What is the definition of twirled superoperator?

Question

I am trying find the definition of twirled (super)operator. One such is Definition 2.3.16 on p. 33 of Christoph Dankert, Efficient Simulation of Random Quantum States and Operators. However, the notations puzzles me.

1. What does $\Lambda(V^\dagger XV)$ mean?
2. How do the two integrals apparently both involving $\Lambda_T$ relate to each other?

Answer 1

1. $X$ is a density matrix. $V^\dagger X V$ is another density matrix. $\Lambda$ is a channel that takes one density matrix to another density matrix. So $\Lambda(V^\dagger X V)$ means take the density matrix $V^\dagger X V$ and apply the channel $\Lambda$ to it.

2. The two integrals are two ways to write the same formula. In the first, they use the Liouville operators $\hat V:=V\otimes V^*$ and $\hat\Lambda$ to define the Liouville operator version of the twirled channel $\hat\Lambda_T$. In the second, they use the channel notation $\Lambda(\cdot)$ to define the twirled channel $\Lambda_T(\cdot)$. If you just follow the definition of the Liouville operator, you can see that the Liouville operator of the thing they define as $\Lambda_T$ is indeed what they define as $\hat \Lambda_T$.