What is the difference between relaxation and dephasing?
T1 characterizes the decay of excited-state population. T2 characterizes coherence decay, which can include both energy relaxation and pure dephasing.
For independent exponential relaxation and pure dephasing, 1/T₂ = 1/(2T₁) + 1/Tφ. The rate relation assumes this decay model.
Source research checked . Teaching calculations are ideal models.
A quantum memory can fail in more than one way. An excited state can relax toward a lower-energy state, changing the populations measured in an energy basis. A relative phase can also become uncertain, weakening interference even when those populations stay the same. Treating these as a single lifetime hides useful information about both the device and the task. A memory designed to preserve an arbitrary qubit needs to preserve the relevant populations and coherence. Which process dominates depends on the hardware, operating conditions and experiment.
A simple Markovian amplitude-damping model reduces the excited population by a factor one minus gamma and the off-diagonal coherence by the square root of that factor. Pure dephasing can instead reduce off-diagonal terms while leaving populations unchanged. Combining exponential relaxation and an additional pure-dephasing process gives the familiar relation one over T2 equals one over twice T1 plus one over Tphi. The relation belongs to the stated model; it is not an excuse to turn every measured lifetime into a universal property of a qubit. Nonexponential noise, sequence-dependent filtering and calibration changes can require a more detailed description.
Starting from the same state, amplitude damping moves population toward the lower level, while phase damping reduces interference contrast without changing those populations. A combined model can contain both processes. For a memory result, ask which coherence was measured, with what sequence and under what control. A long-lived population does not automatically provide a long-lived arbitrary quantum state. Conversely, an echo sequence can improve the response to a particular noise process without changing every operating condition a calculation will encounter. The tested protocol belongs with the time scale.
Amplitude damping can change both populations and coherence. A basis-state test and a superposition test therefore reveal different aspects of the same declared channel. The plus-state example is useful because its output has both an altered diagonal distribution and reduced off-diagonal entries, so the model cannot be summarized completely by one population histogram.
For a real device, ask which measurements were used to fit the channel and how well other measurements agree with its predictions. A convenient model is not established simply because one dataset resembles one of its curves. Energy relaxation, pure dephasing and other processes may need separate parameters or a more appropriate description. The stated relation among T1, T2 and pure-dephasing time belongs to its exponential model assumptions. Keep that condition attached when applying it. The educational value of a small channel calculation is that it shows exactly what is predicted and leaves a clear place for testing the model against evidence.
Relaxation changes populations and coherence
Inputs: Sweep amplitude-damping parameter γ from 0 to 1 for a plus-state input. This figure does not plot the separate phase-damping channel.
Excited population=(1−γ)/2; X expectation=√(1−γ).
| Amplitude-damping parameter γ | Excited population | X expectation |
|---|---|---|
| 0 | 0.5 | 1 |
| 0.12 | 0.44 | 0.938083151965 |
| 0.25 | 0.375 | 0.866025403784 |
| 0.37 | 0.315 | 0.793725393319 |
| 0.5 | 0.25 | 0.707106781187 |
| 0.62 | 0.19 | 0.616441400297 |
| 0.75 | 0.125 | 0.5 |
| 0.87 | 0.065 | 0.360555127546 |
| 1 | 0 | 0 |
Relaxation is directional: it moves population toward the chosen ground state. Depolarizing noise has a different geometry, so one scalar error number cannot describe both processes. An explanatory chart can show population and coherence separately, making clear why a Z-only measurement misses part of the channel’s action.
A distinction to keep
The tempting conclusion: A parameter in an ideal noise channel is automatically the measured infidelity of a physical gate.
What follows instead: Do not interpret the channel parameter as an experimentally measured gate infidelity without a derivation linking the definitions and the physical operation.
Is T2 always twice T1?
Only if the stated exponential model applies and there is no additional pure dephasing. Extra dephasing makes T2 shorter in that model.
Does an echo remove all noise?
No. A sequence filters particular temporal structures and may introduce its own control errors. The sequence and noise assumptions matter.
Which lifetime should a reader prefer?
The relevant experiment depends on the operation or memory task. A single preferred lifetime cannot replace task-specific evidence.
At γ=1, what state emerges from any normalized input? At γ=0, what should a correct implementation do?
At complete damping the output is the ground-state projector, independent of the input. At zero damping the output equals the input. These endpoint identities are strong implementation checks because they test both population transfer and coherence handling.
Follow the complete derivation and reference calculation →
Sources and model boundaries
- IBM Quantum Learning: noise and errors ↗
Primary scientific context for an original ideal teaching example. Source research checked 19 October 2026; site integration inspected 20 October 2026. These are different checks.
- IBM Quantum Learning: quantum channel basics ↗
Primary scientific context for an original ideal teaching example. Source research checked 19 October 2026; site integration inspected 20 October 2026. These are different checks.
Further source for the decay model
Gambetta et al., Qubit-photon interactions in a cavity. Section II.C sets out the Markov approximation. Section V.A gives the intrinsic coherence-decay rate.