  We investigate the inverse problem of recovering the diffusion and absorption
coefficients $(\sigma,q)$ in the nonlocal diffuse optical tomography equation
$(-\text{div}( \sigma \nabla))^s u+q u =0 \text{ in }\Omega$ from the nonlocal
Dirichlet-to-Neumann (DN) map $\Lambda^s_{\sigma,q}$. The purpose of this
article is to establish the following approximation and uniqueness results.
  - Approximation: We show that solutions to the conductivity equation $
\text{div}( \sigma \nabla v)=0 \text{ in }\Omega$ can be approximated in
$H^1(\Omega)$ by solutions to the nonlocal diffuse optical tomography equation
and the DN map $\Lambda_\sigma$ related to conductivity equation can be
approximated by the nonlocal DN map $\Lambda_{\sigma,q}^s$.
  - Local uniqueness: We prove that the absorption coefficient $q$ can be
determined in a neighborhood $\mathcal{N}$ of the boundary $\partial\Omega$
provided $\sigma$ is already known in $\mathcal{N}$.
  - Global uniqueness: Under the same assumptions as for the local uniqueness
result, and if one of the potentials vanishes in $\Omega$, then one can turn
with the help of \ref{item 1 abstract} the local determination into a global
uniqueness result.
  It is worth mentioning that the approximation result relies on the
Caffarelli--Silvestre type extension technique and the geometric form of the
Hahn--Banach theorem.
