A well-defined notion of von Neumann entropy associated to pairs of spatial subregions has been recently proposed both in the holographic context and for general QFTs. I will show that in the case of Gaussian systems ---and similarly to the entanglement entropy (EE)--- this "reflected entropy” can be obtained in terms of correlation functions of the fields. In particular, I will present general formulas valid for free scalars and fermions in arbitrary dimensions. I will apply the results to various free theories in 1+1 and 2+1 dimensions, verifying that the conjectural monotonicity property $R(A,BC)\geq R(A,B)$ and the general inequality $R(A,B)\geq I(A,B)$ hold in all cases. The results obtained suggest that for general regions characterized by length-scales $L_A\sim L_B \sim L$ and separated a distance $\ell$, the reflected entropy in the large-separation regime ($x\equiv L/\ell \ll 1$) is related to the mutual information by: $R(x) \sim −I(x) \log x$ for general CFTs in arbitrary dimensions. Finally, I will argue that the notion of reflected entropy can be canonically generalized in a way which is particularly suitable for theories obtained by restricting the full algebra of operators to those which are neutral under global symmetry groups. A key role in the discussion is played by type-I von Neumann algebras, which differ from the usual type-III algebras associated to spatial subregions in QFT. I will perform various explicit comparisons between both types of algebras.

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