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The problem is similar to the pseudo-telepathy where Alice, Bob and Charlie are given bits $a,b,c \in \{0,1\}$ such that $a \oplus b \oplus c=0$, and their goal is to output the bits $A,B,C$ without communicating such that $A \oplus B \oplus C = a \lor b \lor c$.

I need to thing of a winning strategy for a when they all share the state:

$$\frac{1}{2}(|000\rangle -|011\rangle-|101\rangle-|110\rangle)$$

I know a strategy to solve this problem when they share the GHZ state $\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle)$.

I thought about doing something similar here, but it seems more complicated than just measuring in the Hadamard basis.

Help would be appreciated.

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    $\begingroup$ Hint: players can choose the measurement basis based on the input bit. That is, player measures the corresponding qubit in one basis if the input bit is $0$, and in another basis if the input bit is $1$. $\endgroup$
    – Egretta.Thula
    Commented Feb 25 at 3:28
  • $\begingroup$ I though about doing this with the Hadamard basis for $0$, and the basis $y =\frac{1}{\sqrt{2}} \begin{bmatrix} 1, \\ \pm i \end{bmatrix}$ for $1$, but the calculation didn't work out for me. $\endgroup$
    – Gabi G
    Commented Feb 26 at 6:45

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