Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank

  title={Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank},
  author={Metod Saniga and Henri de Boutray and Fr{\'e}d{\'e}ric Holweck and Alain Giorgetti},
We study certain physically-relevant subgeometries of binary symplectic polar spaces W(2N−1,2) of small rank N, when the points of these spaces canonically encode N-qubit observables. Key characteristics of a subspace of such a space W(2N−1,2) are: the number of its negative lines, the distribution of types of observables, the character of the geometric hyperplane the subspace shares with the distinguished (non-singular) quadric of W(2N−1,2) and the structure of its Veldkamp space. In… 
Contextuality degree of quadrics in multi-qubit symplectic polar spaces
Quantum contextuality takes an important place amongst the concepts of quantum computing that bring an advantage over its classical counterpart. For a large class of contextuality proofs, aka.
Symmetries and Geometries of Qubits, and Their Uses
  • A. Rau
  • Mathematics, Physics
  • 2021
This review brings together the Lie-al algebraic/group-representation perspective of quantum physics and the geometric–algebraic one, as well as their connections to complex quaternions, as a further development of Felix Klein’s Erlangen Program for symmetries and geometries.


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The Veldkamp Space of Two-Qubits
Given a remarkable representation of the generalized Pauli operators of two- qubits in terms of the points of the generalized quadrangle of order two, W(2), it is shown that specific subsets of these
Mermin's pentagram as an ovoid of PG(3, 2)
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Recently, a number of interesting relations have been discovered between genera- lised Pauli/Dirac groups and certain finite geometries. Here, we succeeded in finding a general unifying framework for
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It is found that there are 45 distinct types of Mermin pentagrams in the three-qubit symplectic polar space, with two distinct kinds of negative contexts and as many as four positive ones.
The geometry of generalized Pauli operators of N-qudit Hilbert space, and an application to MUBs
We prove that the set of non-identity generalized Pauli operators on the Hilbert space of N d-level quantum systems, d an odd prime, naturally forms a symplectic polar space of rank N and order d.
Grassmannian connection between three- and four-qubit observables, Mermin’s contextuality and black holes
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Mermin pentagrams arising from Veldkamp lines for three qubits
We study the geometry of the space of Mermin pentagrams, objects that are used to rule out the existence of noncontextual hidden variable theories as alternatives to quantum theory. It is shown that
Geometric Hyperplanes of the Near Hexagon L3 × GQ(2, 2)
Having in mind their potential quantum physical applications, we classify all geometric hyperplanes of the near hexagon that is a direct product of a line of size three and the generalized quadrangle