I watched an IBM quantum tutorial "Hello World" video. It assumed and then demonstrated that the correlation decays with the distance between two Pauli Z observables. I'm not able to derive the decay, even though intuitively it makes sense. (It's a "Hello World" video with qiskit: https://www.youtube.com/watch?v=93-zLTppFZw )
For the ease of notation assume 4 qubits
$$|GHZ \rangle = \frac{1}{\sqrt{2}}(|00000 \rangle +|11111 \rangle)$$
Let's look at the extreme, when two Pauli Zs are furthest apart.
Expectation of sparse Pauli Z-s is defined as $$ \langle GHZ|Z\otimes I \otimes I \otimes Z |GHZ \rangle $$
We have $Z|0 \rangle =|0 \rangle $ and $Z|1 \rangle =-|1 \rangle $, hence:
$$Z\otimes I \otimes I \otimes Z |GHZ \rangle = \frac{1}{\sqrt{2}}(|00000 \rangle +(-1)(-1)|11111 \rangle) = |GHZ \rangle $$
And that results into
$$ \langle GHZ|Z\otimes I \otimes I \otimes Z |GHZ \rangle = \langle GHZ|GHZ \rangle = 1$$
What am I doing wrong?
