# What's the difference between T1 and T2?

I learned that T1 is relaxation time (time from $$|1\rangle$$ to $$|0\rangle$$) and T2 is coherence time. The relaxation is a specific case of decoherence. What's the difference between them and what's the exact meaning of coherence time T2?

T2 is so-called dephasing time.

It describes how long the phase of a qubit stays intact. In your words, it is time from $$|+\rangle= \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$$ to $$|-\rangle= \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)$$, or conversely.

Just note that both T1 and T2 are not actually "time from state x to state y" but rather decay constants. Probability that a qubit will stay in state $$|1\rangle$$ after time $$t$$ is given by formula

$$P(|1\rangle) = \mathrm{e}^{-\frac{t}{T1}}.$$

Similarly for T2.

Both times T1 and T2 are together called decoherence times.

Slight correction to Martin Vesely's answer: $$T_2$$ is not the (decay constant) time after which an initial state $$|+\rangle$$ will necessarily switch to the state $$|-\rangle$$. If it were, then error correction would be easy. Instead, it's the (decay constant) time after which an initial state $$|+\rangle$$ will evolve into an equal classical probabilistic mixture of the $$|+\rangle$$ and $$|-\rangle$$ states, so that you can no longer confidently predict the state. That is, it's the autocorrelation time after which the initial and final states become uncorrelated, not negatively correlated.

• Thanks for clarification. – Martin Vesely Feb 6 '20 at 15:17
• +1 to this answer - a good explanation can be found here as well, I think it is helpful to see the "how do you measure it" and "how the curves typically look like": ocw.mit.edu/courses/mathematics/… – Balint Pato Oct 22 '20 at 18:41