I am trying to read through the proof of ELBO for diffusion models on pg. 8 of this paper. However, I do not see how the author arrived at Eqn (45) from Eqn (44). Specifically, I do not know how they simplified the equation by rewriting it in terms of the KL divergence. I do know the formula for KL divergence but it does not seem like a direct application in this case. Any help is appreciated. I have included an image below of the proof for convenience.
Indeed the notations here are a little involved and confusing since there exist multiple joint conditional random variables here. First please note here all involved conditional random variables are all conditioned only on the fixed value of the input data $x_0$, so we can tentatively and effectively ignore it when considering other (conditional) random variables. Next let's look at the second term in your asked line (44): $$\mathbb{E}_{q(x_{T−1}, x_T | x_0)} [\log \frac{p(x_T)}{q(x_T | x_{T −1})}]$$
Notice compared to the usual KL divergence, there's an additional fixed conditioning value of $x_{T−1}$ as given data involved in the posterior $q$-distribution of the random variable $x_T$ in above term. And since the random variable $x_{T−1}$ is taken jointly with the random variable $x_T$ given $x_0$ over the expectation in above term, we have to take expectation wrt the random variable $x_{T−1}$ after rewriting the above term as KL divergence. With these ideas we can expand the above term as: $$\mathbb{E}_{x_{T−1}|x_0 \sim q(x_{T−1}| x_0)}[\mathbb{E}_{x_T|x_{T-1}, x_0 \sim q(x_T|x_{T-1}, x_0)} [\log \frac{p(x_T)}{q(x_T | x_{T −1})}]] \\= -\mathbb{E}_{q(x_{T−1}| x_0)}[\mathbb{E}_{q(x_T|x_{T-1})} [\log \frac{q(x_T | x_{T −1})}{p(x_T)}]] \\= -\mathbb{E}_{q(x_{T−1}| x_0)}[D_{KL}(q(x_T | x_{T −1})||p(x_T))]$$
Similar process can be applied to the third term to explain the additional expectation operator of the rewritten KL divergence.
