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 $W(3)$.

It is shown that the E 6(6) symmetric entropy formula describing black holes and black strings in D = 5 is intimately tied to the geometry of the generalized quadrangle GQ(2, 4) with automorphism group the Weyl group W (E 6 ). The 27 charges correspond to the points and the 45 terms in the entropy formula to the lines of GQ(2, 4). Different truncations with 15, 11 and 9 charges are represented by three distinguished subconfigurations of GQ(2, 4), well-known to finite geometers; these are the "doily" (i. e. GQ(2, 2)) with 15, the "perp-set" of a point with 11, and the "grid" (i. e. GQ(2, 1)) with 9 points, respectively. In order to obtain the correct signs for the terms in the entropy formula, we use a non-commutative labelling for the points of GQ(2, 4). For the 40 different possible truncations with 9 charges this labelling yields 120 Mermin squares -objects well-known from studies concerning Bell-Kochen-Specker-like theorems. These results are connected to our previous ones obtained for the E 7(7) symmetric entropy formula in D = 4 by observing that the structure of GQ(2, 4) is linked to a particular kind of geometric hyperplane of the split Cayley hexagon of order two, featuring 27 points located on 9 pairwise disjoint lines (a distance-3-spread). We conjecture that the different possibilities of describing the D = 5 entropy formula using Jordan algebras, qubits and/or qutrits correspond to employing different coordinates for an underlying non-commutative geometric structure based on GQ(2, 4).

The projective line over the (non-commutative) ring of two-by-two matrices with coefficients in GF (2) is found to fully accommodate the algebra of 15 operators -generalized Pauli matrices -characterizing two-qubit systems. The relevant sub-configuration consists of 15 points each of which is either simultaneously distant or simultaneously neighbor to (any) two given distant points of the line. The operators can be identified with the points in such a one-to-one manner that their commutation relations are exactly reproduced by the underlying geometry of the points, with the ring geometrical notions of neighbor/distant answering, respectively, to the operational ones of commuting/non-commuting. This remarkable configuration can be viewed in two principally different ways accounting, respectively, for the basic 9+6 and 10+5 factorizations of the algebra of the observables. First, as a disjoint union of the projective line over GF (2) × GF (2) (the "Mermin" part) and two lines over GF (4) passing through the two selected points, the latter omitted. Second, as the generalized quadrangle of order two, with its ovoids and/or spreads standing for (maximum) sets of five mutually non-commuting operators and/or groups of five maximally commuting subsets of three operators each. These findings open up rather unexpected vistas for an algebraic geometrical modelling of finite-dimensional quantum systems and give their numerous applications a wholly new perspective. Projective lines defined over finite associative rings with unity/identity 1−7 have recently been recognized to be an important novel tool for getting a deeper insight into the underlying algebraic geometrical structure of finite dimensional quantum systems. 8−10 Focusing almost uniquely on the two-qubit case, i.e., the set of 15 operators/generalized four-by-four Pauli spin matrices, of particular importance turned out to be the lines defined over the direct product of the simplest Galois fields, GF (2) × GF (2) × . . . × GF (2). Here, the line defined over GF (2) × GF (2) plays a prominent role in grasping qualitatively the basic structure of so-called Mermin squares, 9,10 i. e., three-by-three arrays in certain remarkable 9 + 6 split-ups of the algebra of operators, whereas the line over GF (2) × GF (2) × GF (2) reflects some of the basic features of a specific 8 + 7 ("cubeand-kernel") factorization of the set. 10 Motivated by these partial findings, we started our quest for such a ring line that would provide us with a complete picture of the algebra of all the 15 operators/matrices. After examining a large number of lines defined over commutative rings, 6,7 we gradually realized that a proper candidate is likely to be found in the non-commutative domain and 1

The set of 63 real generalized Pauli matrices of three-qubits can be factored into two subsets of 35 symmetric and 28 antisymmetric elements. This splitting is shown to be completely embodied in the properties of the Fano plane; the elements of the former set being in a bijective correspondence with the 7 points, 7 lines and 21 flags, whereas those of the latter set having their counterparts in 28 anti-flags of the plane. This representation naturally extends to the one in terms of the split Cayley hexagon of order two. 63 points of the hexagon split into 9 orbits of 7 points (operators) each under the action of an automorphism of order 7. 63 lines of the hexagon carry three points each and represent the triples of operators such that the product of any two gives, up to a sign, the third one. Since this hexagon admits a full embedding in a projective 5-space over GF (2), the 35 symmetric operators are also found to answer to the points of a Klein quadric in such space. The 28 antisymmetric matrices can be associated with the 28 vertices of the Coxeter graph, one of two distinguished subgraphs of the hexagon. The P SL 2 (7) subgroup of the automorphism group of the hexagon is discussed in detail and the Coxeter sub-geometry is found to be intricately related to the E 7 -symmetric black-hole entropy formula in string theory. It is also conjectured that the full geometry/symmetry of the hexagon should manifest itself in the corresponding black-hole solutions. Finally, an intriguing analogy with the case of Hopf sphere fibrations and a link with coding theory are briefly mentioned.

It is conjectured that the question of the existence of a set of d + 1 mutually unbiased bases in a ddimensional Hilbert space if d differs from a power of prime is intimatelly linked with the problem whether there exist projective planes whose order d is not a power of prime.Recently, there has been a considerable resurgence of interest in the concept of the so-called mutually unbiased bases [see, e.g., 1-7], especially in the context of quantum state determination, cryptography, quantum information theory and the King's problem. We recall that two different orthonormal bases A and B of a d-dimensional Hilbert space H d are called mutually unbiased if and only if | a|b | = 1/ √ d for all a∈A and all b∈B. An aggregate of mutually unbiased bases is a set of orthonormal bases which are pairwise mutually unbiased. It has been found that the maximum number of such bases cannot be greater than d + 1 [8,9]. It is also known that this limit is reached if d is a power of prime. Yet, a still unanswered question is if there are non-prime-power values of d for which this bound is attained. The purpose of this short note is to draw the reader's attention to the fact that the answer to this question may well be related with the (non-)existence of finite projective planes of certain orders.A finite projective plane is an incidence structure consisting of points and lines such that any two points lie on just one line, any two lines pass through just one point, and there exist four points, no three of them on a line [10]. From these properties it readily follows that for any finite projective plane there exists an integer d with the properties that any line contains exactly d + 1 points, any point is the meet of exactly d + 1 lines, and the number of points is the same as the number of lines, namely d 2 + d + 1. This integer d is called the order of the projective plane. The most striking issue here is that the order of known finite projective planes is a power of prime [10]. The question of which other integers occur as orders of finite projective planes remains one of the most challenging problems of contemporary mathematics. The only "no-go" theorem known so far in this respect is the Bruck-Ryser theorem [11] saying that there is no projective plane of order d if d − 1 or d − 2 is divisible by 4 and d is not the sum of two squares. Out of the first few non-prime-power numbers, this theorem rules out finite projective planes of order 6, 14, 21, 22, 30 and 33. Moreover, using massive computer calculations, it was proved by Lam [12] that there is no projective plane of order ten. It is surmised that the order of any projective plane is a power of a prime.¿From what has already been said it is quite tempting to hypothesize that the above described two problems are nothing but different aspects of one and the same problem. That is, we conjecture that non-existence of a projective plane of the given order d implies that there are less than d + 1 mutually unbiased bases (MUBs) in the corresponding H d , and vice versa. Or, slightly...

We point out an explicit connection between graphs drawn on compact Riemann surfaces defined over the fieldQ of algebraic numbers -so-called Grothendieck's dessins d'enfants -and a wealth of distinguished point-line configurations. These include simplices, cross-polytopes, several notable projective configurations, a number of multipartite graphs and some 'exotic' geometries. Among them, remarkably, we find not only those underlying Mermin's magic square and magic pentagram, but also those related to the geometry of two-and three-qubit Pauli groups. Of particular interest is the occurrence of all the three types of slim generalized quadrangles, namely GQ(2,1), GQ(2,2) and GQ(2,4), and a couple of closely related graphs, namely the Schläfli and Clebsch ones. These findings seem to indicate that dessins d'enfants may provide us with a new powerful tool for gaining deeper insight into the nature of finite-dimensional Hilbert spaces and their associated groups, with a special emphasis on contextuality.

A compact classification of the projective lines defined over (commutative) rings (with unity) of all orders up to thirty-one is given. There are altogether sixty-five different types of them. For each type we introduce the total number of points on the line, the number of points represented by coordinates with at least one entry being a unit, the cardinality of the neighbourhood of a generic point of the line as well as those of the intersections between the neighbourhoods of two and three mutually distant points, the number of 'Jacobson' points per a neighbourhood, the maximum number of pairwise distant points and, finally, a list of representative/base rings. The classification is presented in form of a table in order to see readily not only the fine traits of the hierarchy, but also the changes in the structure of the lines as one goes from one type to the other. We hope this study will serve as an impetus to a search for possible applications of these remarkable geometries in physics, chemistry, biology and other natural sciences as well.

The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are reviewed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite, and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned.