Quantum Mechanics · Chapter 19

Open quantum systems

No real system is alone. Its entanglement with an environment turns superpositions into mixtures, which is decoherence; the maps that describe what happens to the system alone are quantum channels written with Kraus operators; and in continuous time they obey the Lindblad equation, whose relaxation and dephasing are what every experiment with qubits fights.

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

Lessons

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06The Lindblad equationWhen the environment forgets quickly — each record is carried off and replaced by fresh surroundings faster than anything the system does — the channel over a short time $dt$ depends only on the present state, and the density matrix obeys a differential equation of its own: the Lindblad equation. Its first term is the von Neumann commutator; the others, one for each jump operator $\hat{L}_{m}$, describe jumps and the gradual loss of amplitude in not jumping. The lesson derives it from Kraus operators whose jumps have amplitudes of order $\sqrt{dt}$, so that their probabilities grow in proportion to $dt$. It keeps the trace and positivity; decay and dephasing are its two simplest cases.Read lesson →

07Relaxation and dephasingExperiments sum up the noise on a qubit with two times.Read lesson →

08Chapter practiceBring 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.Read lesson →

Mapping the lesson.

Open quantum systems

Build the foundation