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arXiv:2004.07517 · Mathematics

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

Метод Санига

Published Jun 29, 2021• 2 citations
arXivHTMLPDF

Abstract

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 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,…,17) of negative contexts.

DOI
10.3390/math9131524

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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 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,…,17) of negative contexts.
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 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.
Abstract 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 this space of 12 096 possible pentagrams is organized into 1008 families, with each family containing a ‘double-six’ of pentagrams. The 1008 families are connected to a special set of Veldkamp lines in the Veldkamp space for three-qubits an object well-known to finite geometers but has only been introduced to physics recently. Due to the transitive action of the symplectic group on this set of Veldkamp lines it is enough to study only one ‘canonical’ double-six configuration of pentagrams. We prove that the geometry of this double-six configuration is encapsulated in the weight diagram of the 20 dimensional irreducible representation of the group S U ( 6 ) . As an interesting by-product of our approach we show that Mermin pentagrams and a class of Mermin squares labelled by three-qubit Pauli operators are inherently related. We conjecture that by studying the representation theoretic content of other Veldkamp lines of the Veldkamp space for N -qubits makes it possible to find new contextual configurations in a systematic manner.
Observable-based Kochen-Specker proofs are configurations of multi-qubit Pauli observables grouped into contexts whose products are plus or minus the indentity. Their robustness as state-independent contextuality tests can be measured by the tolerated error per context ε=2d/∣H∣\varepsilon = 2d/|H|ε=2d/∣H∣, where ddd is the contextuality degree and ∣H∣|H|∣H∣ the number of contexts. Since the degree depends only on an underlying abstract structure called the hypergram i.e. the pair formed by the context hypergraph and the anticommutation graph, the search for highly contextual proofs can be carried out on them instead, with no reference to qubits or to any particular Pauli labeling. We further exploit this by enumerating anticommutation graphs first, and then by associating to each graph GGG the single hypergram carrying its entire hypergraph support HS(G)HS(G)HS(G), so that exactly one candidate is examined per graph. Applied to the House of Graphs database and to censuses of vertex-transitive graphs on at most 24 vertices, this pipeline recovers the Peres-Mermin square, the doily and the Mermin pentagram, and yields configurations reaching ε=0.707\varepsilon = 0.707ε=0.707, against 0.4240.4240.424 for the previous published record. The best configurations are predominantly those stemming from line graphs and unions of graphs; we explain the former by showing that every perfect matching of a graph is a context of its line graph, which exhibits the Peres-Mermin square and the doily as the first members of two infinite families. We close with finite geometric descriptions of the most striking configurations inside symplectic polar spaces, in terms of ovoids, hyperbolic quadrics and Fano planes.
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 demonstration employs properties of the real three-qubit Pauli group embodied in the geometry of the symplectic polar space W (5, 2) and rests on the facts that: 1) the four observables/operators on any of the five edges of the pentagram can be viewed as points of an affine plane of order two, 2) all the ten observables lie on a hyperbolic quadric of the five-dimensional projective space of order two, PG(5, 2), and 3) that the points of this quadric are in a well-known bijective correspondence with the lines of PG(3, 2).
Abstract 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 sets in explicit forms. This formalism is then employed to shed novel light on recently-discovered structural and cardinality properties of an aggregate of three-qubit Mermin’s “magic” pentagrams. Moreover, some intriguing connections with the so-called black-hole-qubit correspondence are also pointed out.
Abstract 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 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.