While obtaining measurement outcome probabilities gives us useful information about a qubit's state, we're often interested in other measurable quantities that correspond to something physical, such as its energy. In quantum mechanics, and consequently quantum computing, physical quantities are related to a special type of object called an **observable**. Observables correspond to [Hermitian matrices](https://en.wikipedia.org/wiki/Hermitian_matrix) whose eigenvalues represent the possible values of the measurement outcome. *Tip*. A matrix $B$ is **Hermitian** if $B = B^\dagger.$ Hermitian matrices have real eigenvalues, which is a sensible property for measurable physical quantities to have. As an example, let us consider the Pauli $Z$ operation. You can check that $$ Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$ is Hermitian, and has eigenvalues $1$ and $-1$ with corresponding eigenvectors, or eigenstates $\vert 0\rangle$ and $\vert 1\rangle.$ Similarly, Pauli $X$ is Hermitian and has eigenvalues $1$ and $-1,$ but the associated eigenstates are $\vert +\rangle$ and $\vert -\rangle.$ If we were to "measure the observable Pauli $Z$" (or Pauli $X$) for a given qubit state, we would obtain one of the two possible eigenvalues. (This is why in the earlier nodes, we computed the eigenvalues of some of these operations!) Furthermore, following the measurement, the qubit's state will be that of the eigenvector of the corresponding measurement outcome. In order to get a clearer picture of the value of an observable for a given qubit state, we need to measure its **expectation value**. This is essentially the weighted average of what we would see over many experiments. --- ***Exercise I.10.1.*** Suppose we measure an observable $B$ for which there is a $70\%$ chance we obtain eigenvalue $1,$ and $30\%$ chance we obtain the eigenvalue $-1.$ What is the expectation value of $B?$

Solution.

The expectation value of $B$ is $$ \langle B \rangle = 0.7 \times 1 + 0.3 \times (-1) = 0.4. $$ While in each individual experiment we see $1$ or $-1,$ we'll see $1$ more often, and so the expected value is closer to $1$ than it is to $-1.$ ▢

--- Expressed mathematically, the expectation value of some observable $B$ for a state $\vert \psi\rangle$ is given by $$ \langle B \rangle = \langle \psi \vert B \vert \psi \rangle. \tag{1} $$ Namely, we first compute the result of the matrix $B$ applied to $\vert \psi\rangle$ followed by its inner product with $\vert \psi\rangle.$ --- ***Exercise I.10.2.*** Let $$ \begin{equation} \vert \psi\rangle = \frac{4}{5} \vert 0\rangle - \frac{3}{5} e^{i \pi/3} \vert 1\rangle \end{equation}, $$ and suppose we want to measure the observable $$ B = \begin{pmatrix} 1 & -2i \\ 2i & 2 \end{pmatrix}. $$ (a) What are the possible outcomes of the measurement?

Tip.

*Tip*. It is common to specify a measurement by saying "measure in the $X$ basis", or "measure $X$". These statements relate directly back to projective measurements and observables respectively. Measuring in the $X$ basis means to take a projective measurement with respect to the basis composed of the eigenvectors of $X$ (these happen to be $\vert +\rangle$ and $\vert -\rangle$). The output of this measurement is either $\vert +\rangle,$ or $\vert -\rangle.$ Measuring $X,$ on the other hand, means to measure the observable $X,$ which has eigenvalues $+1$ and $-1,$ corresponding to those same eigenstates $\vert +\rangle$ and $\vert -\rangle.$ The output of this measurement is then either $+1,$ or $-1.$ While the two measurements are ultimately the same, the context and language indicate what the output value should look like.

Solution.

To determine the possible measurement outcomes, we must compute the eigenvalues of $B.$ They are $\lambda_1 = 3.56155281$ and $\lambda_2=-0.56155281.$ ▢

(b) For each outcome, what state is the qubit in after the measurement? Express this in the computational basis.

Solution.

The possible outcomes are the associated eigenvectors, $$ \begin{align*} \vert v_1\rangle &= -0.61541221i \vert 0 \rangle + 0.78820544 \vert 1\rangle, \\ \vert v_2\rangle &= 0.78820544 \vert 0 \rangle - 0.61541221i \vert 1 \rangle. \end{align*} $$

▢

Solution.

The analytical expectation value is $$ \langle \psi \vert B \vert \psi \rangle = \frac{1}{25} \begin{pmatrix} 4 & -3 e^{-i\pi/3} \end{pmatrix} \begin{pmatrix} 1 & -2i \\ 2i & 2 \end{pmatrix} \begin{pmatrix} 4 \\ -3 e^{i\pi/3} \end{pmatrix} = -0.302769. $$

▢

(d) Suppose we prepared this state and measured $B$ 1000 times. 54 of the trials yielded the larger of the two possible measurement outcomes, and the remaining tries yielded the smaller one. What is the experimentally-obtained expectation value of $B?$

Solution.

To compute the experimentally obtained eigenvalue, we take the weighted average: $$ \langle \tilde{B} \rangle = \frac{54 \cdot 3.56155281 + 946 \cdot (-0.56155281)}{1000} = -0.338905. $$ The experimentally-obtained value does not exactly match the analytically-obtained one in this case, but will do so in the limit of an infinite number of trials. ▢

Codercise I.10.1 — Measurement of the PauliY observable

Codercise I.10.2 — Setting up the number of experiment shots

Codercise I.10.3 — Evaluating the samples

Codercise I.10.4 — The variance of sample measurements

Observables

Codercise I.10.1 — Measurement of the `PauliY` observable