Multi-qubit doilies: Enumeration for all ranks and classification for ranks four and five

Abstract

For N≥2, an N-qubit doily is a doily living in the N-qubit symplectic polar space. These doilies are related to operator-based proofs of quantum contextuality. Following and extending the strategy of Saniga et al. (Mathematics 9 (2021) 2272) that focused exclusively on three-qubit doilies, we first bring forth several formulas giving the number of both linear and quadratic doilies for any N>2. Then we present an effective algorithm for the generation of all N-qubit doilies. Using this algorithm for N=4 and N=5, we provide a classification of N-qubit doilies in terms of types of observables they feature and number of negative lines they are endowed with. We also list several distinguished findings about N-qubit doilies that are absent in the three-qubit case, point out a couple of specific features exhibited by linear doilies and outline some prospective extensions of our approach.