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. 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 dependence.

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: $Sp({\hat {L}}(t))=\left\lbrace l\in \mathbb {R} |0.1m\leq l\leq 0.3m|\right\rbrace$.
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:
$Sp({\hat {\lambda }}(t))=\left\lbrace \lambda \in \mathbb {Z} |1000mm\leq \lambda \leq 3000mm|\right\rbrace$,
which has a discrete spectrum.

Algebra

A physical system is described by its static constitution, dynamics and expectation function. These three things together completely describe how the observables of physical system evolve in relation to one another throughout time and thus knowledge of them may be used to calculate the outcome of particular experiments or computations.

Expectation value function

The expectation value function identifies the specific trajectory in the multiverse the physical system is following (with reference to the Many Worlds Interpretation as used by Deutsch). It must of course be a real number, as the mean value of the observable as the number of measurements of that observable tends to infinity.
Let us define for all observables of the system $\left\langle {\begin{pmatrix}a&b\\b^{*}&c\end{pmatrix}}\right\rangle =a$, and consider the arbitrary observable: $\left\langle {\hat {Z}}(t)\right\rangle ={\begin{pmatrix}-1&0\\0&1\end{pmatrix}}$.
It is obvious that $\left\langle {\hat {Z}}(t)\right\rangle =-1$.
Let us now consider $\left\langle {\hat {X}}(t)\right\rangle ={\begin{pmatrix}0&1\\1&0\end{pmatrix}}$.
It is again obvious that $\left\langle {\hat {X}}(t)\right\rangle =0$, however, the eigenvalues of ${\hat {X}}(t)$ are -1 and 1 and the expected value will never therefore be observed by an individual measurement. What may therefore be deduced is that 50% of measurements will result in the value -1 being observed and 50% in the value 1.

It is said that the observable $\left\langle {\hat {Z}}(t)\right\rangle$ is sharp, whereas $\left\langle {\hat {X}}(t)\right\rangle$ is not sharp. The statement that not all observables can be simultaneously sharp is one way of expressing Heisenberg's Uncertainty Principle.

References

Lectures on Quantum Computation by David Deutsch