# Pseudo telepathy with 3 people that share the state $\frac{1}{2}(|000\rangle -|011\rangle-|101\rangle-|110\rangle)$

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.

• 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 $1$. Commented Feb 25 at 3:28
• 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. Commented Feb 26 at 6:45