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 Z2xZ3. 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.

arXiv comment8 pages, 6 figures. International Journal of Theoretical Physics (2007) accepté

References in corpus (13)

• Hidden Variables and the Two Theorems of John Bell
• A new proof for the existence of mutually unbiased bases
• Multicomplementary operators via finite Fourier transform
• A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements
• Projective Representations I. Projective lines over rings
• Multiple Qubits as Symplectic Polar Spaces of Order Two
• Projective Ring Line Encompassing Two-Qubits
• The Veldkamp Space of Two-Qubits
• Divisible Designs, Laguerre Geometry, and Beyond
• A Classification of the Projective Lines over Small Rings
• The Projective Line Over the Finite Quotient Ring GF(2)[$x$]/$<x^3-x>$ and Quantum Entanglement II. The Mermin "Magic" Square/Pentagram
• On the Pauli graphs of N-qudits
• Pauli graph and finite projective lines/geometries

Cited by in corpus (15)

• An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group
• Projective Ring Line of an Arbitrary Single Qudit
• Variations on a theme of Heisenberg, Pauli and Weyl
• Pauli graphs when the Hilbert space dimension contains a square: why the Dedekind psi function ?
• Factor-Group-Generated Polar Spaces and (Multi-)Qudits
• Projective Ring Line of a Specific Qudit
• Majorana representation of adiabatic and superadiabatic processes in three-level systems
• Zoology of Atlas-groups: dessins d'enfants, finite geometries and quantum commutation
• Viable entanglement detection of unknown mixed states in low dimensions
• SU(2) and SU(1,1) Approaches to Phase Operators and Temporally Stable Phase States: Applications to Mutually Unbiased Bases and Discrete Fourier Transforms
• On the geometry and invariants of qubits, quartits and octits
• The Veldkamp Space of the Smallest Slim Dense Near Hexagon
• A Sequence of Qubit-Qudit Pauli Groups as a Nested Structure of Doilies
• Generalized spin bases for quantum chemistry and quantum information
• Bases for qudits from a nonstandard approach to SU(2)