  The connectivity index, defined as the number of decoupled components of a
separable quantum system, can change under deformations of the Hamiltonian or
during the dynamical change of the system under renormalization group flow.
Such changes signal a rearrangement of correlations of different degrees of
freedom across spacetime and field theory space. In this paper we quantify such
processes by studying the behavior of entanglement entropy in a specific
example: the RG flow in the Coulomb branch of large-N superconformal field
theories. We find evidence that the transition from the non-separable phase of
the Higgsed gauge theory in the UV to the separable phase of deformed decoupled
CFTs in the IR exhibits sharp features in the middle of the RG flow in the
large-N limit. The entanglement entropy on a sphere with radius $\ell$ exhibits
the formation of a separatrix on the co-dimension-two Ryu-Takayanagi surface in
multi-centered brane geometries above a critical value of $\ell$. We discuss
how other measures of entanglement and separability based on the relative
quantum entropy and quantum mutual information might detect such transitions
between non-separable and separable phases and how they would help describe
some of the key properties of the IR physics of such flows.
