Many-body entanglement and truncated moment problems

Presenting Author: Bharath Hebbe Madhusudhana, Max Planck Institute for Quantum Optics
Contributing Author(s): Grigoriy Blekherman

Preparing, measuring and controlling many-body entangled states is a central challenge in quantum technologies. The problem of deciding whether a quantum state is entangled, in general, is known to be NP hard. Nevertheless, realistically, tomography of large many-body systems are infeasible. Therefore, the mathematical challenge in this problem is to develop criteria to decide whether a many-body state prepared in the lab is entangled based on a small number of observables that could be measured in the lab. We show that this problem is related to the so-called truncated moment problem, well known in convex algebraic geometry. The space of unentangled mixed states is convex and so is the space of the corresponding observable values. Therefore, the problem of determining whether a set of observable values could correspond to an unentangled state is tantamount to checking for membership in a convex set, also known as a moment cone, of a point with coordinates given by the set of observable values. The latter is an instance of the truncated moment problem. Here, we adapt techniques from convex algebraic geometry to develop necessary and sufficient criteria for entanglement in a many-body system of bosonic atoms with a non-zero spin [1]. We also show that these criteria are asymptotically tight, in the number of atoms. [1] G. Blekherman and H. M. Bharath, “Quantum entanglement, nonnegative polynomials and moment problems ”, arXiv: 1904.00072

Read this article online: https://arxiv.org/abs/1904.00072

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