Problem deriving entropic uncertainty relation

Question

In this paper the authors state that the inequality near the bottom of page 2 reduces to inequality (1) when $N=1$. However, I am struggling to get that result, as I have an extra minus sign in front of the integrals. Can anyone try this for themselves and see if they get the correct result?

$$\frac{n}{4}N(1+\ln{\pi})-\frac{1}{2}N^{-1} \int d^n \mathbf{r}|\Psi(\mathbf{r})|^2 \ln{|\Psi(\mathbf{r})}|-\frac{1}{2}N^{-1} \int d^n \mathbf{k}|\tilde{\Psi}(\mathbf{k})|^2 \ln{|\tilde{\Psi}(\mathbf{k})}|+N \ln{N \geq 0}$$

should reduce to

$$-\langle \ln{\rho}\rangle - \langle \ln{\tilde{\rho}}\rangle \geq n(1 + \ln{\pi}) $$

where

$\rho (\mathbf{r})= |\Psi(\mathbf{r})|^2$, $\tilde{\rho}(\mathbf{k})= |\tilde{\Psi}(\mathbf{k})|^2$ and $\langle \rangle$ denotes mean value, so that $\langle \ln{\rho} \rangle = \int d^n \mathbf{r} \,\rho (\mathbf{r})\ln{\rho(\mathbf{r})}$

Answer 1

Due diligence first. The first term of the first equation you wrote is missing a sign, as you should have easily checked yourself, from the coefficient of $\ln \pi$.

That is, write down the parent $W(q)$ right above the equation you start with, around your minimum q =2, i.e. $q\equiv 2(1+\epsilon)$, so that $p= 2(1-\epsilon) +O(\epsilon^2)$, dismissing any higher orders of $\epsilon$.

The relevant term for the derivative w.r.t. $2\epsilon$ then is $\pi^{-n2\epsilon/4} N$, so you may catch that paper's typo.