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

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.

• – glS
Mar 11, 2022 at 23:25