A Class of Three-Qubit Contextual Configurations Located in Fano Pentads

Metod Saniga

doi:10.3390/math9131524

Abstract

Given the symplectic polar space of type W(5,2), let us call a set of five Fano planes sharing pairwise a single point a Fano pentad. Once 63 points of W(5,2) are appropriately labeled by 63 non-trivial three-qubit observables, any such Fano pentad gives rise to a quantum contextual set known as a Mermin pentagram. Here, it is shown that a Fano pentad also hosts another, closely related, contextual set, which features 25 observables and 30 three-element contexts. Out of 25 observables, ten are such that each of them is on six contexts, while each of the remaining 15 observables belongs to two contexts only. Making use of the recent classification of Mermin pentagrams (Saniga et al., Symmetry 12 (2020) 534), it was found that 12,096 such contextual sets comprise 47 distinct types, falling into eight families according to the number (3,5,7,…,17) of negative contexts.

Highlights

Published: 29 June 2021Let us call a set of pairwise commuting observables, whose product is + Id or − Id, Id being the identity, a positive or negative context, respectively
A quantum contextual configuration is a set of contexts such that (i) each observable occurs in an even number of contexts and (ii) the number of negative contexts is odd
There exist a number of proofs of this theorem based on the N-qubit Pauli group, N ≥ 2. Interesting are those where the structure of the underlying symplectic polar space W (2N − 1, 2) can be invoked to better understand their complex nature. This idea was recently employed [8] to achieve a deeper insight into the structure of the aggregate of 12,096 Mermin pentagrams of the three-qubit symplectic polar space

Summary

Introduction

Let us call a set of pairwise commuting observables, whose product is + Id or − Id , Id being the identity, a positive or negative context, respectively. Interesting are those where the structure of the underlying symplectic polar space W (2N − 1, 2) (see, e.g., [5,6,7]) can be invoked to better understand their complex nature. An important structural property of W (5, 2) is a set of five Fano planes such that their pairwise intersections are all different and consist of a single point each, the shared points forming in each Fano plane an affine plane of order two We shall call such a set of Fano planes a Fano pentad, and the affine planes consisting of shared points will be referred to as distinguished ones

A Notable Class of MP-Related Contextual Sets

Conclusions