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

TL;DR

This paper develops formulas and algorithms to enumerate and classify multi-qubit doilies in symplectic polar spaces, extending previous work from three-qubit to higher qubit systems and revealing new structural features.

Contribution

It introduces formulas and an algorithm for enumerating all multi-qubit doilies and classifies them for four and five qubits, extending prior three-qubit focus.

Findings

Formulas for counting linear and quadratic doilies for any N > 2

Classification of 4- and 5-qubit doilies by observable types and negative lines

Identification of features unique to higher-qubit doilies

Abstract

For $N \geq 2$, 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.