Observables
An observable A is a measurable physical property which has a numerical value, for example, spin, position, momentum, or energy.
The term “observable” also extends to any basic measurement in which each outcome has an associated numerical value.
If \lambda_k is the numerical value associated to the outcome |e_k\rangle, then 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 to the projector P_k.
We have already seen the following types of operators:
| normal |
AA^\dagger = A^\dagger A |
| unitary |
A^\dagger = A^{-1} |
| Hermitian (or self-adjoint) |
A^\dagger = A |
| positive semi-definite |
\langle v|A|v\rangle\geqslant 0 for all |v\rangle |
The spectral theorem says that an operator A is normal if and only if it is unitarily diagonalisable: there exists some unitary U and some diagonal D such that A=U^\dagger DU.
Note that unitary, Hermitian, and positive semi-definite operators are all, in particular, normal.
Since (|a\rangle\langle b|)^\dagger=|b\rangle\langle a|, the projectors P_k=|e_k\rangle\langle e_k| are Hermitian, and thus normal, which means that A itself is a normal operator.
Conversely, given 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.
Some textbooks describe observables in terms of Hermitian operators, claiming that the corresponding operators have to be Hermitian “because the outcomes are real numbers”. This is actually a bit backwards. As we say above, the labels can be arbitrary, but, since real number labels are often useful (as we’re about to justify), we tend to only work with Hermitian operators.
For example, the expected value \langle A\rangle (also known as the mean), 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 and the state of the system |\psi\rangle as follows:
\begin{aligned}
\langle A\rangle
&=\sum_k \lambda_k \Pr(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 one particular run of the experiment, but instead a statistical average.
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 expect the average of our measurement results to be roughly \langle A\rangle.
Note that when A is, in particular, a single projector A=\lambda_k|e_k\rangle\langle e_k| then \langle\psi|A|\psi\rangle is the probability of the outcome associated with A.