Suppose Alice and Bob share the quantum state $\frac{1}{\sqrt 2}(|x\rangle + (-1)^b |y\rangle)$ for some $x\neq y \in \{0,1\}^2$ and $b \in \{0,1\}$. They both do not know $x,y$, and use some middlemen who wishes to learn $b$. They are allowed to send only classical messages to him (i.e. not quantum states). The marginal density matrix of Alice, for example, is $$ \rho_A = tr_B(\rho_{AB}) = \frac{1}{2}( |x_1\rangle \langle x_1| +|y_1\rangle \langle y_1| + (-1)^b |x_1 \rangle \langle y_1| \cdot \langle x_2 | y_2 \rangle + (-1)^b |y_1 \rangle \langle x_1| \cdot \langle y_2 | x_2 \rangle ) $$ Note that when $x_2 \neq y_2$, then $\rho_A$ does not depends on $b$. Moreover, if also $x_1 \neq y_1$, from symmetry arguments, $\rho_B$ does not depends on $b$. In such case, it seems to me that they can not send any data to the middleman (such as measurements), to help him deduce $b$, as the marginal density matrix, i.e. their personal view of the system, is oblivous of $b$. Is it correct to say so?
Of course when $x_2 = y_2$ for example, then Alice marginal density matrix does depends on $b$, and she can apply Hadamard gate + measure, to obtain exactly $b$, and send it to the middleman.