Redundant parameters in $2\times 2$ unitary operators?

Question

A complex $n \times n$ unitary operator has $n^2$ free real parameters. For example, a $2 \times 2$ unitary matrix can be parametrized as \begin{equation} \begin{pmatrix} e^{i(\alpha - \beta/2 - \delta/2)} \cos \frac{\gamma}{2} & -e^{i(\alpha - \beta/2 + \delta/2)} \sin \frac{\gamma}{2}\\ e^{i(\alpha + \beta/2 - \delta/2)} \sin \frac{\gamma}{2} & e^{i(\alpha + \beta/2 + \delta/2)} \cos \frac{\gamma}{2} \end{pmatrix}. \end{equation}

However, if a initial state is given as \begin{equation} |\psi_i \rangle= \begin{pmatrix} \cos \frac{\theta_i}{2} \\ \sin \frac{\theta_i}{2} e^{i\phi_i} \end{pmatrix}, \end{equation} and the desired output is \begin{equation} |\psi_f \rangle= \begin{pmatrix} \cos \frac{\theta_f}{2} \\ \sin \frac{\theta_f}{2} e^{i\phi_f} \end{pmatrix}. \end{equation} For both input and output, the global phase is not of interest and therefore omitted.

I would like to find a unitary operator $U$ that converts $|\psi_i \rangle$ to $|\psi_f \rangle$. Since the initial state is known, there are only $2$ parameters underdetermined. This indicates that the $U$ such that $|\psi_i \rangle \xrightarrow{U} |\psi_f \rangle$ contains only two independent parameters.

It seems that two parameters in the general $2\times 2$ unitary matrix are redundant.

Answer 1

Yes, if you only specify one input state and one output state, then you will have free parameters.

One way to think about this is to imagine the space of 2x2 unitary matrices as all the possible ways to rotate a 3 dimensional unit vector. If you only have one input and one output, you can specify the direction of rotation in the plane spanned by the two vectors, and still have free parameters in other directions.

Answer 2

Think about your initial and final states on the Bloch sphere being like your initial and final positions on the globe (say you're going from city A to city B). There are many routes between the two cities, so more than one unitary will take you from A to B! This is the extra freedom that you have found.

Now, if you want to rotate the globe in such a way that city A goes to city B and city B goes somewhere specific, you've lost the extra freedom. This can be thought of as because you need to make sure that continents and oceans etc. all maintain their shapes when all you've done is rotated the globe.

Nailing down these degrees of freedom can be thought of in analogy with determining your position from a GPS satellite. If all you know is your distance from a particular satellite, your position will have many unconstained degrees of freedom. Adding more satellites necessitates that your position be at the intersection of a bunch of spheres, which quickly becomes unique. This is equally true for unitary rotations: the more points whose rotated coordinates you specify, the more you constrain the rotation. You can think about this in higher dimensions too, in that you'll need to specify the unitarily evolved coordinates of more initial points to uniquely determine the unitary.