Entanglement classification with matrix product states

TL;DR

This paper introduces a new entanglement classification method based on matrix product states, linking entanglement families to Hamiltonian interaction length and connecting quantum information with condensed matter physics.

Contribution

It presents a novel entanglement classification scheme using matrix product states, establishing a link between entanglement families and Hamiltonian interaction length.

Findings

Entanglement families relate to Hamiltonian interaction length.

Natural nesting property of entanglement families across particle numbers.

Connection between entanglement classification and condensed matter models.

Abstract

Entanglement is widely considered the cornerstone of quantum information and an essential resource for relevant quantum effects, such as quantum teleportation, quantum cryptography, or the speed-up of quantum computing, as in Shor's algorithm. However, up to now, there is no general characterization of entanglement for many-body systems. In this sense, it is encouraging that quantum states connected by stochastic local operations assisted with classical communication (SLOCC), which perform probabilistically the same quantum tasks, can be collected into entanglement classes. Nevertheless, there is an infinite number of classes for four or more parties that may be gathered, in turn, into a finite number of entanglement families. Unfortunately, we have not been able to relate all classes and families to specific properties or quantum information tasks, although a few of them have certainly raised experimental interest. Here, we present a novel entanglement classification for quantum states according to their matrix-product-state structure, exemplified for the symmetric subspace. The proposed classification relates entanglement families to the interaction length of Hamiltonians, establishing the first connection between entanglement classification and condensed matter. Additionally, we found a natural nesting property in which the families for $N$ parties carry over to the $N+1$ case. We anticipate our proposal to be a starting point for the exploration of the connection between entanglement classification properties and condensed-matter models.