Reading through Nielsen and Chuang, I came across the following example.
Consider a POVM containing three elements, $$ E_1 \equiv \frac{\sqrt{2}}{1 + \sqrt{2}} |1⟩⟨1|,\\ E_2 \equiv \frac{\sqrt{2}}{1+\sqrt{2}}\frac{(|0⟩ - |1⟩)(|0⟩ - |1⟩)}{2}\\ E_3 \equiv I − E_1 − E_2. $$ It is straightforward to verify that these are positive operators which satisfy the completeness relation m Em = I, and therefore form a legitimate POVM.
The point the author makes is that if I'm given one of two states, $|{\psi_1}\rangle=|{0}\rangle$ or $|{\psi_2}\rangle=(|{0}\rangle+|{1}\rangle)/\sqrt{2}$, performing measurements described by this POVM will distinguish the state some of the time but never make a misidentification error.
What does it really mean to perform a measurement described by a POVM? It's clear what I can do experimentally for projective measurements -- in a system like a neutral atom QC I can perform a measurement of some observable, and each projector is uniquely identified with an eigenvalue of the observable. N&C states that for a POVM $\{E_m\}$, the probability of outcome $m$ is $\langle{\psi}|{E_m}|{\psi}\rangle$, which makes sense. However, when we construct POVM elements like in this example though, I don't understand what the corresponding outcomes would be. Since that's not clear, I thus don't see what it means to perform a POVM measurement and how one would distinguish between measurement $E_3$ and $E_2$.
