  Establishing the underlying links between the diverse landscape of
theoretical frameworks for simulating strongly correlated matter is crucial for
advancing our understanding of these systems. In this work, we focus on the
Ghost Gutzwiller Approximation (gGA), an extension of the Gutzwiller
Approximation (GA) based on the variational principle. We derive a framework
called "Ghost Density Matrix Embedding Theory" (gDMET) from quantum embedding
(QE) principles similar to those in Density Matrix Embedding Theory (DMET),
which reproduces the gGA equations for multi-orbital Hubbard models with a
simpler implementation. This derivation highlights the crucial role of the
ghost degrees of freedom, not only as an extension to the GA, but also as the
key element in establishing a consistent conceptual connection between DMET and
the gGA. This connection further elucidates how gGA overcomes the systematic
accuracy limitations of standard GA and achieves results comparable to
Dynamical Mean Field Theory (DMFT). Furthermore, it offers an alternative
interpretation of the gGA equations, fostering new ideas and generalizations.
