Chapter 33 Exercises: Open Quantum Systems and Decoherence

Contributors to Quantum Mechanics: From Wavefunctions to Qubits

Chapter 33 Exercises: Open Quantum Systems and Decoherence

Section 33.1: System-Environment Paradigm

Problem 1 (Conceptual). Explain the distinction between a closed quantum system and an open quantum system. Why is the reduced density operator of an open system generally mixed even if the total system+environment state is pure? Give a concrete physical example.

Problem 2 (Computational). A qubit starts in the state $|\psi\rangle = \frac{1}{\sqrt{3}}|0\rangle + \sqrt{\frac{2}{3}}|1\rangle$ and interacts with a single environmental qubit initially in $|0\rangle_E$. The interaction produces:

(a) Write the full system+environment state after the interaction.

(b) Compute the reduced density matrix $\hat{\rho}_S$ by tracing over the environment.

(c) Compute the purity $\text{Tr}(\hat{\rho}_S^2)$ as a function of $\theta$. For what value of $\theta$ is the purity minimized?

(d) Compute the von Neumann entropy of $\hat{\rho}_S$ for $\theta = \pi/4$.

Problem 3 (Computational). Verify that the Kraus operators for the dephasing channel,

$$\hat{K}_0 = \sqrt{1-p}\,\hat{I}, \quad \hat{K}_1 = \sqrt{p}\,\hat{\sigma}_z$$

satisfy the completeness relation $\hat{K}_0^\dagger\hat{K}_0 + \hat{K}_1^\dagger\hat{K}_1 = \hat{I}$. Then apply the channel to $|\psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ and compute the output density matrix.

Problem 4 (Proof). Prove that the partial trace preserves the trace: if $\hat{\rho}_{\text{total}}$ is a density operator on $\mathcal{H}_S \otimes \mathcal{H}_E$ with $\text{Tr}(\hat{\rho}_{\text{total}}) = 1$, then $\text{Tr}(\hat{\rho}_S) = 1$ where $\hat{\rho}_S = \text{Tr}_E(\hat{\rho}_{\text{total}})$.

Problem 5 (Computational). Consider a system-environment interaction of the form $\hat{H}_{\text{int}} = g\,\hat{\sigma}_z \otimes \hat{B}_E$, where $\hat{B}_E$ is a Hermitian environment operator. Show that the pointer states of this interaction are $|0\rangle$ and $|1\rangle$ (the eigenstates of $\hat{\sigma}_z$), by showing that these states do not get entangled with the environment.

Section 33.2: Lindblad Master Equation

Problem 6 (Computational). For a two-level atom undergoing spontaneous emission with Lindblad operator $\hat{L} = \hat{\sigma}_- = |g\rangle\langle e|$ and rate $\gamma$:

(a) Write out the four coupled differential equations for $\rho_{ee}$, $\rho_{eg}$, $\rho_{ge}$, and $\rho_{gg}$ (ignoring the Hamiltonian part).

(b) Solve for $\rho_{ee}(t)$ and $\rho_{eg}(t)$ given initial conditions $\rho_{ee}(0) = 1$, $\rho_{eg}(0) = 1/2$.

(c) Verify that $\text{Tr}(\hat{\rho}(t)) = 1$ at all times.

(d) Compute the purity $\text{Tr}(\hat{\rho}^2(t))$. At what time is it minimized?

Problem 7 (Conceptual). Explain the physical meaning of the quantum jump interpretation of the Lindblad equation. How does the non-Hermitian effective Hamiltonian $\hat{H}_{\text{eff}} = \hat{H}_S - \frac{i\hbar}{2}\gamma\hat{L}^\dagger\hat{L}$ evolve the system between jumps? What happens to the norm of the state, and how is this related to the jump probability?

Problem 8 (Computational). A qubit has Hamiltonian $\hat{H}_S = \frac{\hbar\omega_0}{2}\hat{\sigma}_z$ and is subject to pure dephasing with Lindblad operator $\hat{L} = \hat{\sigma}_z$ and rate $\gamma_\phi$. Write the Lindblad master equation in full. Show that in the rotating frame (interaction picture with respect to $\hat{H}_S$), the off-diagonal elements decay as $\tilde{\rho}_{01}(t) = \tilde{\rho}_{01}(0)\,e^{-2\gamma_\phi t}$, while the diagonal elements are constant.

Problem 9 (Proof). Prove that the Lindblad master equation preserves the trace of the density operator. That is, show $\frac{d}{dt}\text{Tr}(\hat{\rho}) = 0$ for:

$$\frac{d\hat{\rho}}{dt} = -\frac{i}{\hbar}[\hat{H}, \hat{\rho}] + \gamma\left(\hat{L}\hat{\rho}\hat{L}^\dagger - \frac{1}{2}\{\hat{L}^\dagger\hat{L}, \hat{\rho}\}\right)$$

Hint: Use the cyclic property of the trace.

Problem 10 (Computational). A harmonic oscillator mode (photon in a cavity) decays due to photon loss with Lindblad operator $\hat{L} = \hat{a}$ (the annihilation operator) and rate $\kappa$. Starting from a coherent state $|\alpha\rangle$:

(a) Show that the mean photon number decays as $\langle\hat{n}(t)\rangle = |\alpha|^2 e^{-\kappa t}$.

(b) Show that the state remains a coherent state $|\alpha e^{-\kappa t/2}\rangle$ at all times (for this particular initial state and noise channel).

Hint: Use the equations of motion $\frac{d}{dt}\langle\hat{a}\rangle = -\frac{\kappa}{2}\langle\hat{a}\rangle$ and $\frac{d}{dt}\langle\hat{a}^\dagger\hat{a}\rangle = -\kappa\langle\hat{a}^\dagger\hat{a}\rangle$.

Problem 11 (Computational). Thermal excitation: At finite temperature, a two-level system experiences both spontaneous emission (rate $\gamma(1+\bar{n})$) and absorption (rate $\gamma\bar{n}$), where $\bar{n} = (e^{\hbar\omega/k_BT}-1)^{-1}$ is the Bose-Einstein mean occupation number. The Lindblad operators are:

$$\hat{L}_1 = \sqrt{\gamma(1+\bar{n})}\,\hat{\sigma}_-, \quad \hat{L}_2 = \sqrt{\gamma\bar{n}}\,\hat{\sigma}_+$$

(a) Write the full master equation.

(b) Find the steady-state density matrix $\hat{\rho}_{\text{ss}}$ by setting $\dot{\hat{\rho}} = 0$.

(c) Show that the steady-state population ratio satisfies the Boltzmann distribution: $\rho_{ee}/\rho_{gg} = e^{-\hbar\omega/k_BT}$.

Section 33.3: Decoherence Channels

Problem 12 (Computational). Apply the amplitude damping channel with parameter $\gamma = 0.3$ to the state $\hat{\rho} = |+\rangle\langle +|$, where $|+\rangle = (|0\rangle + |1\rangle)/\sqrt{2}$. Compute the output state, its eigenvalues, and its von Neumann entropy.

Problem 13 (Computational). Show that the depolarizing channel can be written as:

$$\mathcal{E}(\hat{\rho}) = (1-p)\hat{\rho} + p\frac{\hat{I}}{2}$$

starting from the Kraus representation with operators $\hat{K}_0 = \sqrt{1-3p/4}\,\hat{I}$, $\hat{K}_k = \sqrt{p/4}\,\hat{\sigma}_k$ for $k = x, y, z$.

Hint: Use $\hat{\sigma}_x\hat{\rho}\hat{\sigma}_x + \hat{\sigma}_y\hat{\rho}\hat{\sigma}_y + \hat{\sigma}_z\hat{\rho}\hat{\sigma}_z = 2\text{Tr}(\hat{\rho})\hat{I} - \hat{\rho}$ (valid for single-qubit $\hat{\rho}$).

Problem 14 (Computational). A qubit in state $|+\rangle$ is sent through a dephasing channel with parameter $p$, followed by an amplitude damping channel with parameter $\gamma$. Compute the final density matrix. Is the result the same if the channels are applied in the reverse order? If not, explain physically why the order matters.

Problem 15 (Conceptual). For each of the three canonical channels (dephasing, amplitude damping, depolarizing), describe the geometric effect on the Bloch sphere. Include: (a) What shape does the sphere become? (b) Where is the fixed point? (c) Which states are least/most affected?

Problem 16 (Computational). Compute the Choi matrix for the dephasing channel ($\hat{K}_0 = \sqrt{1-p}\,\hat{I}$, $\hat{K}_1 = \sqrt{p}\,\hat{\sigma}_z$) and verify that it is positive semidefinite.

Problem 17 (Computational). A qubit has $T_1 = 100\,\mu$s and $T_\phi = 200\,\mu$s. Compute $T_2$. If the single-qubit gate time is $\tau_g = 20$ ns, what is the approximate error rate per gate due to decoherence?

Section 33.4: Why the World Looks Classical

Problem 18 (Conceptual). Explain the concept of einselection (environment-induced superselection). What determines the pointer basis? Why are macroscopic objects observed in position eigenstates rather than momentum eigenstates?

Problem 19 (Computational). Estimate the decoherence time for a dust grain of mass $m = 10^{-14}$ kg in a superposition of two positions separated by $\Delta x = 10^{-6}$ m, due to scattering of:

(a) Room-temperature photons ($\lambda_{\text{dB}} \approx 8 \times 10^{-6}$ m, photon flux $\sim 10^{18}$ photons/s/m$^2$). Use the scattering-induced decoherence rate $\Lambda \approx \Phi\sigma(\Delta x/\lambda)^2$ where $\Phi$ is the flux, $\sigma$ is the scattering cross-section ($\sim 10^{-12}$ m$^2$ for a dust grain), and $\lambda$ is the photon wavelength.

(b) Air molecules at atmospheric pressure ($\Lambda \sim 10^{36}$ m$^{-2}$s$^{-1}$ for nitrogen at 300 K).

Comment on the observability of macroscopic superpositions.

Problem 20 (Conceptual). What is quantum Darwinism? How does it explain the objectivity of classical information? Why can multiple observers agree on the position of a chair without "collapsing" its quantum state?

Problem 21 (Computational). A Schrodinger-cat state of a harmonic oscillator is $|\psi\rangle = \frac{1}{\sqrt{2}}(|\alpha\rangle + |-\alpha\rangle)$ with $\alpha = 3$. The oscillator loses photons with rate $\kappa$. The decoherence rate for this cat state is $\Gamma_{\text{dec}} = 2\kappa|\alpha|^2$.

(a) Compute $\Gamma_{\text{dec}}/\kappa$.

(b) How many times faster does the cat state decohere compared to a single-photon superposition ($\alpha = 1/\sqrt{2}$)?

(c) If $\kappa = 10^3$ s$^{-1}$ (a high-Q microwave cavity), what is the cat-state decoherence time?

Section 33.5: Connection to Quantum Error Correction

Problem 22 (Computational). In the three-qubit bit-flip code, suppose the second qubit undergoes a bit flip. Starting from the encoded state $|\psi_L\rangle = \alpha|000\rangle + \beta|111\rangle$:

(a) Write the state after the error.

(b) Compute the syndrome measurement outcomes for $\hat{Z}_1\hat{Z}_2$ and $\hat{Z}_2\hat{Z}_3$.

(c) Apply the appropriate correction operation and verify the original state is recovered.

Problem 23 (Conceptual). Explain why the no-cloning theorem does not prevent quantum error correction. How does the three-qubit code encode information without cloning it?

Problem 24 (Computational). Verify the Knill-Laflamme conditions for the three-qubit bit-flip code with error set $\{\hat{I}, \hat{X}_1, \hat{X}_2, \hat{X}_3\}$. Compute $\langle 0_L|\hat{E}_a^\dagger\hat{E}_b|0_L\rangle$ and $\langle 1_L|\hat{E}_a^\dagger\hat{E}_b|1_L\rangle$ for all pairs $(a, b)$ and verify they are equal.

Problem 25 (Proof). Show that the two-qubit subspace spanned by $\{|01\rangle, |10\rangle\}$ is a decoherence-free subspace for collective dephasing noise with $\hat{L} = \hat{\sigma}_z^{(1)} + \hat{\sigma}_z^{(2)}$. That is, show that $\hat{L}|01\rangle = 0$ and $\hat{L}|10\rangle = 0$.

Problem 26 (Computational). The surface code has a threshold error rate of approximately $p_{\text{th}} \approx 0.011$. If the physical error rate is $p = 0.001$, and we use a surface code of distance $d$, the logical error rate scales as:

$$p_L \sim \left(\frac{p}{p_{\text{th}}}\right)^{(d+1)/2}$$

(a) Compute $p_L$ for $d = 3, 5, 7, 9, 11$.

(b) How many physical qubits are needed for distance $d$ (approximately $2d^2$ for the surface code)?

(c) What is the minimum distance needed to achieve $p_L < 10^{-15}$ (the approximate error rate needed for useful quantum algorithms)?

Advanced Problems

Problem 27 (Research/Open-ended). The Caldeira-Leggett model describes a quantum system coupled to a bath of harmonic oscillators. The bath is characterized by its spectral density $J(\omega)$. For ohmic damping, $J(\omega) = \eta\omega\,e^{-\omega/\omega_c}$.

(a) Research and describe the three regimes: sub-ohmic ($J(\omega) \propto \omega^s$, $s < 1$), ohmic ($s = 1$), and super-ohmic ($s > 1$). How does the nature of decoherence differ in each regime?

(b) The spin-boson model has a phase transition at a critical coupling strength. Describe this transition and its physical significance.

Problem 28 (Research/Open-ended). Non-Markovian dynamics and structured environments.

(a) Give a physical example where the Markov approximation breaks down. What is the physical mechanism for information backflow from the environment?

(b) The BLP measure of non-Markovianity is based on increases in the trace distance between two system states. Explain why an increase in trace distance implies information flow from environment to system.

(c) Look up the "hierarchical equations of motion" (HEOM) method. What types of problems can it solve that the Lindblad equation cannot? What are its computational limitations?

(d) Discuss how non-Markovian effects might be exploited as a resource in quantum information processing.

Solutions Hints

Problem 2(c): The purity is $\text{Tr}(\hat{\rho}_S^2) = 1 - \frac{4}{9}\sin^2\theta(1 - \sin^2\theta) \cdot (...)$. Minimize by finding $\theta$ that maximizes the mixedness.

Problem 6(d): The purity has a minimum at a time that depends on the initial conditions; differentiate and set to zero.

Problem 17: $1/T_2 = 1/(2T_1) + 1/T_\phi = 1/200 + 1/200 = 1/100\,\mu$s$^{-1}$, so $T_2 = 100\,\mu$s. Error per gate $\sim \tau_g/T_2 = 20 \times 10^{-9}/(100 \times 10^{-6}) = 2 \times 10^{-4}$.

Problem 21(a): $\Gamma_{\text{dec}}/\kappa = 2|\alpha|^2 = 18$.

Problem 26(c): You need $(p/p_{\text{th}})^{(d+1)/2} < 10^{-15}$. Since $p/p_{\text{th}} \approx 0.091$, take logarithms.