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. 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.
Highlights
It was discovered that there exists a deep connection between the structure of the N-qubit Pauli group and that of the binary symplectic polar space of rank N, W (2N − 1, 2), where commutation relations between elements of the group are encoded in collinearity relations between points of W (2N − 1, 2)
Mermin–Peres magic squares were found to be isomorphic to a special class of geometric hyperplanes of W (3, 2) called grids [11], whereas three-qubit Mermin pentagrams were found to have their natural settings in the magic Veldkamp line of W (5, 2) [12], being isomorphic—under the Grassmannian correspondence of type Gr(2, 4)—to ovoids of
Form theories of gravity seem to indicate that a certain part of the magic Veldkamp line in the four-qubit symplectic polar space, W (7, 2), and the associated extended geometric hyperplanes are of physical relevance as well
Summary
It was discovered (see, e.g., [1,2,3,4]) that there exists a deep connection between the structure of the N-qubit Pauli group and that of the binary symplectic polar space of rank N, W (2N − 1, 2), where commutation relations between elements of the group are encoded in collinearity relations between points of W (2N − 1, 2) This connection has subsequently been used to obtain a deeper insight into, for example, finite geometric nature of observable-based proofs of quantum contextuality (for a recent review, see [5]), properties of certain black-hole entropy formulas [6] and the so-called black-hole/qubit correspondence [7], leading to a finite-geometric underpinning of four distinct Hitchin’s invariants and the Cartan invariant of form theories of gravity [8] and even to an intriguing finite-geometric toy model of space-time [9].
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