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arXiv:1305.5689 · Journal of High Energy Physics

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

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

Published Sep 1, 2013• 24 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1007/jhep09(2013)037

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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).
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.
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.
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.
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)W(7,2)W(7,2) of the real four-qubit Pauli group. Each of the four sets of observables of cardinality five represents an elliptic quadric in the three-dimensional projective space of order two (PG(3,2)(3,2)(3,2)) it spans, whereas the remaining set of cardinality four corresponds to an affine plane of order two. The four ambient PG(3,2)(3, 2)(3,2)s of the quadrics intersect pairwise in a line, the resulting six lines meeting in a point. Projecting the whole configuration from this distinguished point (observable) one gets another, complementary ``magic rectangle" of the same qualitative structure.
We present a geometric approach to describe quantum information systems. Precisely, we elaborate a genuine correspondence between toric geometry and qubit systems using the construction of Clifford graph algebras. This mapping may be explored to investigate qubit information applications using toric geometry considered as a potent tool to understand high energy physics including black holes and string theory. Explicitly, we examine in some details the cases of two and three qubits, and we find that they are associated with certain Clifford graph structures. Mixed states are also discussed.