Symplectic Polar Spaces Research Articles

Exploiting finite geometries for better quantum advantages in Mermin-like games

Quantum games embody non-intuitive consequences of quantum phenomena, such as entanglement and contextuality. The Mermin-Peres game is a simple example, demonstrating how two players can utilise shared quantum information to win a no—communication game with certainty, where classical players cannot. In this paper we look at the geometric structure behind such classical strategies, and borrow ideas from the geometry of symplectic polar spaces to maximise this quantum advantage. We introduce a new game called the Eloily game with a quantum–classical success gap of 0.26― , larger than that of the Mermin-Peres and doily games. We simulate this game in the IBM Quantum Experience and obtain a success rate of 1, beating the classical bound of 0.73― demonstrating the efficiency of the quantum strategy.

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

AbstractWe 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 Theoretical55 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.

Characterizations of symplectic polar spaces

Abstract A polar space 𝒮 is called symplectic if it admits a projective embedding ε : 𝒮 → PG(V) such that the image ε(𝒮) of 𝒮 by ε is defined by an alternating form of V. In this paper we characterize symplectic polar spaces in terms of their incidence properties, with no mention of peculiar properties of their embeddings. This is relevant especially when 𝒮 admits different (non-isomorphic) embeddings, as it is the case when 𝒮 is defined over a field of characteristic 2.

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 . 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.

On large partial ovoids of symplectic and Hermitian polar spaces

AbstractIn this paper we provide constructive lower bounds on the sizes of the largest partial ovoids of the symplectic polar spaces , odd square, , and of the Hermitian polar spaces , even or odd square, , , .

Non-trivial t-intersecting families for symplectic polar spaces

Let P be a symplectic polar space over a finite field Fq, and Pm denote the set of all m-dimensional subspaces in P. We say a t-intersecting subfamily of Pm is trivial if there exists a t-dimensional subspace contained in each member of this family. In this paper, we determine the structure of the maximum sized non-trivial t-intersecting subfamilies of Pm.

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. 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.

Split buildings of type $$\mathsf {F_4}$$ F 4 in buildings of type $$\mathsf {E_6}$$ E 6

A symplectic polarity of a building {\Delta} of type E6 is a polarity whose fixed point structure is a building of type F4 containing residues isomorphic to symplectic polar spaces (i.e., so-called split buildings of type F4). In this paper, we show in a geometric way that every building of type E6 contains, up to conjugacy, a unique class of symplectic polarities. We also show that the natural point-line geometry of each split building of type F4 fully embedded in the natural point-line geometry of {\Delta} arises from a symplectic polarity.

Symplectic polarities of buildings of type E 6

A symplectic polarity of a building Δ of type E 6 is a polarity whose fixed point structure is a building of type F 4 containing residues isomorphic to symplectic polar spaces. In this paper, we present two characterizations of such polarities among all dualities. Firstly, we prove that, if a duality θ of Δ never maps a point to a neighbouring symp, and maps some element to a non-opposite element, then θ is a symplectic duality. Secondly, we show that, if a duality θ never maps a chamber to an opposite chamber, then it is a symplectic polarity. The latter completes the programme for dualities of buildings of type E 6 of determining all domestic automorphisms of spherical buildings, and it also shows that symplectic polarities are the only polarities in buildings of type E 6 for which the Phan geometry is empty.

Modular geometry of the symplectic group attached to a q-level system and to multiple q-dit mixtures

We study the commutation relations within the Pauli groups built on all decompositions of a given Hilbert space dimension q, containing a square, into its factors. Illustrative low dimensional examples are the quartit (q = 4) and two-qubit (q = 22) systems, the octit (q = 8), qubit/quartit (q = 2×4) and three-qubit (q = 23) systems, and so on. In the single qudit case, e.g. q = 4, 8, 12,…, one defines a bijection between the σ(q) maximal commuting sets [with σ[q)the sum of divisors of q] of Pauli observables and the maximal submodules of the modular ring ℤq2, that arrange into the projective line P1(ℤq) and a independent set of size σ(q) − ψ(q) [with ψ(q) the Dedekind psi function]. In the multiple qudit case, e.g. q = 22, 23, 32,…, the Pauli graphs rely on symplectic polar spaces such as the generalized quadrangles GQ(2,2) (if q = 22) and GQ(3,3) (if q = 32). More precisely, in dimension pn (p a prime) of the Hilbert space, the observables of the Pauli group (modulo the center) are seen as the elements of the 2n-dimensional vector space over the field Fp. In this space, one makes use of the commutator to define a symplectic polar space W2n−1(p) of cardinality σ(p2n−1), that encodes the maximal commuting sets of the Pauli group by its totally isotropic subspaces. Building blocks of W2n−1(p) are punctured polar spaces (i.e. a observable and all maximum cliques passing to it are removed) of size given by the Dedekind psi function ψ(p2N−1). For multiple qudit mixtures (e.g. qubit/quartit, qubit/octit and so on), one finds multiple copies of polar spaces, ponctured polar spaces, hypercube geometries and other intricate structures. Such structures play a role in the science of quantum information.

Pauli graphs when the Hilbert space dimension contains a square: Why the Dedekind psi function?

We study the commutation relations within the Pauli groups built on all decompositions of a given Hilbert space dimension q, containing a square, into its factors. Illustrative low-dimensional examples are the quartit (q = 4) and two-qubit (q = 22) systems, the octit (q = 8), qubit/quartit (q = 2 × 4) and three-qubit (q = 23) systems, and so on. In the single qudit case, e.g. q = 4, 8, 12, … , one defines a bijection between the σ(q) maximal commuting sets (with σ[q) being the sum of divisors of q) of Pauli observables and the maximal submodules of the modular ring , that arrange into the projective line and an independent set of size σ(q) − ψ(q) (with ψ(q) being the Dedekind psi function). In the multiple qudit case, e.g. q = 22, 23, 32, … , the Pauli graphs rely on symplectic polar spaces such as the generalized quadrangles GQ(2, 2) (if q = 22) and GQ(3, 3) (if q = 32). More precisely, in the dimension pn (p a prime) of the Hilbert space, the observables of the Pauli group (modulo the center) are seen as the elements of the 2n-dimensional vector space over the field . In this space, one makes use of the commutator to define a symplectic polar space W2n − 1(p) of cardinality σ(p2n − 1) that encodes the maximal commuting sets of the Pauli group by its totally isotropic subspaces. Building blocks of W2n − 1(p) are punctured polar spaces (i.e. a observable and all maximum cliques passing to it are removed) of size given by the Dedekind psi function ψ(p2n − 1). For multiple qudit mixtures (e.g. qubit/quartit, qubit/octit and so on), one finds multiple copies of polar spaces, punctured polar spaces, hypercube geometries and other intricate structures. Such structures play a role in the science of quantum information.

On symplectic polar spaces over non-perfect fields of characteristic 2

Given a field 𝕂 of characteristic 2 and an integer n ≥ 2, let W(2n − 1, 𝕂) be the symplectic polar space defined in PG(2n − 1, 𝕂) by a non-degenerate alternating form of V(2n, 𝕂) and let Q(2n, 𝕂) be the quadric of PG(2n, 𝕂) associated to a non-singular quadratic form of Witt index n. In the literature it is often claimed that W(2n − 1, 𝕂) ≅ Q(2n, 𝕂). This is true when 𝕂 is perfect, but false otherwise. In this article, we modify the previous claim in order to obtain a statement that is correct for any field of characteristic 2. Explicitly, we prove that W(2n − 1, 𝕂) is indeed isomorphic to a non-singular quadric Q, but when 𝕂 is non-perfect the nucleus of Q has vector dimension greater than 1. So, in this case, Q(2n, 𝕂) is a proper subgeometry of W(2n − 1, 𝕂). We show that, in spite of this fact, W(2n − 1, 𝕂) can be embedded in Q(2n, 𝕂) as a subgeometry and that this embedding induces a full embedding of the dual DW(2n − 1, 𝕂) of W(2n − 1, 𝕂) into the dual DQ(2n, 𝕂) of Q(2n, 𝕂).

Tight sets, weighted m-covers, weighted m-ovoids, and minihypers

Minihypers are substructures of projective spaces introduced to study linear codes meeting the Griesmer bound. Recently, many results in finite geometry were obtained by applying characterization results on minihypers (De Beule et al. 16:342–349, 2008; Govaerts and Storme 4:279–286, 2004; Govaerts et al. 28:659–672, 2002). In this paper, using characterization results on certain minihypers, we present new results on tight sets in classical finite polar spaces and weighted m-covers, and on weighted m-ovoids of classical finite generalized quadrangles. The link with minihypers gives us characterization results of i-tight sets in terms of generators and Baer subgeometries contained in the Hermitian and symplectic polar spaces, and in terms of generators for the quadratic polar spaces. We also present extendability results on partial weighted m-ovoids and partial weighted m-covers, having small deficiency, to weighted m-covers and weighted m-ovoids of classical finite generalized quadrangles. As a particular application, we prove in an alternative way the extendability of 53-, 54-, and 55-caps of PG(5,3), contained in a non-singular elliptic quadric Q−(5,3), to 56-caps contained in this elliptic quadric Q−(5,3).

On the Pauli graphs on N-qudits

A comprehensive graph theoretical and finite geometrical study of the commutation relations between the generalized Pauli operators of $N$-qudits is performed in which vertices/points correspond to the operators and edges/lines join commuting pairs of them. As per two-qubits, all basic properties and partitionings of the corresponding {\it Pauli graph} are embodied in the geometry of the generalized quadrangle of order two. Here, one identifies the operators with the points of the quadrangle and groups of maximally commuting subsets of the operators with the lines of the quadrangle. The three basic partitionings are (a) a pencil of lines and a cube, (b) a Mermin's array and a bipartite-part and (c) a maximum independent set and the Petersen graph. These factorizations stem naturally from the existence of three distinct geometric hyperplanes of the quadrangle, namely a set of points collinear with a given point, a grid and an ovoid, which answer to three distinguished subsets of the Pauli graph, namely a set of six operators commuting with a given one, a Mermin's square, and set of five mutually non-commuting operators, respectively. The generalized Pauli graph for multiple qubits is found to follow from symplectic polar spaces of order two, where maximal totally isotropic subspaces stand for maximal subsets of mutually commuting operators. The substructure of the (strongly regular) $N$-qubit Pauli graph is shown to be pseudo-geometric, i.\,e., isomorphic to a graph of a partial geometry. Finally, the (not strongly regular) Pauli graph of a two-qutrit system is introduced; here it turns out more convenient to deal with its dual in order to see all the parallels with the two-qubit case and its surmised relation with the generalized quadrangle $Q(4,3)$, the dual of $W(3)$.

A note on symplectic polar spaces over non-perfect fields of characteristic 2

Abstract Given a field K of characteristic 2, let W ( 2 n − 1 , K ) be the symplectic polar space defined in PG ( 2 n − 1 , K ) by a non-degenerate alternating form of V ( 2 n , K ) and Q ( 2 n , K ) be the quadric of PG ( 2 n , K ) associated to a non-singular quadratic form of Witt index n. In the literature, it is often claimed that W ( 2 n − 1 , K ) ≅ Q ( 2 n , K ) . This is true when K is perfect, but false if otherwise. In this paper we modify the previous claim, so to obtain a statement that is correct for any field of characteristic 2.

A characterization of finite symplectic polar spaces of odd prime order

A sufficient condition for the representation group for a nonabelian representation (Definition 1.1) of a finite partial linear space to be a finite p-group is given (Theorem 2.9). We characterize finite symplectic polar spaces of rank r at least two and of odd prime order p as the only finite polar spaces of rank at least two and of prime order admitting nonabelian representations. The representation group of such a polar space is an extraspecial p-group of order p 1 + 2 r and of exponent p (Theorems 1.5 and 1.6).

{$m$}-systems of polar spaces and maximal arcs in projective planes

Shult and Thas have shown in [13] that m-systems of certain finite classical polar spaces give rise to strongly regular graphs and two weight codes. The main result of this paper is to show that maximal arcs in symplectic translation planes may be obtained from certain m-systems of finite symplectic polar spaces. Many new examples of maximal arcs are then constructed. Examples of m-systems are also constructed in Q(-)(2n + 1,q) and W2n+1(q). A method different from that of Shult and Thas is used to construct strongly regular graphs using differences of m-systems.