# How to understand the three state descriptions from 3 observers of such a system?

Assuming an EPR pair $\psi_{AB}=|00>+|11>$ and three observers Alice, Bob and Charlie, who can communicate with each other but they do not do it for the time being.

Alice with a qubit $|0>$ carries out a CNOT on A and her qubit. So after the operation, the state is a GHZ state as $\psi_{AB,Alice}=|111>+|000>$. Then she will measure her qubit, this lead to $\psi_{AB,Alice}=|111>$ or $|000>$, any way a product state and $AB$ is not entangled.

Bob knows the first operation carried out by Alice but is ignorant of the measurement operation, so for Bob the state should be $\psi_{AB,Alice}=|111>+|000>$, a GHZ state.

Charlie has no knowledge of Alice and Bob at all but he has the information of the original state of AB as $\psi_{AB}=|00>+|11>$.

I am wondering how to understand these three descriptions. Are they compatible with each other? Which description is 'correct'? For example

(1) Are $AB$ entangled or not? For Charlie, it's Yes. For Alice and Bob, it's No.

(2) Can Charlie use AB to achieve a teleportation operation?

(3) Can Bob distill EPR pairs from a collection of such systems?

(4) Is Alice entangled with AB? For Alice and Charlie, it's No. But Yes for Bob.

This is not just ignorance of information or probabilities but we face yes/no problems here.

And what if they start to exchange information? Will a phonecall from Bob to Charlie crashes his hope to teleport a state or he had no such opportunity at all?

Will it result in different output if the same teleportation operation is carried out by Alice , Bob and Charlie?

Thanks.

• @Stephen Powell Thanks for the comment. I tried to fix the problem in my problem. – XXDD Aug 25 '15 at 11:13

## 1 Answer

It's important to remember that the quantum state is physical, and not merely a description of our knowledge or ignorance of a system. At every point during Alice's experiment, the pair has a single quantum state (insofar as the measurement apparatus can be treated as classical), but Alice, Bob and Charlie have different degrees of knowledge about what that state is. In the field of quantum information, this knowledge is described mathematically using a density matrix; a situation where an observer's knowledge is incomplete is called a mixed state.

What you say about Alice is correct. But for Bob, the pair is no longer in a pure state, but rather in a mixed state, with 50% probability of $\lvert 00 \rangle$ and 50% probability of $\lvert 11 \rangle$. This is a very different state from $\lvert 00 \rangle + \lvert 11 \rangle$; for example, it cannot be used for quantum teleportation.

If Charlie has no knowledge of what happened to the pair, then I think it's safe to say simply that he has no knowledge of their current state.

Here are two questions that might help you understand mixed states:

• For Charlie, I mean he does know the existence of Alice and Bob. But he may have the information of the state $\psi_{AB}$. – XXDD Aug 25 '15 at 11:04
• For Bob, maybe you are right. A measurement based scheme will lead to a mixed state. But we can make it more complex. (1)Alice carries out a CNOT on A and her qubit, so she will result in a GHZ state. (2) Then she may measure her qubit. But Bob is only notified about step (1) but not step (2). Then for Alice, she will result in either $|111>$ or $|000>$, but for Bob, it will be a GHZ state. Can this fixed the hole? – XXDD Aug 25 '15 at 11:08