Entanglement classification with matrix product states

摘要

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