Quantum Measurement

In Nielsen and Chuang's QCQI, there are three kinds of measurement: general measurement, projective measurement, positive operator valued measurement(POVM).

The frame of general measurement \(M\) and projective measurement \(P\) is kind of similar. All of them states that after a set of measurement, we will attain state

with probability \(p_m\). The meaning of a set of measurement is \(\sum_mM_mM_m^\dagger=I\). Same as projective measurement. Some misunderstanding might happen here: we cannon decide which state to be after the measurement(a set of measurement operator stands for one-shot measurement), we can only say we have probability \(p_m\) to attain state \(P_m\rho P_m^\dagger/p_m\), we cannot force state to collape into one specific state, except the pose-measurement selection can have similar result.

And the core of POVM is that we don't care about which state did the state collapse into, we only care about the probability to obtain the specific state. So we use \(E_m\) to stands for POVM, and the probability to get the result \(m\) is \(tr(\rho E_m)\), and this probability cannot be changed into form of eq.(1).