Projective Ring Line Encompassing Two-Qubits

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) x 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.Comment: 8 pages, three tables; Version 2 - a few typos and one discrepancy corrected; Version 3: substantial extension of the paper - two-qubits are generalized quadrangles of order two; Version 4: self-dual picture completed; Version 5: intriguing triality found -- three kinds of geometric hyperplanes within GQ and three distinguished subsets of Pauli operator

Geometry of Time and Dimensionality of Space

One of the most distinguished features of our algebraic geometrical, pencil concept of space-time is the fact that spatial dimensions and time stand, as far as their intrinsic structure is concerned, on completely different footings: the former being represented by pencils of lines, the latter by a pencil of conics. As a consequence, we argue that even at the classical (macroscopic) level there exists a much more intricate and profound coupling between space and time than that dictated by (general) relativity theory. It is surmised that this coupling can be furnished by so-called Cremona (or birational) transformations between two projective spaces of three dimensions, being fully embodied in the structure of configurations of their fundamental elements. We review properties of some of the simplest Cremona transformations and show that the corresponding "fundamental" space-times exhibit an intimate connection between the extrinsic geometry of time dimension and the dimensionality of space. Moreover, these Cremonian space-times seem to provide us with a promising conceptual basis for the possible reconciliation between two extreme concepts of (space-)time, viz. physical and psychological. Some speculative remarks in this respect are made

Finite Projective Spaces, Geometric Spreads of Lines and Multi-Qubits

Given a (2N - 1)-dimensional projective space over GF(2), PG(2N - 1, 2), and its geometric spread of lines, there exists a remarkable mapping of this space onto PG(N - 1, 4) where the lines of the spread correspond to the points and subspaces spanned by pairs of lines to the lines of PG(N - 1, 4). Under such mapping, a non-degenerate quadric surface of the former space has for its image a non-singular Hermitian variety in the latter space, this quadric being {\it hyperbolic} or {\it elliptic} in dependence on N being {\it even} or {\it odd}, respectively. We employ this property to show that generalized Pauli groups of N-qubits also form two distinct families according to the parity of N and to put the role of symmetric operators into a new perspective. The N=4 case is taken to illustrate the issue.Comment: 3 pages, no figures/tables; V2 - short introductory paragraph added; V3 - to appear in Int. J. Mod. Phys.

On the Veldkamp Space of GQ(4, 2)

The Veldkamp space, in the sense of Buekenhout and Cohen, of the generalized quadrangle GQ(4, 2) is shown not to be a (partial) linear space by simply giving several examples of Veldkamp lines (V-lines) having two or even three Veldkamp points (V-points) in common. Alongside the ordinary V-lines of size five, one also finds V-lines of cardinality three and two. There, however, exists a subspace of the Veldkamp space isomorphic to PG(3, 4) having 45 perps and 40 plane ovoids as its 85 V-points, with its 357 V-lines being of four distinct types. A V-line of the first type consists of five perps on a common line (altogether 27 of them), the second type features three perps and two ovoids sharing a tricentric triad (240 members), whilst the third and fourth type each comprises a perp and four ovoids in the rosette centered at the (common) center of the perp (90). It is also pointed out that 160 non-plane ovoids (tripods) fall into two distinct orbits -- of sizes 40 and 120 -- with respect to the stabilizer group of a copy of GQ(2, 2); a tripod of the first/second orbit sharing with the GQ(2, 2) a tricentric/unicentric triad, respectively. Finally, three remarkable subconfigurations of V-lines represented by fans of ovoids through a fixed ovoid are examined in some detail.Comment: 6 pages, 7 figures; v2 - slightly polished, subsection on fans of ovoids and three figures adde

The Veldkamp space of multiple qubits

We introduce a point-line incidence geometry in which the commutation relations of the real Pauli group of multiple qubits are fully encoded. Its points are pairs of Pauli operators differing in sign and each line contains three pairwise commuting operators any of which is the product of the other two (up to sign). We study the properties of its Veldkamp space enabling us to identify subsets of operators which are distinguished from the geometric point of view. These are geometric hyperplanes and pairwise intersections thereof. Among the geometric hyperplanes one can find the set of self-dual operators with respect to the Wootters spin-flip operation well-known from studies concerning multiqubit entanglement measures. In the two- and three-qubit cases a class of hyperplanes gives rise to Mermin squares and other generalized quadrangles. In the three-qubit case the hyperplane with points corresponding to the 27 Wootters self-dual operators is just the underlying geometry of the E6(6) symmetric entropy formula describing black holes and strings in five dimensions.Comment: 15 pages, 1 figure; added references, corrected typos; minor change

Bipartite entangled stabilizer mutually unbiased bases as maximum cliques of Cayley graphs

We examine the existence and structure of particular sets of mutually unbiased bases (MUBs) in bipartite qudit systems. In contrast to well-known power-of-prime MUB constructions, we restrict ourselves to using maximally entangled stabilizer states as MUB vectors. Consequently, these bipartite entangled stabilizer MUBs (BES MUBs) provide no local information, but are sufficient and minimal for decomposing a wide variety of interesting operators including (mixtures of) Jamiolkowski states, entanglement witnesses and more. The problem of finding such BES MUBs can be mapped, in a natural way, to that of finding maximum cliques in a family of Cayley graphs. Some relationships with known power-of-prime MUB constructions are discussed, and observables for BES MUBs are given explicitly in terms of Pauli operators.Comment: 8 pages, 1 figur

Hjelmslev Geometry of Mutually Unbiased Bases

The basic combinatorial properties of a complete set of mutually unbiased bases (MUBs) of a q-dimensional Hilbert space H\_q, q = p^r with p being a prime and r a positive integer, are shown to be qualitatively mimicked by the configuration of points lying on a proper conic in a projective Hjelmslev plane defined over a Galois ring of characteristic p^2 and rank r. The q vectors of a basis of H\_q correspond to the q points of a (so-called) neighbour class and the q+1 MUBs answer to the total number of (pairwise disjoint) neighbour classes on the conic.Comment: 4 pages, 1 figure; extended list of references, figure made more illustrative and in colour; v3 - one more figure and section added, paper made easier to follow, references update

An Algorithm for constructing Hjelmslev planes

Projective Hjelmslev planes and Affine Hjelmselv planes are generalisations of projective planes and affine planes. We present an algorithm for constructing a projective Hjelmslev planes and affine Hjelsmelv planes using projective planes, affine planes and orthogonal arrays. We show that all 2-uniform projective Hjelmslev planes, and all 2-uniform affine Hjelsmelv planes can be constructed in this way. As a corollary it is shown that all 2-uniform Affine Hjelmselv planes are sub-geometries of 2-uniform projective Hjelmselv planes.Comment: 15 pages. Algebraic Design Theory and Hadamard matrices, 2014, Springer Proceedings in Mathematics & Statistics 13

New Examples of Kochen-Specker Type Configurations on Three Qubits

A new example of a saturated Kochen-Specker (KS) type configuration of 64 rays in 8-dimensional space (the Hilbert space of a triple of qubits) is constructed. It is proven that this configuration has a tropical dimension 6 and that it contains a critical subconfiguration of 36 rays. A natural multicolored generalisation of the Kochen-Specker theory is given based on a concept of an entropy of a saturated configuration of rays.Comment: 24 page

Black Hole Entropy and Finite Geometry

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$(2,4)$.Comment: 17 pages, 3 figures, v2 a new paragraph added, typos correcte