# What does the outcome $i$ mean when we measuring a quantum system?

The POVM element $$E_{i}$$ is associated with the measurement outcome $$i$$, such that the probability of obtaining it when making a measurement on the quantum state $$\rho$$ is given by: $$p(i)=tr(\rho E_i)$$;

So what does the outcome $$i$$ refer to? It looks like it's just an index symbol and $$\sum_{i=1}^n E_i=I$$.

The outcome $$i$$ is just an index/label. After a POVM measurement corresponding to a set say $$\mathcal{M} = \{ E_i \}\,,$$ where $$|\mathcal{M}|=i$$, the reading on your measurement device could be any of the $$i$$ different outcomes corresponding to this set $$\mathcal{M}$$.
Say you perform POVM measurement on some initial state $$|\psi\rangle$$. Then, after the POVM measurement, you get one of the $$i$$ possible readings on your measurement device, with the probability of getting this $$i^{\text{th}}$$ outcome being $$p_i$$. Where,
$$p_i = \langle \psi | E_i |\psi\rangle\,.$$
• Careful! Your statement would be correct for any measurement operators $\{M_i\}$ such that $\sum_iM_i^\dagger M_i=I$. However, POVMs are not that. They do not have final states associated with the outcomes. One way that you can see this is via the non-uniqueness of $\sqrt{E_i}$: for any $M_i$ such that $E_i=M_i^\dagger M_i$, you can also have $\tilde M_i=UM_i$ for any unitary $U$. Oct 30, 2023 at 7:33
• @DaftWullie Yes I agree. We can write $M_i$ as $$M_i = U_i \sqrt{E_i}$$ upto an arbitrary choice of $U_i$. So, if we only have $\{E_i\}$ specified, then we are free to choose $U_i$ and for every such choice, we would get a different set of $\{M_i\}$, right? and in that case, the post-measurement state would what I have specified above. Is this the correct line of reasoning? Oct 30, 2023 at 9:18
• @DaftWullie But yes, I agree that practically, since this $U_i$ is just an arbitrary choice, the post-measurement state is not uniquely defined; it's upto an arbitrary unitary transformation, so for us in a measurement setup, the post-measurement state after a POVM is unspecified. Is that correct? And also, thanks for pointing out my mistake. Appreciate it! Oct 30, 2023 at 9:22