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Let's say, that we are in the possession of a quantum gate, that is implementing the action of such an operator

$$ \hat{U}|u \rangle = e^{2 \pi i \phi}|u\rangle $$

Moreover, let's say, that this operator has at least two eigenvectors $|u\rangle$ and $|v\rangle$, with the following eigenvalues:

$$ \hat{U}|u \rangle = e^{2 \pi i \phi_0}|u\rangle $$

$$ \hat{U}|v \rangle = e^{2 \pi i \phi_1}|v\rangle $$

If we would like to act with such a quatntum gate on the eigenvector $|u\rangle$, we could write this in the matrix form:

$$ \hat{U}|u\rangle \equiv \begin{bmatrix} e^{2 \pi i \phi_0} & 0 \\ 0 & e^{2 \pi i \phi_0} \\ \end{bmatrix} |u\rangle \equiv e^{2 \pi i \phi_0} |u\rangle $$

What I want to do, is to act with the $\hat{U}$ gate on the superposition of $|u\rangle$ and $|v\rangle$, that is:

$$ \hat{U} [c_0|u\rangle + c_1|v\rangle] = c_0e^{2 \pi i \phi_0}|u\rangle + c_1e^{2 \pi i \phi_1}|v\rangle $$

We could use the following notation to write down eigenvectors $|u\rangle$ and $|v\rangle$ in the matrix form:

$$ |u\rangle = \begin{bmatrix} a_0 \\ a_1 \end{bmatrix}, |v\rangle = \begin{bmatrix} a_2 \\ a_3 \end{bmatrix} $$ Then, we could rewrtie the action $\hat{U} [c_0|u\rangle + c_1|v\rangle]$ as

$$ \hat{U} [c_0|u\rangle + c_1|v\rangle] = c_0e^{2 \pi i \phi_0}|u\rangle + c_1e^{2 \pi i \phi_1}|v\rangle $$

$$ = c_0e^{2 \pi i \phi_0}\begin{bmatrix} a_0 \\ a_1 \end{bmatrix} + c_1e^{2 \pi i \phi_1} \begin{bmatrix} a_2 \\ a_3 \end{bmatrix} = \begin{bmatrix} c_0 e^{2 \pi i \phi_0} a_0 + c_1 e^{2 \pi i \phi_1} a_2 \\ c_0 e^{2 \pi i \phi_0} a_1 + c_1 e^{2 \pi i \phi_1} a_3 \end{bmatrix} $$

Is there any way of writing the $\hat{U}$ gate in the matrix form for the above case? The only thing that comes to my mind, is that it should "at the same time" have values $e^{2 \pi i \phi_0}$ and $e^{2 \pi i \phi_1}$ on its diagonal, but I know that this reasoning is wrong and I was wondering, if there is some official way to write this down.

2There is nothing particularly quantum mechanical about this. To write down U in the standard basis just invert expressions of u and v. This will give you the standard basis vectors in terms of u and v. Once you have this you just have to calculate the four matrix elements. – biryani – 2018-11-17T07:44:25.503