$3 \rightarrow 1$ QRAC encoding for XOR functions

Question

I'm currently working on QRAC and was wondering if there's an encoding protocol in $3 \rightarrow 1$ such that the receiver is able to retrieve any one of the XOR combinations of the bits, along with the original bits itself ( a grand total of 7 functions; if a, b, c are the bit positions, the receiver should be able to guess a, b, c, a+b, b+c, c+a, a+b+c with a probability greater than 1/2).

Is this even possible? My initial guess would be to use some kind of POVM measurements in the decoding part but don't how to proceed with that.

If you have any ideas on this then please let me know. Thank you.

Answer 1

I found a way to do it for the $2\to1$ QRAC. I simply guessed that we could leave the measurement bases as they are, $Z$ for the first bit, and $X$ for the second bit, and added $Y$ as the basis with which to extract the XOR of the two bits. From the guess obtaining the optimal encoding states is then easy, we just need to diagonalize the relevant operators. They are \begin{align*} |\psi_{00}\rangle & = \sqrt{p}|0\rangle + \sqrt{1-p}\ e^{\frac{\pi i}4}|1\rangle \\ |\psi_{01}\rangle & = \sqrt{p}|0\rangle + \sqrt{1-p}\ e^{-\frac{3\pi i }4}|1\rangle \\ |\psi_{10}\rangle & = \sqrt{1-p}|0\rangle + \sqrt{p}\ e^{-\frac{\pi i}4}|1\rangle \\ |\psi_{11}\rangle & = \sqrt{1-p}|0\rangle + \sqrt{p}\ e^{\frac{3\pi i}4}|1\rangle, \end{align*} where $p=\frac{3+\sqrt3}{6}$ is also the probability of success in this QRAC. I don't know whether it is optimal, but I guess it is, since it's so nice and symmetrical.

Now for the $3\to1$ case such a simply idea cannot work, as you want to extract 7 different functions, but there are only these 3 mutually unbiased bases in dimension 2. What I would do is to find some set of 7 bases that is in some sense uniformly distributed on the Bloch sphere and see if it works. Another idea is to simply do a see-saw optimization and see what you get.