Subadditivity of logarithm of violation of geometric Bell inequalities for qudits

Abstract

Geometrical Bell Inequalities (GBIs) are the strongest known Bell inequalities for collections of qubits. However, their generalizations to other systems is not yet fully understood. We formulate GBIs for an arbitrary number $N$ of observers, each of which possesses a particle of an arbitrary dimension $d$. The whole $(d-1)$-parameter family of local observables with eigenbases unbiased to the computational basis is used, but it is immediate to use a discrete subset of them. We argue analytically for qutrits and numerically for other systems that the violations grows exponenetially with $N$. Within the studied range, the violation also grows with $d$. Interestingly, we observe that the logarithm of the violation ratio for ququats grows with $N$ slower than the doubled logarithm of the violation ratio for qubits, which implies a kind of subadditivity.