What does quantum entanglement mean for causality? Due to entanglement, there can be spacelike separated measurements such that there exists a reference frame > where it looks like measurement A precedes and has a causal influence on the outcomes of measurement B, and > also a reference frame where it looks like measurement B precedes and has a causal influence on the outcomes of measurement A.

“Causality” is already a somewhat fraught notion in fundamental physics irrespective of quantum mechanics; it’s not clear that one needs to have some sort of notion of causality in order to do physics, nor that the universe necessarily obeys some underlying causal law. To the extent that quantum mechanics breaks our common-sense notions of causality, it’s only in this very particular sense (where it seems like Alice measuring first “causes” Bob’s measurement to take a certain value, or vice versa), and since neither party can use a measurement scheme like this to send information, the breakage doesn’t invite paradoxes or any sort of other weirdness.

Outside of philosophical musings about causality (which, to be clear, I think are perfectly valid and interesting) it suffices to say that entangled systems exhibit correlations without a common cause, and leave it at that.

If you’re interested in a recent technical discussion of some of these ideas, I recommend the following paper: https://arxiv.org/pdf/2208.02721.pdf

Just to (hopefully) make the distinction a bit more clear:

A true copying operation would take |psi1>|0> to |psi1>|psi1>; that’s to say, it would take as input one qubit in an arbitrary quantum state and a second qubit in |0>, and output two qubits in the same arbitrary quantum state that the first qubit was in. For our example, we’ll take |psi1> to be an equal superposition of 0 and 1: |psi1> = |0> + |1> (ignoring normalization).

If CNOT is a copying operation, it should take (|0> + |1>)|0> to (|0> + |1>)(|0> + |1>) = |00> + |01> + |10> + |11>. But as you noticed, what it actually does is create an entangled state (in this case, a Bell state) that looks like |00> + |11>.

So in some sense yes, the forbidden thing is to have a state copied and not entangled, but more importantly in this case CNOT just doesn’t copy the state, so there’s no tension with the no-cloning theorem.