A \otimes U_B$ where $U_A$ and $U_B$ are unitary acting on Hilbert spaces corresponding to system $A$ and $B$ respectively ?

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No. Consider a unitary acting on the tensor product vector space that leaves that special elementary tensor fixed. Then the unitary has a direct sum decomposition $1\oplus U$ where $U$ is a unitary on a vector space of codimension 1 (i.e. the orthogonal complement of the fixed tensor). For simplicity assume that the same finite dimensional vector spaces, of dimension $n$, are tensored together. Then the unitaries of the form $U_1\otimes U_2$ are parametrized by $2n^2$ real parameters; however those of the form $1\oplus U$ acting on the tensor product vector space have $(2n-1)^2\geq 2n^2$ (for $n\geq2$) parameters.

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