Observable

quantum-computing

Definition

Observable

An observable is a Hermitian operator $M:\mathcal{H}\to \mathcal{H}$ whose eigenvalues are the possible numerical outcomes of a quantum measurement.

For a projective measurement, an observable has a spectral decomposition

$$M=\sum _i{m}_i{P}_i,$$

where $m_i\in \mathbb{R}$ are the possible measurement values and $P_i$ are orthogonal projections onto the corresponding eigenspaces.

If the system is in state $|\psi ⟩$, then outcome $m_i$ occurs with probability

$$p(i)=⟨\psi |P_i|\psi ⟩=||P_i|\psi ⟩|{|}^2.$$

After observing outcome $m_i$, the state becomes

$$|{\psi }_i⟩=\frac{P_i|\psi ⟩}{\sqrt{p(i)}}.$$

The expectation value of observation in state $|\psi ⟩$ is

$$⟨M⟩=⟨\psi |M|\psi ⟩=\sum _i{m}_{i}p(i).$$

Intuitively, the projectors $P_i$ define the measurement branches, while the eigenvalues $m_i$ are the classical numbers reported as outcomes.