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

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'?

Thanks:)

If two systems are coupled, they might become entangled, though this is not necessarily the case. For example, under a coupling term of the form $$\sigma_z\otimes\sigma_z$$, a two-qubit state $$|0,0\rangle$$ will remain separable, but a state $$|+,+\rangle$$ will become entangled.
• @glS Thank you so much for the answer! In your example, is that because $\sigma_z$ has no effect on each of the qubit in $|0⟩$ state? Could you example a bit more about why the state $|++⟩$ will be entangled? Thanks:) – ZR- Oct 28 '20 at 15:44
• @Zhengrong yes it's because $|0\rangle$ is an eigenstate of $\sigma_z$. For the other case you can see what happens by computing $e^{it(\sigma_z\otimes\sigma_z)}|++\rangle$. The gist is that $\sigma_z|+\rangle=|-\rangle$, therefore the coupling changes the input state. – glS Oct 28 '20 at 16:39