arXiv subjects

Metod Saniga

Publications and source records attributed to Metod Saniga.

At least 19 recordsLinked to original sources

arXiv2026

The Finite Geometry of Breaking Quantum Secrets

Using a finite geometric framework for studying the pentagon and heptagon codes we show that the concepts of quantum secret sharing and contextuality can be studied in a nice and unified manner. The basic idea is a careful study of the respective $2+3$ and $3+4$ tensorial factorizations of the elements of the stabilizer groups of these codes. It is demonstrated in detail how finite geometric structures entailing a specific three-qubit (resp. four-qubit) embedding of binary symplectic polar spaces of rank two (resp. three), corresponding to these factorizations, govern issues of contextuality and entanglement needed for a geometric understanding of quantum secret sharing. Using these results for the $(3,5)$ and $(4,7)$ threshold schemes explicit secret breaking protocols are derived. Our results hint at a novel geometric way of looking at contextual configurations.

quant-ph↗

arXiv2024

A new heuristic approach for contextuality degree estimates and its four- to six-qubit portrayals

We introduce and describe a new heuristic method for finding an upper bound on the degree of contextuality and the corresponding unsatisfied part of a quantum contextual configuration with three-element contexts (i.e., lines) located in a multi-qubit symplectic polar space of order two. While the previously used method based on a SAT solver was limited to three qubits, this new method is much faster and more versatile, enabling us to also handle four- to six-qubit cases. The four-qubit unsatisfied configurations we found are quite remarkable. That of an elliptic quadric features 315 lines and has in its core three copies of the split Cayley hexagon of order two having a Heawood-graph-underpinned geometry in common. That of a hyperbolic quadric also has 315 lines but, as a point-line incidence structure, is isomorphic to the dual $\mathcal{DW}(5,2)$ of $\mathcal{W}(5,2)$. Finally, an unsatisfied configuration with 1575 lines associated with all the lines/contexts of the four-qubit space contains a distinguished $\mathcal{DW}(5,2)$ centered on a point-plane incidence graph of PG$(3,2)$. The corresponding configurations found in the five-qubit space exhibit a considerably higher degree of complexity, except for a hyperbolic quadric, whose 6975 unsatisfied contexts are compactified around the point-hyperplane incidence graph of PG$(4,2)$. The most remarkable unsatisfied patterns discovered in the six-qubit space are a couple of disjoint split Cayley hexagons (for the full space) and a subgeometry underpinned by the complete bipartite graph $K_{7,7}$ (for a hyperbolic quadric).

quant-ph↗

arXiv2023

Hexagons govern three-qubit contextuality

Split Cayley hexagons of order two are distinguished finite geometries living in the three-qubit symplectic polar space in two different forms, called classical and skew. Although neither of the two yields observable-based contextual configurations of their own, {\it classically}-embedded copies are found to fully encode contextuality properties of the most prominent three-qubit contextual configurations in the following sense: for each set of unsatisfiable contexts of such a contextual configuration there exists some classically-embedded hexagon sharing with the configuration exactly this set of contexts and nothing else. We demonstrate this fascinating property first on the configuration comprising all 315 contexts of the space and then on doilies, both types of quadrics as well as on complements of skew-embedded hexagons. In connection with the last-mentioned case and elliptic quadrics we also conducted some experimental tests on a Noisy Intermediate Scale Quantum (NISQ) computer to substantiate our theoretical findings.

quant-ph↗

arXiv2023

New and improved bounds on the contextuality degree of multi-qubit configurations

We present algorithms and a C code to reveal quantum contextuality and evaluate the contextuality degree (a way to quantify contextuality) for a variety of point-line geometries located in binary symplectic polar spaces of small rank. With this code we were not only able to recover, in a more efficient way, all the results of a recent paper by de Boutray et al [(2022). Journal of Physics A: Mathematical and Theoretical 55 475301], but also arrived at a bunch of new noteworthy results. The paper first describes the algorithms and the C code. Then it illustrates its power on a number of subspaces of symplectic polar spaces whose rank ranges from 2 to 7. The most interesting new results include: (i) non-contextuality of configurations whose contexts are subspaces of dimension 2 and higher, (ii) non-existence of negative subspaces of dimension 3 and higher, (iii) considerably improved bounds for the contextuality degree of both elliptic and hyperbolic quadrics for rank 4, as well as for a particular subgeometry of the three-qubit space whose contexts are the lines of this space, (iv) proof for the non-contextuality of perpsets and, last but not least, (v) contextual nature of a distinguished subgeometry of a multi-qubit doily, called a two-spread, and computation of its contextuality degree. Finally, in the three-qubit polar space we correct and improve the contextuality degree of the full configuration and also describe finite geometric configurations formed by unsatisfiable/invalid constraints for both types of quadrics as well as for the geometry whose contexts are all 315 lines of the space.

quant-ph↗

arXiv2023

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. observable-based proofs of the Kochen-Specker Theorem, we formulate the contextuality property as the absence of solutions to a linear system and define for a contextual configuration its degree of contextuality. Then we explain why subgeometries of binary symplectic polar spaces are candidates for contextuality proofs. We report the results of a software that generates these subgeometries, decides their contextuality and computes their contextuality degree for some small symplectic polar spaces. We show that quadrics in the symplectic polar space $W_n$ are contextual for $n=3,4,5$. The proofs we consider involve more contexts and observables than the smallest known proofs. This intermediate size property of those proofs is interesting for experimental tests, but could also be interesting in quantum game theory.

quant-ph↗

arXiv2022

Multi-qubit doilies: enumeration for all ranks and classification for ranks four and five

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 number of both linear and quadratic doilies for any $N > 2$. Then we present an effective algorithm for the generation of all $N$-qubit doilies. Using this algorithm for $N=4$ and $N=5$, we provide a classification of $N$-qubit doilies in terms of types of observables they feature and number of negative lines they are endowed with. We also list several distinguished findings about $N$-qubit doilies that are absent in the three-qubit case, point out a couple of specific features exhibited by linear doilies and outline some prospective extensions of our approach.

quant-ph↗

arXiv2022

Three-Qubit-Embedded Split Cayley Hexagon is Contextuality Sensitive

It is known that there are two non-equivalent embeddings of the split Cayley hexagon of order two into $\mathcal{W}(5,2)$, the binary symplectic polar space of rank three, called classical and skew. Labelling the 63 points of $\mathcal{W}(5,2)$ by the 63 canonical observables of the three-qubit Pauli group subject to the symplectic polarity induced by the (commutation relations between the elements of the) group, the two types of embedding are found to be quantum contextuality sensitive. In particular, we show that the complement of a classically-embedded hexagon is not contextual, whereas that of a skewly-embedded one is.

quant-ph↗

arXiv2021

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

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. 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. 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.

quant-ph↗

arXiv2020

$X$-States From a Finite Geometric Perspective

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 to a special class of geometric hyperplanes of the symplectic polar space of order two and rank two. Finally, we introduce the concept of hyperplane-states and briefly address their non-local properties.

math-ph↗

arXiv2020

A Class of Three-Qubit Contextual Configurations Located in Fano Pentads

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 pentad also hosts another, closely related contextual set, which features 25 observables and 30 three-element contexts. Out of 25 observables, ten are such that each of them is on six contexts, while each of the remaining 15 observables belongs to two contexts only. Making use of the recent classification of Mermin pentagrams (Saniga et al., Symmetry 12 (2020) 534), it was found that 12,096 such contextual sets comprise 47 distinct types, falling into eight families according to the number ($3, 5, 7, \ldots, 17$) of negative contexts.

quant-ph↗

arXiv2019

A Finite-Geometric Classification of Three-Qubit Mermin Pentagrams

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 unique (positive or negative) Fano plane. Several intriguing relations between the character of pentagrams' three-qubit observables and `valuedness' of associated Fano planes are pointed out. In particular, we find two distinct kinds of negative contexts and as many as four positive ones.

quant-ph↗

arXiv2019

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

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. The model is based on the facts that: a) the 20 off-doily points of $\mathcal{Q}^+(5,2)$ form ten complementary pairs, each corresponding to a unique grid of the doily; b) the 12 off-doily points of $\mathcal{Q}^-(5,2)$ form six complementary pairs, each corresponding to a unique ovoid of the doily; and c) the 15 off-doily points of $\widehat{\mathcal{Q}}(4,2)$ -- disregarding the nucleus of $\mathcal{Q}(4,2)$ -- 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.

quant-ph↗

arXiv2019

Affine and Projective Planes Linked with Projective Lines over Certain Rings of Lower Triangular Matrices

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 points of such an affine plane are non-free cyclic submodules of ${^2T_n(q)}$ not contained in any non-unimodular free cyclic submodule of ${^2T_n(q)}$ and its lines are points of the projective line. Furthermore, it is demonstrated that each affine plane can be extended to the projective plane of order $q$, with the `line at infinity' being represented by those free cyclic submodules of ${^2T_n(q)}$ that are generated by non-unimodular pairs. Our approach can straightforwardly be adjusted to address the case of arbitrary $n$.

math.RA↗

arXiv2018

Doily as Subgeometry of a Set of Nonunimodular Free Cyclic Submodules

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 relevance for quantum information.

math.CO↗

arXiv2018

Veldkamp Spaces of Low-Dimensional Ternary Segre Varieties

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 binary case, the Veldkamp spaces here feature also non-projective elements. Although for $k=2$ such elements are found only among Veldkamp lines, for $k \geq 3$ they are also present among Veldkamp points of the associated Segre variety. Even if we consider only projective geometric hyperplanes, we find four different types of non-projective Veldkamp lines of $S_3(3)$, having 2268 members in total, and five more types if non-projective ovoids are also taken into account. Sole geometric and combinatorial arguments lead to as many as 62 types of projective Veldkamp lines of $S_3(3)$, whose blowing-ups yield 43 distinct types of projective geometric hyperplanes of $S_4(3)$. As the latter number falls short of 48, the number of different large orbits of $2 \times 2 \times 2 \times 2$ arrays over the three-element field found by Bremner and Stavrou (Lin. Multilin. Algebra 61 (2013) 986), there are five (explicitly indicated) hyperplane types such that each is the merger of two different large orbits. Furthermore, we single out those 22 types of geometric hyperplanes of $S_4(3)$, featuring 7 176 640 members in total, that are in a one-to-one correspondence with the points lying on the unique hyperbolic quadric $\mathcal{Q}_0^{+}(15,3) \subset {\rm PG}(15,3) \subset \mathcal{V}(S_4(3))$; and, out of them, seven ones that correspond bijectively to the set of 91 840 generators of the symplectic polar space $\mathcal{W}(7,3) \subset \mathcal{V}(S_3(3))$. For $k=3$ we also discuss embedding of the binary Veldkamp space into the ternary one.

math.CO↗

arXiv2017

A Combinatorial Grassmannian Representation of the Magic Three-Qubit Veldkamp Line

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 properties of three different types of points and six distinct types of lines of $\mathcal{V}(G_2(7))$, we explicitly show the combinatorial Grassmannian composition of the magic Veldkamp line; we first give representatives of points and lines of its core generalized quadrangle GQ$(2,2)$, and then additional points and lines of a specific elliptic quadric $\mathcal{Q}^{-}$(5,2), a hyperbolic quadric $\mathcal{Q}^{+}$(5,2) and a quadratic cone $\widehat{\mathcal{Q}}$(4,2) that are centered on the GQ$(2,2)$. In particular, each point of $\mathcal{Q}^{+}$(5,2) is represented by a Pasch configuration and its complementary line, the (Schläfli) double-six of points in $\mathcal{Q}^{-}$(5,2) comprise six Cayley-Salmon configurations and six Desargues configurations with their complementary points, and the remaining Cayley-Salmon configuration stands for the vertex of $\widehat{\mathcal{Q}}$(4,2).

math-ph↗

arXiv2017

The magic three-qubit Veldkamp line: A finite geometric underpinning for form theories of gravity and black hole entropy

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 underpinning for understanding the structure of functionals used in form theories of gravity and black hole entropy. We clarify the representation theoretic, finite geometric and physical meaning of the different parts of our MVL. The upshot of our considerations is that the basic finite geometric objects enabling such a diversity of physical applications of the MVL are the unique generalized quadrangles with lines of size three, their one point extensions as well as their other extensions isomorphic to affine polar spaces of rank three and order two. In a previous work we have already connected generalized quadrangles to the structure of cubic Jordan algebras related to entropy fomulas of black holes and strings in five dimensions. In some respect the present paper can be regarded as a generalization of that analysis for also providing a finite geometric understanding of four-dimensional black hole entropy formulas. However, we find many more structures whose physical meaning is yet to be explored. As a familiar special case our work provides a finite geometric representation of the algebraic extension from cubic Jordan algebras to Freudenthal systems based on such algebras.

hep-th↗

arXiv2016

Contextuality with a Small Number of Observables

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, respectively, $9$ and $10$ observables, there being no proofs using less than $9$ observables. We also propose a new proof with $14$ observables forming a `magic' heptagram. On the other hand, some other prominent small-size finite geometries, like the Pasch configuration and the prism, are shown not to be contextual.

quant-ph↗