(Informal) Errata of "A short introduction to the Lindblad Master Equation"

p.8 Equation (24)

　 $\displaystyle\rho_E^{(1)}=\langle 0|_2\rho_E| 0\rangle _2 + \langle 1|_2\rho_E| 1\rangle _2 =\frac{1}{2}(|0\rangle\langle 0|_1+|1\rangle\langle 1|_2)$

p.10 Equation (30)

　 $\displaystyle \rho_B = \pmatrix{ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 }$

p.11 Equation (31)

　 $\displaystyle (\mathbb{1}\otimes T_2)\rho_B = \frac{1}{2}\Bigg\{\pmatrix{1 & 0 \\ 0 & 0}\otimes\pmatrix{0 & 0 \\ 0 & 1} + \pmatrix{0 & 0 \\ 0 & 1}\otimes\pmatrix{1 & 0 \\ 0 & 0}$

　　　　　　　　 $\otimes\pmatrix{0 & 0 \\ 1 & 0}\otimes\pmatrix{0 & 0 \\ 1 & 0} + \pmatrix{0 & 1 \\ 0 & 0}\otimes\pmatrix{0 & 1 \\ 0 & 0}\Bigg\}$

p.13 Derivation of (45) from (44)

First, in (44), we change the integral variable from $s \rightarrow t-s$ (without any approximation) to get:

　 $\displaystyle \dot{\hat\rho}(t) = -\alpha^2\int_0^tds {\rm Tr}_E \Big[ \hat H_I(t) ,\Big[ \hat H_I(t-s),\hat\rho (t)\otimes\hat\rho_E(0) \Big]\Big]$

Then, we assume that the kernel in the integration is dominant where $s \sim 0$ (that is, $\hat H_I(t-s) \sim \hat H_I(t)$ ) and extend the upper limit of the integration to infinity. It results in (45).

p.19 Equation (81)

　 $\displaystyle\frac{d\rho}{dt}=\lim_{\Delta t\to 0}\frac{1}{\Delta t}(\mathcal V(\Delta t)\rho - \rho) = \lim_{\Delta t\to 0}\frac{1}{\Delta t}\left(\sum_{i,j=1}^{d^2}c_{i,j}(\Delta t)F_i\rho F^\dagger_j-\rho\right)$