Matrix product states represent ground states faithfully

TL;DR

This work investigates how faithfully matrix product states (MPS) can represent ground states of 1D quantum spin systems, including critical ones, and provides rigorous bounds on approximation quality.The authors derive bounds that tie MPS truncation errors to the tail of the reduced density operator eigenvalue spectra and to block Renyi entropies, linking entanglement structure to representational efficiency.For translationally invariant 1D systems, the results show that block entropy grows only logarithmically in block size for many cases, implying a bond dimension that grows polynomially with system size to maintain fixed accuracy, hence efficient classical description even at criticality.These findings justify the practical success of DMRG and related renormalization group methods and clarify why quantum speedups are not guaranteed for 1D ground-state problems within this framework.

Abstract

We quantify how well matrix product states approximate exact ground states of 1-D quantum spin systems as a function of the number of spins and the entropy of blocks of spins. We also investigate the convex set of local reduced density operators of translational invariant systems. The results give a theoretical justification for the high accuracy of renormalization group algorithms, and justifies their use even in the case of critical systems.

Figures (3)

• Figure 1: Convex sets of the possible reduced density operators of translational invariant states in the XX-ZZ plane: the big triangle represents all positive density operators; the inner parallellogram represents the separable states; the union of the separable cone and the convex hull of the full curved line is the complete convex set in the case of a 1-D geometry, and the dashed lines represent extreme points in the 2-D case of a square lattice. The singlet corresponds to the point with coordinates (-1,-1).
• Figure 2: A one dimensional spin chain with finite correlation length $\xi_{corr}$; $l_{AB}$ denotes the distance between the block $A$ (left) and $B$ (right). Because $\l_{AB}$ is much larger than the correlation length $\xi_{corr}$, the state $\rho_{AB}$ is essentially a product state.
• Figure 3: Convex sets in the XXZ-plane: the inner diamond borders the set of separable states (see Fig. 1). Dash-dotted: extreme points of the convex set produced by MPS of $D=2$.

Theorems & Definitions (3)

• Lemma 1
• Lemma 2
• Lemma 3