I am taking a Quantum Information course and I stumbled onto this problem, which I am unsure about. The situation assumes BB84 protocol, but with Eve doing a cloning operation (let's assume that she's doing it in states $|+>$ and $|->$ which are orthogonal) instead of the measurement. She keeps the cloned qubit. Would it mean that Bob and Alice won't know if the qubit was intercepted?

Also, if Bob and Alice are working in the same basis, would it actually mean that Eve intercepted the qubit without them knowing and has perfect knowledge of the qubit's state?

  • $\begingroup$ can you provide more details about the protocol you are talking about? If Eve can clone |±⟩, she cannot clone other states. Indeed, such "cloning operation" essentially amounts to Eve performing a CNOT operation (plus a few Hadamards) between the intercepted qubit and her qubit. Is this the sort of thing you are talking about? also, I wouldn't use the word "cloning" to refer to such a situation. This is more like Eve entangling her qubit to the transmitted one $\endgroup$ – glS Nov 5 '18 at 15:08
  • $\begingroup$ If this is the sort of thing you mean by "cloning", then yes Alice and Bob will know that the qubit was intercepted, because Eve being entangled to it implies that A&B's state is not pure anymore. In other words, this sort of thing breaks the shared correlations, which A&B will be able to realise when they compare their measurement results $\endgroup$ – glS Nov 5 '18 at 15:12

Cloning an arbitrary quantum state is forbidden. See here for a proof https://en.wikipedia.org/wiki/No-cloning_theorem

It is not clear what you mean by "assume she does it (cloning) in the $\vert +\rangle, \vert -\rangle$ basis" but she is unable to measure (as that is detectable) and unable to clone as that is forbidden by quantum mechanics.

  • $\begingroup$ it is not forbidden to clone two fixed orthogonal states, as long as one does not also want to clone superpositions of them. If what the OP means is that Eve is able to only clone $\lvert\pm\rangle$ that is fine, although of course it will imply that Eve will not be able to clone without error any other state $\endgroup$ – glS Nov 5 '18 at 10:40
  • $\begingroup$ Yes of course. Perhaps it's just a matter of terminology but if I know that I have a fixed pair of orthogonal states like $\vert+\rangle$ and $\vert -\rangle$, cloning is equivalent to simply measuring in that basis and then output as many copies as I like. $\endgroup$ – user1936752 Nov 5 '18 at 15:00
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    $\begingroup$ or even more simply applying the equivalent of a CNOT operation in the given basis, which "clones" the state without interfering with it. But yes I agree that talking of "cloning" for this sort of thing is misleading $\endgroup$ – glS Nov 5 '18 at 15:03

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