Suppose Alice and Bob hold one qubit each of an arbitrary two-qubit state $|\psi \rangle$ that is possibly entangled. They can apply local operations and are allowed classical communication. Their goal is to apply the CNOT gate to their state $| \psi \rangle$. How can they achieve this using two ebits of communication?
I'm pretty lost on where to start this problem. I would assume that if Alice has the control bit for the CNOT, I would need to follow a protocol something like:
- tensor $|\psi\rangle$ with the EPR pairs
- Alice applies some local operations and measures the EPR pairs
- Alice classically communicates something to Bob
- Bob applies some local operations with the information sent classically by Alice
- CNOT completed.
However, it is possible that Alice would only use one ebit, and Bob would use one as well to communicate something back. I'm not sure.
Really I just feel like I have no clue where to start this problem. Initially, some guidance on how to approach this would perfect, because right now i'm just poking around at a 6 qubit state.
A second idea I just had: might it be beneficial to instead of looking at the complete state, look at Alice and Bob's local density matrices?