R\'enyi Entanglement of Purification and Half R\'enyi Reflected Entropy in Free Scalar Theory

In the AdS/CFT context, the entanglement of purification (EoP, denoted as $E_{P}$) of CFT is conjectured to be dual to the entanglement wedge cross section (EWCS) in bulk. However, another quantity called reflected entropy $S_{R}$ is also supposed to be dual to two times the EWCS. A natural question is whether they are the same in holographic CFTs even though they are different in general. Previous studies have shown $E_{P} \ge \frac{1}{2} S_{R}^{(n)}, n \ge2$ for random tensor networks. In this paper, we study this inequality beyond $n \ge 2$, and we focus on the range $0 < n < 2$. However, the calculations of EoP are notoriously difficult in general. Thus, our calculations mainly focus on the free scalar theory which is close to the holographic CFTs. We generalized the previous strategy for EoP in \cite{Takayanagi:2018sbw} to the R\'enyi case. And we have also presented two methods for R\'enyi reflected entropy, one is using correlators, the other one is Gaussian wavefunction ansatz. Our calculations show that the inequality still holds for $0 < n < 2$, and it may give us some insights into the equivalence of EoP and half reflected entropy in holographic CFTs. As byproducts of our research, we have also demonstrated the positivity of the R\'enyi Markov gap and the monotonicity of the R\'enyi reflected entropy in the free scalar theory.