I believe that any unit length vector $|\Psi\rangle$, whose components are complex numbers, can be transformed into any other vector $|\Phi\rangle$, of the same dimension, same length, and also composed of complex numbers, by a unitary operator, $\hat{U}$, such that $\hat{U}|\Psi\rangle=|\Phi\rangle$ and $\hat{U^{\dagger}}|\Phi\rangle=|\Psi\rangle$. Is that true and, if so, where can I find a proof of that?
I can't seem to prove it or find such an operator even for simple cases such as $(a_1^2 + a_2^2 + a_3^2 + a_4^2)^{-1/2}\hat{U}\begin{bmatrix} a_1 \\ a_2 \\ a_3 \\ a_4 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 1 \end{bmatrix}2^{-1/2}$, where each $a_i$ is a complex number and $(a_1^2 + a_2^2 + a_3^2 + a_4^2)^{-1/2}\begin{bmatrix} a_1 \\ a_2 \\ a_2 \\ a_4 \end{bmatrix} = \begin{bmatrix} \alpha_1 \\ \alpha_2 \end{bmatrix} \otimes \begin{bmatrix} \beta_1 \\ \beta_2 \end{bmatrix}$, the tensor product of two other unit length vectors (if the tensor product aspect makes it easier...).
In Quantum Mechanics, the interpretation of the example, above, is that I am looking to prove that any pure state $|a\rangle$ (which we can assume is a tensor product of two systems and so is not entangled) can be transformed into any pure entangled state (for instance the vector in the example that contains two $0$s and two $1s$), by a unitary transformation.
Thanks.
It suffices to construct a unitary matrix whose first row is an arbitrary unit vector $v$. Take any non-singular matrix with first row $v$ and apply the Gram-Schmidt process to its rows. The vectors one gets form the rows of a unitary matrix with top row $v$.
In more detail. Such a matrix $M$ has $e_1M=v$ where $e_1= \pmatrix{1&0&\cdots&0}$. If you have two unit vectors, $v$ and $w$ there are unitary matrices $M$ and $N$ with $e_1M=v$ and $e_2N=w$. Then $M^{-1}N$ is unitary and $v(M^{-1}N)=w$.