Quantum Mechanics · Lesson 08

Chapter practice

Bring the chapter together: open systems and purification, decoherence, channels and POVMs, Kraus operators, the standard noise models, the Lindblad equation, and the two times that sum up a qubit’s noise. No worked example sits above the answers.

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ρ^_{S} = tr_{E} ρ^_{S E}

ρ^\mapsto E (ρ^)

E (ρ) = k \sum E_{k} ρ E_{k}^{†}

k \sum E_{k}^{†} E_{k} = I

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Intuition

Bring the chapter together: open systems and purification, decoherence, channels and POVMs, Kraus operators, the standard noise models, the Lindblad equation, and the two times that sum up a qubit’s noise. No worked example sits above the answers.

A sailor reads the sea by its swell, its gusts and its currents, each with its own effect on the boat. These are the sea’s effects on a qubit.

Before you practise

$\rho_{01}\to\rho_{01}\braket{e_{1}|e_{0}}$ ; $\mathcal{E}(\hat{\rho})=\sum\hat{M}_{l}\hat{\rho}\hat{M}_{l}^{\dagger}$ with $\sum\hat{M}_{l}^{\dagger}\hat{M}_{l}=\hat{1}$ ; depolarising $\mathbf{r}\to(1-p)\mathbf{r}$ ; $1/T_{2}=1/2T_{1}+1/T_{\varphi}$ .

\frac{d\hat{\rho}}{dt}=-\frac{i}{\hbar}[\hat{H},\hat{\rho}]+\sum_{m}\bigl(\hat{L}_{m}\hat{\rho}\hat{L}_{m}^{\dagger}-\tfrac12\{\hat{L}_{m}^{\dagger}\hat{L}_{m},\hat{\rho}\}\bigr)

Reminders

The whole is unitary; the part is not.
Decay: populations $e^{-\gamma t}$ , coherences $e^{-\gamma t/2}$ .
A POVM is positive operators adding to the identity.

Applications

Practice

Try it

A coherence of $\tfrac12$ meets an environment whose records overlap by $0.6$ . What is $|\rho_{01}|$ afterwards?