# Metod SanigaAstronomical Institute of the Slovak Academy of Sciences

Metod Saniga

DrSc.

My current activity is centered around the doily -- a rising star of the Quantum Universe.

## About

209

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Introduction

**Skills and Expertise**

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March 2014 - June 2015

## Publications

Publications (209)

For $N \geq 2$, an $N$-qubit doily is a doily living in the $N$-qubit symplectic polar space. These doilies are related to operator-based proofs of quantum contextuality. Following and extending the strategy of Saniga et al. (Mathematics 9 (2021) 2272) that focused exclusively on three-qubit doilies, we first bring forth several formulas giving the...

In this article, we show that sets of three-qubit quantum observables obtained by considering both the classical and skew embeddings of the split Cayley hexagon of order two into the binary symplectic polar space of rank three can be used to detect quantum state-independent contextuality. This reveals a fundamental connection between these two appe...

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. observable-based proofs of the Kochen-Specker Theorem, we formulate the con-textuality property as the absence of solutions to a linear system and define for...

It is known that there are two non-equivalent embeddings of the split Cayley hexagon of order two into W(5, 2), the binary symplectic polar space of rank three, called classical and skew. Labelling the 63 points of W(5, 2) by the 63 canonical observables of the three-qubit Pauli group subject to the symplectic polarity induced by the (commutation r...

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 ob-servables. 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 t...

Given the symplectic polar space of type W(5,2), let us call a set of five Fano planes sharing pairwise a single point a Fano pentad. Once 63 points of W(5,2) are appropriately labeled by 63 non-trivial three-qubit observables, any such Fano pentad gives rise to a quantum contextual set known as a Mermin pentagram. Here, it is shown that a Fano pen...

We study doilies (i. e., W(3; 2)'s) living in W(5; 2), when the points of
the latter space are parametrized by canonical three-fold products of Pauli matrices and the associated identity matrix (i. e., by three-qubit observables). Key characteristics of such a doily are: the number of its negative lines, distribution of types of observables, charac...

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...

It is found that 15 different types of two-qubit X-states split naturally into two sets (of cardinality 9 and 6) once their entanglement properties are taken into account. We characterize both the validity and entangled nature of the X-states with maximally-mixed subsystems in terms of certain parameters and show that their properties are related t...

It is found that $15$ different types of two-qubit $X$-states split naturally into two sets (of cardinality $9$ and $6$) once their entanglement properties are taken into account. We {characterize both the validity and entangled nature of the $X$-states with maximally-mixed subsystems in terms of certain parameters} and show that their properties a...

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 t...

Given the symplectic polar space of type $W(5,2)$, let us call a set of five Fano planes sharing pairwise a single point a Fano pentad. Once 63 points of $W(5,2)$ are appropriately labeled by 63 non-trivial three-qubit observables, any such Fano pentad gives rise to a quantum contextual set known as Mermin pentagram. Here, it is shown that a Fano p...

Given the fact that the three-qubit symplectic polar space features three different kinds of observables and each of its labeled Fano planes acquires a definite sign, we found that there are 45 distinct types of Mermin pentagrams in this space. A key element of our classification is the fact that any context of such pentagram is associated with a u...

Let Tn(q) be the ring of lower triangular matrices of order n≥2 with entries from the finite field F(q) of order q≥2 and let Tn2(q) denote its free left module. For n=2,3 it is shown that the projective line over Tn(q) gives rise to a set of (q+1)(n−1)q3(n−1)(n−2)2 affine planes of order q. The points of such an affine plane are non-free cyclic sub...

Given the facts that the three-qubit symplectic polar space features three different kinds of observables and each of its labeled Fano planes acquires a definite sign, we found that there are 45 distinct types of Mermin pentagrams in this space. A key element of our classification is the fact that any context of such pentagram is associated with a...

Among finite geometries relevant for the theory of quantum information, the unique {\it triangle-free} $15_3$-configuration -- the doily -- has been recognized to play the foremost role. First, being isomorphic to the symplectic polar space of type $W(3,2)$, it underlies the commutation relations between the elements of the two-qubit Pauli group an...

A magic three-qubit Veldkamp line of $W(5,2)$, i.\,e. the line comprising a hyperbolic quadric $\mathcal{Q}^+(5,2)$, an elliptic quadric $\mathcal{Q}^-(5,2)$ and a quadratic cone $\widehat{\mathcal{Q}}(4,2)$ that share a parabolic quadric $\mathcal{Q}(4,2)$, the doily, is shown to provide an interesting model for the Veldkamp space of the latter. T...

Let $T_n(q)$ be the ring of lower triangular matrices of order $n \geq 2$ with entries from the finite field $F(q)$ of order $q \geq 2$ and let ${^2T_n(q)}$ denote its free left module. For $n=2,3$ it is shown that the projective line over $T_n(q)$ gives rise to a set of $(q+1)^{(n-1)}q^{\frac{3(n-1)(n-2)}{2}}$ affine planes of order $q$. The point...

In this paper, it is shown that there exists a particular associative ring with unity of order 16 such that the relations between non-unimodular free cyclic submodules of its two-dimensional free left module can be expressed in terms of the structure of the generalized quadrangle of order two. Such a doily-centered geometric structure is surmised t...

Making use of the ‘Veldkamp blow-up’ recipe, introduced by Saniga et al. (Ann Inst H Poincaré D 2:309–333, 2015) for binary Segre varieties, we study geometric hyperplanes and Veldkamp lines of Segre varieties S k (3) , where S k (3) stands for the k-fold direct product of projective lines of size four and k runs from 2 to 4. Unlike the binary case...

It is shown that there exists a particular associative ring with unity of order 16 such that the relations between nonunimodular free cyclic submodules of its two-dimensional free left module can be expressed in terms of the structure of the generalized quadrangle of order two. Such a doily-centered geometric structure is surmised to be of relevanc...

Making use of the `Veldkamp blow-up' recipe, introduced by Saniga and others (Ann. Inst. H. Poincar\' e D2 (2015) 309) for binary Segre varieties, we study geometric hyperplanes and Veldkamp lines of Segre varieties $S_k(3)$, where $S_k(3)$ stands for the $k$-fold direct product of projective lines of size four and $k$ runs from 2 to 4. Unlike the...

It is demonstrated that the magic three-qubit Veldkamp line occurs naturally within the Veldkamp space of combinatorial Grassmannian of type $G_2(7)$, $\mathcal{V}(G_2(7))$. The lines of the ambient symplectic polar space are those lines of $\mathcal{V}(G_2(7))$ whose cores feature an odd number of points of $G_2(7)$. After introducing basic proper...

The talk will highlight the role of finite geometry in the theory of
quantum information. In the first part of the talk, I will demonstrate how finite (symplectic and orthogonal) polar spaces provide a natural "geometrization" of the structure of the N-qubit Pauli group; here, the cases of N = 2, 3 and 4 will be discussed in detail. In the second p...

We investigate the structure of the three-qubit magic Veldkamp line (MVL). This mathematical notion has recently shown up as a tool for understanding the structures of the set of Mermin pentagrams, objects that are used to rule out certain classes of hidden variable theories. Here we show that this object also provides a unifying finite geometric u...

Using a standard technique sometimes (inaccurately) known as Burnside's Lemma, it is shown that the Veldkamp space of the near hexagon L3×GQ(2, 2) features 156 different types of lines. We also give an explicit description of each type of a line by listing the types of the three geometric hyperplanes it consists of and describing the properties of...

We investigate small geometric configurations that furnish observable-based proofs of the Kochen-Specker theorem. Assuming that each context consists of the same number of observables and each observable is shared by two contexts, it is proved that the most economical proofs are the famous Mermin-Peres square and the Mermin pentagram featuring, res...

Regarding a Dynkin diagram as a specific point-line incidence structure (where each line has just two points), one can associate with it a Veldkamp space. Focusing on extended Dynkin diagrams of type $\widetilde{D}_n$, $4 \leq n \leq 8$, it is shown that the corresponding Veldkamp space always contains a distinguished copy of the projective space P...

Cosmological and spatio-temporal aspects/implications of near-death and other extraordinary experiences; a collection of excerpts from freely-available first-person accounts. (Unfortunately, some older links are already broken.)

Given a seven-element set $X = \{1,2,3,4,5,6,7\}$, there are 30 ways to
define a Fano plane on it. Let us call a line of such Fano plane, that is to
say an unordered triple from $X$, ordinary or defective according as the sum of
two smaller integers from the triple is or is not equal to the remaining one,
respectively. A point of the labeled Fano p...

We point out an explicit connection between graphs drawn on compact Riemann
surfaces defined over the field $\bar{\mathbb{Q}}$ of algebraic numbers ---
so-called Grothendieck's {\it dessins d'enfants} --- and a wealth of
distinguished point-line configurations. These include simplices,
cross-polytopes, several notable projective configurations, a n...

Continuing our series of observations of the motion and dynamics of the solar
corona over the solar-activity cycle, we observed the corona from sites in
Queensland, Australia, during the 13 (UT)/14 (local time) November 2012 total
solar eclipse. The corona took the low-ellipticity shape typical of solar
maximum (flattening index {\epsilon} = 0.01),...

We discuss the large-scale structure of the solar corona, in particular its helmet streamers, as observed during total solar eclipses around maxima of solar cycles and make its comparison with solar polar magnetic field strength as observed by the Wilcox Solar Observatory (WSO) since 1976. Even though the magnetic field strength at the solar poles...

Given a hyperbolic quadric of PG(5,2), there are 28 points off this quadric
and 56 lines skew to it. It is shown that the $(28_6, 56_3)$-configuration
formed by these points and lines is isomorphic to the combinatorial
Grassmannian of type $G_2(8)$. It is also pointed out that a set of seven
points of $G_2(8)$ whose labels share a mark corresponds...

Given a $2^N$-dimensional Cayley-Dickson algebra, where $3 \leq N \leq 6$, we
first observe that the multiplication table of its imaginary units $e_a$, $1
\leq a \leq 2^N -1$, is encoded in the properties of the projective space
PG$(N-1,2)$ if one regards these imaginary units as points and distinguished
triads of them $\{e_a, e_b, e_c\}$, $1 \leq...

Let $S_{(N)} \equiv PG(1,\,2) \times PG(1,\,2) \times \cdots \times
PG(1,\,2)$ be a Segre variety that is $N$-fold direct product of projective
lines of size three. Given two geometric hyperplanes $H'$ and $H''$ of
$S_{(N)}$, let us call the triple $\{H', H'', \overline{H' \Delta H''}\}$ the
Veldkamp line of $S_{(N)}$. We shall demonstrate, for the...

Our studies of the solar chromosphere and corona at the 2012 and 2013
eclipses shortly after cycle maximum 24 (2011/2012) of solar activity
(see: http://www.swpc.noaa.gov/SolarCycle/) involved radio observations
of the 2012 annular eclipse with the Jansky Very Large Array, optical
observations of the 2012 total eclipse from Australia, optical
obser...

Employing the fact that the geometry of the $N$-qubit ($N \geq 2$) Pauli
group is embodied in the structure of the symplectic polar space
$\mathcal{W}(2N-1,\,2)$ and using properties of the Lagrangian Grassmannian
$LGr(N,\,2N)$ defined over the smallest Galois field, it is demonstrated that
there exists a bijection between the set of maximum sets o...

Using a standard technique sometimes (inaccurately) known as Burnside's
Lemma, it is shown that the Veldkamp space of the near hexagon L_3 times GQ(2,
2) features 158 different types of lines. We also give an explicit description
of each type of a line by listing the types of the three geometric hyperplanes
it consists of and describing the propert...

A total eclipse swept across Queensland and other sites in northeastern
Australia on the early morning of 14 November 2012, local time. We
mounted equipment to observe coronal images and spectra during the
approximately 2 minutes of totality, the former for comparison with
spacecraft images and to fill in the doughnut of imaging not well
covered wi...

We invoke some ideas from finite geometry to map bijectively 135 heptads of
mutually commuting three-qubit observables into 135 symmetric four-qubit ones.
After labeling the elements of the former set in terms of a seven-dimensional
Clifford algebra, we present the bijective map and most pronounced actions of
the associated symplectic group on both...

The white-light eclipse solar corona shows a plethora of structures of varying size and shape. A prominent type of them, very bright and far elongated of the solar limb, are the so-called helmet streamers, which connect regions of opposite magnetic polarity. We tried to derive their angular width from a series of eclipse observations. Our analysis...

Given a (2N − 1)-dimensional projective space over GF(2), PG(2N − 1, 2), and its geometric spread of lines, there exists a remarkable mapping of this space onto PG(N − 1, 4) where the lines of the spread correspond to the points and subspaces spanned by pairs of lines to the lines of PG(N − 1, 4). Under such mapping, a nondegenerate quadric surface...

Disregarding the identity, the remaining 63 elements of the generalized
three-qubit Pauli group are found to contain 12096 distinct copies of Mermin's
magic pentagram. Remarkably, 12096 is also the number of automorphisms of the
smallest split Cayley hexagon. We give a few solid arguments showing that this
may not be a mere coincidence. These argum...

We study dynamics of polar plumes observed during the 2008 eclipse from
three ground-based sites and the Hinode satellite. The speed of apparent
upward propagation, as inferred from the changes of brightness within
each plume, is found to lie in the range from 30 to 100 km
s-1. Some white-light plumes located in polar coronal holes
were identified...

We reveal an intriguing connection between the set of 27 (disregarding the identity) invertible symmetric 3 × 3 matrices over GF(2) and the points of the generalized quadrangle GQ(2, 4). The 15 matrices with eigenvalue one correspond to a copy of the subquadrangle GQ(2, 2), whereas the 12 matrices without eigenvalues have their geometric counterpar...

The geometry of the real four-qubit Pauli group, being embodied in the structure of the symplectic polar space W(7, 2), is analyzed in terms of ovoids of a hyperbolic quadric of PG(7, 2), the seven-dimensional projective space of order 2. The quadric is selected in such a way that it contains all 135 symmetric elements of the group. Under such circ...

Recently Waegell and Aravind [J. Phys. A: Math. Theor. 45 (2012), 405301, 13
pages] have given a number of distinct sets of three-qubit observables, each
furnishing a proof of the Kochen-Specker theorem. Here it is demonstrated that
two of these sets/configurations, namely the $18_{2} - 12_{3}$ and $2_{4}14_{2}
- 4_{3}6_{4}$ ones, can uniquely be e...

Employing five commuting sets of five-qubit observables, we propose specific
160-661 and 160-21 state proofs of the Bell-Kochen-Specker theorem that are
also proofs of Bell's theorem. A histogram of the 'Hilbert-Schmidt' distances
between the corresponding maximal bases shows in both cases a noise-like
behaviour. The five commuting sets are also as...

It is shown that the Veldkamp space of the unique generalized quadrangle GQ(2, 4) is isomorphic to PG(5, 2). Since the GQ(2, 4) features only two kinds of geometric hyperplanes, namely point's perp-sets and GQ(2, 2)s, the 63 points of PG(5, 2) split into two families; 27 being represented by perp-sets and 36 by GQ(2, 2)s. The 651 lines of PG(5, 2)...

A "magic rectangle" of eleven observables of four qubits, employed by Harvey
and Chryssanthacopoulos (2008) to prove the Bell-Kochen-Specker theorem in a
16-dimensional Hilbert space, is given a neat finite-geometrical
reinterpretation in terms of the structure of the symplectic polar space $W(7,
2)$ of the real four-qubit Pauli group. Each of the...

The corona is the uppermost part of the Sun's atmosphere and, up to now,
total solar eclipses provide the best conditions for observing it from
Earth's surface. The white-light corona (WLC) is the scattered light of
the photosphere off free electrons and it dominates in the regions up to
2-3 solar radii. As the motion of electrons is governed by ma...

Mermin's pentagram, a specific set of ten three-qubit observables arranged in
quadruples of pairwise commuting ones into five edges of a pentagram and used
to provide a very simple proof of the Kochen-Specker theorem, is shown to be
isomorphic to an ovoid (elliptic quadric) of the three-dimensional projective
space of order two, PG(3,2). This demon...

The white-light corona (WLC) during the total solar eclipse of 2009 July 22 was observed by several teams in the Moon's shadow stretching from India and China across the Pacific Ocean with its many isolated islands. We present a comparison of the WLC as observed by eclipse teams located in China (Shanghai region) and on the Enewetak Atoll in the Ma...

Time-latitude and time-longitude distributions of solar surface magnetic
fields are analyzed employing classical and modified synoptic charts.
The equatorial branch in solar cycle 23 in the southern hemisphere
lasted two years longer than its northern counterpart. The most
remarkable finding is that in the present solar cycle a pole-ward
migrating...

The green coronal line Fe XIV 530.3 nm ranks amongst the most pronounced
emission lines in the visible part of the solar spectrum. Its
observations outside solar eclipses started sporadically in 1939 (the
Arosa coronal station), being extended, in 1946, to more coronal
stations. It was found that the green corona intensities vary with solar
cycle,...

Predicting maxima and minima of solar activity cycles, including their
magnitude, is important not only for a better understanding of the
underlying physical processes on the Sun, but also from the point of
view of solar-terrestrial relations. Such predictions employ a variety
of well-know relations like those between even and odd cycles, or
betwee...

Given a finite associative ring with unity, $R$, and its two-dimensional left
module, $^{2}R$, the following two problems are addressed: 1) the existence of
vectors of $^{2}R$ that do not belong to any free cyclic submodule (FCS)
generated by a unimodular vector and 2) conditions under which such
(non-unimodular) vectors generate FCSs. The main res...

The white-light corona (WLC) during the total solar eclipse on 2010 July 11 was observed by several teams in the Moon's shadow stretching across the Pacific Ocean and a number of isolated islands. We present a comparison of the WLC as observed by eclipse teams located on the Tatakoto Atoll in French Polynesia and on Easter Island, 83 minutes later,...

The 11 July 2010 total solar eclipse was observed on the ground from
French Polynesia and, 83 minutes later, from Easter Island, and
near-simultaneous images were made with spacecraft instruments including
AIA/SDO, HMI/SDO, EUVI/STEREO, SWAP/PROBA2, EIT/SOHO, and LASCO/SOHO. We
report on changes in the corona detectable with high-resolution image
p...

Metod Saniga Astronomical Institute Slovak Academy of Sciences SK-05960 Tatranská Lomnica Slovak Republic (msaniga@astro.sk)

Following the spirit of a recent work of one of the authors (J. Phys. A:
Math. Theor. 44 (2011) 045301), the essential structure of the generalized
Pauli group of a qubit-qu$d$it, where $d = 2^{k}$ and an integer $k \geq 2$, is
recast in the language of a finite geometry. A point of such geometry is
represented by the maximum set of mutually commut...

The Veldkamp space, in the sense of Buekenhout and Cohen, of the generalized quadrangle GQ(4, 2) is shown not to be a (partial) linear space by simply giving several examples of Veldkamp lines (V-lines) having two or even three Veldkamp points (V-points) in common. Alongside the ordinary V-lines of size five, one also finds V-lines of cardinality t...