  We investigate whether pure entangled states can be associated to a
measurement basis in which all vectors are local unitary transformations of the
original state. We prove that for bipartite states with a local dimension that
is either $2, 4$ or $8$, every state corresponds to a basis. Via numerics we
strongly evidence the same conclusion also for two qutrits and three qubits.
However, for some states of four qubits we are unable to find a basis, leading
us to conjecture that not all quantum states admit a corresponding measurement.
Furthermore, we investigate whether there can exist a set of local unitaries
that transform \textit{any} state into a basis. While we show that such a
state-independent construction cannot exist for general quantum states, we
prove that it does exist for real-valued $n$-qubit states if and only if
$n=2,3$, and that such constructions are impossible for any multipartite system
of an odd local dimension. Our results suggest a rich relationship between
entangled states and iso-entangled measurements with a strong dependence on
both particle numbers and dimension.
