The Veldkamp space of multiple qubits

Abstract

We introduce a point-line incidence geometry in which the commutation relations of the real Pauli group of multiple qubits are fully encoded. Its points are pairs of Pauli operators differing in sign and each line contains three pairwise commuting operators any of which is the product of the other two (up to sign). We study the properties of its Veldkamp space enabling us to identify subsets of operators which are distinguished from the geometric point of view. These are geometric hyperplanes and pairwise intersections thereof. Among the geometric hyperplanes one can find the set of self-dual operators with respect to the Wootters spin-flip operation well-known from studies concerning multiqubit entanglement measures. In the two- and three-qubit cases a class of hyperplanes gives rise to Mermin squares and other generalized quadrangles. In the three-qubit case the hyperplane with points corresponding to the 27 Wootters self-dual operators is just the underlying geometry of the E6(6) symmetric entropy formula describing black holes and strings in five dimensions.

15 pages, 1 figure; added references, corrected typos; minor changes

References in corpus (7)

• Black Holes, Qubits and Octonions
• Three-Qubit Operators, the Split Cayley Hexagon of Order Two and Black Holes
• Implementation of Clifford gates in the Ising-anyon topological quantum computer
• Black Hole Entropy and Finite Geometry
• Projective Ring Line Encompassing Two-Qubits
• Factor-Group-Generated Polar Spaces and (Multi-)Qudits
• Clifford groups of quantum gates, BN-pairs and smooth cubic surfaces

Cited by in corpus (17)

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• Stochastic Local Operations with Classical Communication of Absolutely Maximally Entangled States
• From qubits to E7
• Mermin pentagrams arising from Veldkamp lines for three qubits
• The magic three-qubit Veldkamp line: A finite geometric underpinning for form theories of gravity and black hole entropy
• Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank
• Contextuality degree of quadrics in multi-qubit symplectic polar spaces
• The Veldkamp Space of the Smallest Slim Dense Near Hexagon
• A Sequence of Qubit-Qudit Pauli Groups as a Nested Structure of Doilies
• 'Magic' Configurations of Three-Qubit Observables and Geometric Hyperplanes of the Smallest Split Cayley Hexagon
• Charting the Real Four-Qubit Pauli Group via Ovoids of a Hyperbolic Quadric of PG(7,2)
• A Combinatorial Grassmannian Representation of the Magic Three-Qubit Veldkamp Line
• Hexagons govern three-qubit contextuality
• Invertible Symmetric 3 x 3 Binary Matrices and GQ(2,4)
• A new heuristic approach for contextuality degree estimates and its four- to six-qubit portrayals
• Finite Projective Spaces, Geometric Spreads of Lines and Multi-Qubits
• Veldkamp Spaces: From (Dynkin) Diagrams to (Pauli) Groups