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Suppose Alice and Bob share the $2n$-qubits state $$|\phi _{x,y,b} \rangle = \frac{1}{2}(|x\rangle + (-1)^b |y\rangle)$$

where $x,y$ are $2n$-length strings of $0$ and $1$, $b$ is a bit, and $x \ne y$.

Alice holds the first $n$ qubits, and bob holds the other $n$ qubits, but they do not know what state they are sharing (they don't know $x,y,b$).

Furthermore, there is Charlie who knows $x,y$ but wants to learn $b$.

Alice and Bob can each send one classical message to Charlie.

I want to think about a protocol that helps Charlie learn $b$.

It seems that I can use super-dense coding in some sort of way, but I'm having trouble to understand how.

Help would be appreciated.

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Get Alice and Bob to measure all their qubits in the $X$ basis, and send the $\pm 1$ results to Charlie. The combined answer is a bit string $w$.

Charlie discards all the results associated with qubits $i$ where $x_i=y_i$. On the remaining $k$ qubits, Alice and Bob shared $$ |\psi\rangle=|z\rangle+(-1)^b|\bar z\rangle $$ for $z\in\{0,1\}^k$. What's the result of an $X$ measurement? \begin{align*} \langle+|^{\otimes k}Z^{w_k}|\psi\rangle&=\langle+|^{\otimes k}Z^{w_k}Z_1^b(|z\rangle+X^{\otimes k}|z\rangle) \\ &=\langle+|^{\otimes k}(I+(-1)^{|w_k|+b}X^{\otimes k})Z^{w_k}Z_1^b|z\rangle \\ &=(-1)^{z\cdot w_k+z_1}\langle+|^{\otimes k}(I+(-1)^{|w_k|+b}X^{\otimes k})|z\rangle \\ &=(-1)^{z\cdot w_k+z_1}\langle+|^{\otimes k}z\rangle(1+(-1)^{|w_k|+b}) \end{align*} You can only get an answer $w_k$ (the relevant $k$ bits of $w$) if $|w_k|+b$ is even. In other words, $b=|w_k|\text{ mod }2$.

For example, imagine that Alice and Bob initially share $$ |0011\rangle+(-1)^b|1101\rangle. $$ and assume they get the answers +++-, which they send to Charlie. Charlie knows $w_k=000$ and $z=001$, ignoring the last bit because they're the same for $x$ and $y$. Since $w_k$ is even, $b=0$. Basically, we're claiming that in this case $$ \langle +++|(|001\rangle-|110\rangle)=0, $$ i.e. we could never have got the +++ answer if $b=1$.

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  • $\begingroup$ Isn't there a (vanishingly small) probability that after Hadamarding, all of Alice and Bob's measurements are +1? $\endgroup$ Feb 26 at 16:09
  • $\begingroup$ yes. It shouldn't be a problem. Or at last wouldn't be if I hadn't missed out a term... $\endgroup$
    – DaftWullie
    Feb 26 at 16:20
  • $\begingroup$ Oh! Charlie gets to ignore when $x\oplus y=0$ because he knows $x$ and $y$; he just doesn't know the phase. Unlike in the examples that came to my mind initially, where $x$ and $y$ are not known and one only had access to $d$ such that $d\cdot (x\oplus y)=0$. $\endgroup$ Feb 26 at 17:13
  • $\begingroup$ Could you elaborate on how you calculated $\langle+|^{\otimes k}Z^{w_k}|\psi\rangle$, because I'm not sure I understand. $\endgroup$
    – Gabi G
    Feb 27 at 20:16

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