Exercises: Chapter 4 — Measurement — The Born Rule, Projection, Collapse, and Why Observing a Qubit Changes It

Exercise 4.1 — Born Rule Calculations. A qubit is prepared in the state $|\psi\rangle = \sqrt{0.3}|0\rangle + \sqrt{0.7}e^{i\pi/4}|1\rangle$. Compute the probabilities of obtaining $|0\rangle$ and $|1\rangle$ when measuring in (a) the computational basis, (b) the Hadamard basis $\{|+\rangle, |-\rangle\}$.

Exercise 4.2 — Collapse and Repeated Measurement. A qubit starts in $|0\rangle$. We apply a Hadamard gate, measure in the computational basis, then measure again in the computational basis. What is the probability that both measurements yield the same result? What if we measure in the $X$-basis between the two $Z$-measurements?

Exercise 4.3 — POVM for State Discrimination. Construct a POVM $\{E_0, E_1, E_?\}$ that unambiguously discriminates between $|0\rangle$ and $|+\rangle$ with no errors (but possibly inconclusive outcomes). Verify that your POVM elements are positive semi-definite and sum to the identity. Calculate the probability of an inconclusive outcome.

Exercise 4.4 — No-Cloning Corollary. Prove that the no-cloning theorem implies that it is impossible to perfectly delete an unknown quantum state (the no-deleting theorem). That is, there is no unitary $U$ such that $U(|\psi\rangle \otimes |\psi\rangle) = |\psi\rangle \otimes |0\rangle$ for all $|\psi\rangle$.

Exercise 4.5 — Tomography Simulation. Using Qiskit, prepare the state $|\psi\rangle = \cos(\theta/2)|0\rangle + e^{i\phi}\sin(\theta/2)|1\rangle$ for $\theta = \pi/3$, $\phi = \pi/4$. Perform single-qubit tomography and compare the reconstructed Bloch vector to the theoretical one. Compute the fidelity between the reconstructed and theoretical states.

Exercise 4.6 — Helstrom Bound. Two states $|\psi\rangle = \cos(\pi/8)|0\rangle + \sin(\pi/8)|1\rangle$ and $|\phi\rangle = \cos(\pi/8)|0\rangle - \sin(\pi/8)|1\rangle$ are prepared with equal probability. (a) Compute the overlap $|\langle\psi|\phi\rangle|$. (b) Calculate the Helstrom bound for minimum-error discrimination. (c) Design a projective measurement that achieves this bound.

Exercise 4.7 — Expectation Values and Uncertainty. For the state $|\psi\rangle = \frac{3}{5}|0\rangle + \frac{4}{5}|1\rangle$: (a) Compute $\langle X \rangle$, $\langle Y \rangle$, $\langle Z \rangle$. (b) Compute $\Delta X$, $\Delta Y$, $\Delta Z$. (c) Verify the uncertainty relation $\Delta X \cdot \Delta Y \geq |\langle Z \rangle|$. (d) Find a state for which $\Delta X \cdot \Delta Y = |\langle Z \rangle|$ (saturation of the uncertainty bound).

Exercise 4.8 — POVM Verification. Verify that the POVM elements $E_0 = (2 - \sqrt{2})|1\rangle\langle 1|$, $E_+ = (2 - \sqrt{2})|-\rangle\langle -|$, and $E_? = I - E_0 - E_+$ form a valid POVM (positive semi-definite, sum to identity). Calculate the probability of each outcome when the input is $|0\rangle$ and when it is $|+\rangle$.

Exercise 4.9 — Measurement in a Rotated Basis. A qubit is in state $|\psi\rangle = |+\rangle$. Define the basis $\{|v_0\rangle, |v_1\rangle\}$ where $|v_0\rangle = \cos(\pi/6)|0\rangle + \sin(\pi/6)|1\rangle$ and $|v_1\rangle = -\sin(\pi/6)|0\rangle + \cos(\pi/6)|1\rangle$. (a) Compute $P(v_0)$ and $P(v_1)$. (b) What unitary $U$ maps $\{|0\rangle, |1\rangle\}$ to $\{|v_0\rangle, |v_1\rangle\}$? (c) Construct a quantum circuit to measure in this basis.

Exercise 4.10 — CNOT Is Not a Cloner. Show explicitly that applying a CNOT gate to $|\psi\rangle \otimes |0\rangle$ for $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ produces an entangled state, not a product state $|\psi\rangle \otimes |\psi\rangle$. Compute the density matrix of each qubit separately (partial trace) and show they are both mixed states.