Self-Assessment Quiz: Measurement

Twenty questions on the Born rule, projection, collapse, expectation values, and what measurement can and cannot do. Answer each before opening the key. Aim for 16+.

Question 1

The Born rule states that the probability of outcome $m$ when measuring $|\psi\rangle$ is:

A) $\langle\psi|M_m|\psi\rangle$ for any operator $M_m$ B) $|\langle m|\psi\rangle|^2$ for a projective measurement onto $|m\rangle$ C) $\langle m|\psi\rangle$ D) $|\langle m|\psi\rangle|$

Question 2

Measuring $|\psi\rangle = 0.6|0\rangle + 0.8|1\rangle$ in the computational basis gives outcome $1$ with probability:

A) $0.8$ B) $0.64$ C) $0.36$ D) $0.64$ for outcome 0

Question 3

After that measurement returns $1$, the post-measurement state is:

A) Unchanged B) $|1\rangle$ C) $0.8|1\rangle$ D) $|+\rangle$

Question 4

Measuring the same qubit again immediately, in the same basis, yields:

A) A fresh random outcome B) The same outcome with certainty C) The opposite outcome D) An error

Question 5

The projector onto outcome $m$ satisfies:

A) $P_m^2 = P_m$ and $P_m^\dagger = P_m$ B) $P_m^2 = I$ C) $P_m$ is unitary D) $\mathrm{Tr}(P_m) = 1$ always

Question 6

The projectors of a complete projective measurement satisfy:

A) $\sum_m P_m = 0$ B) $\sum_m P_m = I$ C) $\prod_m P_m = I$ D) $P_m P_{m'} = I$

Question 7

The post-measurement state after outcome $m$ is:

A) $P_m|\psi\rangle$ B) $P_m|\psi\rangle / \sqrt{\langle\psi|P_m|\psi\rangle}$ C) $|\psi\rangle - P_m|\psi\rangle$ D) $P_m$

Question 8

The expectation value $\langle Z\rangle$ for $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ is:

A) $|\alpha|^2 + |\beta|^2$ B) $|\alpha|^2 - |\beta|^2$ C) $\alpha\beta^*$ D) $0$

Question 9

$\langle Z\rangle$ is estimated experimentally by:

A) A single shot B) $(N_0 - N_1)/(N_0 + N_1)$ over many shots C) $N_0/N_1$ D) Measuring in the $X$ basis

Question 10

Measuring in the $X$ basis on hardware that only measures in $Z$ is done by:

A) Applying $H$ before the measurement B) Applying $X$ before the measurement C) Applying $S$ before the measurement D) It is impossible

Question 11

Measuring in the $Y$ basis requires applying, before a $Z$ measurement:

A) $H$ B) $S^\dagger$ then $H$ C) $T$ D) $Z$ then $H$

Question 12

Shot noise on an expectation value estimated from $N$ shots scales as:

A) $1/N$ B) $1/\sqrt{N}$ C) $1/N^2$ D) Constant

Question 13

To halve the error bar on an expectation value you must:

A) Double the shots B) Quadruple the shots C) Halve the shots D) Add one qubit

Question 14

Measurement of one half of an entangled pair:

A) Sends information to the other half instantly B) Changes the local statistics observable at the other half C) Correlates outcomes but transmits no information D) Breaks unitarity globally

Question 15

The reason measurement cannot be represented by a unitary on the system alone is:

A) It is not linear B) It is irreversible and does not preserve distinguishability of the input C) Its matrix is not square D) It is basis-dependent

Question 16

True or false: Measurement always destroys the qubit physically.

Question 17

True or false: Two non-orthogonal quantum states can be distinguished with certainty by a single measurement.

Question 18

True or false: Choosing a different measurement basis can change what information you extract from the same state.

Question 19

Short answer. You measure 10,000 shots and get 5,120 zeros. Estimate $\langle Z\rangle$ and its uncertainty, and say whether this is consistent with $|+\rangle$.

Question 20

Short answer. Explain why a quantum algorithm's final step is almost always designed so that the correct answer has high amplitude, rather than merely "present" in the superposition.

Answer Key