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I have a few questions in understanding the difference between coupling and entanglement in quantum systems: Is there a clear boundary between quantum coupling and quantum entanglement? If two quantum systems are coupled, do they need to be restricted to a certain distance? Is there a difference between 'coupling the two qubits' and 'entangling' them using a Hadamard Gate? In Schrodinger's cat thought experiment, are we saying the cat and radioactive source are 'entangled' or 'coupled'?

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Is there a clear boundary between quantum coupling and quantum entanglement?

If two qubits (or more) are coupled, they can influence each other. This means that they are entangled.

If two quantum systems are coupled, do they need to be restricted to a certain distance?

Generally not, but increasing the distance increases probability that the system is influenced by external environment and the entanglement is disrupted. If you be able to perfectly isolate the system, the distance can be arbitrary.

Is there a difference between 'coupling the two qubits' and 'entangling' them using a Hadamard Gate?

Hadamard gate does not produce entangled state. It changes qubit in state $|0\rangle$ to $\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$ and state $|1\rangle$ to $\frac{1}{\sqrt{2}}(|0\rangle-|1\rangle)$. This means that it creates equally distributed superposition of two states for one qubit, not the entanglement. Hadamard gate is also inverse gate to itself, i.e. $HH=I$, so if it is applied on qubit in state $\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$, a result is $|0\rangle$.

In Schrodinger's cat thought experiment, are we saying the cat and radioactive source are 'entangled' or 'coupled'?

See the answer to the first question.

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