Quantum data compression, quantum information generation, and the density-matrix renormalization group method

TL;DR

This paper explores quantum data compression and information generation using the DMRG method for finite quantum systems, revealing relationships between entropy and Hilbert space dimensions, and proposing error control techniques.

Contribution

It introduces a relationship between block entropy and Hilbert space size in DMRG for non-independent site density matrices and proposes a new error control based on Kholevo's theory.

Findings

A simple relationship between block entropy and Hilbert space dimension.

A new method for controlling information loss during RG.

Validation of results with quantum chemistry DMRG on molecules.

Abstract

We have studied quantum data compression for finite quantum systems where the site density matrices are not independent, i.e., the density matrix cannot be given as direct product of site density matrices and the von Neumann entropy is not equal to the sum of site entropies. Using the density-matrix renormalization group (DMRG) method for the 1-d Hubbard model, we have shown that a simple relationship exists between the entropy of the left or right block and dimension of the Hilbert space of that block as well as of the superblock for any fixed accuracy. The information loss during the RG procedure has been investigated and a more rigorous control of the relative error has been proposed based on Kholevo's theory. Our results are also supported by the quantum chemistry version of DMRG applied to various molecules with system lengths up to 60 lattice sites. A sum rule which relates site entropies and the total information generated by the renormalization procedure has also been given which serves as an alternative test of convergence of the DMRG method.