  We study the concept of entanglement distance between two quantum states
which quantifies the amount of information shared between their reduced density
matrices (RDMs). Using analytical arguments combined with
density-matrix-renormalization-group (DMRG) and exact diagonalization (ED)
calculations, we show that for gapless systems the entanglement distance has
power law dependence on the energy separation and subsystem size with
$\alpha_E$ and $\alpha_{\ell}$ exponents, respectively. Using conformal field
theory (CFT) we find $\alpha_E = 2$ and $\alpha_{\ell} = 4$ for Abelian
theories with $c=1$ such as free fermions. For non-Abelian CFTs $\alpha_E = 0$
, and $\alpha_{\ell}$ is twice the conformal dimension of the thermal primary
fields. For instance for $Z_3$ parafermion CFT $\alpha_E = 1$ and
$\alpha_{\ell} = 4/5$. For gapped 1+1D fermion systems, we show that the
entanglement distance divides the low energy excitations into two branches with
different values of $\alpha_E$ and $\alpha_{\ell}$. These two branches are
related to momentum transfers near zero and $\pi$. We also demonstrate that the
entanglement distance reaches its maximum for degenerate states related through
nonlocal operators such as Wilson loops. For example, degenerate ground-states
(GSs) of 2+1 D topological states have maximum entanglement distance. On the
contrary, degenerate GSs related through confined anyon excitations such as
genons have minimum entanglement distance. Various implications of this concept
for quantum simulations are discussed. Finally, based on the ideas developed we
discuss the computational complexity of DMRG algorithms that are capable of
finding all degenerate GSs.
