bibkey: mullersanigagiorgettideboutrayholweck2022doilies
authors: Axel Muller; Metod Saniga; Alain Giorgetti; Henri de Boutray; Frédéric Holweck
year: 2022
title: "Multi-qubit doilies: enumeration for all ranks and classification for ranks four and five"
doi: 10.1016/j.jocs.2022.101853
url: https://arxiv.org/abs/2206.03599v2
claim: "N-qubit doilies are copies of the generalized quadrangle of order two whose points are N-qubit observables and whose lines are commuting triples; a line is negative if its three observables multiply to minus the identity. The paper counts linear and quadratic doilies for all N, classifies them for N = 4 and 5, and conjectures that every linear doily has an odd number of negative lines for all N ≥ 2."
strata_touched:
  - D5/S3/Quantum/Contextuality/LinearDoilyNegativeLineParity
license: citation-only
triage: anchor

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

A. Muller, M. Saniga, A. Giorgetti, H. de Boutray and F. Holweck, arXiv:2206.03599v2 (2022); J. Comput. Sci. 64, 101853 (2022). Quotations are from the v2 TeX source; source macros are expanded.

The symplectic polar space:

The \emph{symplectic polar space} $\mathcal{W}(2N-1,2)$, $N\ge 1$, consists of all the points of PG$(2N-1,2)$, {(x1​,x2​,…,x2N​):xj​∈{0,1},j∈{1,2,…,2N}}</span>{(0,0,…,0)}, together with the totally isotropic subspaces with respect to the standard symplectic form σ(x,y)=x1​yN+1​−xN+1​y1​+x2​yN+2​−xN+2​y2​+⋯+xN​y2N​−x2N​yN​.

Doilies and their line signs:

an $N$-qubit doily is a doily whose points are bijectively identified with 15 specific observables from ${\mathcal{S}}_N$, such that any two commuting observables share the same line, and, given any line, the product of (any) two observables lying on it is, up to a sign, equal to the remaining observable on it. A line of an $N$-qubit doily will be called positive (resp. negative) if the product of its three observables is $+{\mathcal{I}}_N$ (resp. $-{\mathcal{I}}_N$).

Linear doilies:

A linear $N$-qubit doily spans a PG$(3,2)$ of the ambient PG$(2N-1,2)$.

The conjecture:

From our results on three-, four- and five-qubit cases it follows that a linear doily (a) always features an odd number of negative lines, and (b) does not share a perp-set with the distinguished quadric. We conjecture that Property (a) holds for any number of qubits $N\ge 2$, but we have not yet found of proof of it

Verified locator

Source: https://arxiv.org/abs/2206.03599v2

• URL: https://arxiv.org/abs/2206.03599v2 (source xiv-MSGDH2.tex, md5 ba1ee66980eb180ebdcbe7fcd5d8a411):
  • the symplectic polar space, l. 222–231;
  • N-qubit doilies and line signs, l. 306–314;
  • linear doilies, l. 1086–1087;
  • the number of linear doilies, l. 1574–1586;
  • the conjecture, l. 2254–2259.
• DOI: 10.1016/j.jocs.2022.101853 (https://doi.org/10.1016/j.jocs.2022.101853).