Observables
An observable A is a measurable physical property which has a numerical value, for example, spin or momentum or energy.
The term “observable” also extends to any basic measurement in which each outcome is associated with a numerical value.
If \lambda_k is the numerical value associated with outcome |e_k\rangle then we say that the observable A is represented by the operator
\begin{aligned}
A
&= \sum_k \lambda_k |e_k\rangle\langle e_k|
\\&= \sum_k \lambda_k P_k,
\end{aligned}
where \lambda_k is now the eigenvalue corresponding to the eigenvector |e_k\rangle, or the projector P_k.
Recall the following types of operator:
| normal |
AA^\dagger = A^\dagger A |
| unitary |
AA^\dagger = A^\dagger A = \mathbf{1} |
| Hermitian, or self-adjoint |
A^\dagger = A |
| positive semi-definite |
\langle v|A|v\rangle\geqslant 0 for all |v\rangle |
Note that an operator A is normal if and only if it is unitarily diagonalisable, and that both unitary and Hermitian operators are normal.
Conversely, to any normal operator A we can associate a measurement defined by the eigenvectors of A, which form an orthonormal basis, and use the eigenvalues of A to label the outcomes of this measurement.
If we choose the eigenvalues to be real numbers then A becomes a Hermitian operator.
For example, the standard measurement on a single qubit is often called the Z-measurement, because the Pauli Z operator can be diagonalised in the standard basis and written as Z = (+1)|0\rangle\langle 0| + (-1)|1\rangle\langle 1|.
The two outcomes, |0\rangle and |1\rangle, are now labelled as +1 and -1, respectively.
Using the same association we also have the X- and the Y-measurements, defined by the Pauli X and Y operators, respectively.
The outcomes can be labelled by any symbols of your choice; it is the decomposition of the Hilbert space into mutually orthogonal subspaces that defines a measurement, not the labels.
This said, labelling outcomes with real numbers is very useful.
For example, the expected value \langle A\rangle, which is the average of the numerical values \lambda_k weighted by their probabilities, is a very useful quantity and can be easily expressed in terms of the operator A as \langle\psi|A|\psi\rangle, as follows:
\begin{aligned}
\sum_k \lambda_k \Pr (\text{outcome k})
&= \sum_k \lambda_k |\langle e_k|\psi\rangle|^2
\\&= \sum_k\lambda_k \langle\psi|e_k\rangle\langle e_k|\psi\rangle
\\&= \langle\psi| \left( \sum_k\lambda_k|e_k\rangle\langle e_k| \right)|\psi\rangle
\\&= \langle\psi|A|\psi\rangle.
\end{aligned}
To be clear, this is not a value we expect to see in any particular experiment.
Instead, imagine a huge number of quantum objects, all prepared in the state |\psi\rangle and think about the observable A being measured on each of the objects.
Statistically we then expect the average of our measurement results to be roughly \langle A\rangle.
Note that when A is a projector then \langle\psi|A|\psi\rangle is the probability of the outcome associated with A.