Decoherence

Abstract

In this chapter, we ask what happens when a quantum system is in contact with its environment. To describe this, we need a more general concept of the quantum state, namely the density matrix. We study the density matrix, and how it describes both classical and quantum uncertainty. This also naturally leads to a description of quantum systems at a non-zero temperature. We conclude this chapter with a short discussion on how entropy arises in quantum mechanics.

Exercises

• 1. Calculate the purity of the state

  $$ \rho = \frac{1}{3} \begin{pmatrix} 1 &{} 1 \\ 1 &{} 2 \end{pmatrix} . $$

  Is it a pure state or a mixed state?

• 2. Two qubits, 1 and 2, are prepared in the initial states

  $$|\psi \rangle _1=\frac{1}{\sqrt{2}}\left( |\mathsf {0}\rangle _1+|\mathsf {1}\rangle _1\right) \qquad \text {and}\qquad |\phi \rangle _2=|\mathsf {0}\rangle _2\, ,$$

  and the interaction between the two qubits is described by the Hamiltonian

  $$ H = \hbar g |\mathsf {1}\rangle _1\!\langle \mathsf {1}| \otimes \left( |\mathsf {1}\rangle _2\!\langle \mathsf {0}| + |\mathsf {0}\rangle _2\!\langle \mathsf {1}|\right) \, .$$

  • (a) Calculate the state of the joint two-qubit system after an interaction time T.

  • (b) What are the quantum states of the individual qubits after they have interacted for a time T?

  • (c) Calculate the entropy $S(\rho ) = - \text {Tr}{\left[ {\rho \log _2\rho }\right] }$ of the individual qubit states $\rho$.

  • (d) For what value of T are the two qubits maximally entangled?

• 3.

  • (a) A spin-$\frac{1}{2}$ particle is in the state $\rho = \frac{1}{3} |\uparrow \rangle \langle \uparrow | + \frac{2}{3} |\downarrow \rangle \langle \downarrow |$. Calculate the purity of the spin, given by $\text {Tr}{\left[ {\rho ^2}\right] }$.

  • (b) The spin evolves in time according to the Hamiltonian

    $$H = \hbar \omega \, X \qquad \text {with}\qquad X = |\uparrow \rangle \langle \downarrow | + |\downarrow \rangle \langle \uparrow |\, .$$

    Calculate the state $\rho (t)$ at time t.

  • (c) Sketch the probability of finding the measurement outcome “$\uparrow$” in a measurement of the spin as a function of time.

  • (d) How will the purity of the spin change when all spin coherence is lost?

• 4. Quantum teleportation with three entangled qubits.

  • (a) A so-called GHZ state for three qubits can be written as

    $$ |\mathrm {GHZ}\rangle = \frac{|\mathsf {000}\rangle +|\mathsf {111}\rangle }{\sqrt{2}}\, . $$

    After losing the third qubit, what is the state of the remaining two qubits? Calculate the entropy $S(\rho ) = -\text {Tr}{\left[ {\rho \log _2 \rho }\right] }$ of this state.

  • (b) A so-called W state for three qubits can be written as

    $$ |\mathrm {W}\rangle = \frac{|\mathsf {001}\rangle +|\mathsf {010}\rangle +|\mathsf {100}\rangle }{\sqrt{3}}\, . $$

    After losing the third qubit, what is the state of the remaining two qubits? Calculate the entropy of this state.

  • (c) The first qubit of the GHZ state is used as part of the entanglement channel for teleporting a fourth qubit in the state $\alpha |\mathsf {0}\rangle + \beta |\mathsf {1}\rangle$. Calculate the remaining two-qubit state after applying the teleportation protocol. Assuming perfect equipment, can the remaining two qubits retrieve the original qubit with certainty?

  • (d) Instead of the GHZ state, use the W state for the teleportation protocol in part (c). Can the remaining two qubits retrieve the original qubit with certainty?

• 5. The density matrix.

  • (a) Show that $\frac{1}{2}|\mathsf {0}\rangle + \frac{1}{2}|+\rangle$ is not a properly normalized state.

  • (b) Show that $\text {Tr}{\left[ {\rho }\right] }=1$, and then prove that any density operator has unit trace and is Hermitian.

  • (c) Show that density operators are convex, i.e., that $\rho = w_1 \rho _1 + w_2 \rho _2$ with $w_1 + w_2 =1$ ($w_1,w_2\ge 0$), and $\rho _1$, $\rho _2$ again density operators.

• 6. Two qubits, labeled A and B, are prepared in the entangled state $|\Psi \rangle = \frac{3}{5}|\mathsf {00}\rangle + \frac{4}{5} |\mathsf {11}\rangle$.

  • (a) Show that the correlations in the $\{|\mathsf {0}\rangle ,|\mathsf {1}\rangle \}$ basis are perfect, but the correlations in the $\{|+\rangle ,|-\rangle \}$ basis are not (think first carefully how you would define a correlation). The states $|+\rangle$ and $|-\rangle$ are defined by

    $$|\pm \rangle = \frac{|\mathsf {0}\rangle \pm |\mathsf {1}\rangle }{\sqrt{2}}\, .$$

  • (b) Calculate the entanglement entropy $\mathscr {E}$ of $|\Psi \rangle$. This is a measure of the amount of entanglement in $|\Psi \rangle$, and is given by

    $$\mathscr {E} = S(\text {Tr}{\left[ {[}\right] }B]{|\Psi \rangle \langle \Psi |}) \qquad \text {with}\qquad S(\rho ) = -\text {Tr}{\left[ {\rho \log _2\rho }\right] }\, .$$

  • (c) We teleport a state $\alpha |\mathsf {0}\rangle + \beta |\mathsf {1}\rangle$ using the entangled state $|\Psi \rangle$. What is the best and worst average teleportation fidelity $\langle F\rangle$, using otherwise ideal components? The fidelity F is defined by $F = |{\langle \psi _\mathrm{ideal}|\psi _\mathrm{teleported}\rangle }|^2$.

• 7. The partial transpose of a two-qubit state is calculated by writing the density matrix $\rho$ as

  $$ \rho = \sum _{jklm} \rho _{jk,lm} |j,k\rangle \langle l,m| \qquad \text {with}\qquad j,k,l,m = \mathsf {0},\; \mathsf {1} $$

  and swapping k and m in the kets and bras (the full transpose would be swapping both k and m and j and l). If the eigenvalues of the resulting matrix include negative values, then the original state was entangled. Calculate the partial transpose of the states given in Exercise 1 of Chap. 6 (making the identification $|\mathsf {0}\rangle \leftrightarrow |\uparrow \rangle$ and $|\mathsf {1}\rangle \leftrightarrow |\downarrow \rangle$) and determine their eigenvalues. Which states are entangled? Does it agree with your previous answer?

• 8. An electron with spin 1/2 is prepared in a state

  $$ \rho = \frac{1}{2} \begin{pmatrix} 1 &{} \quad e^{-i\omega t - \gamma t} \\ e^{i\omega t - \gamma t} &{} \quad 1 \end{pmatrix} , $$

  where $\omega$ and $\gamma$ are real positive numbers, and t denotes time. Sketch the time evolution of the state in the Bloch sphere. What is the physical interpretation of $\omega$ and $\gamma$?

• 9. Any single-qubit mixed state can be written in terms of the Pauli matrices as

  $$ \rho = \frac{\mathbb {I} + {\mathbf {r}}\cdot {\varvec{\sigma }}}{2}\, , $$

  where $\varvec{\sigma }= (\sigma _x,\sigma _y,\sigma _z)$ and $\mathbf {r} = (r_x,r_y,r_z)$, $|{\mathbf {r}}|\le 1$. Moreover, any single qubit observable can be expressed as

  $$ A = a_0 \mathbb {I} + \mathbf {a}\cdot \varvec{\sigma }\, , $$

  with $a_0$ and $\mathbf {a}=(a_x,a_y,a_z)$ real numbers. Show that the expectation value of A with respect to $\rho$ can be written as

  $$ \langle A\rangle = \frac{1}{2} a_0 + \frac{1}{2} \mathbf {a}\cdot \mathbf {r}\, . $$

• 10. The Boltzmann entropy of a system is given by the partition function and the average energy $\langle E\rangle$ as

  $$ S = k_\mathrm{B} \left( \ln Z + \beta \langle E\rangle \right) \, . $$

  Show that the Shannon entropy in the second line of Eq. (7.64) for a thermal state corresponds to the Boltzmann entropy when we choose $k_\mathrm{B} = 1$.

• 11. A two-level atom has energy separation $\hbar \omega$ between the ground state and the excited state. The atom is brought into thermal equilibrium with a bath at temperature T.

  • (a) Find the thermal state of the atom.

  • (b) Calculate the average energy and the entropy of the atom.

  • (c) Let’s assume that we prepare the atom in the excited state, and we do not let it thermalise with the bath. What would be the effective temperature of the atom? You can use the expression for the Boltzmann entropy in Exercise 10.

• 12. Does a thermal state evolve over time?

• 13. Prove Eq. (7.57).

• 14. Express the variance of an observable A in terms of the trace and the density operator, similar to Eq. (7.27).