Self-Testing multipartite entangled states through projections onto two systems

Abstract

Finding ways to test the behaviour of quantum devices is a timely enterprise, especially in light of the rapid development of quantum technologies. Device-independent self-Testing is one desirable approach, as it makes minimal assumptions on the devices being tested. In this work, we address the question of which states can be self-Tested. This has been answered recently in the bipartite case (Coladangelo et al 2017 Nat. Commun. 8 15485), while it is largely unexplored in the multipartite case, with only a few scattered results, using a variety of different methods: maximal violation of a Bell inequality, numerical SWAP method, stabiliser self-Testing etc. In this work, we investigate a simple, and potentially unifying, approach: combining projections onto two-qubit spaces (projecting parties or degrees of freedom) and then using maximal violation of the tilted CHSH inequalities. This allows one to obtain self-Testing of Dicke states and partially entangled GHZ states with two measurements per party, and also to recover self-Testing of graph states (previously known only through stabiliser methods). Finally, we give the first self-Test of a class of multipartite qudit states: we generalise the self-Testing of partially entangled GHZ states by adapting techniques from (Coladangelo et al 2017 Nat. Commun. 8 15485), and show that all multipartite states which admit a Schmidt decomposition can be self-Tested with few measurements.