Finite
Geometry Notes |

The Geometry of Qubits

by Steven H. Cullinane

August, 2007

In the context of quantum
information theory, the following structure seems to be of
interest--

"... the full two-by-two matrix ring with entries in*GF*(2), *M*_{2}(*GF*(2))--
the unique simple non-commutative ring of order 16 featuring six units
(invertible elements) and ten zero-divisors."

-- "Geometry of Two-Qubits," by Metod Saniga (pdf, 17 pp.), Jan. 25, 2007

This ring is another way of looking at the 16 elements of the affine 4-space*A*_{4}(*GF*(2))
over the 2-element field. (Arrange the four coordinates of each
element-- 1's and 0's-- into a square instead of a straight line, and
regard the resulting squares as matrices.) (For more on *A*_{4}(*GF*(2)),
see Finite Relativity and
related notes at Finite
Geometry of the Square and Cube.) Using the above ring,
Saniga constructs a system of 35 objects (not unlike the 35 lines of the
finite geometry *PG*(3,2)) that he calls a "projective line"
over the ring. This system of 35 objects has a subconfiguration
isomorphic to
the (2,2) generalized quadrangle *W*_{2}
(which occurs naturally as a subconfiguration of *PG*(3,2)-- see Inscapes.) The connection of this
generalized quadrangle with *PG*(3,2) is noted in a later paper
by Saniga and others ("The Veldkamp Space of Two-Qubits," cited below).

Saniga concludes "Geometry of Two-Qubits" as follows:

"... the full two-by-two matrix ring with entries in

-- "Geometry of Two-Qubits," by Metod Saniga (pdf, 17 pp.), Jan. 25, 2007

This ring is another way of looking at the 16 elements of the affine 4-space

Saniga concludes "Geometry of Two-Qubits" as follows:

"We have demonstrated that the basic properties of a system of two interacting spin-1/2 particles are uniquely embodied in the (sub)geometry of a particular projective line, found to be equivalent to the generalized quadrangle of order two. As such systems are the simplest ones exhibiting phenomena like quantum entanglement and quantum non-locality and play, therefore, a crucial role in numerous applications like quantum cryptography, quantum coding, quantum cloning/teleportation and/or quantum computing to mention the most salient ones, our discovery thus

- not only offers a principally new geometrically-underlined insight into their intrinsic nature,
- but also gives their applications a wholly new perspective
- and opens up rather unexpected vistas for an algebraic geometrical modelling of their higher-dimensional counterparts."

is not without relevance to the physics of quantum
theory.

Other material related to the above Saniga paper:

Quantum Geometry, a list of recent work in this area by Metod Saniga

The Veldkamp Space of Two-Qubits, by Metod Saniga, Michel Planat, Petr Pracna, and Hans Havlicek,*Symmetry, Integrability and
Geometry: Methods and Applications* (SIGMA 3 (2007), 075) (pdf, June
18, 2007, 7 pp.), and the following cited papers:

Other material related to the above Saniga paper:

Quantum Geometry, a list of recent work in this area by Metod Saniga

The Veldkamp Space of Two-Qubits, by Metod Saniga, Michel Planat, Petr Pracna, and Hans Havlicek,

Pauli Operators ofN-Qubit Hilbert Spaces and the Saniga–Planat Conjecture, by K. Thas,Chaos Solitons Fractals, to appear (as of June 18, 2007)

The Geometry of Generalized Pauli Operators ofN-Qudit Hilbert Space, by K. Thas,Quantum Information and Computation, submitted (as of June 18, 2007)

Page created Aug. 12, 2007; last modified Aug. 15, 2007.