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.


Introduction
A Mermin pentagram, first introduced by Mermin in 1993 [1] to furnish an observable-based proof of quantum contextuality [2], is a set of ten observables of a three-qubit system with eigenvalues ±1 that are arranged into five four-element sets (contexts) that lie along the five edges of a pentagram in such a way that four observables in the same set mutually commute, their product is +1 or −1, and the number of sets where the latter holds is odd. Some years ago, using computer, it was shown [3] that the symplectic polar space associated with the three-qubit Pauli group contains as many as 12 096 such pentagrams, forming three distinct families according as the number of negative contexts is five (108 members), three (4104 members) or one (7884 members). More recently, Lévay and Szabó [4] analyzed this symplectic polar space in terms of its magic Veldkamp line and discovered that Mermin pentagrams form naturally aggregates of six pairs, each such double-six 'cell' being intricately related to six spreads of the core doily (generalized quadrangle of order two). In this short note we shall reveal further interesting traits in the structure of Mermin pentagrams by taking into account that the elements of the three-qubit Pauli group (three-qubit observables) are of three different kinds and employing an important earlier observation [5] that each edge of a Mermin pentagram corresponds to a unique Fano plane of the associated symplectic polar space with a fixed/prescribed three-qubit labeling of its points.

3-Qubit Observables and Positive/Negative Fano Planes
The (generalized) three-qubit Pauli group, P 3 , is generated by three-fold tensor products of the matrices Explicitly, Here, we will only be dealing with its factored version P 3 ≡ P 3 /Z(P 3 ), where the center Z(P 3 ) consists of ±I ⊗ I ⊗ I and ±iI ⊗ I ⊗ I, 1 and whose geometry is that of the symplectic polar space W (5, 2) [6]. This space is, freely speaking, a collection of all totally isotropic subspaces of the ambient five-dimensional binary projective space, PG(5, 2), equipped with a non-degenerate alternating bilinear form.
The 63 non-trivial elements of the group are in a bijective correspondence with the 63 points of W (5, 2) in such a way that two commuting elements correspond to two points joined by a totally isotropic line and a maximum set of mutually commuting elements of the group has its counterpart in a maximal totally isotropic subspace, which is a projective plane of order two, the Fano plane. Let us assume that W (5, 2) has its points labeled by the elements of P 3 . A line/plane of such a space is called positive or negative according as the product of the group elements located in it is +III or −III, respectively. Next, let us call an element of P 3 to be of type A, B or C in dependence on whether, respectively, it features two I's, one I or no I. With this last notion at hand, we will find that there are four different types of three-qubit-labeled Fano planes: one negative type and three positive ones. A negative Fano plane consists of three concurrent negative lines. If a positive Fano plane contains negative lines (type a), there are always four of them, forming the Pasch configuration. If a positive Fano plane is devoid of negative lines, then it has either one element of P 3 of type A and three of type C (type b), or vice versa (type c). The situation is schematically illustrated in Figure 1.
Let us highlight some of the most interesting properties of Mermin pentagrams stemming from our analysis. Obviously, there are only two types where all five Fano planes are negative (types 1 and 4) or positive (types 41 and 42). No pentagram possesses less than two observables of type C, or has just eight of them. If a pentagram features observables of type B, there are four or five of them; and if an edge of such a pentagram contains one such observable, it must contain one more. Next, if a pentagram contains just three observables of type A, they are situated on the same edge; the only exception are types 41 and 42, since they feature no negative Fano plane. There are no pentagrams featuring more than six observables of type A. Any pentagram that contains positive Fano planes of type b and c, but none of type a, is that with a single negative context; the only exception to this rule is type 12. One also sees that a pentagram with no observable of type A is also devoid of positive Fano planes of type c. Further, if a pentagram is associated with Fano planes featuring all the three positive types, then this pentagram also exhibits all the three kinds of observables, with a single exception (type 19). It is also worth mentioning that there are three types of pentagrams having positive Fano planes of type c only (types 13, 27 and 45), as well as two types with all positive Fano types being equally represented (types 15 and 32). From the physical point of view, the most important finding is certainly the existence of two different types of negative contexts of Mermin pentagrams and as many as four distinct kinds of positive ones; the former case involves negative Fano planes and positive Fano planes of type a, whereas the latter one entails all the four types of valued Fano planes. A particularly nice example of these properties is furnished by a pentagram of types 28 or 36, whose single negative context is associated with a positive Fano plane and the remaining four positive contexts are all of different characters.
As a particular task, we also analyzed case by case all 336 pentagrams lying on that hyperbolic quadric (Klein quadric) of W (5, 2) that accommodates all 35 symmetric elements of P 3 , and found out that they only fall into 34 distinct types; interestingly enough, the eleven missing types are types 2, 3, 4, 6, 8, 9, 11, 14, 17, 21 and 31, none of them being associated with a positive Fano plane of type c (see the last column of Table 1).

Concluding Remarks
We have introduced a remarkable finite-geometric classification of three-qubit Mermin pentagrams that intricately combines the character of three-qubit observables with the properties of positive/negative-valued Fano planes of the associated symplectic polar space and reveals important finer structure of three-qubit (observable-based) quantum contexts, distinguishing between two negative and as many as four positive ones. We believe that such classification can be of relevance in any branch of quantum information theory (quantum protocols) where a Mermin pentagram is an essential element; it can also be helpful in revealing finer traits of the so-called black-hole/qubit correspondence (see, e. g., [8]) in those of its aspects that are linked to the structure of the three-qubit symplectic polar space.