The Dephasing Channel

Dephasing destroys phase coherence while preserving populations, the loss mechanism behind T2 and the classical limit.

Loss of coherence, not energy

The dephasing channel (phase damping) erases the relative phase between |0> and |1> without changing their populations. In Kraus form it can be written with K0 = sqrt(1 - p/2) I and K1 = sqrt(p/2) Z, or as a probabilistic Z flip: E(rho) = (1 - p) rho + p Z rho Z. The diagonal of rho is untouched; only the off-diagonal coherences shrink.

Effect on the state

On the Bloch sphere, dephasing collapses the x and y components toward zero while leaving z fixed. The sphere flattens into the z-axis. Repeated dephasing drives any state toward a classical mixture diagonal in the computational basis. This is decoherence in its purest form: the state loses its ability to interfere while its measurement statistics in the preferred basis stay intact.

The off-diagonal terms carrying relative phase are suppressed by a factor (1 - p), heading to zero and leaving only populations.

T2 and why it dominates

Dephasing sets the T2 coherence time, typically shorter than the T1 relaxation time because phase is disturbed by many low-energy environmental fluctuations that do not exchange energy. Magnetic field noise, charge noise, and slow drifts all dephase without relaxing. Since interference is the resource behind quantum algorithms, dephasing is often the most damaging error to fight.

Mitigation

Because pure dephasing frequently comes from slow noise, it can be partly reversed by dynamical decoupling: sequences of pulses that refocus accumulated phase, echoing away slow fluctuations. Operating at coherence sweet spots, where the qubit frequency is first-order insensitive to a noise source, also extends T2. What cannot be echoed away is handled by error correction, where phase-flip codes specifically target dephasing.