Projective Ring Line Encompassing Two-Qubits

Abstract

The projective line over the (non-commutative) ring of two-by-two matrices with coefficients in GF(2) is found to fully accommodate the algebra of 15 operators - generalized Pauli matrices - characterizing two-qubit systems. The relevant sub-configuration consists of 15 points each of which is either simultaneously distant or simultaneously neighbor to (any) two given distant points of the line. The operators can be identified with the points in such a one-to-one manner that their commutation relations are exactly reproduced by the underlying geometry of the points, with the ring geometrical notions of neighbor/distant answering, respectively, to the operational ones of commuting/non-commuting. This remarkable configuration can be viewed in two principally different ways accounting, respectively, for the basic 9+6 and 10+5 factorizations of the algebra of the observables. First, as a disjoint union of the projective line over GF(2) x GF(2) (the "Mermin" part) and two lines over GF(4) passing through the two selected points, the latter omitted. Second, as the generalized quadrangle of order two, with its ovoids and/or spreads standing for (maximum) sets of five mutually non-commuting operators and/or groups of five maximally commuting subsets of three operators each. These findings open up rather unexpected vistas for an algebraic geometrical modelling of finite-dimensional quantum systems and give their numerous applications a wholly new perspective.

8 pages, three tables; Version 2 - a few typos and one discrepancy corrected; Version 3: substantial extension of the paper - two-qubits are generalized quadrangles of order two; Version 4: self-dual picture completed; Version 5: intriguing triality found -- three kinds of geometric hyperplanes within GQ and three distinguished subsets of Pauli operators

References in corpus (10)

• Mutually unbiased binary observable sets on N qubits
• Projective Representations I. Projective lines over rings
• Multiple Qubits as Symplectic Polar Spaces of Order Two
• 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
• A Classification of the Projective Lines over Small Rings II. Non-Commutative Case
• 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
• On the Fine Structure of the Projective Line Over GF(2) x GF(2) x GF(2)

Cited by in corpus (27)

• Three-Qubit Operators, the Split Cayley Hexagon of Order Two and Black Holes
• An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group
• Black Hole Entropy and Finite Geometry
• Multiple Qubits as Symplectic Polar Spaces of Order Two
• Projective Ring Line of an Arbitrary Single Qudit
• Variations on a theme of Heisenberg, Pauli and Weyl
• Factor-Group-Generated Polar Spaces and (Multi-)Qudits
• Generalized X states of N qubits and their symmetries
• Parity proofs of the Kochen-Specker theorem based on 60 complex rays in four dimensions
• The Veldkamp space of multiple qubits
• Multi-Line Geometry of Qubit-Qutrit and Higher-Order Pauli Operators
• The Veldkamp Space of Two-Qubits
• Mermin's Pentagram as an Ovoid of PG(3,2)
• Quantum contextual finite geometries from dessins d'enfants
• On the Pauli graphs of N-qudits
• Twin "Fano-Snowflakes" Over the Smallest Ring of Ternions
• The Veldkamp Space of the Smallest Slim Dense Near Hexagon
• Pauli graph and finite projective lines/geometries
• 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 $\mathrm{LGr}(N,2N)$
• Geometric Hyperplanes of the Near Hexagon L_3 times GQ(2, 2)
• Tensor Commutation Matrices and Some Generalizations of the Pauli Matrices
• Charting the Real Four-Qubit Pauli Group via Ovoids of a Hyperbolic Quadric of PG(7,2)
• Space versus Time: Unimodular versus Non-Unimodular Projective Ring Geometries?
• On the Veldkamp Space of GQ(4, 2)
• Affine and Projective Planes Linked with Projective Lines over Certain Rings of Lower Triangular Matrices
• Free Cyclic Submodules and Non-Unimodular Vectors