1
$\begingroup$

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 ?

$\endgroup$

1 Answer 1

2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.