# Amplitude damping channel

The amplitude damping channel is a quantum channel that models the loss of energy from a qubit to its environment, most commonly spontaneous emission, in which the excited state |1⟩ decays to the ground state |0⟩ with probability γ.<sup>[1](https://quantumchannelzoo.org/channel/amplitude-damping)</sup> It is one of the standard examples of quantum noise, alongside depolarizing and phase damping channels, and it is the canonical model for energy relaxation (T1 decay) in quantum computing hardware.<sup>[2](https://quantumcomputingcourses.com/tutorials/quantum-channels-kraus-operators)</sup>

| Key fact | Detail |
|---|---|
| Physical model | With probability γ an excited atom decays to its ground state by emitting a photon; the environment changes from "no photon" to "one photon" when emission occurs<sup>[3](https://www.preskill.caltech.edu/ph219/Ph-CS-219A-Slides-2020/Ph-CS-219A-Lecture-5-Qubit-Channels.pdf)</sup> |
| Kraus operators | \( K_0 = \begin{pmatrix} 1 & 0 \\ 0 & \sqrt{1-\gamma} \end{pmatrix} \), \( K_1 = \begin{pmatrix} 0 & \sqrt{\gamma} \\ 0 & 0 \end{pmatrix} \)<sup>[1](https://quantumchannelzoo.org/channel/amplitude-damping)</sup> |
| Degradability | Degradable for \( \gamma \in [0, 1/2] \); anti-degradable for \( \gamma \in [1/2, 1] \), where the quantum capacity is zero<sup>[1](https://quantumchannelzoo.org/channel/amplitude-damping)</sup><sup> • </sup><sup>[4](https://www.nature.com/articles/srep05720)</sup> |
| Quantum capacity | In the degradable regime it has a single-letter expression maximized over the excited-state population of the input<sup>[5](https://zenodo.org/records/22922992)</sup> |
| Classical capacity | Equals the single-use Holevo information, attained by a binary pure-state ensemble with collective decoding<sup>[6](https://arxiv.org/abs/2609.28592)</sup> |
| Composition rule | Two channels with transmissions \( 1-\gamma_1 \) and \( 1-\gamma_2 \) compose to one channel with transmission \( (1-\gamma_1)(1-\gamma_2) \)<sup>[7](https://quantum.nschuch.fastmail.fm.user.fm/teaching/ss15/ss15-quantuminformation-exercise02.pdf)</sup> |
| Hardware relevance | The generalized version is a noise source in superconducting-circuit quantum computing and the qubit analog of the bosonic thermal channel<sup>[8](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.102.012401)</sup> |

## Kraus representation

A quantum channel maps an input density matrix ρ to an output \( \rho' = \sum_j K_j \rho K_j^{\dagger} \), where the [Kraus operators](https://www.edgepedia.org/kraus-operators) satisfy the completeness relation \( \sum_j K_j^{\dagger} K_j = I \).<sup>[9](https://repository.hkust.edu.hk/ir/bitstream/1783.1-141033/1/991013231858103412.pdf)</sup> For the amplitude damping channel with damping probability γ, the two Kraus operators are the "no-jump" operator \( K_0 = |0\rangle\langle 0| + \sqrt{1-\gamma}\,|1\rangle\langle 1| \) and the "jump" operator \( K_1 = \sqrt{\gamma}\,|0\rangle\langle 1| \).<sup>[1](https://quantumchannelzoo.org/channel/amplitude-damping)</sup><sup> • </sup><sup>[3](https://www.preskill.caltech.edu/ph219/Ph-CS-219A-Slides-2020/Ph-CS-219A-Lecture-5-Qubit-Channels.pdf)</sup> The jump operator is not Hermitian, unlike the Kraus operators of Pauli-type noise channels.<sup>[10](https://courses.cs.washington.edu/courses/cse599d/06wi/lecturenotes15.pdf)</sup>

Applied to a general state, the channel transforms the density matrix entries as

\[ \rho' = \begin{pmatrix} \rho_{00} + \gamma\,\rho_{11} & \sqrt{1-\gamma}\,\rho_{01} \\ \sqrt{1-\gamma}\,\rho_{10} & (1-\gamma)\,\rho_{11} \end{pmatrix}. \]

Spontaneous decay therefore affects both the diagonal and the off-diagonal entries: population flows from |1⟩ to |0⟩, and coherence is reduced by the factor \( \sqrt{1-\gamma} \).<sup>[3](https://www.preskill.caltech.edu/ph219/Ph-CS-219A-Slides-2020/Ph-CS-219A-Lecture-5-Qubit-Channels.pdf)</sup> This distinguishes it from phase damping, which preserves the diagonal elements and only degrades coherence.<sup>[11](https://homel.vsb.cz/~bil0127/subjects/koch/exercise-5/)</sup>

## Physical origin

The channel arises from a unitary interaction with a vacuum environment:<sup>[3](https://www.preskill.caltech.edu/ph219/Ph-CS-219A-Slides-2020/Ph-CS-219A-Lecture-5-Qubit-Channels.pdf)</sup>

\[ |0\rangle_A \otimes |0\rangle_E \mapsto |0\rangle_A \otimes |0\rangle_E, \qquad |1\rangle_A \otimes |0\rangle_E \mapsto \sqrt{1-p}\,|1\rangle_A \otimes |0\rangle_E + \sqrt{p}\,|0\rangle_A \otimes |1\rangle_E. \]

Tracing over the environment yields the Kraus form. The same dynamics follows from the Lindblad master equation with jump operator \( L = \sqrt{\Gamma_1}\,\sigma_- \), where \( \sigma_- = |0\rangle\langle 1| \) is the lowering operator and \( \Gamma_1 \) the spontaneous emission rate; over a time interval t the damping parameter is \( \gamma = 1 - e^{-\Gamma_1 t} \).<sup>[12](https://cqdmqd.yale.edu/sites/default/files/2024-04/quantum_dynamics_tutorials_PartII.pdf)</sup><sup> • </sup><sup>[13](https://everettyou.github.io/teaching/PHYS130B/notes/ch6_quantum-foundations/6-4-2-lindblad-master-equation.html)</sup> Beyond spontaneous emission, the channel models inelastic scattering, thermalization of spins to a lattice, and other processes in which energy is exchanged between system and environment.<sup>[14](https://openlearninglibrary.mit.edu/courses/course-v1%3AMITx%2B8.371.1x%2B2T2018/jump_to/block-v1%3AMITx%2B8.371.1x%2B2T2018%2Btype%40sequential%2Bblock%40ps3)</sup>

**Geometric picture.** On the [Bloch sphere](https://www.edgepedia.org/bloch-sphere) the channel acts as \( r'_x = \sqrt{1-\gamma}\,r_x \), \( r'_y = \sqrt{1-\gamma}\,r_y \), \( r'_z = (1-\gamma)\,r_z + \gamma \): the equatorial axes shrink by \( \sqrt{1-\gamma} \), the polar axis shrinks by \( 1-\gamma \) and is translated up by γ, and the image of the sphere is an oblate spheroid whose unique fixed point is the north pole |0⟩.<sup>[15](https://www.physik.fu-berlin.de/en/einrichtungen/ag/ag-eisert/teaching/ss26/sheet_05_solutions.pdf)</sup> The channel is non-unital, meaning it does not leave the maximally mixed state invariant; it contracts every state toward |0⟩.<sup>[16](https://qiqc-op.com/problem/op_fcd21a1a5021e464/)</sup>

## Capacities

The quantum capacity of a memoryless channel is given by a regularized coherent information, generally requiring an optimization over arbitrarily many channel uses.<sup>[17](https://citeseerx.ist.psu.edu/document?doi=78ec92f656a96b3381fdeb62148d60abf43308c1&repid=rep1&type=pdf)</sup> For the amplitude damping channel the degradability structure simplifies this: the channel is degradable for \( \gamma \le 1/2 \), where the quantum capacity equals a single-letter coherent information maximized over the excited-state population of the input, and anti-degradable for \( \gamma \ge 1/2 \), where the quantum capacity is zero.<sup>[1](https://quantumchannelzoo.org/channel/amplitude-damping)</sup><sup> • </sup><sup>[4](https://www.nature.com/articles/srep05720)</sup><sup> • </sup><sup>[5](https://zenodo.org/records/22922992)</sup> Because the channel is non-unital, the maximizing input population is generally not 1/2.<sup>[16](https://qiqc-op.com/problem/op_fcd21a1a5021e464/)</sup> Recent work has evaluated this optimization analytically, giving an explicit absolutely convergent series for the capacity in the interior of the degradable regime, with no residual input-state optimization.<sup>[5](https://zenodo.org/records/22922992)</sup>

**Classical capacity.** A 2025 result removed the regularization for the classical capacity: for the amplitude damping channel with damping probability p, the unassisted classical capacity equals the single-use Holevo information, attained by a binary pure-state ensemble with collective decoding, and the entanglement cost of simulating the channel is \( h_2((1+\sqrt{p})/2) \) ebits per use, where \( h_2 \) is the binary entropy.<sup>[6](https://arxiv.org/abs/2609.28592)</sup>

**Correlated variants.** For a fully correlated amplitude damping channel acting on two qubits, D'Arrigo, Benenti, Falci, and Macchiavello derived the single-shot classical capacity and showed that entanglement is needed to achieve the channel's best performance; they also evaluated the quantum capacity for any value of the noise parameter and computed the entanglement-assisted classical capacity.<sup>[18](https://arxiv.org/abs/1309.2219)</sup>

## Generalized and multilevel versions

The generalized amplitude damping channel (GADC) adds a parameter for the excited-state population at thermal equilibrium, modeling a qubit coupled to a thermal bath at nonzero temperature, where the qubit can absorb energy from the environment as well as emit.<sup>[19](https://docs.quantum.ibm.com/api/qiskit/0.42/qiskit_aer.noise.amplitude_damping_error)</sup><sup> • </sup><sup>[2](https://quantumcomputingcourses.com/tutorials/quantum-channels-kraus-operators)</sup> The GADC is one of the sources of noise in superconducting-circuit-based quantum computing and can be viewed as the qubit analog of the bosonic thermal channel, used to model lossy processes in the presence of background noise for low-temperature systems.<sup>[8](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.102.012401)</sup> Its equilibrium state after infinitely many applications is diagonal with populations \( 1-p_1 \) and \( p_1 \), the excited-state thermal population.<sup>[19](https://docs.quantum.ibm.com/api/qiskit/0.42/qiskit_aer.noise.amplitude_damping_error)</sup> Information-theoretic work has determined the parameter ranges for which the GADC is entanglement breaking and antidegradable, and established upper bounds on its classical, quantum, private, and two-way assisted capacities that tighten earlier bounds over the entire parameter range.<sup>[8](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.102.012401)</sup>

Multilevel amplitude damping channels generalize the qubit channel to quantum systems of finite dimension d, modeling noise and decoherence in a qudit coupled to a thermal environment.<sup>[20](https://arxiv.org/abs/2008.00477)</sup><sup> • </sup><sup>[21](https://arxiv.org/abs/2605.27369)</sup>

## Use in simulation and benchmarking

The channel is a built-in noise model in standard quantum computing frameworks. Qiskit provides `amplitude_damping_error` and a combined `phase_amplitude_damping_error` parameterized by a damping parameter and an excited-state thermal population, and a `thermal_relaxation_error` parameterized by the relaxation times \( T_1 \), \( T_2 \), gate time, and thermal population.<sup>[22](https://eu-de.quantum.cloud.ibm.com/docs/en/guides/build-noise-models)</sup> In device benchmarking, energy decay of a qubit, including spontaneous emission and similar non-radiative energy-loss processes, is typically modeled as an amplitude damping error, and decay also destroys phase coherence since a qubit that decays to the ground state loses all information about its prior phase.<sup>[23](https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuantum.6.030202)</sup>

## References

1. [Amplitude Damping Channel, The Quantum Channel Zoo](https://quantumchannelzoo.org/channel/amplitude-damping)
2. [Quantum Channels and Kraus Operators Explained](https://quantumcomputingcourses.com/tutorials/quantum-channels-kraus-operators)
3. [Ph/CS 219A Lecture 5: Qubit Channels, John Preskill, Caltech](https://www.preskill.caltech.edu/ph219/Ph-CS-219A-Slides-2020/Ph-CS-219A-Lecture-5-Qubit-Channels.pdf)
4. [Non-Markovianity and reservoir memory of quantum channels, Scientific Reports](https://www.nature.com/articles/srep05720)
5. [Exact series formula for the quantum capacity of the amplitude damping channel, S. Pirandola, Zenodo](https://zenodo.org/records/22922992)
6. [Classical Capacity and Entanglement Cost of the Amplitude Damping Channel, arXiv](https://arxiv.org/abs/2609.28592)
7. [Quantum information exercise sheet 2 (amplitude damping channel)](https://quantum.nschuch.fastmail.fm.user.fm/teaching/ss15/ss15-quantuminformation-exercise02.pdf)
8. [Information-theoretic aspects of the generalized amplitude-damping channel, Phys. Rev. A 102, 012401 (2020)](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.102.012401)
9. [HKUST thesis: Kraus operators and noise channels](https://repository.hkust.edu.hk/ir/bitstream/1783.1-141033/1/991013231858103412.pdf)
10. [CSE 599d Lecture 15, University of Washington](https://courses.cs.washington.edu/courses/cse599d/06wi/lecturenotes15.pdf)
11. [Exercise Sheet 5: Error Correction Introduction](https://homel.vsb.cz/~bil0127/subjects/koch/exercise-5/)
12. [Quantum dynamics tutorials, Part II, Yale](https://cqdmqd.yale.edu/sites/default/files/2024-04/quantum_dynamics_tutorials_PartII.pdf)
13. [Lindblad Master Equation, Quantum Mechanics notes](https://everettyou.github.io/teaching/PHYS130B/notes/ch6_quantum-foundations/6-4-2-lindblad-master-equation.html)
14. [Two-qubit amplitude damping code, MIT Open Learning Library](https://openlearninglibrary.mit.edu/courses/course-v1%3AMITx%2B8.371.1x%2B2T2018/jump_to/block-v1%3AMITx%2B8.371.1x%2B2T2018%2Btype%40sequential%2Bblock%40ps3)
15. [Exercise Sheet 5: Quantum channels, FU Berlin](https://www.physik.fu-berlin.de/en/einrichtungen/ag/ag-eisert/teaching/ss26/sheet_05_solutions.pdf)
16. [Capacity-achieving codes for amplitude damping, QIQCOP Zoo](https://qiqc-op.com/problem/op_fcd21a1a5021e464/)
17. [Quantum capacity of an amplitude-damping channel, CiteSeerX](https://citeseerx.ist.psu.edu/document?doi=78ec92f656a96b3381fdeb62148d60abf43308c1&repid=rep1&type=pdf)
18. [Classical and quantum capacities of a fully correlated amplitude damping channel, D'Arrigo et al., arXiv](https://arxiv.org/abs/1309.2219)
19. [amplitude_damping_error, Qiskit Aer documentation](https://docs.quantum.ibm.com/api/qiskit/0.42/qiskit_aer.noise.amplitude_damping_error)
20. [Quantum capacity analysis of multi-level amplitude damping channels, arXiv](https://arxiv.org/abs/2008.00477)
21. [Generalized multilevel amplitude damping channels and their thermodynamic performances, arXiv](https://arxiv.org/abs/2605.27369)
22. [Build noise models, IBM Quantum Documentation](https://eu-de.quantum.cloud.ibm.com/docs/en/guides/build-noise-models)
23. [Practical Introduction to Benchmarking and Characterization of Quantum Computers, PRX Quantum 6, 030202](https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuantum.6.030202)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum information science › Quantum communication and information theory › Quantum information theory › Quantum channels and capacity › Noise models and channel families*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license