Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank

Abstract

We study certain physically-relevant subgeometries of binary symplectic polar spaces $W(2N-1,2)$ of small rank $N$, when the points of these spaces canonically encode $N$-qubit observables. Key characteristics of a subspace of such a space $W(2N-1,2)$ 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 $W(2N-1,2)$ and the structure of its Veldkamp space. In particular, we classify and count polar subspaces of $W(2N-1,2)$ whose rank is $N-1$. $W(3,2)$ features three negative lines of the same type and its $W(1,2)$'s are of five different types. $W(5,2)$ is endowed with 90 negative lines of two types and its $W(3,2)$'s split into 13 types. 279 out of 480 $W(3,2)$'s with three negative lines are composite, i.\,e. they all originate from the two-qubit $W(3,2)$. Given a three-qubit $W(3,2)$ and any of its geometric hyperplanes, there are three other $W(3,2)$'s possessing the same hyperplane. The same holds if a geometric hyperplane is replaced by a `planar' tricentric triad. A hyperbolic quadric of $W(5,2)$ is found to host particular sets of seven $W(3,2)$'s, each of them being uniquely tied to a Conwell heptad with respect to the quadric. There is also a particular type of $W(3,2)$'s, a representative of which features a point each line through which is negative. Finally, $W(7,2)$ is found to possess 1908 negative lines of five types and its $W(5,2)$'s fall into as many as 29 types. 1524 out of 1560 $W(5,2)$'s with 90 negative lines originate from the three-qubit $W(5,2)$. Remarkably, the difference in the number of negative lines for any two distinct types of four-qubit $W(5,2)$'s is a multiple of four.

20 pages, 5 figures and 5 tables; Version 2 - a new paragraph, new section (Sec. 5), four more figures and five more references

References in corpus (6)

• Black Hole Entropy and Finite Geometry
• Multiple Qubits as Symplectic Polar Spaces of Order Two
• Factor-Group-Generated Polar Spaces and (Multi-)Qudits
• Mermin pentagrams arising from Veldkamp lines for three qubits
• A finite geometric toy model of space-time as an error correcting code
• A Finite-Geometric Classification of Three-Qubit Mermin Pentagrams

Cited by in corpus (6)

• Contextuality degree of quadrics in multi-qubit symplectic polar spaces
• Multi-qubit doilies: enumeration for all ranks and classification for ranks four and five
• New and improved bounds on the contextuality degree of multi-qubit configurations
• An abstract structure determines the contextuality degree of observable-based Kochen-Specker proofs
• A new heuristic approach for contextuality degree estimates and its four- to six-qubit portrayals
• Hexagons govern three-qubit contextuality