Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Distinguished three-qubit ‘magicity’ via automorphisms of the split Cayley hexagon

Michel R. P. Planat, Метод Санига, Slovak Republic

Published Aug 14, 2016• 21 citations
OpenAlex

Abstract

Abstract. Disregarding the identity, the remaining 63 elements of the generalized three-qubit Pauli group are found to contain 12 096 distinct copies of Mermin’s magic pentagram. Remarkably, 12 096 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 arguments are mainly tied to the structure of certain types of geometric hyperplanes of the hexagon. It is further demonstrated that also an (182, 123)-type of magic configurations, recently proposed by Waegell and Aravind (J. Phys. A: Math. Theor. 45 (2012) 405301), seems to be intricately linked with automorphisms of the hexagon. Finally, the entanglement properties exhibited by edges of both pentagrams and these particular Waegell-Aravind configurations are addressed. PACS numbers: 03.65.Aa, 03.65.Fd, 03.65.Ud, 02.10.Ox 1.

Related papers

'Magic' Configurations of Three-Qubit Observables and Geometric Hyperplanes of the Smallest Split Cayley Hexagon

Метод Санига

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 extended into geometric hyperplanes of the split Cayley hexagon of order two, namely into those of types V 22 (37; 0, 12, 15, 10) and V 4 (49; 0, 0, 21, 28) in the classification of Frohardt and Johnson [Comm.Algebra 22 (1994), 773-797].Moreover, employing an automorphism of order seven of the hexagon, six more replicas of either of the two configurations are obtained.

arXiv:1206.3436•Symmetry Integrability and Geometry Methods and Applications•Nov 6, 2012•4 citations

Distinguished three-qubit ‘magicity’ via automorphisms of the split Cayley hexagon

Michel R. P. Planat, Метод Санига, Frédéric Holweck

arXiv:1212.2729•Quantum Information Processing•Feb 23, 2013•22 citations

Mermin pentagrams arising from Veldkamp lines for three qubits

Péter Lévay, Z. Szabó

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.

arXiv:1608.03400•Journal of Physics A Mathematical and Theoretical•Jan 4, 2017•14 citations

Grassmannian connection between three- and four-qubit observables, Mermin’s contextuality and black holes

Péter Lévay, Michel R. P. Planat, Метод Санига

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.

arXiv:1305.5689•Journal of High Energy Physics•Sep 1, 2013•24 citations

Mermin's pentagram as an ovoid of PG(3, 2)

Метод Санига, Péter Lévay

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).

arXiv:1111.5923•Europhysics Letters (EPL)•Mar 1, 2012•18 citations

Hexagons govern three-qubit contextuality

Метод Санига, Frédéric Holweck, Colm Kelleher +3 more

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, 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.

arXiv:2312.07738•Quantum•Jan 20, 2025•1 citation
Abstract. Disregarding the identity, the remaining 63 elements of the generalized three-qubit Pauli group are found to contain 12 096 distinct copies of Mermin’s magic pentagram. Remarkably, 12 096 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 arguments are mainly tied to the structure of certain types of geometric hyperplanes of the hexagon. It is further demonstrated that also an (182, 123)-type of magic configurations, recently proposed by Waegell and Aravind (J. Phys. A: Math. Theor. 45 (2012) 405301), seems to be intricately linked with automorphisms of the hexagon. Finally, the entanglement properties exhibited by edges of both pentagrams and these particular Waegell-Aravind configurations are addressed. PACS numbers: 03.65.Aa, 03.65.Fd, 03.65.Ud, 02.10.Ox 1.
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 extended into geometric hyperplanes of the split Cayley hexagon of order two, namely into those of types V 22 (37; 0, 12, 15, 10) and V 4 (49; 0, 0, 21, 28) in the classification of Frohardt and Johnson [Comm.Algebra 22 (1994), 773-797].Moreover, employing an automorphism of order seven of the hexagon, six more replicas of either of the two configurations are obtained.
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.
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.
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).
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, 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.