Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank
Metod Saniga, Henri de Boutray, Frederic Holweck, Alain Giorgetti

TL;DR
This paper classifies and analyzes subgeometries of binary symplectic polar spaces related to multi-qubit systems, revealing their structure, types, and relationships with hyperplanes and quadrics, with implications for quantum observables.
Contribution
It provides a detailed classification and counting of polar subspaces of small rank in symplectic polar spaces, linking geometric structures to multi-qubit observables and hyperplanes.
Findings
Classified polar subspaces of W(2N-1,2) with rank N-1.
Identified types and counts of negative lines in W(3,2), W(5,2), and W(7,2).
Established relationships between hyperplanes, geometric hyperplanes, and subspace types.
Abstract
We study certain physically-relevant subgeometries of binary symplectic polar spaces of small rank , when the points of these spaces canonically encode -qubit observables. Key characteristics of a subspace of such a space are: the number of its negative lines, the distribution of types of observables, the character of the geometric hyperplane the subspace shares with the distinguished (non-singular) quadric of and the structure of its Veldkamp space. In particular, we classify and count polar subspaces of whose rank is . features three negative lines of the same type and its 's are of five different types. is endowed with 90 negative lines of two types and its 's split into 13 types. 279 out of 480 's with three negative lines are composite, i.\,e. they all originate from the two-qubit…
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