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7

As Davit Khachatryan's answer points out, the task is impossible / ill-defined, since the desired target state is generally not normalized and it depends on the relative global phases of the two initial states. However, it is possible to rephrase the question so that is meaningful and has an interesting answer. The two problems -- sensitivity to the global ...

2

Here's one way to do it. Let's start with some assumptions: here, I'm assuming your circuits $C_1$ and $C_2$ use the same number of qubits. In the drawing, I have used four qubits to illustrate the concept, but that doesn't matter. The answer below does not care about the number of qubits (which I call $n$), just that the two circuits have the same number of ...

5

A counterexample that shows that this is not possible in the general case (here I am neglecting post-selection possibility discussed in the comments of the question and in the accepted answer): $$C_1 = X \qquad C_2 = -X$$ Or one can take $C_2 = R_y(- \pi)$ and all mentioned below equations will reamin true. So: C_1 |0\rangle = |1\rangle = |\psi \rangle \...

5

To talk about entanglement, you have to first identify subsystems. In your $d=4$ example, you defined an isomorphism $\mathbb{C}^4\simeq \mathbb{C}^2\otimes\mathbb{C}^2$ via the identification of basis states. Whether this is meaningful, depends on the context/the physical scenario you have in mind. But it definitely can be. For $d=3$, this is never possible....

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