Observable (quantum computation)

In quantum mechanics, an observable is a property of the system, whose value may be determined by performing physical operations on the system. An observable is equivalent to a degree of freedom in classical physics. To every observable of the system, there is a corresponding self-adjoint operator, that is to say one whose matrix is Hermitian. Upon measurement, the value of the observable must become sharp. This means that the observable takes a value which is one of the eigenvalues of the Hermitian matrix. This set of values is the observable's spectrum. It is the case that for certain observables the expectation value of it (notated by angular brackets) may not be one of the eigenvalues of the matrix.

Simple Example

Let us, in order to demonstrate the concept, examine one observable of a physical system: the length of a spring. We denote this observable ${\hat {L}}(t)$, allowing for the possibility of time dependance.

The Spectrum

If our spring's natural length is 0.1m and its maximum length before it ceases to be Hookean is 0.3m, we may say: $\left\langle {\hat {L}}(t)\right\rangle =\left\lbrace l\in \mathbb {R} |0.1m\leq l\leq 0.3m|\right\rbrace$.
The spectrum is continuous. However, we could also define a new observable, ${\hat {\lambda }}(t)$, which measures the springs length only to the nearest millimeter:
$\left\langle {\hat {\lambda }}(t)\right\rangle =\left\lbrace \lambda \in \mathbb {Z} |1000mm\leq \lambda \leq 3000mm|\right\rbrace$,
which has a discrete spectrum.

References

Lectures on Quantum Computation by David Deutsch