Magic three-qubit Veldkamp line: A finite geometric underpinning for form theories of gravity and black hole entropy

We investigate the structure of the three-qubit magic Veldkamp line (MVL). This mathematical notion has recently shown up as a tool for understanding the structures of the set of Mermin pentagrams, objects that are used to rule out certain classes of hidden variable theories. Here we show that this object also provides a unifying finite geometric underpinning for understanding the structure of functionals used in form theories of gravity and black hole entropy. We clarify the representation theoretic, finite geometric and physical meaning of the different parts of our MVL. The upshot of our considerations is that the basic finite geometric objects enabling such a diversity of physical applications of the MVL are the unique generalized quadrangles with lines of size three, their one-point extensions as well as their other extensions isomorphic to affine polar spaces of rank 3 and order 2. In a previous work we have already connected generalized quadrangles to the structure of cubic Jordan algebras related to entropy fomulas of black holes and strings in five dimensions. In some respect the present paper can be regarded as a generalization of that analysis for also providing a finite geometric understanding of four-dimensional black hole entropy formulas. However, we find many more structures whose physical meaning is yet to be explored. As a familiar special case our work provides a finite geometric representation of the algebraic extension from cubic Jordan algebras to Freudenthal systems based on such algebras.

Figure 1

The doily with the duad labeling.

Figure 2

The doily labeled by nontrivial two-qubit Pauli observables. Inside the doily a geometric hyperplane (see definition 2), or grid is shown.

Figure 3

The full set of Mermin squares living inside the doily. The ten copies of relevant grids can be identified after successive rotations by 72 degrees of the embedded patterns of grids seen in the second and fourth columns. The first Mermin square, up to an automorphism, is the grid of Fig. 2. It is embedded in the doily after a counter clockwise rotation by 72 degrees of the second pattern seen in the lower right corner.

Figure 4

The structure of the magic Veldkamp line. The colored parallelograms are geometric hyperplanes with characteristic cardinalities (number of points) as follows: red 31 (perp-set), blue 27 (elliptic quadric), and green 35 (hyperbolic quadric). Their common intersection is the core set of 15 points, which forms a doily. The three hyperplanes satisfy the properties of Eqs. (22) and (23). The red dot on the top of the triangle corresponds to the special point defining the perp-set. For the canonical magic Veldkamp line this point is labeled by the observable $YYY$.

Figure 5

Decomposition of the magic Veldkamp line into triangles of characteristic substructure via Clifford labeling. In this decomposition the basis vector ${\mathrm{\Gamma }}_7$ corresponding to the red dot enjoys a special status. It belongs to the perp-set part of the Veldkamp line.

Figure 6

The $A_5$ Dynkin diagram labeled by duads.

Figure 7

The weight diagram for the 15 of $A_5$. The weights labeled by three-element subsets as $\{ab7\}$ correspond to the core set of Fig. 5 of our Veldkamp line labeled by ${\mathrm{\Gamma }}_a{\mathrm{\Gamma }}_b{\mathrm{\Gamma }}_7$.

Figure 8

The twin Steiner-Plücker configurations illustrating the 30 blocks of $EGQ(2,1)$ related to the 20 of $A_5$. Our $EGQ(2,1)$, which is the affine polar space of order 2 and type $D_2^{+}$ [41], can be viewed as the union of twin Steiner-Plücker configurations. The two configurations are identical as point sets, their points being represented by unordered triples of elements from the set $S=\{1,2,3,4,5,6\}$. Lines (blocks) of the configurations are represented by four points that pairwise share one element. The name Steiner-Plücker configuration comes from the fact [47] that it is a $(20_3,15_4)$ configuration that consists of 20 Steiner points and 15 Plücker lines of the famous hexagrammum mysticum of Pascal (see e.g., [48, 49]).

Figure 9

The $D_6$ Dynkin diagram with our labeling convention shown.

Figure 10

The weight diagram for the 32 of $D_6$ labeled by one-, three-, and five-element subsets of $S$.

Figure 11

The doily corresponding to the residue ${\mathcal{T}}_{123}$.

Figure 12

The $E_6$ Dynkin diagram labeled by subsets.

Figure 13

The weight diagram of the 27 of $E_6$ labeled by the subsets $\{a\},\{a7\}$, and $\{ab7\}$.

Figure 14

Different extensions $EGQ(2,2)$ of the doily $GQ(2,2)$ living inside our magic Veldkamp line.

Figure 15

The characteristic numbers (top) and finite-geometric structures (bottom) for the decomposition of the four-qubit MVL (compare with Figs. 5 and 14). Both the extended generalized quadrangle $EGQ(2,4)$ [of type $D_2^{-}$; see example 9.6(i) in 40] and the extended Klein quadric $E{Q}^{+}(5,2)$ [alias the extended dual 2 design with residues isomorphic to the duals of $PG(3,2)$; see example 7.3(b) in 91] have the property that for every point there exists a unique antipodal point; hence, they both have unique quotients isomorphic, respectively, to a one-point extension of $GQ(2,4)$ (see, e. g., example 9.7 in [40]) and a one-point extension of the Klein quadric $Q^{+}(5,2)$ (see, e. g., [87]). For the three extensions of symplectic polar space $\mathcal{W}(5,2)$ we use our own symbols, reflecting whether the point set of the extension lies in the complement of the hyperbolic quadric [$E{W}^{(+)}(5,2)$] of the elliptic quadric [$E{W}^{(-)}(5,2)$] or of the quadratic cone [$EW(5,2)$] of $\mathcal{W}(7,2)$. The symbol $\{\mathcal{W}(5,2)\}$ stands for the projection of the core $W(5,2)$ from the vertex of the cone.