Amplitude Damping Channel

The amplitude damping channel is the mathematical fingerprint of energy loss. Every real qubit — a superconducting transmon, a trapped ion, an electron spin — sits coupled to a colder world that will, sooner or later, drain its excitation away. The canonical picture is an atom in its excited state $|1\rangle$ that spontaneously emits a photon and drops to the ground state $|0\rangle$ ; a single number $\gamma\in[0,1]$ records the probability of that decay. Because excitation flows one way — from system to bath — the channel breaks the symmetry between $|0\rangle$ and $|1\rangle$ , exactly what sets it apart from the even-handed depolarizing channel and from pure dephasing. This article is the deep-tier synthesis of the open-systems thread, taking the operator-sum machinery from aq-qci-kraus-representation-q9, the complete-positivity constraints from aq-qci-cptp-maps-quantum-operations, and the partial-trace construction from aq-qci-open-quantum-systems-reduced-dynamics and assembling from them the most-used noise model in quantum computing. Three consequences organize everything that follows. First, damping is non-unital: it does not fix the maximally mixed state, and its unique fixed point is the pure ground state, so it is the prototype of a relaxing, zero-temperature-thermalizing process. Second, even the branch in which no photon is emitted disturbs the qubit — the absence of a click is evidence, and $K_0$ is not the identity. Third, iterating an infinitesimal version of the channel produces exponential decay with rate $1/T_1$ , which is how the abstract $\gamma$ becomes the relaxation time an experimentalist measures. The finite-temperature generalization, the Choi picture, and the tie to $T_2$ all fall out of the same two Kraus operators.

Key takeaways

The amplitude damping channel models irreversible energy loss — spontaneous emission being the archetype — driving a qubit from $|1\rangle$ toward $|0\rangle$ with probability $\gamma$ .
It is a single-qubit CPTP map with just two Kraus operators, $K_0$ (no-jump) and $K_1$ (the jump $|1\rangle\to|0\rangle$ ), satisfying $K_0^\dagger K_0+K_1^\dagger K_1=I$ .
It is non-unital: unlike the depolarizing or dephasing channels it does not fix $I/2$ ; its one fixed point is the pure ground state $|0\rangle\langle0|$ .
Populations decay by $1-\gamma$ but coherences only by $\sqrt{1-\gamma}$ — the mismatch is the microscopic origin of the bound $T_2\le 2T_1$ .
In continuous time $\gamma(t)=1-e^{-t/T_1}$ , so the discrete channel parameter is a direct readout of the relaxation time $T_1$ .
On the Bloch ball the action is an affine contraction that squashes the equator by $\sqrt{1-\gamma}$ , shrinks the $z$ axis by $1-\gamma$ , and pushes the center up by $\gamma$ toward the north pole.
Key idea: amplitude damping is the channel $\mathcal{E}(\rho)=K_0\rho K_0^\dagger+K_1\rho K_1^\dagger$ with $K_0=|0\rangle\langle0|+\sqrt{1-\gamma}\,|1\rangle\langle1|$ and $K_1=\sqrt{\gamma}\,|0\rangle\langle1|$ : it transfers excited-state population to the ground state, decays coherences by $\sqrt{1-\gamma}$ , and has the ground state as its unique fixed point.

The core insight

Amplitude damping is the CPTP map $\mathcal{E}(\rho)=K_0\rho K_0^\dagger+K_1\rho K_1^\dagger$ with $K_0=|0\rangle\langle0|+\sqrt{1-\gamma}\,|1\rangle\langle1|$ and $K_1=\sqrt{\gamma}\,|0\rangle\langle1|$ : it moves excited-state population to the ground state, decays coherences by $\sqrt{1-\gamma}$ and the excited population by $1-\gamma$ , and has the pure ground state $|0\rangle\langle0|$ as its unique fixed point.

Where this lives today

Amplitude damping is not a shelf idealization; it is the noise model compiled into every serious quantum-computing stack. When a hardware team reports a transmon $T_1$ of a few hundred microseconds, they are reporting the parameter that fixes $\gamma(t)=1-e^{-t/T_1}$ for every idle and every gate. Circuit simulators such as Qiskit and Cirq insert exactly the two Kraus operators of equation (1) at each time step to predict how a device degrades a computation, and randomized-benchmarking and gate-set-tomography pipelines fit the observed decay back to an amplitude-damping-plus-dephasing model to separate the $T_1$ contribution from the $T_\phi$ contribution to $T_2$ . On the error-correction side the channel's non-Pauli, non-unital structure is a genuine frontier: because relaxation is not a Pauli error it is not automatically handled by the stabilizer picture, which motivated the four-qubit approximate amplitude-damping code of Leung, Nielsen, Chuang, and Yamamoto and the biased-noise codes now studied for surface-code and cat-qubit architectures. The finite-temperature generalization — the generalized amplitude damping channel — describes qubits that thermalize to a Gibbs state rather than a pure ground state, and is the model of record for hot or weakly isolated hardware. From a chip datasheet to a threshold proof, the same two operators keep reappearing.

The physical picture

Put an excited atom in an empty cavity. Over some window of time it either emits a photon and falls to the ground state — probability $\gamma$ — or it does not. But "no emission" is not "nothing happened." Because the ground state cannot emit, the very fact that no photon appeared makes the ground-state hypothesis more likely, so even the silent branch reshapes the qubit. That is the physical content of the $\sqrt{1-\gamma}$ in the no-jump operator $K_0$ : measurement back-action from a detector that clicked zero times. The environment need not be watched — tracing it out has the same effect as an unread record. Geometrically the two branches together squash the Bloch ball and shove it north. The equatorial components — the coherences — shrink by $\sqrt{1-\gamma}$ , while the polar coordinate is dragged toward $|0\rangle$ at the north pole. The asymmetry between the poles is the whole story: a zero-temperature bath can absorb energy but never supply it, so population only flows downhill. This one-way flow is why the channel is non-unital and why its fixed point is a pure state rather than the center of the ball. The first figure shows both halves at once — the emission on the left, the deformed ball on the right — for $\gamma=0.5$ .

Figure

An excited two-level system decays to its ground state by emitting one photon into the environment with probability γ, while the Bloch ball is squashed toward the north pole — the equator contracts by √(1−γ) and the z axis by 1−γ with an upward shift γ, drawn here for γ = 0.5 so the equatorial factor is 0.707 and the center rises 0.5 toward the ground state

The mismatch between how fast populations and coherences fall drives everything downstream: populations lose a factor $1-\gamma$ , coherences only its square root $\sqrt{1-\gamma}$ . In continuous time that becomes a factor-of-two difference in rates and yields the relaxation floor $1/T_2=1/(2T_1)$ before any dephasing is added. The second figure makes this concrete with the number quoted on a chip spec sheet: the excited-state population of a transmon with $T_1=100\ \mu\mathrm{s}$ decaying as a clean exponential, and $\gamma(t)$ climbing to $0.632$ after one lifetime.

Figure

Excited-state population of a transmon qubit under amplitude damping, ρ₁₁ following e to the minus t over T₁ with T₁ = 100 μs — one lifetime leaves 0.368 of the population, two lifetimes leave 0.135, and the channel parameter γ rises to 0.632 at t = T₁

The core equations

Everything is encoded in two Kraus operators: $K_0$ is the branch in which no quantum is emitted, $K_1$ the jump that lowers the qubit. Writing them in the $\{|0\rangle,|1\rangle\}$ basis makes the structure transparent.

K_0=\begin{pmatrix}1&0\\0&\sqrt{1-\gamma}\end{pmatrix},\qquad K_1=\begin{pmatrix}0&\sqrt{\gamma}\\0&0\end{pmatrix} \qquad (1)

where: $\gamma$ — decay probability over the interval, $\gamma\in[0,1]$ ; $K_0$ — no-jump operator, leaving $|0\rangle$ alone and damping the excited amplitude by $\sqrt{1-\gamma}$ ; $K_1$ — jump operator implementing $|1\rangle\to|0\rangle$ with amplitude $\sqrt{\gamma}$ . Physically $K_1$ is a pure lowering event, $K_0$ the "nothing emitted" branch that nonetheless shrinks the excited amplitude. For a legitimate trace-preserving channel the operators must resolve the identity — the single algebraic constraint CPTP maps impose on a Kraus set, and here it fixes the $\sqrt{1-\gamma}$ in $K_0$ given the $\sqrt{\gamma}$ in $K_1$ .

K_0^\dagger K_0+K_1^\dagger K_1=\begin{pmatrix}1&0\\0&1-\gamma\end{pmatrix}+\begin{pmatrix}0&0\\0&\gamma\end{pmatrix}=I \qquad (2)

where: $I$ — the qubit identity. The excited-state entries $1-\gamma$ and $\gamma$ add to $1$ , so probability is conserved; if $K_0$ were the identity this sum would exceed $I$ and trace would not be preserved. Applying $\mathcal{E}(\rho)=K_0\rho K_0^\dagger+K_1\rho K_1^\dagger$ to a general density matrix shows both effects at once: population is pumped from the $\rho_{11}$ corner into the $\rho_{00}$ corner, and the coherences are attenuated.

\mathcal{E}_{\mathrm{AD}}\!\begin{pmatrix}\rho_{00}&\rho_{01}\\\rho_{10}&\rho_{11}\end{pmatrix}=\begin{pmatrix}\rho_{00}+\gamma\rho_{11}&\sqrt{1-\gamma}\,\rho_{01}\\\sqrt{1-\gamma}\,\rho_{10}&(1-\gamma)\rho_{11}\end{pmatrix} \qquad (3)

where: $\rho_{11}$ — excited population, multiplied by $1-\gamma$ ; $\rho_{00}$ — ground population, gaining exactly the lost $\gamma\rho_{11}$ ; $\rho_{01}$ — coherence, multiplied by $\sqrt{1-\gamma}$ . The trace is preserved because $\gamma\rho_{11}$ leaves the excited corner and arrives in the ground corner. Because energy only flows downhill, the ground state is untouched — it has nothing to emit. This is the defining non-unital feature: the fixed point is a pure state at a pole, not the center.

\mathcal{E}_{\mathrm{AD}}(|0\rangle\langle0|)=|0\rangle\langle0|,\qquad \mathcal{E}_{\mathrm{AD}}(I/2)=\tfrac12\begin{pmatrix}1+\gamma&0\\0&1-\gamma\end{pmatrix}\neq \tfrac{I}{2} \qquad (4)

where: $|0\rangle\langle0|$ — the unique fixed point; $I/2$ — the maximally mixed state, not fixed for $\gamma>0$ . The failure to fix $I/2$ is the precise statement that amplitude damping is non-unital, unlike depolarizing. Parametrizing $\rho=\tfrac12(I+\vec r\cdot\vec\sigma)$ turns the action into an affine map on the Bloch vector — the cleanest way to see the geometry of the first figure.

\vec r=(r_x,r_y,r_z)\;\longmapsto\;\big(\sqrt{1-\gamma}\,r_x,\;\sqrt{1-\gamma}\,r_y,\;(1-\gamma)r_z+\gamma\big) \qquad (5)

where: the equatorial radius contracts by $\sqrt{1-\gamma}$ ; the polar component contracts by $1-\gamma$ then shifts by $+\gamma$ . The additive $+\gamma$ is the signature of a non-unital map — a unital channel would have no shift. Finally, the continuous-time face is a Lindblad (GKSL) generator with a single jump operator — the bridge from the discrete $\gamma$ to the measured relaxation rate.

\frac{\mathrm{d}\rho}{\mathrm{d}t}=\Gamma\Big(\sigma_-\rho\,\sigma_+-\tfrac12\{\sigma_+\sigma_-,\rho\}\Big),\qquad \sigma_-=|0\rangle\langle1| \qquad (6)

where: $\Gamma=1/T_1$ — spontaneous-emission rate; $\sigma_-=|0\rangle\langle1|$ — lowering (jump) operator; $\{\cdot,\cdot\}$ — anticommutator, supplying the no-jump back-action. Integrating over time $t$ returns exactly the channel of equation (3) with $\gamma(t)=1-e^{-t/T_1}$ .

Derivation AKraus operators from a system plus environment unitary

The goal is to build the two Kraus operators from first principles rather than postulate them, by coupling the qubit to one environment mode, evolving under a joint unitary, and tracing the environment away. This is the reduced-dynamics recipe owned by the open-quantum-systems article, applied to the simplest bath: a single mode initialized in its vacuum $|0\rangle_E$ that can absorb at most one quantum. The payoff is that the $\sqrt{1-\gamma}$ in the no-jump branch appears automatically from unitarity, not by hand.

Step 1. Fix the joint unitary by what it must do physically: the ground state cannot deposit energy, so $U|0\rangle_S|0\rangle_E=|0\rangle_S|0\rangle_E$ , while the excited state emits a quantum with amplitude $\sqrt{\gamma}$ , giving $U|1\rangle_S|0\rangle_E=\sqrt{1-\gamma}\,|1\rangle_S|0\rangle_E+\sqrt{\gamma}\,|0\rangle_S|1\rangle_E$ . Why: this is a beam-splitter-like rotation on the one-excitation subspace, and its two output amplitudes are forced to be $\sqrt{1-\gamma}$ and $\sqrt{\gamma}$ so the state stays normalized.

Step 2. Reduce to the qubit by tracing over the environment: $\mathcal{E}(\rho)=\mathrm{Tr}_E[U(\rho\otimes|0\rangle\langle0|_E)U^\dagger]=\sum_k \langle k|_E U|0\rangle_E\,\rho\,\langle0|_E U^\dagger|k\rangle_E$ , defining $K_k=\langle k|_E\,U\,|0\rangle_E$ . Why: the partial trace is a sum of "sandwiches" of $U$ , each an operator on the qubit alone — the operator-sum representation.

Step 3. Project onto the no-photon outcome for $K_0=\langle0|_E U|0\rangle_E=|0\rangle\langle0|+\sqrt{1-\gamma}\,|1\rangle\langle1|$ , and the one-photon outcome for $K_1=\langle1|_E U|0\rangle_E=\sqrt{\gamma}\,|0\rangle\langle1|$ . Why: the $\sqrt{1-\gamma}$ is exactly the "no click, so the qubit is a little more likely in the non-emitting state" back-action — unitarity bookkeeping, not an extra assumption.

Step 4. Confirm it is a legitimate channel by checking $\sum_k K_k^\dagger K_k=\mathrm{diag}(1,1-\gamma)+\mathrm{diag}(0,\gamma)=I$ , equivalent to $U$ acting as an isometry on the relevant subspace. Why: completeness is trace preservation, the abstract shadow of $U$ being unitary — CPTP is guaranteed once the dilation is unitary. *Verified: reproduces Nielsen and Chuang Eqs. (8.103)–(8.104); the completeness sum equals the identity for every γ in [0,1].*

Go deeper · Derivation B · 1 minThe Bloch-ball affine map and why the channel is non-unital

Derivation BThe Bloch-ball affine map and why the channel is non-unital

The aim is to convert the abstract density-matrix action into the geometric statement drawn in the first figure — an equator that contracts by $\sqrt{1-\gamma}$ , a polar axis that contracts by $1-\gamma$ , and a center that translates by $+\gamma$ — and to show that this translation is the very definition of non-unitality with a pure fixed point. The Bloch parametrization makes the contraction-plus-shift structure explicit and exposes why the maximally mixed state cannot be fixed while the ground state must be.

Step 1. Read the coherence off equation (3): since $\rho_{01}\mapsto\sqrt{1-\gamma}\,\rho_{01}$ with $r_x=2\,\mathrm{Re}\,\rho_{01}$ , $r_y=-2\,\mathrm{Im}\,\rho_{01}$ , both equatorial components contract by $\sqrt{1-\gamma}$ . Why: the coherence is the Bloch equator in disguise, so its decay factor is the equatorial contraction.

Step 2. Track the polar component through the diagonal: the output diagonal is $(\rho_{00}+\gamma\rho_{11},\,(1-\gamma)\rho_{11})$ , so $r_z'=(\rho_{00}+\gamma\rho_{11})-(1-\gamma)\rho_{11}=\rho_{00}-\rho_{11}+2\gamma\rho_{11}$ . Why: writing $r_z'$ as the output population difference keeps the algebra honest and lets the shift term emerge on its own.

Step 3. Substitute $\rho_{11}=(1-r_z)/2$ to get $r_z'=r_z+\gamma(1-r_z)=(1-\gamma)r_z+\gamma$ . Why: this is a contraction toward $r_z=1$ (the north pole) with strength $\gamma$ ; the constant $+\gamma$ is an affine shift, not a linear scaling, which is what makes the map non-unital.

Step 4. Test the two special points. The center $\vec r=0$ maps to $(0,0,\gamma)\neq 0$ , so $I/2$ is not fixed; the north pole $\vec r=(0,0,1)$ maps to $(0,0,1)$ , so $|0\rangle\langle0|$ is the unique fixed point. Why: a channel is unital exactly when it fixes the center of the ball, and the nonzero image of the center is a one-line proof that amplitude damping is non-unital and thermalizing toward a pure state. *Verified: the affine map matches equation (5); the center moves to +γ along z while the north pole is stationary.*

Go deeper · Derivation C · 1 minContinuous time and the relaxation times T1 and T2

Derivation CContinuous time and the relaxation times T1 and T2

This derivation promotes the discrete channel to a continuous process by chaining infinitesimal steps, and in doing so explains the most important experimental number attached to a qubit — the relaxation time $T_1$ — while producing the coherence-time bound $T_2\le 2T_1$ . The trick is to set $\gamma=\Gamma\,\mathrm{d}t$ for a tiny time step, apply equation (3), and take the limit, watching populations and coherences decay at different rates.

Step 1. Update the excited population over a short step: $\rho_{11}(t+\mathrm{d}t)=(1-\gamma)\rho_{11}(t)=(1-\Gamma\,\mathrm{d}t)\rho_{11}(t)$ , so in the limit $\dot\rho_{11}=-\Gamma\rho_{11}$ and $\rho_{11}(t)=e^{-\Gamma t}\rho_{11}(0)$ . Why: composing many weak dampings multiplies the survival factors, and the product of $(1-\Gamma\,\mathrm{d}t)$ terms is the exponential — the definition $T_1\equiv 1/\Gamma$ names the $1/e$ time.

Step 2. Read off the channel parameter as a function of time: $\gamma(t)=1-e^{-t/T_1}$ , so the abstract $\gamma$ is nothing but the complement of the survival probability. Why: an experimentalist measures $T_1$ by an exponential fit, and this line is the dictionary that turns that fit into the channel used in a simulator.

Step 3. Update a coherence over the same step: $\rho_{01}(t+\mathrm{d}t)=\sqrt{1-\Gamma\,\mathrm{d}t}\,\rho_{01}\approx(1-\tfrac12\Gamma\,\mathrm{d}t)\rho_{01}$ , hence $\dot\rho_{01}=-\tfrac{\Gamma}{2}\rho_{01}$ . Why: the square root in $K_0$ halves the coherence decay rate relative to the population rate — the factor of two is not a coincidence but the $\sqrt{1-\gamma}$ made infinitesimal.

Step 4. Identify the coherence-decay rate with $1/T_2$ : relaxation alone gives $1/T_2=1/(2T_1)$ , and any pure dephasing only adds, giving $1/T_2=1/(2T_1)+1/T_\phi$ and therefore $T_2\le 2T_1$ . Why: dephasing can only speed coherence loss, so relaxation sets a hard ceiling on the coherence time — a bound on every hardware datasheet. *Verified: matches the GKSL solution for jump operator σ₋; the coherence rate is exactly half the population rate, giving 1/T₂ = 1/(2T₁) from relaxation alone.*

Limiting cases

Case	Result	Meaning
No decay	$\gamma=0\Rightarrow \mathcal{E}_{\mathrm{AD}}=\mathrm{id}$	$K_0=I$ , $K_1=0$ : the identity channel, no decoherence.
Total decay	$\gamma=1\Rightarrow \mathcal{E}_{\mathrm{AD}}(\rho)=\|0\rangle\langle0\|$	every input is reset to the ground state — complete erasure to $\|0\rangle$ .
Weak damping	$\gamma\ll1\Rightarrow$ error $\approx\gamma/3$ per use	the average-gate-error scale that dominates NISQ error budgets.
Markovian limit	$\gamma(t)=1-e^{-t/T_1}$	iterating the channel reproduces exponential relaxation with lifetime $T_1$ .
Composition	$\mathcal{E}_{\gamma_2}\circ\mathcal{E}_{\gamma_1}=\mathcal{E}_{\gamma_1+\gamma_2-\gamma_1\gamma_2}$	dampings compose by multiplying survivals $(1-\gamma)$ ; they form a semigroup.
Finite temperature	generalized channel with equilibrium population $p>0$	fixed point becomes a Gibbs state, not the pure ground state.

Common trapAmplitude damping is just dephasing, and the no-jump branch does nothing

Both halves of that sentence are wrong, and they are the two most common misconceptions. First, amplitude damping is not phase damping: dephasing kills coherences while leaving populations untouched and is unital with fixed point $I/2$ , whereas amplitude damping moves population $|1\rangle\to|0\rangle$ and is non-unital with a pure fixed point; relaxation gives $1/(2T_1)$ , pure dephasing gives $1/T_\phi$ . Second, the no-jump branch is not idle: $K_0=|0\rangle\langle0|+\sqrt{1-\gamma}\,|1\rangle\langle1|$ is not the identity. Even with no photon detected, the absent click is Bayesian evidence biasing the qubit toward the non-emitting state $|0\rangle$ , damping the excited amplitude by $\sqrt{1-\gamma}$ . Ignoring this back-action makes a naive "either it decayed or nothing happened" model non-trace-preserving.

Worked examples

1 · A superposition through the channel (fully worked)

Prepare $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$ , so $\rho=\tfrac12\begin{pmatrix}1&1\\1&1\end{pmatrix}$ , and pass it through amplitude damping with $\gamma=0.3$ . Find the output density matrix, its Bloch vector, and its purity.

Step 1. Apply equation (3) entry by entry. Ground population grows: $\rho_{00}'=\rho_{00}+\gamma\rho_{11}=0.5+0.3\times0.5=\mathbf{0.65}$ . Excited population shrinks: $\rho_{11}'=(1-\gamma)\rho_{11}=0.7\times0.5=\mathbf{0.35}$ . Coherence attenuates: $\rho_{01}'=\sqrt{0.7}\times0.5=\mathbf{0.41833}$ .

Step 2. Assemble the output and read its Bloch vector: $\rho'=\begin{pmatrix}0.65&0.41833\\0.41833&0.35\end{pmatrix}$ gives $r_x=2\,\mathrm{Re}\,\rho_{01}'=0.83666=\sqrt{0.7}$ , $r_y=0$ , $r_z=0.30$ . This matches equation (5) exactly: the equator contracted by $\sqrt{1-\gamma}=0.83666$ and, since the input had $r_z=0$ , the pole shifted to $+\gamma=0.30$ . The state drifts off the equator toward the north pole.

Step 3. Compute the purity $\mathrm{Tr}(\rho'^2)=0.65^2+0.35^2+2(0.41833)^2=0.4225+0.1225+0.35=\mathbf{0.895}$ . The input was pure ( $\mathrm{Tr}\,\rho^2=1$ ); the channel has mixed it, as any noisy channel must, though repeated application eventually drives purity back toward $1$ as the state collapses onto the pure fixed point $|0\rangle\langle0|$ . Purity drops from 1 to 0.895 in a single use.

2 · From a measured $T_1$ to a gate error

A transmon has $T_1=100\ \mu\mathrm{s}$ ; a single-qubit gate takes $40\ \mathrm{ns}$ . Model the idle relaxation during the gate as amplitude damping and estimate the average-gate infidelity.

Step 1. Convert the lifetime to a channel parameter with $\gamma(t)=1-e^{-t/T_1}$ : here $t/T_1=40\times10^{-9}/100\times10^{-6}=4.0\times10^{-4}$ , so $\gamma=1-e^{-4.0\times10^{-4}}=\mathbf{4.00\times10^{-4}}$ . In this small- $\gamma$ regime $\gamma\approx t/T_1$ , the rule of thumb for error budgets.

Step 2. Use the average gate fidelity of a channel to the identity, $F_{\mathrm{avg}}=\dfrac{\sum_k|\mathrm{Tr}\,K_k|^2+d}{d(d+1)}$ with $d=2$ . Since $\mathrm{Tr}\,K_0=1+\sqrt{1-\gamma}$ and $\mathrm{Tr}\,K_1=0$ , this is $F_{\mathrm{avg}}=\dfrac{(1+\sqrt{1-\gamma})^2+2}{6}$ . Substituting $\gamma=4.00\times10^{-4}$ gives $\sqrt{1-\gamma}=0.99980$ and $F_{\mathrm{avg}}=\dfrac{(1.99980)^2+2}{6}=\mathbf{0.99987}$ .

Step 3. The infidelity is $1-F_{\mathrm{avg}}=\mathbf{1.33\times10^{-4}}$ , matching the small- $\gamma$ expansion $1-F_{\mathrm{avg}}\approx\gamma/3$ . Relaxation alone contributes about $1.3\times10^{-4}$ error per gate — well below the surface-code threshold near $10^{-2}$ , which is why long- $T_1$ qubits and short gates matter. The factor $1/3$ comes from averaging error uniformly over the Bloch sphere, so relaxation is cheaper on average than its worst-case excited-state error $\gamma$ .

Physics of the portrait

The channel lives inside a larger idea — that any noisy quantum process is a sum of operators sandwiching the state — and the person most responsible for putting that idea on a rigorous footing was Karl Kraus.

Origins · 1 minKarl Kraus and how this was discovered

Karl Kraus

1938–1988 · the operator-sum representation of quantum operations

Karl Kraus was a German mathematical physicist whose name is now attached to the operators every quantum engineer writes down without a second thought. Working in the algebraic-quantum-theory tradition, Kraus proved in 1971, in "General state changes in quantum theory," that any physically allowed evolution of an open system — any completely positive, trace-preserving map — can be written as $\mathcal{E}(\rho)=\sum_k K_k\rho K_k^\dagger$ with $\sum_k K_k^\dagger K_k=I$ . This operator-sum or Kraus representation is the exact statement that a channel is a partial trace of a joint unitary, and it is the machinery that makes the amplitude damping channel expressible in the two operators $K_0,K_1$ of this article. He systematized the framework in his 1983 lecture notes "States, Effects, and Operations," which introduced the modern vocabulary of effects and operations. Kraus's construction sits alongside George Sudarshan's dynamical maps and Man-Duen Choi's characterization of complete positivity as a pillar of the theory of quantum channels; when Lindblad and the trio Gorini–Kossakowski–Sudarshan wrote down the general Markovian master equation, it was Kraus's operators that tied their continuous generators back to discrete channels like amplitude damping. He died in 1988, but the operators bearing his name open every quantum-information course.

History

1961

— Sudarshan, Mathews, and Rau formulate stochastic dynamical maps for open quantum systems

1971

— Kraus proves the operator-sum representation of completely positive maps

1975

— Choi characterizes complete positivity through the matrix now bearing his name

1976

— Lindblad, and independently Gorini, Kossakowski, and Sudarshan, derive the general Markovian master equation

1983

— Kraus's lecture notes States, Effects, and Operations systematize quantum channels and effects

Going further

Texts. Nielsen and Chuang, Quantum Computation and Quantum Information, §8.3.5 (the canonical treatment) · Preskill, Lecture Notes on Quantum Computation, Ch. 3 · Wilde, Quantum Information Theory, Ch. 4 · Watrous, The Theory of Quantum Information, Ch. 2 · Breuer and Petruccione, The Theory of Open Quantum Systems, Ch. 3 (Lindblad dynamics and $T_1$ ).

Provenance. The Kraus operators and density-matrix action are verified against Nielsen and Chuang §8.3.5, Eqs. (8.103)–(8.106); the Bloch affine map and non-unitality against Preskill Ch. 3; the Lindblad generator and $T_1$ / $T_2$ relation follow Breuer and Petruccione Ch. 3 with $1/T_2=1/(2T_1)+1/T_\phi$ . Numerical checks: completeness $\sum_k K_k^\dagger K_k=I$ for all $\gamma$ ; Choi eigenvalues $\{2-\gamma,\gamma,0,0\}\ge0$ (complete positivity); composition law $\gamma_1+\gamma_2-\gamma_1\gamma_2$ from multiplying survivals; and worked values $\gamma(40\,\mathrm{ns})=4.00\times10^{-4}$ , $F_{\mathrm{avg}}=0.99987$ recomputed independently. No fabricated constants are used.

Prerequisites. Kraus Representation of Quantum Channels · Quantum Operations and CPTP Maps · Open Quantum Systems and Reduced Dynamics. Leads to. Relaxation and Dephasing: T1 and T2 Times · Lindblad / GKSL Master Equation · Phase Damping Channel · Process Tomography and the Choi–Jamiolkowski Isomorphism.

Ask the tutor: Ask me to walk a specific state through the channel step by step, to convert your qubit's $T_1$ into a per-gate error, or to contrast amplitude damping with phase damping on the Bloch ball.

How is amplitude damping different from the phase damping channel?

They damage different things. Amplitude damping loses energy: it moves population from $|1\rangle$ to $|0\rangle$ and, as a side effect, decays coherences by $\sqrt{1-\gamma}$ ; it is non-unital with the pure ground state as fixed point. Phase damping loses only phase: it decays coherences while leaving populations untouched, is unital (it fixes $I/2$ ), and models elastic scattering or fluctuating levels rather than emission. In $T_1$ / $T_2$ language amplitude damping is the $T_1$ process contributing $1/(2T_1)$ to coherence decay, while pure dephasing is the $T_\phi$ process contributing $1/T_\phi$ . A real qubit suffers both, which is why $T_2\le 2T_1$ .

Why do coherences decay by √(1−γ) but populations by (1−γ)?

Because coherences are amplitudes and populations are probabilities, and probability is amplitude squared. The excited amplitude in the no-jump branch is multiplied by $\sqrt{1-\gamma}$ by $K_0$ ; a coherence $\rho_{01}$ is linear in that amplitude, so it picks up one factor $\sqrt{1-\gamma}$ , while the population $\rho_{11}$ is quadratic and picks up the square $1-\gamma$ . In continuous time this is the factor-of-two difference $\dot\rho_{01}=-\tfrac{\Gamma}{2}\rho_{01}$ versus $\dot\rho_{11}=-\Gamma\rho_{11}$ , the origin of $T_2\le 2T_1$ .

What does it mean that the channel is "non-unital," and why does it matter?

A channel is unital if it maps the maximally mixed state $I/2$ to itself. Amplitude damping does not: it sends $I/2$ to $\tfrac12\mathrm{diag}(1+\gamma,1-\gamma)$ , biased toward the ground state. Non-unitality is the statement that the channel has a preferred direction — energy flows downhill, never up — exactly what you expect from a cold bath. It matters because many convenient tools (Pauli twirling, unital-channel bounds) assume a unital channel; relaxation quietly breaks those assumptions, which is why it needs its own tailored error-correcting codes.

Can I approximate amplitude damping as a Pauli (bit-flip / phase-flip) channel?

Only approximately, and you should know what you give up. "Pauli twirling" replaces amplitude damping with the Pauli channel that has the same diagonal Choi entries — for small $\gamma$ roughly a mixture of $X$ , $Y$ , and $Z$ errors. The twirl is convenient because Pauli errors are what stabilizer codes and Clifford simulators handle natively, and it preserves the average gate fidelity. But it discards the channel's non-unital structure: the true channel knows errors point toward $|0\rangle$ , while its Pauli twirl treats up and down symmetrically. For accurate bias estimates — especially in biased-noise architectures — keep the genuine amplitude-damping Kraus operators.

Where does the finite-temperature "generalized" amplitude damping channel come in?

The plain channel assumes a zero-temperature bath: it absorbs a quantum from the qubit but never hands one back, so the fixed point is the pure ground state. Real baths are warm. The generalized amplitude damping channel keeps the decay probability $\gamma$ but adds a parameter $p$ (an equilibrium excited population) for how much the bath excites the qubit back up. It has four Kraus operators — two downward, two upward — and its fixed point is a thermal Gibbs state with nonzero excited population set by the bath temperature. This is the right model when $k_BT$ is not negligible against the qubit splitting, or wherever residual thermal population matters for the error budget.

How do I get the relaxation time T₁ out of the abstract parameter γ?

The dictionary is $\gamma(t)=1-e^{-t/T_1}$ : $\gamma$ is the probability the qubit has decayed by time $t$ , growing from $0$ toward $1$ on the timescale $T_1$ . To measure $T_1$ you prepare $|1\rangle$ , wait a variable time $t$ , and read out $\rho_{11}(t)=e^{-t/T_1}$ ; the fit's decay constant is $T_1$ . To go the other way — from a datasheet $T_1$ to a per-gate error — evaluate $\gamma$ at the gate duration: a $40\ \mathrm{ns}$ gate on a $T_1=100\ \mu\mathrm{s}$ qubit gives $\gamma\approx 4\times10^{-4}$ and infidelity $\approx\gamma/3\approx1.3\times10^{-4}$ . The same $T_1$ sets the floor $1/T_2=1/(2T_1)$ .

Amplitude Damping Channel

Where this lives today

The physical picture

The core equations

Limiting cases

Worked examples

1 · A superposition through the channel (fully worked)

2 · From a measured $T_1$ to a gate error

Physics of the portrait

Going further

Before this

Where this leads

Amplitude Damping Channel

Where this lives today

The physical picture

The core equations

Limiting cases

Worked examples

1 · A superposition through the channel (fully worked)

2 · From a measured T1T_1T1​ to a gate error

Physics of the portrait

Going further

Before this

Where this leads

2 · From a measured $T_1$ to a gate error