Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 34 / 5 / a

The generalized measurement postulate specifies Kraus operators $M_{i\alpha }$, where $i$ is the recorded outcome and $\alpha$ permits unobserved alternatives for that outcome, satisfyingFor a density operator $\rho$, outcome $i$ has probability and conditional stateWith one operator per outcome, this is the usual formula $p_i=\mathrm{Tr}(M_i\rho M_i^{†})$, ${\rho }_i=M_i\rho M_i^{†}/p_i$. The positive operators $E_i={\sum }_{\alpha }{M}_{i\alpha }^{†}{M}_{i\alpha }$ form a POVM, while the state-update maps form a quantum instrument.

The full postulate reduces to the usual projective measurement with the Lüders update when there is one operator per outcome and $M_i=P_i$, with $P_i=P_i^{†}=P_i^2$, $P_i{P}_j=0$ for $i\ne j$, and ${\sum }_i{P}_i=I$. Then $p_i=\mathrm{Tr}(P_i\rho )$ and ${\rho }_i=P_i\rho P_i/p_i$. Projective effects $E_i=P_i$ alone ensure these probabilities but do not fix the state update: an additional outcome-dependent unitary could still be applied. The choice $M_i=P_i$ makes the required projective state-update condition explicit.