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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'?

Thanks:)

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Generally "coupling" simply means that the Hamiltonian with which the system evolves contains interaction terms between the systems (though I guess this might also depend on the context).

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.

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    $\begingroup$ As an add-on to this excellent answer, two systems that have become entangled (perhaps due to coupling in the past) can remain entangled even after the coupling has long since stopped. This is the basic settup in Bell inequality experiments for example. en.wikipedia.org/wiki/Bell%27s_theorem $\endgroup$ – Dast Oct 28 '20 at 11:35
  • $\begingroup$ @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:) $\endgroup$ – ZR- Oct 28 '20 at 15:44
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    $\begingroup$ @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. $\endgroup$ – glS Oct 28 '20 at 16:39

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