Kwek, Leong Chuan

Self-testing is a technology to certify states and measurements using only the statistics of the experiment. Self-testing is possible if some extremal points in the set B_Q of quantum correlations for a Bell experiment are achieved, up to isometries, with specific states and measurements. However, B_Q is difficult to characterize, so it is also difficult to prove whether or not a given matrix of quantum correlations allows for self-testing. Here, we show how some tools from graph theory can help to address this problem. We observe that B_Q is strictly contained in an easy-to-characterize set associated with a graph, Θ(G). Therefore, whenever the optimum over B_Q and the optimum over Θ(G)) coincide, self-testing can be demonstrated by simply proving self-testability with Θ(G). Interestingly, these maxima coincide for the quantum correlations that maximally violate many families of Bell-like inequalities. Therefore, we can apply this approach to prove the self-testability of many quantum correlations, including some that are not previously known to allow for self-testing. In addition, this approach connects self-testing to some open problems in discrete mathematics. We use this connection to prove a conjecture [M. Araujo et a/., Phys. Rev. A, 88, 022118 (2013)] about the closed-form expression of the Lovasz theta number for a family of graphs called the Mobius ladders. Although there are a few remaining issues (e.g., in some cases, the proof requires the assumption that measurements are of rank 1). this approach provides an alternative method to self-testing and draws interesting connections between quantum mechanics and discrete mathematics.

The idea of self-testing is to render guarantees concerning the inner workings of a device based on the measurement statistics. It is one of the most formidable quantum certification and benchmarking schemes. Recently it was shown [A. Coladangelo, K. T. Goh, and V. Scarani, Nat. Commun. 8, 15485 (2017)] that all pure bipartite entangled states can be self-tested in the device-independent scenario by employing subspace methods introduced by Yang and Navascués [Phys. Rev. A 87, 050102(R) (2013)]. Here, we have adapted their method to show that any bipartite pure entangled state can be certified in the semi-device-independent scenario through quantum steering. Analogous to the tilted Clauser-Horne-Shimony-Holt inequality, we use a steering inequality called the tilted steering inequality for certifying any pure two-qubit entangled state. Furthermore, we use this inequality to certify any bipartite pure entangled state by certifying two-dimensional subspaces of the qudit state by observing the structure of the set of assemblages obtained on the trusted side after measurements are made on the untrusted side. As a feature of quantum state certification via steering, we use the notion of assemblage-based robust state certification to provide robustness bounds for the certification result in the case of pure maximally entangled states of any local dimension.