Graph-Theoretic Framework for Self-Testing in Bell Scenarios

Abstract

Quantum self-testing is the task of certifying quantum states and measurements using the output statistics solely, with minimal assumptions about the underlying quantum system. It is based on the observation that some extremal points in the set of quantum correlations can only be achieved, up to isometries, with specific states and measurements. Here, we present a new approach for quantum self-testing in Bell non-locality scenarios, motivated by the following observation: the quantum maximum of a given Bell inequality is, in general, difficult to characterize. However, it is strictly contained in an easy-to-characterize set: the theta body of a vertex-weighted induced subgraph (Gex, w) of the graph in which vertices represent the events and edges join mutually exclusive events. This implies that, for the cases where the quantum maximum and the maximum within the theta body (known as the Lovász theta number) of (Gex, w) coincide, self-testing can be demonstrated by just proving self-testability with the theta body of Gex. This graph-theoretic framework allows us to: (i) recover the self-testability of several quantum correlations that are known to permit selftesting (like those violating the Clauser-Horne-Shimony-Holt (CHSH) and three-party Mermin Bell inequalities for projective measurements of arbitrary rank, and chained Bell inequalities for rankone projective measurements), (ii) prove the self-testability of quantum correlations that were not known using existing self-testing techniques (e.g., those violating the Abner Shimony Bell inequality for rank-one projective measurements). Additionally, the analysis of the chained Bell inequalities, along with prior results in Bell non-locality literature, gives us a closed form expression of the Lovász theta number for a family of well studied graphs known as the Möbius ladders, which might be of independent interest in the community of discrete mathematics.

Robust self-testing of multipartite GHZ-state measurements in quantum networks

“…Self-testing, acting as a device-independent certification method, has attracted lots of attention since the pioneer works of Mayers and Yao [2]. It can be used to certify entangled pure states and measurements [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. Up to now, a wide range of entangled quantum states are proved to be self-testable, such as the elegant results for all pure bipartite entangled states [23], three-qubit W states [24], and graph states [25].…”