# How to experimentally perform a POVM measurement?

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 $$E_2$$.

• Please focus on one question. May 28, 2020 at 15:55
• I have edited the question accordingly. May 29, 2020 at 9:31
• In experiments, you rarely perform perfect projective measurements. Rather, you perform some POVM which is somewhat close to a projective measurement. May 29, 2020 at 13:12
• But if you want to start from projective measurements, I recommend Preskill's lecture notes. They take more the physicist's approach to teaching QI. May 29, 2020 at 13:13
• Could you provide an example of a POVM in an experiment? May 29, 2020 at 14:25