Another Look at Averages in Quantum Mechanics

There is a nice way of writing averages of measurements in quantum mechanics using the Dirac bracket notation. This method often shows up in discussions of QC. The basic idea is that the average can be written, as we saw in the previous section, as a sum of products of probabilities and measurement outcome values. We also know that the probabilities can be written as the squares of the amplitudes associated with a particular state, and that the sum of those squares shows up when we multiply a right (column) state vector by the corresponding left (row) state vector. For example, for a two-qubit system with state amplitudes $c$, $d$, $e$, and $f$, we have

$$\left(c \hspace{0.2cm} d \hspace{0.2cm} e \hspace{0.2cm} f \right) \begin{pmatrix} c \\ d \\ e \\ f \end{pmatrix} = c^2+d^2+e^2+f^2.$$

The final ingredient is the fact that every measurable property $M$ can be represented by an operator and that the appropriate measurement basis states are the states that satisfy

$$M |α⟩ = λ_α |α⟩ \\ M |β⟩ = λ_β |β⟩,$$

where $λ_α$ and $λ_β$ are the numerical values associated with the operator $M$. States that satisfy these conditions are called eigenstates or eigenvectors associated with the operator $M$. So, $λ_α$ and $λ_β$ are called the eigenvalues for $M$.

How does this tie into average values? For a single qubit, the quantum state vector can always be written as a sum of $M$’s eigenvectors:

$$|ψ⟩ = a_α |α⟩ + a_β |β⟩.$$

When the operator $M$ acts on this state, we get

$$M |ψ⟩ = a_αM |α⟩ + a_βM |β⟩ = a_αλ_α |α⟩ + a_βλ_β |β⟩.$$

In matrix form, the equation above becomes

$$\left( M \right) \begin{pmatrix} a_α \\ a_β \\ \end{pmatrix} = \begin{pmatrix} a_α λ_α\\ a_β λ_β\\ \end{pmatrix}.$$

The next step might not be obvious, but it turns out to be just what we need to calculate the average of a sequence of measurements of $M$. We multiply this equationEq_14_21 on the left by the left vector associated with the state $|ψ⟩$:

$$|ψ | M | ψ⟩ \Rightarrow (a_α \hspace{0.2cm} a_β) \begin{pmatrix} a_α λ_α\\ a_β λ_β\\ \end{pmatrix} = a_α^2 λ_α a^2_β λ_β.$$

Surprisingly, the result is just the average of the $M$ measurements, a generalization of the average value expression in this equationEq_14_9. The Dirac bracket is a reminder that the left side of the equation above is indeed an average of the measurements of the property represented by the operator $M$ given that the qubit has been prepared in the state $|ψ⟩$. Some authors like to call the bracket $⟨ψ| M |ψ⟩$ a “sandwich”—the operator $M$ is sandwiched between two slices of $ψ$ bread.