Multiple Qubits as Symplectic Polar Spaces of Order Two

TL;DR

This paper explores the deep geometric structure underlying the algebra of Pauli operators in N-qubit systems, modeling it with symplectic polar spaces to better understand quantum commutation relations.

Contribution

It establishes a correspondence between Pauli operators and points in symplectic polar spaces, revealing geometric insights into their commutation properties and partitions.

Findings

Pauli operators correspond to points in symplectic polar spaces

Maximally commuting subsets relate to maximal totally isotropic subspaces

Commutation translates to collinearity in the geometric model

Abstract

It is surmised that the algebra of the Pauli operators on the Hilbert space of N-qubits is embodied in the geometry of the symplectic polar space of rank N and order two, W_{2N - 1}(2). The operators (discarding the identity) answer to the points of W_{2N - 1}(2), their partitionings into maximally commuting subsets correspond to spreads of the space, a maximally commuting subset has its representative in a maximal totally isotropic subspace of W_{2N - 1}(2) and, finally, "commuting" translates into "collinear" (or "perpendicular").