Systematic construction of ROCN Bell-inequalities

TL;DR

This work develops a constructive, device-independent framework for self-testing using ROCN Bell inequalities by leveraging symmetric subspaces. It proves a sufficient condition: if the column vectors of an ROCN matrix form a symmetric spanning set, the maximal quantum value self-tests the canonical strategy, enabling explicit construction of self-testing ROCN inequalities for any even number of Clifford generators with local dimension $d=2^r$. The authors present a concrete construction by concatenating orthogonal blocks to form ROCN matrices whose columns span the symmetric subspace, and they illustrate the method with $m=2$ and $m=4$ examples, including CHSH-embedded structures. While the approach is general and physically transparent, it is not optimally economical in the number of columns, indicating avenues for more efficient designs with preserved analytic structure. Overall, the work strengthens the link between symmetry, ROCN structure, and self-testing, expanding the toolkit for robust, device-independent quantum certification.

Abstract

Self-testing constitutes one of the most powerful forms of device certification, enabling a complete and device-independent characterization of a quantum apparatus solely from the observed correlations. In recent work by the authors [23], a general framework was introduced for constructing Bell inequalities that self-test entire families of Clifford generators. In this manuscript, we develop an alternative and complementary self-testing criterion based on symmetric spanning sets. This formulation provides an explicit and constructive route to designing self-testing Bell inequalities in arbitrary dimensions.