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3

Two antipodal states in the Bloch sphere (note that $0 \leq \theta \leq \pi$): \begin{equation} |\psi \rangle = \cos \frac{\theta}{2} |0 \rangle + e^{i\varphi}\sin \frac{\theta}{2} |1 \rangle \\ |\psi^\perp \rangle = \cos \frac{\pi - \theta}{2} |0 \rangle + e^{i\varphi + \pi}\sin \frac{\pi - \theta}{2} |1 \rangle = \sin \frac{\theta}{2} |0 \rangle - e^{...

1

If that is all you want to do then SciPy is really the way to go. Indeed, qiskit just wraps that functionality when requesting a classical answer. In SciPy you can use scipy.linalg.eig. You can find examples of using this function in the documentation.

2

Let $H_1,H_2$ be two Hilbert spaces and for unit vectors $u_1,v_1 \in H_1$ and $u_2,v_2 \in H_2$ we have $$u_1 \otimes u_2 = v_1 \otimes v_2 \in H_1 \otimes H_2.$$ Then it must be $$v_1 = \lambda u_1, ~~ v_2 = \frac{1}{\lambda} u_2,$$ for some $\lambda \in \mathbb{C}$, $|\lambda|=1$. To see why consider a scalar product $$1 = (u_1\otimes u_2, u_1\... 3 Let's say I know that$$ U_1|\Psi_1\rangle\otimes U_2|\Psi_2\rangle=e^{i\theta}|\Psi_1\rangle\otimes|\Psi_2\rangle. $$Now, let's imagine that U_1|\Psi_1\rangle=|\phi_1\rangle and U_2|\Psi_2\rangle=|\phi_2\rangle. So,$$ |\phi_1\rangle\otimes |\phi_2\rangle=e^{i\theta}|\Psi_1\rangle\otimes|\Psi_2\rangle.  Now, I would just read off your desired ...

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