# Calculate the product state/quantum register back into its tensor product

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?

## 1 Answer

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: $$a_1=\frac{1}{\sqrt{2}}$$ $$a_2=\frac{1}{\sqrt{2}}$$ $$b_1=\frac12+\frac{\mathrm{i}}{4}$$ $$b_2=\frac12+\mathrm{i}\frac{\sqrt{7}}{4}$$

• i dont get the part with \theta_1 + \phi_1 .... Like were do the 1/2 or sqrt(7)/2 come from? May 2 at 19:27
• @ChristianBernhard If a complex number $z=a+\mathrm{i}b$ has a real part that is positive, that is if $a\geqslant0$, then its argument is given by $\arctan\left(\frac{b}{a}\right)$. The $\frac12$ comes from $\frac12=\frac{\frac14}{\frac12}$, since we want to compute the argument of $\frac12+\mathrm{i}\frac14$, while the other one comes from $\frac{\sqrt{7}}{2}=\frac{\sqrt{\frac{7}{16}}}{\frac12}$, since we want to compute the argument of $\frac12+\mathrm{i}\sqrt{\frac{7}{16}}$. May 3 at 9:01