A Sequence of Qubit-Qudit Pauli Groups as a Nested Structure of Doilies

TL;DR

This paper explores the geometric structure of the generalized Pauli group for qubit-qudit systems with dimensions as powers of two, revealing a nested configuration of doilies and patterns of commuting elements.

Contribution

It introduces a novel finite geometric framework for the qubit-qudit Pauli group, highlighting a nested structure of doilies and an alternating pattern of commuting sets, extending previous computational observations.

Findings

Identifies a nested configuration of doilies in the geometry of the Pauli group.

Discovers an isomorphism between exceptional points in higher cases and lower-dimensional geometries.

Highlights the computational nature of the current results and the challenge of a rigorous proof for larger cases.

Abstract

Following the spirit of a recent work of one of the authors (J. Phys. A: Math. Theor. 44 (2011) 045301), the essential structure of the generalized Pauli group of a qubit-qu$d$it, where $d = 2^{k}$ and an integer $k \geq 2$, is recast in the language of a finite geometry. A point of such geometry is represented by the maximum set of mutually commuting elements of the group and two distinct points are regarded as collinear if the corresponding sets have exactly $2^{k} - 1$ elements in common. The geometry comprises $2^{k} - 1$ copies of the generalized quadrangle of order two ("the doily") that form $2^{k-1} - 1$ pencils arranged into a remarkable nested configuration. This nested structure reflects the fact that maximum sets of mutually commuting elements are of two different kinds (ordinary and exceptional) and exhibits an intriguing alternating pattern: the subgeometry of the exceptional points of the $(k+2)$-case is found to be isomorphic to the full geometry of the $k$-case. It should be stressed, however, that these generic properties of the qubit-qudit geometry were inferred from purely computer-handled cases of $k = 2, 3, 4$ and 5 only and, therefore, their rigorous, computer-free proof for $k \geq 6$ still remains a mathematical challenge.