Entanglement classification with matrix product states

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.

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Cited by in corpus (14)

• Geometry of spin coherent states
• Classification of four-qubit entangled states via Machine Learning
• The quantum marginal problem for symmetric states: applications to variational optimization, nonlocality and self-testing
• Matrix Product States: Entanglement, symmetries, and state transformations
• Entanglement Classification with Algebraic Geometry
• Comment: Inductive entanglement classification of four qubits under stochastic local operation and classical communication
• Dicke states as matrix product states
• Frustration-free Hamiltonians supporting Majorana zero edge modes
• Classification and measurement of multipartite entanglement by reconstruction of correlation tensors on an NMR quantum processor
• Classification of measurement-based quantum wire in stabilizer PEPS
• Pairwise Concurrence in Cyclically Symmetric Quantum States
• Multipartite States under Elementary Local Operations
• Multipartite entanglement classes of a multiport beam-splitter
• Clustering and enhanced classification using a hybrid quantum autoencoder