- #1
Ackbach
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So in quantum mechanics, there are at least three different kinds of measurement operators: the General, the Projective, and the Positive Operator-Valued (POVM). They have different properties and relationships. In a typical QM book, these are not delineated, but in Quantum Computing they are, since we want to have much more fine control over measurements. So here is a table comparing the three different kinds of measurement operators.
$$
\begin{array}{|c|c|c|c|}\hline
\textbf{Type} &\text{General} &\text{Projective} &\text{POVM} \\ \hline
\textbf{Basic Symbol} &\text{Measurement Op} \; M_m &\text{Observable} \; M=\sum_m m P_m &\text{Measurement Op} \; M_m \\ \hline
\textbf{Special Property} & &P_m \; \text{a projector:} \; P_m^2=P_m &\text{POVM element} \; E_m:=M_m^{\dagger}M_m \\ \hline
\textbf{Probability} &p(m)=\langle\psi|M_m^{\dagger}M_m|\psi\rangle &p(m)=\langle\psi|P_m|\psi\rangle &p(m)=\langle\psi|E_m|\psi\rangle \\ \hline
\textbf{State After Measurement} &\dfrac{M_m|\psi\rangle}{\sqrt{\langle\psi|M_m^{\dagger}M_m|\psi\rangle}}
&\dfrac{P_m|\psi\rangle}{\sqrt{\langle\psi|P_m|\psi\rangle}} &\text{not of interest} \\ \hline
\textbf{Completeness} &\sum_mM_m^{\dagger}M_m=I &\sum_m P_m=I &\sum_m E_m=I \\ \hline
\textbf{Hermitian} &M_{m}^{\dagger}=M_m &P_{m}^{\dagger}=P_m &E_{m}^{\dagger}=E_m \\ \hline
\textbf{Other Special} & &E(M)=\langle\psi|M|\psi\rangle &\text{If} \; M_m^2=M_m \; \text{then} \; M_m=P_m \\ \hline
& &M_mM_{m'}=\delta_{m,m'}I &\text{Otherwise,} \; M_m=\sqrt{E_m} \\ \hline
\end{array}
$$
Also note that, by definition, in the POVM case, $\{E_m\}$ is the POVM.
$$
\begin{array}{|c|c|c|c|}\hline
\textbf{Type} &\text{General} &\text{Projective} &\text{POVM} \\ \hline
\textbf{Basic Symbol} &\text{Measurement Op} \; M_m &\text{Observable} \; M=\sum_m m P_m &\text{Measurement Op} \; M_m \\ \hline
\textbf{Special Property} & &P_m \; \text{a projector:} \; P_m^2=P_m &\text{POVM element} \; E_m:=M_m^{\dagger}M_m \\ \hline
\textbf{Probability} &p(m)=\langle\psi|M_m^{\dagger}M_m|\psi\rangle &p(m)=\langle\psi|P_m|\psi\rangle &p(m)=\langle\psi|E_m|\psi\rangle \\ \hline
\textbf{State After Measurement} &\dfrac{M_m|\psi\rangle}{\sqrt{\langle\psi|M_m^{\dagger}M_m|\psi\rangle}}
&\dfrac{P_m|\psi\rangle}{\sqrt{\langle\psi|P_m|\psi\rangle}} &\text{not of interest} \\ \hline
\textbf{Completeness} &\sum_mM_m^{\dagger}M_m=I &\sum_m P_m=I &\sum_m E_m=I \\ \hline
\textbf{Hermitian} &M_{m}^{\dagger}=M_m &P_{m}^{\dagger}=P_m &E_{m}^{\dagger}=E_m \\ \hline
\textbf{Other Special} & &E(M)=\langle\psi|M|\psi\rangle &\text{If} \; M_m^2=M_m \; \text{then} \; M_m=P_m \\ \hline
& &M_mM_{m'}=\delta_{m,m'}I &\text{Otherwise,} \; M_m=\sqrt{E_m} \\ \hline
\end{array}
$$
Also note that, by definition, in the POVM case, $\{E_m\}$ is the POVM.
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