Chapter 23 Exercises: The Density Matrix and Mixed States

Part A: Conceptual Questions (⭐)

These questions test your understanding of the core ideas. No calculations required unless specified.

A.1 A friend says: "A 50/50 mixture of $|{+z}\rangle$ and $|{-z}\rangle$ is the same as the superposition $|{+x}\rangle = \frac{1}{\sqrt{2}}(|{+z}\rangle + |{-z}\rangle)$, because in both cases you get 50% spin-up and 50% spin-down if you measure along $z$." Identify the flaw in this reasoning. What measurement would distinguish these two states?

A.2 Explain why the global phase of a ket $|\psi\rangle \to e^{i\phi}|\psi\rangle$ is unobservable, and show explicitly that the density operator $\hat{\rho} = |\psi\rangle\langle\psi|$ is invariant under this transformation. Why is this considered an advantage of the density operator formalism?

A.3 Two spin-1/2 particles are in the state $|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|{+z}\rangle|{-z}\rangle + |{-z}\rangle|{+z}\rangle)$. Without doing any calculation, what do you expect the reduced density matrix of particle $A$ to be? Justify your answer using the symmetry of the problem.

A.4 A qubit density matrix has Bloch vector $\vec{r} = (0.3, 0, 0.4)$. Is this state pure or mixed? What is its purity? What direction on the Bloch sphere is it "closest to"?

A.5 Explain in your own words why the von Neumann entropy of a pure state is zero. Why does this make physical sense in terms of information content?

A.6 The decoherence time for a dust grain in air is approximately $10^{-31}$ s. Explain qualitatively why: (a) larger objects decohere faster, (b) objects in denser environments decohere faster, and (c) superpositions of more widely separated positions decohere faster.

A.7 A colleague claims: "Decoherence solves the measurement problem." Evaluate this claim. What does decoherence accomplish, and what does it leave unresolved?

A.8 Explain why the density operator formalism is essential for describing thermal equilibrium states. Could you describe a system at temperature $T > 0$ using a single ket? Why or why not?

Part B: Applied Problems (⭐⭐)

These problems require direct application of the chapter's key equations and techniques.

B.1: Constructing and Analyzing Density Matrices

Consider the ensemble where a spin-1/2 particle is in state $|{+z}\rangle$ with probability $2/3$ and state $|{+x}\rangle$ with probability $1/3$.

(a) Write the density matrix $\hat{\rho}$ in the $\{|{+z}\rangle, |{-z}\rangle\}$ basis.

(b) Verify that $\hat{\rho}$ is Hermitian, has trace 1, and is positive semi-definite (find its eigenvalues).

(c) Compute the purity $\text{Tr}(\hat{\rho}^2)$. Is this a pure or mixed state?

(d) Compute $\langle\hat{S}_z\rangle$ and $\langle\hat{S}_x\rangle$.

(e) Find the Bloch vector $\vec{r}$ and verify that $|\vec{r}| < 1$.

B.2: Von Neumann Entropy Calculations

For each of the following density matrices, calculate the von Neumann entropy $S = -\text{Tr}(\hat{\rho}\ln\hat{\rho})$:

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

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

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

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

Rank them from most to least entropic and explain the pattern.

B.3: Partial Trace Computation

Two qubits are in the state:

$$|\psi\rangle = \frac{1}{\sqrt{3}}|00\rangle + \frac{1}{\sqrt{3}}|01\rangle + \frac{1}{\sqrt{3}}|11\rangle$$

(a) Write the density matrix $\hat{\rho}_{AB} = |\psi\rangle\langle\psi|$ as a $4\times 4$ matrix in the $\{|00\rangle, |01\rangle, |10\rangle, |11\rangle\}$ basis.

(b) Compute the reduced density matrix $\hat{\rho}_A = \text{Tr}_B(\hat{\rho}_{AB})$.

(c) Compute the reduced density matrix $\hat{\rho}_B = \text{Tr}_A(\hat{\rho}_{AB})$.

(d) Verify that $\text{Tr}(\hat{\rho}_A) = \text{Tr}(\hat{\rho}_B) = 1$.

(e) Compute the von Neumann entropy of $\hat{\rho}_A$. Is the state entangled?

B.4: Expectation Values via Trace

A spin-1/2 particle has density matrix $\hat{\rho} = \frac{1}{4}\begin{pmatrix} 3 & 1-i \\ 1+i & 1 \end{pmatrix}$.

(a) Verify this is a valid density matrix (Hermitian, unit trace, positive semi-definite).

(b) Compute $\langle\hat{S}_x\rangle$, $\langle\hat{S}_y\rangle$, $\langle\hat{S}_z\rangle$ using $\langle\hat{A}\rangle = \text{Tr}(\hat{\rho}\hat{A})$.

(c) Determine the Bloch vector and the purity.

(d) Is this state pure or mixed?

B.5: Thermal Density Matrix of a Harmonic Oscillator

A quantum harmonic oscillator with frequency $\omega$ is in thermal equilibrium at temperature $T$.

(a) Write the density matrix $\hat{\rho}$ in the energy eigenbasis $\{|n\rangle\}$, with $E_n = (n + 1/2)\hbar\omega$.

(b) Show that $\hat{\rho}$ is diagonal in the energy basis and compute the diagonal elements $p_n$.

(c) Compute the partition function $Z = \text{Tr}(e^{-\beta\hat{H}})$ and show that $Z = \frac{e^{-\beta\hbar\omega/2}}{1 - e^{-\beta\hbar\omega}}$.

(d) Compute $\langle\hat{H}\rangle = \text{Tr}(\hat{\rho}\hat{H})$ and show that it reduces to $\langle E\rangle = \hbar\omega\left(\frac{1}{2} + \frac{1}{e^{\beta\hbar\omega} - 1}\right)$, recognizing the Planck/Bose-Einstein mean occupation number.

(e) Compute the von Neumann entropy and verify that $S \to 0$ as $T \to 0$.

B.6: Dephasing Dynamics

A qubit starts in the state $|{+x}\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ and undergoes pure dephasing with rate $\Gamma$.

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

(b) Write $\hat{\rho}(t)$ under pure dephasing.

(c) Compute the purity $\text{Tr}(\hat{\rho}(t)^2)$ as a function of time.

(d) Compute the von Neumann entropy $S(t)$ as a function of time. (Hint: diagonalize $\hat{\rho}(t)$ first.)

(e) At what time does the purity reach $3/4$ (halfway between pure and maximally mixed)?

(f) Sketch the trajectory of the Bloch vector as a function of time. Where does it start? Where does it end up?

Part C: Advanced Problems (⭐⭐⭐)

These problems require synthesis of multiple concepts or advanced mathematical reasoning.

C.1: Ambiguity of Ensemble Decomposition

Consider the density matrix $\hat{\rho} = \frac{1}{2}\hat{I}$ (maximally mixed qubit state).

(a) Show that this can be written as $\hat{\rho} = \frac{1}{2}|{+z}\rangle\langle{+z}| + \frac{1}{2}|{-z}\rangle\langle{-z}|$.

(b) Show that it can also be written as $\hat{\rho} = \frac{1}{2}|{+x}\rangle\langle{+x}| + \frac{1}{2}|{-x}\rangle\langle{-x}|$.

(c) Show that it can also be written as $\hat{\rho} = \frac{1}{3}|{+z}\rangle\langle{+z}| + \frac{1}{3}|{+x}\rangle\langle{+x}| + \frac{1}{3}|{-y}\rangle\langle{-y}|$. (Hint: compute the sum explicitly.)

Wait — does part (c) actually work? Verify by explicit calculation. If it does not give $\hat{I}/2$ exactly, find the correct three-state decomposition using states uniformly distributed on the Bloch sphere (at $120°$ intervals in the $xz$-plane).

(d) What physical principle guarantees that many different ensembles produce the same density matrix?

C.2: Purification Theorem

Any mixed state $\hat{\rho}_A$ of system $A$ can be obtained as the reduced density matrix of some pure state $|\Psi\rangle_{AB}$ of a larger system $AB$. This pure state is called a purification of $\hat{\rho}_A$.

(a) Prove this for an arbitrary density matrix $\hat{\rho}_A = \sum_i \lambda_i |i\rangle\langle i|$ by constructing $|\Psi\rangle_{AB}$ explicitly. (Hint: let $\mathcal{H}_B$ have the same dimension as $\mathcal{H}_A$, and try $|\Psi\rangle_{AB} = \sum_i \sqrt{\lambda_i}\;|i\rangle_A|i\rangle_B$.)

(b) Verify that $\text{Tr}_B(|\Psi\rangle_{AB}\langle\Psi|) = \hat{\rho}_A$.

(c) Is the purification unique? If not, how are different purifications related?

(d) Apply this to the thermal state of a qubit: $\hat{\rho} = p_0|0\rangle\langle 0| + p_1|1\rangle\langle 1|$. Write down the purification and interpret the ancilla system $B$ physically.

C.3: Entanglement Entropy and the Schmidt Decomposition

A bipartite pure state $|\Psi\rangle_{AB}$ can be written in its Schmidt decomposition (see Chapter 11):

$$|\Psi\rangle_{AB} = \sum_k \sqrt{\lambda_k}\;|\alpha_k\rangle_A|\beta_k\rangle_B$$

where $\lambda_k \geq 0$, $\sum_k \lambda_k = 1$, and $\{|\alpha_k\rangle\}$, $\{|\beta_k\rangle\}$ are orthonormal.

(a) Show that the reduced density matrices are $\hat{\rho}_A = \sum_k \lambda_k |\alpha_k\rangle\langle\alpha_k|$ and $\hat{\rho}_B = \sum_k \lambda_k |\beta_k\rangle\langle\beta_k|$.

(b) Prove that $S(\hat{\rho}_A) = S(\hat{\rho}_B)$.

(c) Show that the state is a product state (not entangled) if and only if there is exactly one nonzero Schmidt coefficient.

(d) For two qubits in the state $|\psi\rangle = \cos\theta|00\rangle + \sin\theta|11\rangle$, compute the entanglement entropy as a function of $\theta$. For what value of $\theta$ is the entanglement maximized?

C.4: Von Neumann Equation Derivation

Starting from the Schrodinger equation $i\hbar\frac{d}{dt}|\psi_k(t)\rangle = \hat{H}|\psi_k(t)\rangle$:

(a) Derive the von Neumann equation $i\hbar\frac{d\hat{\rho}}{dt} = [\hat{H}, \hat{\rho}]$ by taking the time derivative of $\hat{\rho} = \sum_k p_k |\psi_k(t)\rangle\langle\psi_k(t)|$.

(b) Show that $\text{Tr}(\hat{\rho})$ is conserved under von Neumann evolution.

(c) Show that $\text{Tr}(\hat{\rho}^2)$ is conserved under von Neumann evolution. What does this imply about entropy?

(d) Explain physically why unitary evolution cannot change a pure state into a mixed state. Why is this significant for the measurement problem?

C.5: Decoherence of a Superposition of Coherent States

Consider a quantum harmonic oscillator prepared in a "Schrodinger cat state" — a superposition of two coherent states $|\alpha\rangle$ and $|-\alpha\rangle$ with $\alpha$ real and large:

$$|\psi_{\text{cat}}\rangle = \frac{1}{\sqrt{2(1 + e^{-2\alpha^2})}}(|\alpha\rangle + |-\alpha\rangle)$$

(a) Write the density matrix $\hat{\rho}_{\text{cat}}$ and identify the diagonal and off-diagonal terms.

(b) The overlap between coherent states is $\langle\beta|\alpha\rangle = e^{-|\alpha - \beta|^2/2}$. For large $\alpha$, show that $\langle -\alpha|\alpha\rangle \approx e^{-2\alpha^2} \approx 0$, so the two components are nearly orthogonal.

(c) Under amplitude damping (coupling to a zero-temperature bath), the coherent state amplitudes decay: $|\alpha\rangle \to |{\alpha e^{-\kappa t/2}}\rangle$. Show that the off-diagonal coherences decay as $e^{-2\alpha^2(1 - e^{-\kappa t})}$, which for short times is approximately $e^{-2\alpha^2 \kappa t}$.

(d) The decoherence rate is proportional to $\alpha^2$ — more macroscopic superpositions ($|\alpha|$ large) decohere faster. Explain physically why this is the case.

(e) For $\alpha = 5$ and $\kappa = 10^6\;\text{s}^{-1}$ (typical cavity decay rate), estimate the decoherence time. How many oscillation periods $2\pi/\omega$ does this correspond to for $\omega = 10^{10}$ rad/s?

C.6: Quantum Mutual Information

For a bipartite system $AB$, the quantum mutual information is defined as:

$$I(A:B) = S(\hat{\rho}_A) + S(\hat{\rho}_B) - S(\hat{\rho}_{AB})$$

(a) Show that for a product state $\hat{\rho}_{AB} = \hat{\rho}_A \otimes \hat{\rho}_B$, the mutual information vanishes.

(b) For a pure entangled state, show that $I(A:B) = 2S(\hat{\rho}_A)$.

(c) Compute $I(A:B)$ for the singlet state $|\Psi^-\rangle$. Express your answer in nats and in bits.

(d) Argue that $I(A:B) \geq 0$ always (this is the quantum analogue of the subadditivity of entropy). What would negative mutual information mean, and why is it forbidden?

Part D: Computational Problems (⭐⭐)

These problems should be solved using the Python code from this chapter.

D.1 Use the DensityMatrix class from code/example-01-density.py to verify your answers to Problem B.1. Construct the density matrix from the ensemble, compute purity, entropy, and expectation values programmatically.

D.2 Modify the decoherence simulation in code/example-01-density.py to simulate a qubit starting in the state $\hat{\rho}(0) = |{+y}\rangle\langle{+y}|$ (i.e., initial Bloch vector $\vec{r} = (0, 1, 0)$) undergoing simultaneous dephasing ($T_2 = 10\;\mu$s) and amplitude damping ($T_1 = 50\;\mu$s). Plot: (a) the three Bloch vector components vs. time, (b) the purity vs. time, (c) the von Neumann entropy vs. time.

D.3 Write a function that takes a random $4\times 4$ density matrix for two qubits and computes both reduced density matrices and the entanglement entropy. Test it on: (a) the four Bell states, (b) the state $|\psi\rangle = \cos\theta|00\rangle + \sin\theta|11\rangle$ for $\theta = 0, \pi/8, \pi/4, 3\pi/8, \pi/2$, and (c) five random pure states generated from random unitary matrices acting on $|00\rangle$.

D.4 Reproduce the thermal qubit entropy plot from Section 23.6 using Python. Plot $S(T)$ as a function of $k_BT/\Delta E$ from $0.01$ to $10$ on a semi-log scale. Mark the points where $S = 0.1\ln 2$, $0.5\ln 2$, and $0.9\ln 2$.

D.5 Implement the Luders measurement update rule from Section 23.5. Write a function measure(rho, projectors) that takes a density matrix and a list of projection operators, randomly selects an outcome according to the Born rule probabilities, and returns the post-measurement state. Test it by starting in $|{+x}\rangle\langle{+x}|$, measuring $\hat{S}_z$ 10,000 times, and verifying that the statistics match $P(\pm\hbar/2) = 1/2$.

D.6 Simulate the decoherence of a "Schrodinger cat" state of a harmonic oscillator. Represent the cat state $|\psi_{\text{cat}}\rangle \propto |\alpha\rangle + |-\alpha\rangle$ in the Fock basis (truncated to $n_{\max} = 40$), using $\alpha = 3$. Apply amplitude damping by evolving the coherent state amplitudes as $\alpha \to \alpha e^{-\kappa t/2}$ and compute the density matrix at each time step. Plot the Wigner function at $t = 0$, $t = 1/(4\kappa)$, and $t = 1/\kappa$, showing the disappearance of the interference fringes.

Part E: Challenge Problems (⭐⭐⭐⭐)

E.1: The No-Broadcasting Theorem

The no-cloning theorem (Chapter 11) states that an unknown pure state cannot be copied. The no-broadcasting theorem generalizes this to mixed states: there is no quantum operation that takes $\hat{\rho}$ to $\hat{\rho} \otimes \hat{\rho}$ for an arbitrary unknown mixed state.

Prove a simplified version: for a qubit, show that the map $\hat{\rho} \to \hat{\rho} \otimes \hat{\rho}$ is not linear. (Hint: consider $\hat{\rho}_1 = |0\rangle\langle 0|$ and $\hat{\rho}_2 = |1\rangle\langle 1|$ and their mixture.)

E.2: Strong Subadditivity

The von Neumann entropy satisfies the strong subadditivity inequality for three systems $A$, $B$, $C$:

$$S(\hat{\rho}_{ABC}) + S(\hat{\rho}_B) \leq S(\hat{\rho}_{AB}) + S(\hat{\rho}_{BC})$$

This is one of the deepest results in quantum information theory. Verify it numerically for a random three-qubit state (use QuTiP or NumPy). Generate 1000 random states and check that the inequality is never violated.

Hints and Selected Answers

A.1: Measure along $x$. The superposition $|{+x}\rangle$ gives spin-up with 100% probability. The mixture gives 50%.

B.1(a): $\hat{\rho} = \frac{1}{6}\begin{pmatrix} 5 & 1 \\ 1 & 1 \end{pmatrix}$

B.1(c): $\text{Tr}(\hat{\rho}^2) = 27/36 = 3/4$

B.2(c): $S = -\frac{3}{4}\ln\frac{3}{4} - \frac{1}{4}\ln\frac{1}{4} \approx 0.562$ nats

B.3(b): $\hat{\rho}_A = \frac{1}{3}\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$

B.5(d): $\langle E\rangle = \frac{\hbar\omega}{2}\coth\!\left(\frac{\beta\hbar\omega}{2}\right)$

B.6(e): $t = \frac{\ln 2}{2\Gamma}$

C.3(d): $S(\theta) = -\cos^2\theta\ln\cos^2\theta - \sin^2\theta\ln\sin^2\theta$, maximized at $\theta = \pi/4$.

C.6(c): $I(A:B) = 2\ln 2 \approx 1.386$ nats $= 2$ bits.