Entanglement distance between quantum states and its implications for density-matrix-renormalization-group study of degenerate ground-states

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.