# Trace calculation from the basis to |+> and |-> states

I was reading the paper; https://arxiv.org/abs/2002.00055 and going through some of the formulas below and I am a bit stuck between (1) and (2).

How the equation (1) turns into (2) is not clear to me.. Any point or help would be appreciated.

## 1 Answer

Let the output state (last line of equation 1) be $$\sigma$$. You're going to measure using projectors (corresponding to a standard basis measurement on the first qubit) $$P_+=|0\rangle\langle 0|\otimes I, \qquad P_-=|1\rangle\langle 1|\otimes I$$ and you want to know the probability of getting the two outcomes $$y=\pm$$ for a fixed value of $$\theta$$, so you could write this as $$p(y|\theta)=\text{Tr}(P_y\sigma).$$

• Thank you, Are you assuming $+$ is $|0\rangle$ and $-$ is $|1\rangle$ and not the superposition of 0 and 1 states? Commented Aug 18, 2021 at 15:36
• I'm not assuming. They say "if we interpret the 0/1 outcomes of the measurement as +/- respectively" Commented Aug 18, 2021 at 16:02
• @ Daftwullie, Sorry I missed that part ! :) Commented Aug 18, 2021 at 16:04