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So let's asume I have a product state/quantum register as a result of a tensor product of two qubits.
Lets take a "hard" product state matrix like: $$\frac{1}{\sqrt{2}} \begin{bmatrix} \frac12 + \frac{i}{4} \\\frac12 + \sqrt{\frac{7}{16}}i \\ \frac12 + \frac{i}{4} \\ \frac12 + \sqrt{\frac{7}{16}}i \end{bmatrix}$$
How would I decompose it back to the tensor product of two qubits?
You know that: $$\begin{pmatrix}a_1\\a_2\end{pmatrix}\otimes\begin{pmatrix}b_1\\b_2\end{pmatrix}=\begin{pmatrix}a_1b_1\\a_1b_2\\a_2b_1\\a_2b_2\end{pmatrix}=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac12+\frac{\mathrm{i}}{4}\\\frac12+\mathrm{i}\sqrt{\frac{7}{16}}\\\frac12+\frac{\mathrm{i}}{4}\\\frac12+\mathrm{i}\sqrt{\frac{7}{16}}\end{pmatrix}$$ Assuming you don't see any obvious structure (which isn't the case here, so we could speed up the computation a bit in this case), you can proceed using the fact that: $$\left|a_1b_1\right|^2+\left|a_1b_2\right|^2=\left|a_1\right|^2$$ Using similar equations, you can find the squared modulus of each coefficient. You then want to find their argument. Let us the arguments of the $a_i$ as $\theta_i$ and those of $b_i$ as $\varphi_1$. The argument of $a_ib_j$ is then $\theta_i+\varphi_j\pmod{2\pi}$. You can thus deduce each phase from the equations.
Applying this to this example, we find: $$\left|a_1\right|^2=\frac12\left|\frac12+\frac{\mathrm{i}}{4}\right|^2+\frac12\left|\frac12+\mathrm{i}\sqrt{\frac{7}{16}}\right|^2=\frac{5}{32}+\frac{11}{32}=\frac12$$ Computing $\left|a_2\right|^2$ leads to the exact same computation. We also have: $$\left|b_1\right|^2=\frac12\left|\frac12+\frac{\mathrm{i}}{4}\right|^2+\frac12\left|\frac12+\frac{\mathrm{i}}{4}\right|^2=\frac{5}{16}$$ We could perform a similar computation for $\left|b_2\right|^2$, or simply use the fact that $\left|b_1\right|^2+\left|b_2\right|^2=1$ to find $\left|b_2\right|^2=\frac{11}{16}$. We now want to compute the associated phases. We have: $$\begin{align} \theta_1+\varphi_1&=\arctan\left(\frac12\right)\\ \theta_1+\varphi_2&=\arctan\left(\frac{\sqrt{7}}{2}\right)\\ \theta_2+\varphi_1&=\arctan\left(\frac12\right)\\ \theta_2+\varphi_2&=\arctan\left(\frac{\sqrt{7}}{2}\right) \end{align}$$ We thus find $\theta_1=\theta_2$, which simplifies the equations to: $$\begin{align} \theta_1+\varphi_1&=\arctan\left(\frac12\right)\\ \theta_1+\varphi_2&=\arctan\left(\frac{\sqrt{7}}{2}\right)\end{align}$$ Note that we won't have enough equations to fully determine all the variables. This is expected: note that if you apply a global phase of $\theta_1$ on the first qubit and a global phase of $-\theta_1$ on the second one, you will end up with the same vector. We can thus consider $\theta_1$ as a variable and express $\varphi_1$ and $\varphi_2$ as functions of $\theta_1$, which finally fives us all of our solutions: $$a_1=\frac{1}{\sqrt{2}}\mathrm{e}^{\mathrm{i}\theta_1}$$ $$a_2=\frac{1}{\sqrt{2}}\mathrm{e}^{\mathrm{i}\theta_1}$$ $$b_1=\frac{\sqrt{5}}{4}\mathrm{e}^{\mathrm{i}\left(\arctan\left(\frac12\right)-\theta_1\right)}$$ $$b_2=\frac{\sqrt{11}}{4}\mathrm{e}^{\mathrm{i}\left(\arctan\left(\frac{\sqrt{7}}{2}\right)-\theta_1\right)}$$ In particular, for $\theta_1=0$, we have: $$a_1=\frac{1}{\sqrt{2}}$$ $$a_2=\frac{1}{\sqrt{2}}$$ $$b_1=\frac12+\frac{\mathrm{i}}{4}$$ $$b_2=\frac12+\mathrm{i}\frac{\sqrt{7}}{4}$$
