Twirling of quantum states: Maximally entangled states

Question

I have been reading the paper "Resource theory of unextendibility and non-asymptotic quantum capacity" (https://arxiv.org/pdf/1803.10710.pdf) by Kaur et.al, I have two questions specifically. How does the inequality (114) come about and how does this inequality transform into (115)?

The authors mention "The first equality (115) is due to the “transpose trick” property of the maximally entangled state, which leads to its $U\otimes U^{*}$ invariance." Firstly, I don't understand why $\Phi_{RA}$ can be written in this way( as an integral), after some preliminary reading I have found that this is related to Haar measures and twirls. I am completely unfamiliar with the subject. The references that I have looked at online treat the topic from a pure mathematics perspective, being a physics student I find it hard to grasp it fully. I am looking for a treatment of this subject from a quantum information point of view. Any help will be appreciated. Basically, I am confused about how (115) is coming at all.

Answer 1

Let $|\psi\rangle = \frac{1}{\sqrt{d}}\sum_{i=0}^{d-1} |ii\rangle_{AB}$ be a maximally entangled state on a bipartite system $AB$.

Such states satisfy the transpose property $$ (X \otimes I) |\psi\rangle_{AB} = (I \otimes X^T) |\psi\rangle_{AB}. $$ This means for any unitary $U$ we have $$ |\psi\rangle_{AB} = (UU^\dagger \otimes I)|\psi\rangle_{AB} = (U \otimes U^*)|\psi\rangle_{AB} $$ where $U^*$ is the entrywise complex conjugate of $U$. What we have shown is that $|\psi\rangle_{AB}$ does not change (invariant) under the application of $U\otimes U^*$. As a density matrix it means $$ |\psi\rangle\langle\psi| = (U\otimes U^*) |\psi\rangle\langle\psi|(U\otimes U^*)^\dagger $$ and so twirling over unitaries we have $$ \int \mathrm{d}U (U\otimes U^*) |\psi\rangle\langle\psi|(U\otimes U^*)^\dagger = \int \mathrm{d}U |\psi\rangle\langle \psi| = |\psi\rangle \langle \psi|. $$