As it is stated in this post: Transforming the first Bell state into the other Bell states, one can transform the $|\phi^{+}\rangle$ state to the $|\psi ^{-}\rangle$ by applying the tensor product of the X and Z Pauli gates, $X \otimes Z$.

Could you explain how this transformation is derived? More in general, given two arbitrary (entangled) states, how can I derive the transformation between them?

Furthermore, if I have two non-entangled many-qubit states, and I am trying to find the transformation between them, is it true that I can just find the transformations qubit by qubit and then combined these by taking their tensor product to get the overall transformation?


1 Answer 1


Given two arbitrary bipartite pure states (of the same dimensions), you can find local unitary operations sending one in the other iff they have the same Schmidt coefficients. In particular, all maximally entangled states are locally equivalent.

More precisely, let $|\psi\rangle$ and $|\phi\rangle$ be states with Schmidt decompositions $$|\psi\rangle = \sum_k \sqrt{p_k} (|u_k\rangle\otimes|v_k\rangle), \\ |\phi\rangle = \sum_k \sqrt{p_k} (|u_k'\rangle\otimes|v_k'\rangle).$$ Observe that the operator $U$ such that $U|u_k\rangle=|u_k'\rangle$ for all $k$ is unitary, and so is $V$ such that $V|v_k\rangle=|v_k'\rangle$. It follows that $|\psi\rangle=(U\otimes V)|\phi\rangle$.

In other words, given pure states with the same Schmidt coefficients, the local unitaries sending one in the other can be built as $$ U = \sum_k |u_k'\rangle\!\langle u_k|, \qquad V = \sum_k |v_k'\rangle\!\langle v_k|.$$

I'm not quite sure about the more general case of multipartite states. Schmidt coefficients only characterise bipartite entanglement. You can therefore apply the above argument to any bipartition in a multipartite state. You can certainly say that if two states are such that their Schmidt coefficients are the same with respect to any bipartition, then they are locally equivalent (and that this is necessary and sufficient). I don't know if there's a simpler way to state this condition (as in, a way to not have to check every single subset of spaces).


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