Quantum Cookbook

Part II · Chapter 13 · Final chapter of Part II

Amplitude-Amplifying Channel

Read ~16 min Prerequisites Ch. 12 Tools generalized damping widget, code cell Live
Basic

Ch. 12's amplitude damping only ever pushed a qubit toward |0⟩ — energy only ever leaks out, never in, as long as the environment is colder than the qubit. But what if the environment isn't cold? A hot, or "inverted," environment can actually hand energy to the qubit, pumping it from |0⟩ up to |1⟩. That's the amplifying channel — Ch. 12's process, run in reverse.

The Kraus operators, mirrored

Swap the roles of |0⟩ and |1⟩ in Ch. 12's amplitude-damping operators and you get the amplifying channel exactly:

$$K_0=\begin{bmatrix}\sqrt{1-\gamma}&0\\0&1\end{bmatrix} \qquad K_1=\begin{bmatrix}0&0\\\sqrt{\gamma}&0\end{bmatrix}$$

Where Ch. 12's K₁ moved population from |1⟩ down to |0⟩, this K₁ moves it from |0⟩ up to |1⟩. Same structure, opposite direction.

Intermediate

The real picture: generalized amplitude damping

In practice, an environment is rarely purely hot or purely cold — it has some excited-state population p. The honest, general channel blends damping and amplifying together, with 4 Kraus operators:

$$M_0=\sqrt{p}\begin{bmatrix}1&0\\0&\sqrt{1-\gamma}\end{bmatrix},\ M_1=\sqrt{p}\begin{bmatrix}0&\sqrt{\gamma}\\0&0\end{bmatrix},\ M_2=\sqrt{1-p}\begin{bmatrix}\sqrt{1-\gamma}&0\\0&1\end{bmatrix},\ M_3=\sqrt{1-p}\begin{bmatrix}0&0\\\sqrt{\gamma}&0\end{bmatrix}$$

Set p=1 and M₂=M₃=0 — you're back to Ch. 12's plain amplitude damping. Set p=0 and M₀=M₁=0 — you get the purely amplifying channel from this chapter. Everything in between is a genuine physical blend.

Fig. 13.1 — slide p from amplifying to dampingone family, two extremes
Start θ=0° (the input is |0⟩) and set p=0: watch the output get pulled down toward |1⟩ — the opposite of everything Ch. 12 showed you. Drag p up to 1 with the same θ=0° and the point stops moving at all, since plain damping has nothing to do to a qubit that's already in the ground state.
Advanced

Why this matters: it's not just a toy mirror image

Generalized amplitude damping with p≠1 is the correct model for a qubit in thermal equilibrium with a finite-temperature bath — p is set by the Boltzmann distribution at that temperature. p=1 is the idealized zero-temperature limit everyone uses for simplicity (and what Ch. 12 presented), but it's genuinely an approximation; real superconducting qubits sit at a small but nonzero temperature, and the honest noise model is this 4-operator generalized channel, not the simpler 2-operator one. This is also the mechanism behind population-inverted systems used to build masers and lasers — the same "pump from ground to excited" process, just engineered deliberately instead of arriving as unwanted noise.

Fig. 13.2 — code, runs in your browser
At p=1, |0⟩ stays exactly |0⟩ (damping has nothing to do). At p=0, population genuinely moves into |1⟩. p=0.5 sits exactly between them — the full sweep Fig. 13.1's slider lets you feel directly.

End of Part II

Density matrices, purity, partial trace, Kraus operators, and these two damping channels are the complete toolkit for describing any quantum system that isn't perfectly isolated — which, in practice, is every quantum system anyone has ever built. Part III picks up the entanglement thread from Ch. 10's Bell-state example and runs much further with it: teleportation, GHZ and W states, and genuinely multipartite entanglement.

Check your understanding

What does the amplifying channel do that plain amplitude damping (Ch. 12) cannot?

It pumps population from |0⟩ up to |1⟩ — energy flowing into the qubit from a hot/inverted environment, the reverse direction from ordinary decay.

Check your understanding

In the generalized amplitude damping channel, what do the two extreme values of p correspond to?

p=1 reduces to plain amplitude damping (Ch. 12); p=0 reduces to the purely amplifying channel. Values in between are a genuine physical blend, corresponding to a finite-temperature environment.

Check your understanding

If a qubit starts in |0⟩ and you apply plain amplitude damping (p=1), what happens?

Nothing — |0⟩ stays exactly |0⟩. Amplitude damping only describes decay away from |1⟩; a qubit already at the ground state has nowhere to decay to.

Check your understanding

Why is the 4-operator generalized channel the more physically honest model than the simpler 2-operator amplitude damping from Ch. 12?

Because real environments sit at some nonzero temperature, meaning they always have at least a little excited-state population — the p=1 idealization assumes a perfectly cold (zero-temperature) bath, which is a convenient approximation but not literally true for real hardware.

Check your understanding

Name one engineered (non-noise) application of the same "pump from ground to excited" mechanism.

Lasers and masers — both rely on deliberately creating population inversion, the same physical process the amplifying channel describes when it shows up as unwanted environmental noise instead.

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