Applications of Quantum Information in the Density-Matrix Renormalization Group

Abstract

In the past few years, there has been an increasingly active exchange of ideas and methods between the formerly rather disjunct fields of quantum information and many-body physics. This has been due, on the one hand, to the growing sophisti- cation of methods and the increasing complexity of problems treated in quantum information theory, and, on the other, to the recognition that a number of central issues in many-body quantum systems can fruitfully be approached from the quan- tum information point of view. Nowhere has this been more evident than in the context of the family of numerical methods that go under the rubric density-matrix renormalization group. In particular, the concept of entanglement and its definition, measurement, and manipulation lies at the heart of much of quantum information theory (1). The density-matrix renormalization group (DMRG) methods use proper- ties of the entanglement of a bipartite system to build up an accurate approximation to particular many-body wave functions. The cross-fertilization between the two fields has led to improvements in the understanding of interacting quantum systems in general and the DMRG method in particular, has led to new algorithms related to and generalizing the DMRG, and has opened up the possibility of studying many new physical problems, ones of interest both for quantum information theory and for understanding the behavior of strongly correlated quantum systems (2). In this line, we discuss some relevant concepts in quantum information theory, including the relation between the DMRG and data compression and entanglement. As an application, we will use the quantum information entropy calculated with the DMRG to study quantum phase transitions, in particular in the bilinear-biquadratic spin-one chain and in the frustrated spin-1/2 Heisenberg chain.