Chapter 11 Exercises: Tensor Products and Composite Systems

Section A: Tensor Product Foundations (Problems 1–8)

Problem 1. (Warm-up) Compute the tensor product $|\psi\rangle \otimes |\phi\rangle$ where $|\psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle + i|1\rangle)$ and $|\phi\rangle = \frac{1}{\sqrt{3}}(|0\rangle + \sqrt{2}|1\rangle)$. Express your answer as a 4-component column vector and verify that it is normalized.

Problem 2. (Kronecker product) Compute the Kronecker product $\hat{\sigma}_x \otimes \hat{\sigma}_z$ explicitly as a $4 \times 4$ matrix. Verify that it is Hermitian and find its eigenvalues.

Problem 3. (Dimension counting) (a) What is the dimension of the Hilbert space for a system of 3 qubits? (b) For a system consisting of a spin-1/2 particle and a spin-1 particle? (c) For 10 qubits? Compare to the number of atoms on Earth ($\sim 10^{50}$) — at what number of qubits does the Hilbert space dimension exceed this?

Problem 4. (Basis construction) Write out all 6 computational basis states for the tensor product of a qubit ($\{|0\rangle, |1\rangle\}$) and a qutrit ($\{|0\rangle, |1\rangle, |2\rangle\}$). Verify orthonormality for three pairs of your choice.

Problem 5. (Operator action) Given the two-qubit state $|\Psi\rangle = \frac{1}{2}(|00\rangle + |01\rangle + |10\rangle + |11\rangle)$, compute: (a) $(\hat{\sigma}_z \otimes \hat{I})|\Psi\rangle$ (b) $(\hat{I} \otimes \hat{\sigma}_x)|\Psi\rangle$ (c) $(\hat{\sigma}_z \otimes \hat{\sigma}_x)|\Psi\rangle$ (d) Verify that $(c) = (a)$ acted upon by $(\hat{I} \otimes \hat{\sigma}_x)$.

Problem 6. (Commutativity of product operators) Prove that $(\hat{A} \otimes \hat{I})$ and $(\hat{I} \otimes \hat{B})$ always commute, for any operators $\hat{A}$ and $\hat{B}$ acting on their respective spaces.

Problem 7. (Inner products) Compute the following inner products on the two-qubit space: (a) $\langle \Psi | \Phi \rangle$ where $|\Psi\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ and $|\Phi\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)$ (b) $\langle \Psi | \hat{\sigma}_z \otimes \hat{\sigma}_z | \Psi \rangle$ for $|\Psi\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ (c) $\langle \Psi | \hat{\sigma}_x \otimes \hat{\sigma}_x | \Psi \rangle$ for the same state

Problem 8. (Three-qubit tensor product) Consider the state $|+\rangle \otimes |0\rangle \otimes |-\rangle$ where $|\pm\rangle = \frac{1}{\sqrt{2}}(|0\rangle \pm |1\rangle)$. (a) Expand this as a sum of computational basis states $|ijk\rangle$. (b) What is the probability of measuring the third qubit in the state $|0\rangle$?

Section B: Entanglement Detection (Problems 9–14)

Problem 9. (Coefficient matrix method) For each of the following two-qubit states, form the coefficient matrix and determine whether the state is entangled or separable: (a) $|\Psi\rangle = \frac{1}{2}|00\rangle + \frac{1}{2}|01\rangle + \frac{1}{2}|10\rangle - \frac{1}{2}|11\rangle$ (b) $|\Psi\rangle = \frac{1}{\sqrt{3}}|00\rangle + \frac{1}{\sqrt{3}}|01\rangle + \frac{1}{\sqrt{3}}|11\rangle$ (c) $|\Psi\rangle = \frac{1}{\sqrt{2}}|00\rangle + \frac{i}{\sqrt{2}}|11\rangle$

For any separable states, find the explicit factorization.

Problem 10. (Parametric entanglement) Consider $|\Psi(\theta)\rangle = \cos\theta|00\rangle + \sin\theta|11\rangle$ where $0 \leq \theta \leq \pi/2$. (a) For what values of $\theta$ is this state separable? (b) For what value of $\theta$ is entanglement maximized? (c) Compute the entanglement entropy $S(\theta) = -\cos^2\theta\log_2\cos^2\theta - \sin^2\theta\log_2\sin^2\theta$ and sketch it.

Problem 11. (Entanglement under local unitaries) Let $|\Psi\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ and let $\hat{U} = \hat{U}_A \otimes \hat{U}_B$ be a local unitary. (a) Show that $\hat{U}|\Psi\rangle$ has the same Schmidt coefficients as $|\Psi\rangle$. (b) Conclude that local unitaries cannot create or destroy entanglement. (c) What about non-local unitaries like CNOT?

Problem 12. (Entanglement of superpositions) (a) Is the sum of two product states always entangled? Give an example or counterexample. (b) Is the sum of two entangled states always entangled? Give an example or counterexample.

Problem 13. (Qubit-qutrit system) Consider the state $|\Psi\rangle = \frac{1}{\sqrt{3}}(|0,0\rangle + |0,1\rangle + |1,2\rangle)$ in $\mathbb{C}^2 \otimes \mathbb{C}^3$. Is this state entangled? Find its coefficient matrix and determine its rank.

Problem 14. (GHZ vs. W states) Consider the three-qubit states: $$|GHZ\rangle = \frac{1}{\sqrt{2}}(|000\rangle + |111\rangle), \quad |W\rangle = \frac{1}{\sqrt{3}}(|001\rangle + |010\rangle + |100\rangle)$$ (a) Show that neither can be written as a product state. (b) Trace out qubit 3 from each state. Which reduced two-qubit state is entangled? (c) What does this tell you about the robustness of entanglement in GHZ vs. W states?

Section C: Schmidt Decomposition (Problems 15–20)

Problem 15. (Direct Schmidt decomposition) Find the Schmidt decomposition of $|\Psi\rangle = \frac{1}{\sqrt{2}}|00\rangle + \frac{1}{2}|01\rangle + \frac{1}{2}|10\rangle$.

Problem 16. (Schmidt decomposition via SVD) For the state $|\Psi\rangle = \frac{1}{2}|00\rangle + \frac{1}{2}|01\rangle + \frac{1}{2}|10\rangle - \frac{1}{2}|11\rangle$: (a) Form the coefficient matrix $C$. (b) Compute $CC^\dagger$ and find its eigenvalues. (c) Find the Schmidt decomposition explicitly. (d) Is this state entangled?

Problem 17. (Schmidt rank) Without performing the full decomposition, determine the Schmidt rank of: (a) $|\Psi\rangle = |01\rangle$ (b) $|\Psi\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |01\rangle)$ (c) $|\Psi\rangle = \frac{1}{\sqrt{3}}(|00\rangle + |01\rangle + |10\rangle)$ (d) $|\Psi\rangle = \frac{1}{2}(|00\rangle + |01\rangle + |10\rangle + |11\rangle)$

Problem 18. (Uniqueness of Schmidt decomposition) Consider $|\Psi\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$. (a) Verify that this is already in Schmidt form. (b) Since both Schmidt coefficients are equal ($\lambda_1 = \lambda_2 = 1/\sqrt{2}$), the Schmidt bases are not unique. Find a different Schmidt decomposition of the same state. Hint: try the $|\pm\rangle$ basis.

Problem 19. (Schmidt decomposition in higher dimensions) Consider the state in $\mathbb{C}^3 \otimes \mathbb{C}^3$: $$|\Psi\rangle = \frac{1}{\sqrt{6}}(|00\rangle + |01\rangle + |10\rangle + |11\rangle + |20\rangle + |21\rangle)$$ (a) Write the coefficient matrix. (b) What is the maximum possible Schmidt rank? (c) Find the Schmidt decomposition. Hint: factor the coefficient matrix.

Problem 20. (Schmidt coefficients and entanglement measures) A two-qutrit state has Schmidt coefficients $\lambda_1 = \sqrt{0.6}$, $\lambda_2 = \sqrt{0.3}$, $\lambda_3 = \sqrt{0.1}$. (a) Verify normalization. (b) Compute the entanglement entropy $S = -\sum_k \lambda_k^2 \log_2(\lambda_k^2)$. (c) What are the maximum and minimum possible values of $S$ for a two-qutrit system? (d) Compute the Schmidt number (Schmidt rank). Is this state maximally entangled?

Section D: Partial Trace and Reduced Density Matrices (Problems 21–26)

Problem 21. (Partial trace of a product state) For the product state $|+\rangle|0\rangle$: (a) Compute $\hat{\rho}_{AB} = |+,0\rangle\langle +,0|$ as a $4 \times 4$ matrix. (b) Compute $\hat{\rho}_A = \text{Tr}_B(\hat{\rho}_{AB})$. (c) Compute $\hat{\rho}_B = \text{Tr}_A(\hat{\rho}_{AB})$. (d) Verify that $\hat{\rho}_A$ is a pure state and $\hat{\rho}_A = |+\rangle\langle +|$.

Problem 22. (Partial trace of an entangled state) For $|\Psi\rangle = \cos\theta|00\rangle + \sin\theta|11\rangle$: (a) Compute $\hat{\rho}_A$ as a function of $\theta$. (b) Compute $\text{Tr}(\hat{\rho}_A^2)$ and show it equals 1 only when $\theta = 0$ or $\theta = \pi/2$. (c) Interpret the result: when is subsystem $A$ in a pure state?

Problem 23. (Partial trace with off-diagonal terms) Compute the reduced density matrix $\hat{\rho}_A$ for $|\Psi\rangle = \frac{1}{\sqrt{3}}(|00\rangle + |01\rangle + |10\rangle)$. Is $\hat{\rho}_A$ a pure or mixed state? Compute $\text{Tr}(\hat{\rho}_A^2)$.

Problem 24. (Reduced density matrix of Bell states) Show that for all four Bell states, $\hat{\rho}_A = \frac{1}{2}\hat{I}_A$. What does this tell us about making local measurements on one half of any Bell pair?

Problem 25. (Partial trace of a mixed state) Consider the mixed state $\hat{\rho}_{AB} = \frac{1}{2}|\Phi^+\rangle\langle\Phi^+| + \frac{1}{2}|00\rangle\langle 00|$. Compute $\hat{\rho}_A = \text{Tr}_B(\hat{\rho}_{AB})$.

Problem 26. (Three-body partial trace) For the three-qubit GHZ state $|GHZ\rangle = \frac{1}{\sqrt{2}}(|000\rangle + |111\rangle)$: (a) Compute $\hat{\rho}_{AB} = \text{Tr}_C(|GHZ\rangle\langle GHZ|)$. (b) Is $\hat{\rho}_{AB}$ a pure state? (c) Compute $\hat{\rho}_A = \text{Tr}_B(\hat{\rho}_{AB})$. (d) Compare $\hat{\rho}_A$ to the reduced state of one qubit from a Bell pair. Are they the same?

Section E: Bell States and Applications (Problems 27–30)

Problem 27. (Bell basis expansion) Express the following states in the Bell basis: (a) $|00\rangle$ (b) $|+\rangle|+\rangle$ (c) $\frac{1}{\sqrt{2}}(|00\rangle + i|11\rangle)$

Problem 28. (Bell state measurement correlations) For the state $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$: (a) What is the probability that both qubits yield outcome 0 when measured in the $\{|0\rangle, |1\rangle\}$ basis? (b) What is the probability that both yield outcome $+$ when measured in the $\{|+\rangle, |-\rangle\}$ basis? Hint: rewrite $|\Phi^+\rangle$ in the $\{|+\rangle, |-\rangle\}$ basis. (c) Compute $\langle \hat{\sigma}_z \otimes \hat{\sigma}_z \rangle$ and $\langle \hat{\sigma}_x \otimes \hat{\sigma}_x \rangle$ for $|\Phi^+\rangle$.

Problem 29. (Quantum teleportation circuit) Alice and Bob share $|\Phi^+\rangle_{23}$ and Alice has an unknown qubit $|\psi\rangle_1 = \alpha|0\rangle + \beta|1\rangle$. (a) Write the full three-qubit state $|\psi\rangle_1 \otimes |\Phi^+\rangle_{23}$. (b) Expand qubit 1 and 2 in the Bell basis (rewrite as a sum over Bell states of qubits 1 and 2, times states of qubit 3). (c) Show that after Alice measures qubits 1 and 2 in the Bell basis, Bob's qubit 3 is always proportional to a Pauli transformation of $|\psi\rangle$. (d) What classical information must Alice send Bob? How many classical bits?

Problem 30. (Superdense coding) Alice and Bob share $|\Phi^+\rangle$. Alice can apply one of $\{\hat{I}, \hat{\sigma}_x, \hat{\sigma}_z, i\hat{\sigma}_y\}$ to her qubit before sending it to Bob. (a) What state does Bob receive in each of the four cases? (b) Can Bob distinguish these four states? Hint: are they orthogonal? (c) How many classical bits of information has Alice sent by transmitting one qubit? Why is this called "superdense" coding?

Solutions Hints

Problem 1: Use the Kronecker product formula. The result should have $|\alpha|^2 = 1$ verified by summing squared moduli.

Problem 10: $S(\theta)$ is maximized at $\theta = \pi/4$ and vanishes at $\theta = 0, \pi/2$.

Problem 16: The state is separable — check if $\det(C) = 0$. It factors as $|+\rangle|-\rangle$ where $|-\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)$.

Problem 29: The key identity is $|0\rangle|0\rangle = \frac{1}{\sqrt{2}}(|\Phi^+\rangle + |\Phi^-\rangle)$ and similar for other product states.