Could someone please ELI5 why using a CNOT gate (if the target qubit was initially zero) does not violate the no-cloning theorem?
EDIT:
Oh, I think I got it. The forbidden thing is to have a state “copied and not entangled”. CNOT gate creates a state that is “copied and entangled”, which is okay, because you can only measure it once (if you measure either the original or the copy, the state of the other one collapses). The forbidden thing is to have a copy that you could measure independently (e.g. you could measure the copy without collapsing the original).
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.
Some context: I am a “quantum autodidact”, and I am currently reading a book Q is for Quantum, which is a very gentle, beginner-friendly introduction to quantum computing. I was thinking how it relates to the things I have read before, and then I noticed that I was confused. I looked at Wikipedia, which said that CNOT does not violate the no-cloning theorem… but I didn’t understand the explanation why.
I think I get it now. |00> + |11> is not a copy (looking at one qubit collapses the other), |00> + |01> + |10> + |11> would be a copy (looking at one qubit would still leave the other as |0> + |1>).
Could someone please ELI5 why using a CNOT gate (if the target qubit was initially zero) does not violate the no-cloning theorem?
EDIT:
Oh, I think I got it. The forbidden thing is to have a state “copied and not entangled”. CNOT gate creates a state that is “copied and entangled”, which is okay, because you can only measure it once (if you measure either the original or the copy, the state of the other one collapses). The forbidden thing is to have a copy that you could measure independently (e.g. you could measure the copy without collapsing the original).
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.
Thank you!
Some context: I am a “quantum autodidact”, and I am currently reading a book Q is for Quantum, which is a very gentle, beginner-friendly introduction to quantum computing. I was thinking how it relates to the things I have read before, and then I noticed that I was confused. I looked at Wikipedia, which said that CNOT does not violate the no-cloning theorem… but I didn’t understand the explanation why.
I think I get it now. |00> + |11> is not a copy (looking at one qubit collapses the other), |00> + |01> + |10> + |11> would be a copy (looking at one qubit would still leave the other as |0> + |1>).
I recommend this article by the discoverers of the no-cloning theorem for a popular science magazine over the Wikipedia page for anyone trying to understand it.