# Entanglement of the $i$-th qubit with the $j$-th qubit for this specific quantum circuit

Suppose that I have a Quantum Circuit with $$N$$ qubits initialized to $$|0\rangle$$. The Quantum circuit is described by the followin Figure:

The angles of the rotations $$R_y$$ are random. At which point of the circuit I can say that in general the generic qubit $$i$$ is entangled with the generic qubit $$j$$ (Regardless of $$N$$) ? How can I prove it?

(Of course there are come specific case where we have trivial state due to some particular rotations, but I'm not interested in them)

• Your statement will be probabilistic. I.e. with high probability i and j are entangled and a necessary condition is that there is some path in the circuit that connects qubits i and j where the cnot gates connect the different wires. I guess how deep the circuit needs to be will depend on how close the qubits initially were. Dec 3, 2022 at 13:59