# What is the difference between relaxation, dephasing, and decoherence?

Many sources seem to loosely use the terms "relaxation", "dephasing", and "decoherence" interchangeably, while others seem to treat certain of them as special cases of another terms, but I can't find any statements that explictly distinguish them.

To me, "decoherence" refers to the process of a pure-state density operator evolving into a mixed state, typically via the off-diagonal entries of the density matrix in some semiclassical "pointer basis" decaying to zero via uncontrolled interactions with the environment. I beleive that "relaxation" and "dephasing" are special cases of decoherence, but what are their definitions (as contrasted with general decoherence)?

## 1 Answer

Decoherence is the very general term which, more or less, is anything resulting in a loss of purity during the evolution of a system. Sometimes, when people are being a bit non-specific, they might be thinking of a particular type of decoherence such as dephasing (or perhaps depolarising) when they use the term decoherence.

Relaxation and dephasing are two very special cases of decoherence. In relaxation, we generally think of the qubits as being two-level systems where one level (say $$|1\rangle$$) is at a higher energy than the other ($$|0\rangle$$). Over time, there is the tendency of the $$|1\rangle$$ to 'relax' back to the state $$|0\rangle$$. If you want to visualise this, think of the Bloch Sphere. The action of the relaxation map is to contract the sphere towards to $$|0\rangle$$ point (so $$|0\rangle$$ stays as $$|0\rangle$$). This is Figure 8.14 in Nielsen and Chuang.

Dephasing is the same as "Z noise". This is the process that tends to reduce the off-diagonal entries of the density matrix, basically defining the "pointer basis". To visualise the effect on the Bloch sphere, it's a contraction of the sphere where every point moves towards the Z axis (every point on the Z axis is preserved). It's figure 8.9 in Nielsen & Chuang.

Their proper definitions can be given mathematically as Lindblad operators. The main equation is 8.134 in Nielsen and Chuang, with the operators (effectively) being defined in 8.96 (dephasing) and 8.108 (relaxation, also referred to as amplitude damping)

• In the case of relaxation, does the transition from an arbitrary pure state to the pure state $|0\rangle$ really constitute "loss of purity", or just non-unitary time evolution? – tparker Oct 1 '19 at 3:12
• While it is true that in the long time limit everything heads to a pure state (and so you’re right that non-unitary might be a better description that loss of purity), the process that it goes through changes any pure state (except 0) into a mixed state, hence the description I chose. – DaftWullie Oct 1 '19 at 5:34