My question is second part of exercise 2.70 page 98 of the book Quantum Computation and Quantum Information written by Michael A. Nielsen and Isaac L. Chuang.
Assume Bob and Alice share each one qubit of the entangled state $$|\psi\rangle=\dfrac{|00\rangle+|11\rangle}{\sqrt{2}}$$ Alice for sending a 2-bits data operate one of the Pauli single-qubit gets on her qubit then sends her qubit to Bob, then bob with a projective measurement find out what was the message of Alice with their pre-agreement as follow. $$\begin{array}{l} 00\;:\;\dfrac{|00\rangle+|11\rangle}{\sqrt{2}}\\ 01\;:\;\dfrac{|00\rangle-|11\rangle}{\sqrt{2}}\\ 10\;:\;\dfrac{|01\rangle+|10\rangle}{\sqrt{2}}\\ 11\;:\;\dfrac{|01\rangle-|10\rangle}{\sqrt{2}} \end{array}$$ The exercise says;
Exercise 2.70: Suppose $E$ is any positive operator acting on Alice's qubit. Show that $\langle \psi|E\otimes I|\psi\rangle$ takes the same value when $|\psi\rangle$ is any of the four Bell states.
Suppose some malevolent third party ('Eve') intercepts Alice's qubit on the way to Bob in the superdense coding protocol. Can Eve infer anything about which of the four possible bit strings $00,\,01,\,10,\,11$ Alice is trying to send? If so, how, or if not, why not?
I did the first part and actually that same value is $\frac{1}{2}tr(E)$. It's obvious or at least I think, I have to use the first part of the question for rejecting that Eve can understand anything but I don't know how.
You've yourself said that the expectation value of Eve's operator does not depend on what Alice is trying to send, so you're most of the way there.
What's the reduced density operator for half of a maximally entangled Bell state? Does it depend on which Bell state it is made from?