Geometric Constructions over ℂ $${\mathbb {C}}$$ and 𝔽 2 $${\mathbb {F}}_2$$ for Quantum Information

Holweck, Frédéric

doi:10.1007/978-3-030-06122-7_5

Cited by 4 publications

(15 citation statements)

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“…quadratic doily of W(5, 2) defines a line resp. plane in the ambient PG (5,2). The latter type of a triad is found to be shared by four quadratic doilies.…”

Section: T Cmentioning

confidence: 87%

“…As a doily is also a subgeometry of W(5, 2), it either lies fully in Q − (YYY) (5, 2) (Type 8), or shares with Q − (YYY) (5, 2) a set of points that form a geometric hyperplane; an ovoid (Types 3, 4, 6 and 11), a perp-set (Types 1, 5, 9 and 12) and a grid (Types 2, 7, 10 and 13). One also observes that no quadratic doily shares a grid with Q − (YYY) (5,2). In addition to the distinguished elliptic quadric, there are also three distinguished hyperbolic quadrics in W(5, 2), namely: the quadric whose 35 observables feature either two X s or no X,…”

Section: T Cmentioning

confidence: 97%

“…Some fifteen years ago, it was discovered (see, e.g., [1][2][3][4]) that there exists a deep connection between the structure of the N-qubit Pauli group and that of the binary symplectic polar space of rank N, W(2N − 1, 2), where commutation relations between elements of the group are encoded in collinearity relations between points of W(2N − 1, 2). This connection has subsequently been used to obtain a deeper insight into, for example, finite geometric nature of observable-based proofs of quantum contextuality (for a recent review, see [5]), properties of certain black-hole entropy formulas [6] and the so-called black-hole/qubit correspondence [7], leading to a finite-geometric underpinning of four distinct Hitchin's invariants and the Cartan invariant of form theories of gravity [8] and even to an intriguing finite-geometric toy model of space-time [9]. This group-geometric connection was further strengthened by making use of the concept of geometric hyperplane and that of the Veldkamp space of W(2N − 1, 2) [10].…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2021

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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 particular, we classify and count polar subspaces of W(2N−1,2) whose rank is N−1. W(3,2) features three negative lines of the same type and its W(1,2)’s are of five different types. W(5,2) is endowed with 90 negative lines of two types and its W(3,2)’s split into 13 types. A total of 279 out of 480 W(3,2)’s with three negative lines are composite, i.e., they all originate from the two-qubit W(3,2). Given a three-qubit W(3,2) and any of its geometric hyperplanes, there are three other W(3,2)’s possessing the same hyperplane. The same holds if a geometric hyperplane is replaced by a ‘planar’ tricentric triad. A hyperbolic quadric of W(5,2) is found to host particular sets of seven W(3,2)’s, each of them being uniquely tied to a Conwell heptad with respect to the quadric. There is also a particular type of W(3,2)’s, a representative of which features a point each line through which is negative. Finally, W(7,2) is found to possess 1908 negative lines of five types and its W(5,2)’s fall into as many as 29 types. A total of 1524 out of 1560 W(5,2)’s with 90 negative lines originate from the three-qubit W(5,2). Remarkably, the difference in the number of negative lines for any two distinct types of four-qubit W(5,2)’s is a multiple of four.

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“…quadratic doily of W(5, 2) defines a line resp. plane in the ambient PG (5,2). The latter type of a triad is found to be shared by four quadratic doilies.…”

Section: T Cmentioning

confidence: 87%

Section: T Cmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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

et al. 2021

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“…The tangential variety X q of Y q (see also Section 6.6) has dimension 6q + 6 and is defined by the quartic polynomial q (cf. [11,20,27]).…”

Section: Groups Of Type Ementioning

confidence: 99%

Lagrangian Grassmannians and Spinor Varieties in Characteristic Two

Geemen

Marrani

2019

SIGMA

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The vector space of symmetric matrices of size n has a natural map to a projective space of dimension 2 n − 1 given by the principal minors. This map extends to the Lagrangian Grassmannian LG(n, 2n) and over the complex numbers the image is defined, as a set, by quartic equations. In case the characteristic of the field is two, it was observed that, for n = 3, 4, the image is defined by quadrics. In this paper we show that this is the case for any n and that moreover the image is the spinor variety associated to Spin(2n + 1). Since some of the motivating examples are of interest in supergravity and in the blackhole/qubit correspondence, we conclude with a brief examination of other cases related to integral Freudenthal triple systems over integral cubic Jordan algebras.

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“…Within the last ten to fifteen years it has been gradually realized that finite geometries represent key mathematical concepts of QIT. We here mention finite projective (Hjelmslev) planes, projective lines over certain modular rings and rings of ternions, small generalized polygons (in particular the split Cayley hexagon of order two), factor-group-generated symplectic and orthogonal polar spaces, affine polar spaces of rank three and order two (in particular extended generalized quadrangles with lines of size three), combinatorial Grassmannians, binary Segre varieties and, last but not least, Veldkamp spaces of certain point-line incidence structures with three points per line; for the relevant literature, see recent reviews by Holweck [1] and Keppens [2] aimed, respectively, at both physicists and mathematicians. Among them, the unique triangle-free 15 3 -configuration (out of 245,342 ones), also known as the Cremona-Richmond configuration and in the sequel referred to as the doily, acquires a special footing.…”

Section: Introductionmentioning

confidence: 99%

Magic Three-Qubit Veldkamp Line and Veldkamp Space of the Doily

Санига

Szabó

2020

Symmetry

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A magic three-qubit Veldkamp line of W ( 5 , 2 ) , i.e., the line comprising a hyperbolic quadric Q + ( 5 , 2 ) , an elliptic quadric Q − ( 5 , 2 ) and a quadratic cone Q ^ ( 4 , 2 ) that share a parabolic quadric Q ( 4 , 2 ) , the doily, is shown to provide an interesting model for the Veldkamp space of the doily. The model is based on the facts that: (a) the 20 off-doily points of Q + ( 5 , 2 ) form ten complementary pairs, each corresponding to a unique grid of the doily; (b) the 12 off-doily points of Q − ( 5 , 2 ) form six complementary pairs, each corresponding to a unique ovoid of the doily; and (c) the 15 off-doily points of Q ^ ( 4 , 2 ) , disregarding the nucleus of Q ( 4 , 2 ) , are in bijection with the 15 perp-sets of the doily. These findings lead to a conjecture that also parapolar spaces can be relevant for quantum information.

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Geometric Constructions over ℂ $${\mathbb {C}}$$ and 𝔽 2 $${\mathbb {F}}_2$$ for Quantum Information

Cited by 4 publications

References 66 publications

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

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

Lagrangian Grassmannians and Spinor Varieties in Characteristic Two

Magic Three-Qubit Veldkamp Line and Veldkamp Space of the Doily

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