Why DMRG Works: Information Theory Thomas E. Baker & Benedikt Bruognuolo—October 21, 2015 The Density Matrix Renormalization Group was originally derived without the use of [[Matrix Product States|MPS]] from arguments in the broader theory of renormalization group. The formulation in terms of MPSs shows why the algorithm works so well: the DMRG algorithm correctly calculates the correct entanglement entropy of the system. This article provides a basic survey of information theory and its connection to quantum entanglement. ### Why DMRG works: Information Theory and Entanglement When receiving a message that is known to have some characters distorted, it is necessary to determine what the uncertainty of a particular character @@x@@ might be (especially if some of the characters are garbled). Otherwise, how sure are we that this was the original character sent? We therefore must define a mathematically rigorous quantity which is the uncertainty of obtaining the message @@x@@ and connecting this to the the probability of choosing the correct @@x@@. The quantity we are searching for is most critical to the understanding of how much information can be sent though the communication channel is known as the information entropy (also known as the entanglement entropy). The first step is to quantify how much uncertainty there is in the message. Beginning from the most general statements, there are four conditions we can expect on any measure of information passing through a channel [1]. * The probability, @@\rho@@, should increase with more possibilities, @@M@@ available to sample (i.e., @@\rho(M)<\rho(M')@@ if @@M