Finite Projective Spaces, Geometric Spreads of Lines and Multi-Qubits

TL;DR

This paper explores the geometric relationship between projective spaces over GF(2) and GF(4), revealing how these mappings relate to the structure of multi-qubit Pauli groups and their classification based on parity.

Contribution

It introduces a novel geometric framework connecting projective space mappings with the structure of multi-qubit Pauli groups, highlighting parity-dependent properties.

Findings

Mapping of PG(2N - 1, 2) to PG(N - 1, 4) preserves geometric structures.

Non-degenerate quadrics correspond to Hermitian varieties, differing by parity of N.

Provides new insights into symmetric operators in multi-qubit systems.

Abstract

Given a (2N - 1)-dimensional projective space over GF(2), PG(2N - 1, 2), and its geometric spread of lines, there exists a remarkable mapping of this space onto PG(N - 1, 4) where the lines of the spread correspond to the points and subspaces spanned by pairs of lines to the lines of PG(N - 1, 4). Under such mapping, a non-degenerate quadric surface of the former space has for its image a non-singular Hermitian variety in the latter space, this quadric being {\it hyperbolic} or {\it elliptic} in dependence on N being {\it even} or {\it odd}, respectively. We employ this property to show that generalized Pauli groups of N-qubits also form two distinct families according to the parity of N and to put the role of symmetric operators into a new perspective. The N=4 case is taken to illustrate the issue.