Multi-Line Geometry of Qubit–Qutrit and Higher-Order Pauli Operators

Abstract

The commutation relations of the generalized Pauli operators of a qubit-qutrit system are discussed in the newly established graph-theoretic and finite-geometrical settings. The dual of the Pauli graph of this system is found to be isomorphic to the projective line over the product ring Z 2 × Z 3 . A "peculiar" feature in comparison with two-qubits is that two distinct points/operators can be joined by more than one line. The multi-line property is shown to be also present in the graphs/geometries characterizing two-qutrit and three-qubit Pauli operators' space and surmised to be exhibited by any other higher-level quantum system.

Projective ring line of an arbitrary single qudit

“…Here, the d = 4 case can serve as an elementary illustration of this fact; while our single 4-qudit is characterized by two non-trivial layers (disregarding the trivial Z(G)-layer) which are embodied in the structure of the projective line over Z 4 , a two-qubit features just a single layer since the geometry behind the corresponding tensor products of the classical Pauli matrices is that of the generalized quadrangle of order two [10]- [12]. Similar comparisons can also be made for several other low-dimensional quantum systems [9,13,14]. These should prove helpful when extending this group-geometrical approach to the most general case of multiple qudits.…”

Zoology of Atlas-Groups: Dessins D’enfants, Finite Geometries and Quantum Commutation

“…The group of automorphisms of G is isomorphic to Z 4 2 P 36 , where P 36 was encountered in Figure 1 as the symmetry group of the Mermin square. The complement of the collinearity graph of G is the (3 × 4)-grid that physically corresponds to the geometry of the 12 maximum sets of commuting operators in a qubit-qutrit system [24]. Two points on a line of the grid correspond to maximum sets having one point in common, while the triangles in (k) correspond to maximum sets of (three) mutually unbiased bases.…”

A sequence of qubit–qudit Pauli groups as a nested structure of doilies

“…Again, to facilitate our reasoning, we number these elements from 1 to 255 consecutively as they are listed in the last expression. Using computer, we find out that they form 87 maximum sets of mutually commuting guys, each of cardinality 15 4,6,32,34,36,38,192,194,196,198,224,226,228 16,20,32,36,48,52,128,132,144,148,160,164,176, 180}, 16,20,32,36,48,52,192,196,208,212,224,228,240 One observes that each twelve-element set features three common elements and the two sets in a pair share a single element, this being the element 4 = I (2) ⊗ X 4 (8) , 32 = I (2) ⊗ Z 4 (8) and 36 = I (2) ⊗ Z 4 (8) X 4 (8) , respectively. Paralleling the preceding case, we again form the point-line incidence geometry where points are the 87 maximum sets, but where two points are now collinear if the corresponding sets have exactly seven (=2 3 −1) elements in common.…”