projective measurement

The crucial concept is the observable which is a Hermitian operator (just a Hermitian matrix?) \(M\) which has the following spectral decomposition: \[M = \sum m P_m\] where \(P_m\) is a projector onto the eigenspace associated with the eigenvalue \(m\).

For state \(\psi\), the probability of getting result \(m\) is \(\bra{\psi}P_m\ket{\psi}\) and the resulting state will be \(\frac{P_m\ket{\psi}}{\bra{\psi}P_m\ket{\psi}^{1/2}}\). Note that the \(ket{\psi}\) has been projected onto the eigenspace. So for example, if \(P_m=\ket{0}\bra{0}\), then the resulting state will be \(\braket{0 \mid \psi}\ket{0}\).

1. sources

Nielsen and Chuang

1.1. multiple qubits

What happens when we measure one qubit out of a multi-qubit system?
See here, the probability of the outcome \(i\) for qubit \(j\) is a marginalization over all basis states where the \(i\) is true of \(j\)
- The resulting state is a weighted sum of the basis states, weighted by these probabilities