Chapter 23 Quiz: The Density Matrix and Mixed States

Instructions: This quiz covers the core concepts from Chapter 23. For multiple choice, select the single best answer. For true/false, provide a brief justification (1-2 sentences). For short answer, aim for 3-5 sentences. For applied scenarios, show your work.

Multiple Choice (10 questions)

Q1. The density operator for a pure state $|\psi\rangle$ is:

(a) $|\psi\rangle + \langle\psi|$ (b) $|\psi\rangle\langle\psi|$ (c) $\langle\psi|\psi\rangle$ (d) $\hat{I} - |\psi\rangle\langle\psi|$

Q2. A valid density matrix must satisfy all of the following EXCEPT:

(a) $\hat{\rho}^\dagger = \hat{\rho}$ (b) $\text{Tr}(\hat{\rho}) = 1$ (c) $\hat{\rho}^2 = \hat{\rho}$ (d) $\langle\phi|\hat{\rho}|\phi\rangle \geq 0$ for all $|\phi\rangle$

Q3. The purity $\text{Tr}(\hat{\rho}^2)$ of the maximally mixed state of a three-level system (qutrit) equals:

(a) 0 (b) $1/3$ (c) $1/2$ (d) 1

Q4. Two particles are in an entangled state $|\Psi\rangle_{AB}$, and the reduced density matrix of particle $A$ has von Neumann entropy $S(\hat{\rho}_A) = 0.5$ nats. Which statement is correct?

(a) The composite state $|\Psi\rangle_{AB}$ has entropy 0.5 nats (b) The composite state $|\Psi\rangle_{AB}$ has zero entropy (c) Particle $B$ has zero entropy (d) The entanglement cannot be quantified from this information

Q5. In the Bloch ball representation, a qubit density matrix with $\vec{r} = (0, 0, 0)$ corresponds to:

(a) The spin-up state $|{+z}\rangle$ (b) The spin-down state $|{-z}\rangle$ (c) The superposition $|{+x}\rangle$ (d) The maximally mixed state $\hat{I}/2$

Q6. The von Neumann entropy of a $d$-dimensional maximally mixed state $\hat{\rho} = \hat{I}/d$ is:

(a) $0$ (b) $1/d$ (c) $\ln d$ (d) $d \ln d$

Q7. Under pure dephasing, which of the following is true?

(a) Both diagonal and off-diagonal elements of $\hat{\rho}$ decay (b) Only diagonal elements decay; off-diagonal elements are preserved (c) Only off-diagonal elements decay; diagonal elements are preserved (d) The trace of $\hat{\rho}$ decreases over time

Q8. The partial trace $\hat{\rho}_A = \text{Tr}_B(\hat{\rho}_{AB})$ is best described as:

(a) Setting the $B$ degrees of freedom to zero (b) Projecting the composite state onto a specific state of $B$ (c) Summing over all possible states of subsystem $B$ (d) Discarding the $A$ degrees of freedom

Q9. Which of the following density matrices represents a pure state?

(a) $\hat{\rho} = \begin{pmatrix} 1/2 & 0 \\ 0 & 1/2 \end{pmatrix}$

(b) $\hat{\rho} = \begin{pmatrix} 1/2 & 1/2 \\ 1/2 & 1/2 \end{pmatrix}$

(c) $\hat{\rho} = \begin{pmatrix} 3/4 & 0 \\ 0 & 1/4 \end{pmatrix}$

(d) $\hat{\rho} = \begin{pmatrix} 1/3 & 0 \\ 0 & 2/3 \end{pmatrix}$

Q10. The expectation value of an observable $\hat{A}$ in a state described by density operator $\hat{\rho}$ is given by:

(a) $\langle\psi|\hat{A}|\psi\rangle$ (b) $\text{Tr}(\hat{\rho}\hat{A})$ (c) $\text{Tr}(\hat{A})$ (d) $\hat{\rho}\hat{A}$

True/False (4 questions)

Q11. TRUE or FALSE: The von Neumann entropy of a closed quantum system (evolving under the Schrodinger equation with no environmental interaction) can increase over time.

Q12. TRUE or FALSE: A density matrix can always be decomposed into a unique ensemble of pure states with definite classical probabilities.

Q13. TRUE or FALSE: If the reduced density matrix $\hat{\rho}_A$ is a pure state, then the composite state $|\Psi\rangle_{AB}$ is a product state (not entangled).

Q14. TRUE or FALSE: Decoherence requires a deviation from unitary quantum mechanics — it cannot be explained within standard Schrodinger evolution of the system-plus-environment.

Short Answer (4 questions)

Q15. Explain the physical significance of the off-diagonal elements (coherences) of a density matrix. What happens to these elements during decoherence, and what observable consequences does this have?

Q16. Why is the density operator formalism necessary for describing the state of a subsystem that is entangled with another system? Illustrate with the example of the singlet state.

Q17. State the relationship between the $T_1$ and $T_2$ timescales. What physical processes do they describe, and why must $T_2 \leq 2T_1$?

Q18. What is the "pointer basis" in the context of decoherence? What determines which basis becomes the pointer basis, and why does this matter for the quantum-to-classical transition?

Applied Scenarios (2 questions)

Q19. A quantum computing lab prepares superconducting qubits with $T_1 = 80\;\mu$s and $T_2 = 50\;\mu$s. A single-qubit gate takes 20 ns to execute.

(a) What is the pure dephasing time $T_\phi$?

(b) Approximately how many gates can be performed before the qubit coherence drops to $1/e$ of its initial value?

(c) If the qubit starts in state $|{+x}\rangle$ and sits idle for time $t = T_2$, write the density matrix $\hat{\rho}(T_2)$ assuming pure dephasing only.

(d) What is the purity of $\hat{\rho}(T_2)$?

Q20. A spin-1/2 particle is prepared in the state $|\psi\rangle = \frac{\sqrt{3}}{2}|{+z}\rangle + \frac{1}{2}|{-z}\rangle$ and then enters a decohering environment.

(a) Write the initial density matrix $\hat{\rho}(0)$.

(b) After complete dephasing ($t \to \infty$), the off-diagonal elements vanish. Write $\hat{\rho}(\infty)$.

(c) Compute the von Neumann entropy before and after decoherence.

(d) The system was initially pure ($S = 0$). Where did the entropy come from? Is entropy conserved in the total system-plus-environment?

Answer Key

Q1: (b) — The density operator for a pure state is the outer product $|\psi\rangle\langle\psi|$.

Q2: (c) — The condition $\hat{\rho}^2 = \hat{\rho}$ (idempotency) holds only for pure states, not for all valid density matrices. It is a diagnostic, not a requirement.

Q3: (b) — For $d = 3$, $\hat{\rho} = \hat{I}/3$, so $\text{Tr}(\hat{\rho}^2) = \text{Tr}(\hat{I}/9) = 3/9 = 1/3$.

Q4: (b) — A pure composite state always has zero von Neumann entropy: $S(|\Psi\rangle\langle\Psi|) = 0$, regardless of the entanglement entropy of its subsystems.

Q5: (d) — $\vec{r} = 0$ gives $\hat{\rho} = \hat{I}/2$, the maximally mixed state at the center of the Bloch ball.

Q6: (c) — All eigenvalues are $1/d$, so $S = -d \cdot (1/d)\ln(1/d) = \ln d$.

Q7: (c) — Pure dephasing preserves populations (diagonal elements) and exponentially damps coherences (off-diagonal elements).

Q8: (c) — The partial trace sums over all possible states of the traced-out subsystem, not projecting onto any specific one.

Q9: (b) — $\text{Tr}(\hat{\rho}^2) = \text{Tr}\begin{pmatrix} 1/2 & 1/2 \\ 1/2 & 1/2 \end{pmatrix} = 1$. This is $|{+x}\rangle\langle{+x}|$.

Q10: (b) — The trace formula $\text{Tr}(\hat{\rho}\hat{A})$ works for both pure and mixed states.

Q11: FALSE — Unitary evolution preserves the eigenvalues of $\hat{\rho}$, so $S = -\sum_i \lambda_i \ln \lambda_i$ is constant. The entropy of a closed system never changes.

Q12: FALSE — A single density matrix can correspond to infinitely many different ensemble decompositions. For example, $\hat{I}/2$ can be decomposed using any orthonormal basis.

Q13: TRUE — If $\hat{\rho}_A$ is pure for a pure composite state $|\Psi\rangle_{AB}$, then the Schmidt decomposition has exactly one term, meaning $|\Psi\rangle_{AB} = |\psi\rangle_A|\phi\rangle_B$.

Q14: FALSE — Decoherence is entirely within standard unitary quantum mechanics. The system + environment evolves unitarily; decoherence emerges from tracing over the environment.

Q15: The off-diagonal elements encode quantum coherence — the ability of the system to exhibit interference effects. During decoherence, these elements decay exponentially toward zero. Observable consequences include the loss of interference fringes, the disappearance of Ramsey oscillations in atomic clocks, and the decay of qubit fidelity in quantum computers.

Q16: When subsystem $A$ is entangled with $B$, no ket $|\psi\rangle_A$ describes $A$ alone. For the singlet state, $\hat{\rho}_A = \hat{I}/2$: a maximally mixed state that can only be expressed as a density operator, not a ket. This is because entanglement introduces correlations that make the subsystem description inherently probabilistic.

Q17: $1/T_2 = 1/(2T_1) + 1/T_\phi$. $T_1$ is the relaxation time (energy exchange with environment, population decay). $T_2$ is the dephasing time (coherence decay). $T_2 \leq 2T_1$ because energy relaxation always causes phase randomization at rate $1/(2T_1)$, and additional pure dephasing ($T_\phi$) can only make $T_2$ shorter.

Q18: The pointer basis is the preferred basis selected by the system-environment interaction in which the density matrix becomes diagonal after decoherence. It is determined by the form of $\hat{H}_{\text{int}}$ — specifically, the eigenstates of the system observable that couples to the environment. This matters because it determines which states survive decoherence (appear "classical") and which superpositions are destroyed.

Q19: (a) $1/T_\phi = 1/T_2 - 1/(2T_1) = 1/50 - 1/160 = (160 - 50)/(50 \times 160) = 110/8000$. So $T_\phi = 8000/110 \approx 72.7\;\mu$s. (b) The coherence drops to $1/e$ after time $T_2 = 50\;\mu$s. Number of gates: $T_2/(20\;\text{ns}) = 50{,}000/20 = 2{,}500$ gates. (c) $\hat{\rho}(T_2) = \begin{pmatrix} 1/2 & (1/2)e^{-1} \\ (1/2)e^{-1} & 1/2 \end{pmatrix} \approx \begin{pmatrix} 0.500 & 0.184 \\ 0.184 & 0.500 \end{pmatrix}$ (d) $\text{Tr}(\hat{\rho}^2) = 1/2 + 2(1/2)^2 e^{-2} = 1/2 + e^{-2}/2 \approx 0.500 + 0.068 = 0.568$.

Q20: (a) $\hat{\rho}(0) = \begin{pmatrix} 3/4 & \sqrt{3}/4 \\ \sqrt{3}/4 & 1/4 \end{pmatrix}$ (b) $\hat{\rho}(\infty) = \begin{pmatrix} 3/4 & 0 \\ 0 & 1/4 \end{pmatrix}$ (c) Before: $S = 0$ (pure state). After: $S = -\frac{3}{4}\ln\frac{3}{4} - \frac{1}{4}\ln\frac{1}{4} \approx 0.562$ nats. (d) The entropy "came from" the entanglement with the environment — the information about the relative phase was transferred to the system-environment correlations. The total entropy of system + environment is conserved (it remains zero if the composite started pure). The subsystem entropy increased because information was delocalized into inaccessible environmental degrees of freedom.