Hjelmslev Geometry of Mutually Unbiased Bases

Abstract

The basic combinatorial properties of a complete set of mutually unbiased bases (MUBs) of a q-dimensional Hilbert space H\_q, q = p^r with p being a prime and r a positive integer, are shown to be qualitatively mimicked by the configuration of points lying on a proper conic in a projective Hjelmslev plane defined over a Galois ring of characteristic p^2 and rank r. The q vectors of a basis of H\_q correspond to the q points of a (so-called) neighbour class and the q+1 MUBs answer to the total number of (pairwise disjoint) neighbour classes on the conic.

4 pages, 1 figure; extended list of references, figure made more illustrative and in colour; v3 - one more figure and section added, paper made easier to follow, references updated

References in corpus (6)

• Constructions of Mutually Unbiased Bases
• Mutually Unbiased Bases and Finite Projective Planes
• The limitations of nice mutually unbiased bases
• Mutually unbiased phase states, phase uncertainties, and Gauss sums
• Sets of Mutually Unbiased Bases as Arcs in Finite Projective Planes?
• Mutually Unbiased Bases and Orthogonal Decompositions of Lie Algebras

Cited by in corpus (12)

• A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements
• The Frobenius formalism in Galois quantum systems
• Weak mutually unbiased bases
• The Projective Line Over the Finite Quotient Ring GF(2)[$x$]/$<x^3-x>$ and Quantum Entanglement II. The Mermin "Magic" Square/Pentagram
• The Projective Line Over the Finite Quotient Ring GF(2)[$x$]/$<x^3-x>$ and Quantum Entanglement I. Theoretical Background
• From pentacyclic coordinates to chain geometries, and back
• SU(2) nonstandard bases: the case of mutually unbiased bases
• An analytic function approach to weak mutually unbiased bases
• Hexagons govern three-qubit contextuality
• An Algorithm for constructing Hjelmslev planes
• Mutually unbiased bases as submodules and subspaces
• Harmonic analysis on a galois field and its subfields