Abstract
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 and exponents, respectively. Using conformal field theory (CFT) we find and for Abelian theories with , as in the case of free fermions. For non-Abelian CFTs , and is twice the conformal dimension of the thermal primary fields. For instance, for parafermion CFT and . For gapped 1+1 dimensional (1+1D) fermion systems, we show that the entanglement distance divides the low energy excitations into two branches with different values of and . These two branches are related to momentum transfers near zero and . 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+1D topological states have maximum entanglement distance. In contrast, 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.
- Received 11 October 2016
- Revised 4 October 2017
DOI:https://doi.org/10.1103/PhysRevB.96.165129
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