There's an exercise in Nielsen & Chuang asking us to show that if some third party intercepts Alice's qubit she's sending to Bob while attempting to transmit (2 bits of) classical information, then this third party can't actually infer anything about the information she's trying to communicate. Effectively, this comes down to showing that $\langle\psi|E_A \otimes I_B|\psi\rangle$ is the same whenever $\psi$ is any of the Bell states and $E$ is a positive operator.
This is simple to show, but I wonder if this is an incomplete concept, since it supposedly requires $E$ to be a positive operator. If we just check one of them:
$$\langle\Phi^+|E \otimes I|\Phi^+\rangle = \langle 00 + 11|E \otimes I|00 + 11 \rangle = \langle 0|E|0 \rangle\langle 0|0 \rangle + \langle 0|E|1 \rangle\langle 0|1 \rangle + \langle 1|E|0 \rangle\langle 1|0 \rangle + \langle 1|E|1 \rangle\langle 1|1 \rangle = \langle 0|E|0 \rangle + \langle 1|E|1 \rangle.$$
This is the same for each Bell state. Couldn't $E$ just be Hermitian and we would obtain the same result?