Factor-Group-Generated Polar Spaces and (Multi-)Qudits

Abstract

Recently, a number of interesting relations have been discovered between generalised Pauli/Dirac groups and certain finite geometries. Here, we succeeded in finding a general unifying framework for all these relations. We introduce gradually necessary and sufficient conditions to be met in order to carry out the following programme: Given a group $\vG$, we first construct vector spaces over $\GF(p)$, $p$ a prime, by factorising $\vG$ over appropriate normal subgroups. Then, by expressing $\GF(p)$ in terms of the commutator subgroup of $\vG$, we construct alternating bilinear forms, which reflect whether or not two elements of $\vG$ commute. Restricting to $p=2$, we search for ``refinements'' in terms of quadratic forms, which capture the fact whether or not the order of an element of $\vG$ is $\le 2$. Such factor-group-generated vector spaces admit a natural reinterpretation in the language of symplectic and orthogonal polar spaces, where each point becomes a ``condensation'' of several distinct elements of $\vG$. Finally, several well-known physical examples (single- and two-qubit Pauli groups, both the real and complex case) are worked out in detail to illustrate the fine traits of the formalism.

20 pages, 6 figures, 1 table; Version 2 - slightly polished, updated references; Version 3 - published version in SIGMA

References in corpus (10)

• Three-Qubit Operators, the Split Cayley Hexagon of Order Two and Black Holes
• Black Hole Entropy and Finite Geometry
• Multiple Qubits as Symplectic Polar Spaces of Order Two
• Projective Ring Line of an Arbitrary Single Qudit
• Projective Ring Line Encompassing Two-Qubits
• Projective Ring Line of a Specific Qudit
• Multi-Line Geometry of Qubit-Qutrit and Higher-Order Pauli Operators
• 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
• Two-qubit analogs of octonions and projective planes

Cited by in corpus (22)

• Entanglement can increase asymptotic rates of zero-error classical communication over classical channels
• Pauli graphs when the Hilbert space dimension contains a square: why the Dedekind psi function ?
• Grassmannian Connection Between Three- and Four-Qubit Observables, Mermin's Contextuality and Black Holes
• The Veldkamp space of multiple qubits
• Mermin's Pentagram as an Ovoid of PG(3,2)
• Quantum contextual finite geometries from dessins d'enfants
• Five-Qubit Contextuality, Noise-Like Distribution of Distances Between Maximal Bases and Finite Geometry
• Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank
• Aspects of the Segre variety S_{1,1,1}(2)
• The Veldkamp Space of the Smallest Slim Dense Near Hexagon
• A Sequence of Qubit-Qudit Pauli Groups as a Nested Structure of Doilies
• A Notable Relation between $N$-Qubit and $2^{N-1}$-Qubit Pauli Groups via Binary $\mathrm{LGr}(N,2N)$
• A Finite-Geometric Classification of Three-Qubit Mermin Pentagrams
• 'Magic' Configurations of Three-Qubit Observables and Geometric Hyperplanes of the Smallest Split Cayley Hexagon
• The Complement of Binary Klein Quadric as a Combinatorial Grassmannian
• A Class of Three-Qubit Contextual Configurations Located in Fano Pentads
• On the Veldkamp Space of GQ(4, 2)
• Hexagons govern three-qubit contextuality
• A new heuristic approach for contextuality degree estimates and its four- to six-qubit portrayals
• Moebius Pairs of Simplices and Commuting Pauli Operators
• Graph States and the Variety of Principal Minors
• Veldkamp Spaces: From (Dynkin) Diagrams to (Pauli) Groups