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

We study certain physically-relevant subgeometries of binary symplectic polar spaces W ( 2 N − 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 ( 2 N − 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 ( 2 N − 1 , 2 ) and the structure of its Veldkamp space. In particular, we classify and count polar subspaces of W ( 2 N − 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. A total of 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. A total of 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.