I was reading the article "Multiparty quantum secret sharing". It had 3 parties Bob, Alice and Charlie. They agree on sharing a classical secret quantum mechanically. The steps they follow are for a particular case $N=3$, say

  1. Bob prepares $3$ states say $|0\rangle$, $|1\rangle$, $\dfrac{|0\rangle+|1\rangle}{2}$. sends them to Alice.

  2. Charlie chooses randomly to apply the operator $I$, $U$ on these received states, where $I$ is the identity operator, and $U|0\rangle=-|1\rangle$, $U|1\rangle=|0\rangle$. So let the choices he makes is $I|0\rangle=|0\rangle$, $U|1\rangle=|0\rangle$ and $U\left(\dfrac{|0\rangle+|1\rangle}{2}\right)=\left(\dfrac{-|1\rangle+|0\rangle}{2}\right)$. After applying these operators he sends these to Alice.

  3. Alice stores the third particle and selects the first and second particle and announces their position in public.

  4. For each selected photon Alice randomly selects one action from the following two choices. One is that Alice lets Bob first tell her the initial state of the photon and then lets Charlie tell her which unitary operation he has performed on it; Then Alice first performs the same unitary operation as Charlie has performed on the photon and then measures the photon by using the basis the initial state belongs to. After her measurements, Alice can determine the error rate. If the error rate exceeds the threshold, the process is aborted. Otherwise, the process continues and Alice performs unitary operations either $I$ or $U$ on the stored photons to encode her secret messages. That is, if Alice wants to encode a bit $0$ she performs the identity unitary operation I; if Alice wants to encode a bit $1$” she performs the unitary operation U. Alice sends these encoded photons to Charlie.

  5. After Charlie receives these encoded photons, if Bob and Charlie collaborate, both Bob and Charlie can obtain Alice’s secret message by using the correct measuring basis for each encoded photon.

My question is why are both Bob and Charlie needed for knowing what Alice sent. Since for example Alice wanted to send a $1$, so he operates the stored qubit with $U$, i.e $$U\left(\dfrac{-|1\rangle+|0\rangle}{2}\right)= \left(\dfrac{-|0\rangle-|1\rangle}{2}\right)$$. Now when Charlie gets this, he knows before hand that third particle that Bob sent was $\left(\dfrac{|0\rangle+|1\rangle}{2}\right)$, so he measures in this basis, to get $1$, the message Alice wanted to send, where is Bob needed in this protocol? Can somebody help?


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