MULTIPARTITE QUANTUM ENTANGLEMENT

Work package:
In the recent years, several questions arising in quantum information theory [AS2017] have been tackled through asymptotic geometric analysis, i.e. through the use of probabilistic techniques in the study of finite but high dimensional Banach spaces. This high dimensional regime is indeed a natural one in quantum physics, since the dimension of a system grows exponentially with the number of subsystems composing it. As for the probabilistic techniques, they are useful either to identify the typical properties of such high dimensional quantum systems or to prove the existence of quantum systems having certain properties through random constructions. In quantum physics, most objects of interest are mathematically represented by operators on complex Hilbert spaces, i.e. in finite dimension by complex matrices. Studying the typical behavior of such objects thus boils down to identifying typical features of random matrices [Aub2012]. However, the crucial underlying tensor structure calls for tools that go beyond the traditional framework of random matrix theory, and requires to view the considered objects as random tensors. Being able to certify that a given many-body quantum system is in an entangled state is a key question, as it is what guarantees that it exhibits genuinely quantum correlations and can thus provide an advantage over a classical system in information processing tasks. However, beyond the case of bipartite pure states, entanglement is in general difficult to characterize and quantify, both mathematically and experimentally [DVC2000]. The general goal of this internship is to understand if this can however be done for generic instances (rather than worst-case ones), i.e. for multipartite states that are picked at random. There will be two main research axes in this internship: • Design sufficient conditions for the entanglement of bipartite mixed states, a.k.a. entanglement criteria, which are easier to verify than entanglement itself, and estimate their typical performance. • Study the generic amount of entanglement (and more generally the behavior of entanglement-related properties) in specific classes of multipartite pure states that are physically relevant, such as tensor network states.