Positive Operator Value Measurement Question

Question

I'm attempting to understand some of the characteristics of Posiitive Operator Value Measurement (POVM). For instance in Nielsen and Chuang, they obtain a set of measurement operators $\{E_m\}$ for states $|\psi_1\rangle = |0\rangle, |\psi_2\rangle = (|0\rangle + |1\rangle)/\sqrt{2}$. The end up obtaining the following set of operators:

\begin{align*} E_1 &\equiv \frac{\sqrt{2}}{1+\sqrt{2}} |1\rangle \langle 1 |, \\ E_2 &\equiv \frac{\sqrt{2}}{1+\sqrt{2}} \frac{(|0\rangle - |1\rangle) (\langle 0 | - \langle 1 |)}{2}, \\ E_3 &\equiv I - E_1 - E_2 \end{align*}

Basically, I'm oblivious to how they were able to obtain these. I thought that perhaps they found $E_1$ by utilizing the formula:

\begin{align*} E_1 = \frac{I - |\psi_2\rangle \langle \psi_2|}{1 + |\langle \psi_1|\psi_2\rangle|} \end{align*}

However, when working it out, I do not obtain the same result. I'm sure it's something dumb and obvious I'm missing here. Any help on this would be very much appreciated.

Thanks.

Answer 1

Yes, those are the results but you have the subindexes swaped.