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

@article{Planat2008MultiLineGO, title={Multi-Line Geometry of Qubit–Qutrit and Higher-Order Pauli Operators}, author={Michel Planat and Anne-C'eline Baboin and Metod Saniga}, journal={International Journal of Theoretical Physics}, year={2008}, volume={47}, pages={1127-1135} }

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
$\mathcal{Z}_{2}\times\mathcal{Z}_{3}$
. 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… Expand

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#### References

SHOWING 1-10 OF 21 REFERENCES

Pauli graph and finite projective lines/geometries

- Mathematics, Engineering
- SPIE Optics + Optoelectronics
- 2007

The commutation relations between the generalized Pauli operators of N-qudits (i. e., N p-level quantum systems), and the structure of their maximal sets of commuting bases, follow a nice graph… Expand

Projective ring line encompassing two-qubits

- Mathematics, Physics
- 2006

We find that the projective line over the (noncommutative) ring of 2×2 matrices with coefficients in GF(2) fully accommodates the algebra of 15 operators (generalized Pauli matrices) characterizing… Expand

Quantum Entanglement and Projective Ring Geometry

- Mathematics, Physics
- 2006

The paper explores the basic geometrical properties of the observables charac- terizing two-qubit systems by employing a novel projective ring geometric approach. After introducing the basic facts… Expand

On the Pauli graphs on N-qudits

- Computer Science, Mathematics
- Quantum Inf. Comput.
- 2008

The Pauli graph of a two-qutrit system is introduced and it turns out more convenient to deal with its dual in order to see all the parallels with the two-qubit case and its surmised relation with the generalized quadrangle Q(4; 3), the dual of W(3). Expand

Multiple Qubits as Symplectic Polar Spaces of Order Two

- Mathematics, Physics
- 2006

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… Expand

The Veldkamp Space of Two-Qubits

- Mathematics, Physics
- 2007

Given a remarkable representation of the generalized Pauli operators of two- qubits in terms of the points of the generalized quadrangle of order two, W(2), it is shown that specific subsets of these… Expand

Multicomplementary operators via finite Fourier transform

- Mathematics, Physics
- 2005

A complete set of d + 1 mutually unbiased bases exists in a Hilbert space of dimension d, whenever d is a power of a prime. We discuss a simple construction of d + 1 disjoint classes (each one having… Expand

Projective line over the finite quotient ring GF(2)[x]/〈x3 ™ x〉 and quantum entanglement: The Mermin “magic” square/pentagram

- Mathematics, Physics
- 2007

In 1993, Mermin gave surprisingly simple proofs of the Bell-Kochen-Specker (BKS) theorem in Hilbert spaces of dimensions four and eight respectively using what has since been called the Mermin-Peres… Expand

A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements

- Mathematics, Physics
- 2004

The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are reviewed and an emerging link between them is outlined. It is shown… Expand

A Classification of the Projective Lines over Small Rings II. Non-Commutative Case

- Mathematics, Physics
- 2006

A list of different types of a projective line over non-commutative rings with unity of order up to thirty-one inclusive is given. Eight different types of such a line are found. With a single… Expand