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I have a question regarding a wording from an exercise book: “Two states psi and phi of a composite system are said to be 'LOCC equivalent' if each can be converted to the other using only local operations and classical communication. Local operations act only on one subsystem at a time, and can include measurements.”

Now I am wondering (e.g. to show that one bell state can be transferred into another via. the LOCC), what the local operations and classical communications are exactly. So from the question I read, that an operation is a local operation if it only acts on one Subsystem at a time —> does this mean, that all one is allowed to apply all (unitary) operators on one state of one Hilbert space at a time, while leaving the other untouched, or applying two operators individually to e.g. two different states from 2 Hilbert spaces. But that excludes the possibility of an operator, that would act on both states together, like a CNOT e.g., right?

Then I am wondering, what the consequence of “can be converted by classical communication” is? I unfortunately didn’t find an explanation (that I understood) on how this can affect the states.

If anybody has a tip on what the consequence of the classical communication part is, or can clarify, if my understanding of local operations is correct, I would be very grateful!

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I think you are basically right. If Alice on Earth has in her possession a number of qubits $|\psi\rangle$, and Bob on Mars has a number of qubits $|\phi\rangle$ in his possession, then even if $|\psi\rangle$ and $|\phi\rangle$ are not initially entangled with each other Alice and Bob can meet to perform entangling gates (such as a CNOT gate) between them to learn more about $|\psi\rangle\otimes|\phi\rangle$ than they would if Alice acted only on $|\psi\rangle$ and Bob acted only on $|\phi\rangle$.

Otherwise the requirement of "local operations" means that Alice can only act on her qubits and Bob on his, and the requirement of "classical communications" means that Alice can report on the results of her measurements to Bob and Bob the results of his to Alice but they do not otherwise knowingly pre-share any entanglement.

Indeed Alice could locally perform a partial measurement on some of her qubits, and send the results of her measurement to Bob, who could adjust his measurement of his qubits and send the results back to Alice, and so on ad infinitum, but there are some states where Alice and Bob can learn much more about their own register only if they are able to meet up to do the entangling.

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