# 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)?

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.
• 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