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I'm referring to an earlier question. It involves secret sharing based on the different measurement directions 3 people i.e Alice Bob and Charlie do. Now there is a block in the referred paper which I want to understand in order to proceed further and here it goes assuming the GHZ state they share is $$ |\psi\rangle=\dfrac{|000\rangle+|111\rangle}{\sqrt{2}}$$

Suppose that Bob is dishonest and that he has managed to get a hold of Charlie’s particle as well as his own. He then measures the two particles and sends one of them on to Charlie. His object is to discover what Alice’s bit is, without any assistance from Charlie, and to do so in a way that cannot be detected. Alice has measured her particle in either the $x$ or $y$ direction, but Bob does not know which. He would like to measure the quantum state of his two-particle system, but because he does not know what measurement Alice made, he does not know whether to make his in the $\dfrac{|00\rangle ± |11\rangle}{\sqrt{2}}$ basis or in the $\dfrac{|00\rangle ± i|11\rangle}{\sqrt{2}}$ basis. Choosing at random he has a probability of $1/2$ of making a mistake. If he chooses correctly, he will know, for valid combinations of measurement axes, what the result of Charlie’s measurement is from the result of his own, and this means he will then know what Alice’s bit is. For example, if Alice measured in the $x$ direction and found $| + x\rangle$, then the state Bob receives is $\dfrac{|00\rangle + |11\rangle}{\sqrt{2}}$. If Bob now measures in the $\dfrac{|00\rangle \pm |11\rangle}{\sqrt{2}}$ basis, he knows what the two- particle state is, and because $$\dfrac{|00\rangle + |11\rangle}{\sqrt{2}}= \dfrac{|+x\rangle|+x\rangle + |-x\rangle|-x\rangle}{\sqrt{2}}$$ Bob knows that Charlie’s measurement will produce a result identical to his

The questions I have is

  1. How do they accomplish the sharing of particles of this GHZ state? What particle is Alice given of this GHZ state I know it is given that the first particle but what does "first" mean?

  2. It says Bob catches hold of Charlie's particle measures it and then sends to Charlie, but isn't Charlie already holding his particle when they decided to share the GHZ state or does Charlie get his particle after Bob measures both his and Charlie's particle?

  3. As given if Bob receives $\dfrac{|00\rangle + |11\rangle}{\sqrt{2}}$ then after measuring in the basis given he will get $p_{++}=1$ and $p_{--}=0$, where $p_{++}=\dfrac{|00\rangle + |11\rangle}{\sqrt{2}}$ and $p_{--}=\dfrac{|00\rangle - |11\rangle}{\sqrt{2}}$, what does this have to do with the text that follows this measurement outcome?

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