Entanglement estimation in tensor network states via sampling

TL;DR

This paper presents a sampling-based method to estimate entanglement measures in tensor network states, reducing computational demands compared to traditional techniques and enabling analysis of larger subsystems.

Contribution

The authors introduce a stochastic sampling approach for entanglement estimation in tensor networks that avoids explicit density matrix reconstruction and replica contraction.

Findings

Method accurately estimates entanglement in tensor networks.

Applicable to large subsystems with moderate computational cost.

Validated on 1D XX chain and 2D toric code.

Abstract

We introduce a method for extracting meaningful entanglement measures of tensor network states in general dimensions. Current methods require the explicit reconstruction of the density matrix, which is highly demanding, or the contraction of replicas, which requires an effort exponential in the number of replicas and which is costly in terms of memory. In contrast, our method requires the stochastic sampling of matrix elements of the classically represented reduced states with respect to random states drawn from simple product probability measures constituting frames. Even though not corresponding to physical operations, such matrix elements are straightforward to calculate for tensor network states, and their moments provide the R\'enyi entropies and negativities as well as their symmetry-resolved components. We test our method on the one-dimensional critical XX chain and the two-dimensional toric code in a checkerboard geometry. Although the cost is exponential in the subsystem size, it is sufficiently moderate so that - in contrast with other approaches - accurate results can be obtained on a personal computer for relatively large subsystem sizes.