Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small
Rank

Metod Saniga1, Henri de Boutray2, Frédéric Holweck3 and

Alain Giorgetti2

1Astronomical Institute of the Slovak Academy of Sciences,

SK-05960 Tatranská Lomnica, Slovak Republic

<EMAIL_ADDRESS>
2Institut FEMTO-ST, DISC $-$ UFR-ST, Université Bourgogne Franche-Comté,
F-25030 Besançon, France

<EMAIL_ADDRESS><EMAIL_ADDRESS>
and

3Laboratoire Interdisciplinaire Carnot de Bourgogne, ICB/UTBM, UMR 6303 CNRS,
Université Bourgogne Franche-Comté, F-90010 Belfort, France

<EMAIL_ADDRESS>
Abstract

We study certain physically-relevant subgeometries of binary symplectic polar
spaces $W(2N-1,2)$ of small rank $N$, when the points of these spaces
canonically encode $N$-qubit observables. Key characteristics of a subspace of
such a space $W(2N-1,2)$ are: the number of its negative lines, the
distribution of types of observables, the character of the geometric
hyperplane the subspace shares with the distinguished (non-singular) quadric
of $W(2N-1,2)$ and the structure of its Veldkamp space. In particular, we
classify and count polar subspaces of $W(2N-1,2)$ whose rank is $N-1$.
$W(3,2)$ features three negative lines of the same type and its $W(1,2)$’s are
of five different types. $W(5,2)$ is endowed with 90 negative lines of two
types and its $W(3,2)$’s split into 13 types. 279 out of 480 $W(3,2)$’s with
three negative lines are composite, i. e. they all originate from the two-
qubit $W(3,2)$. Given a three-qubit $W(3,2)$ and any of its geometric
hyperplanes, there are three other $W(3,2)$’s possessing the same hyperplane.
The same holds if a geometric hyperplane is replaced by a ‘planar’ tricentric
triad. A hyperbolic quadric of $W(5,2)$ is found to host particular sets of
seven $W(3,2)$’s, each of them being uniquely tied to a Conwell heptad with
respect to the quadric. There is also a particular type of $W(3,2)$’s, a
representative of which features a point each line through which is negative.
Finally, $W(7,2)$ is found to possess 1908 negative lines of five types and
its $W(5,2)$’s fall into as many as 29 types. 1524 out of 1560 $W(5,2)$’s with
90 negative lines originate from the three-qubit $W(5,2)$. Remarkably, the
difference in the number of negative lines for any two distinct types of four-
qubit $W(5,2)$’s is a multiple of four.

Keywords: $N$-Qubit Observables – Binary Symplectic Polar Spaces –
Distinguished Sets of Doilies – Geometric Hyperplanes – Veldkamp lines

##  1 Introduction

Some fifteen years ago it was discovered (see, e.g., [1, 2, 3, 4]) that there
exists a deep connection between the structure of the $N$-qubit Pauli group
and that of the binary symplectic polar space of rank $N$, $W(2N-1,2)$, where
commutation relations between elements of the group are encoded in
collinearity relations between points of $W(2N-1,2)$. This connection has
subsequently been used to get a deeper insight into, for example, finite
geometric nature of observable-based proofs of quantum contextuality (for a
recent review, see [5]), properties of certain black-hole entropy formulas [6]
and the so-called black-hole/qubit correspondence [7], leading to finite-
geometric underpinning four distinct Hitchin’s invariants and the Cartan
invariant of form theories of gravity [8] and even to an intriguing finite-
geometric toy model of space-time [9]. This group-geometric connection was
further strengthened by making use of the concept of geometric hyperplane and
that of the Veldkamp space of $W(2N-1,2)$ [10]. As per quantum contextuality,
famous two-qubit Mermin-Peres magic squares were found to be isomorphic to a
special class of geometric hyperplanes of $W(3,2)$ called grids [11], whereas
three-qubit Mermin pentagrams were found to have their natural settings in the
magic Veldkamp line of $W(5,2)$ [12], being also isomorphic – under the
grassmannian correspondence of type Gr$(2,4)$ – to ovoids of $W(3,2)$ [13].
Concerning the black-hole/qubit correspondence, here a key role is played by
the geometric hyperplane isomorphic to an elliptic quadric of $W(5,2)$.
Interestingly, form theories of gravity seem to indicate that a certain part
of the magic Veldkamp line in the four-qubit symplectic polar space, $W(7,2)$,
and the associated extended geometric hyperplanes are of physical relevance as
well.

From the preceding paragraph it is obvious that revealing finer traits of the
structure of binary symplectic polar spaces of small rank can be vital for
further physical applications of these spaces. Having this in view, we will
focus on sets of $W(2N-3,2)$’s located in $W(2N-1,2)$, for $N=2,3,4$,
providing their comprehensive observable-based taxonomy. Key parameters of our
classification of such subspaces of $W(2N-1,2)$ will be: the number of
negative lines they contain (which is also an important parameter when it
comes to quantum contextuality), the distribution of different types of
observables they feature, the character of the geometric hyperplane a subspace
of a given type shares with the distinguished (non-singular) quadric of
$W(2N-1,2)$ and, in the case of refined ‘decomposition’ of three-qubit
$W(3,2)$’s, also the very structure of their Veldkamp lines.

The paper is organized as follows. Sec. 2 provides the reader with the
necessary finite-geometric background and notation. Sec. 3 deals with $W(3,2)$
and the hierarchy of its triads. Sec. 4 addresses the three-qubit $W(5,2)$ and
its $W(3,2)$’s; here we classify $W(3,2)$’s in two distinct ways and
illustrate the fact that there are four $W(3,2)$’s sharing a geometric
hyperplane, or a specific tricentric triad. Sec. 5 focuses on prominent
septuplets of $W(3,2)$’s that are closely related to Conwell heptads with
respect to a hyperbolic quadric of $W(5,2)$. Sec. 6 classifies $W(5,2)$’s
living in the four-qubit $W(7,2)$ and furnishes a couple of examples of their
composite types. Finally, Sec. 7 is devoted to concluding remarks.

##  2 Finite Geometry Background

Given a $d$-dimensional projective space PG$(d,2)$ over GF$(2)$, a polar space
${\cal P}$ in this projective space consists of the projective subspaces that
are totally isotropic/singular with respect to a given non-singular bilinear
form; PG$(d,2)$ is called the ambient projective space of ${\cal P}$. A
projective subspace of maximal dimension in ${\cal P}$ is called a generator;
all generators have the same (projective) dimension $r-1$. One calls $r$ the
rank of the polar space.

Polar spaces of relevance for us are of three types (see, for example, [14,
15]): symplectic, hyperbolic and elliptic. The symplectic polar space
$W(2N-1,2)$, $N\geq 1$, consists of all the points of PG$(2N-1,2)$,
$\\{(x_{1},x_{2},\ldots,x_{2N}):x_{j}\in\\{0,1\\},j\in\\{1,2,\ldots,2N\\}\\}\backslash\\{(0,0,\ldots,0)\\}$,
together with the totally isotropic subspaces with respect to the standard
symplectic form

$\sigma(x,y)=x_{1}y_{N+1}-x_{N+1}y_{1}+x_{2}y_{N+2}-x_{N+2}y_{2}+\dots+x_{N}y_{2N}-x_{2N}y_{N}.$
(1)

This space features

$|W|_{p}=4^{N}-1$ (2)

points and

$|W|_{g}=(2+1)(2^{2}+1)\cdots(2^{N}+1)$ (3)

generators. The hyperbolic orthogonal polar space $\mathcal{Q}^{+}(2N-1,2)$,
$N\geq 1$, is formed by all the subspaces of PG$(2N-1,2)$ that lie on a given
non-singular hyperbolic quadric, with the standard equation

$x_{1}x_{N+1}+x_{2}x_{N+2}\ldots+x_{N}x_{2N}=0.$ (4)

Each $\mathcal{Q}^{+}(2N-1,2)$ contains

$|Q^{+}|_{p}=(2^{N-1}+1)(2^{N}-1)$ (5)

points and there are

$|W|_{Q^{+}}=|Q^{+}|_{p}+1=(2^{N-1}+1)(2^{N}-1)+1$ (6)

copies of them in $W(2N-1,2)$. Finally, the elliptic orthogonal polar space
$\mathcal{Q}^{-}(2N-1,2)$, $N\geq 2$, comprises all points and subspaces of
PG$(2N-1,2)$ satisfying the standard equation

$f(x_{1},x_{N+1})+x_{2}x_{N+2}+\cdots+x_{N}x_{2N}=0,$ (7)

where $f$ is an irreducible polynomial over GF$(2)$. Each
$\mathcal{Q}^{-}(2N-1,2)$ contains

$|Q^{-}|_{p}=(2^{N-1}-1)(2^{N}+1)$ (8)

points and there are

$|W|_{Q^{-}}=|Q^{-}|_{p}+1=(2^{N-1}-1)(2^{N}+1)+1$ (9)

copies of them in $W(2N-1,2)$. For both symplectic and hyperbolic polar spaces
$r=N$, whereas for the elliptic one $r=N-1$. The smallest non-trivial
symplectic polar space is the $N=2$ one, $W(3,2)$, often referred to as the
doily. It features 15 points (see eq. (2)) and the same number of lines (that
are also its generators, see eq. (3)), with three points per line and three
lines through a point; it is a self-dual $15_{3}$-configuration and the only
one out of 245 342 such configurations that is triangle-free, being, in fact,
isomorphic to the generalized quadrangle of order two (GQ$(2,2)$). This
symplectic polar space features ten $\mathcal{Q}^{+}(3,2)$’s (by eq. (6)) and
six $\mathcal{Q}^{-}(3,2)$’s (by eq. (9)). A $\mathcal{Q}^{+}(3,2)$ contains
nine points and six lines forming a $3\times 3$ grid, so it is also called a
grid. A $\mathcal{Q}^{-}(3,2)$ features five pairwise non-collinear points,
hence it is an ovoid. A triple of mutually non-collinear points of $W(3,2)$ is
called a triad and a point collinear with all the three points of a triad is
called a center of the triad; $W(3,2)$ contains 60 unicentric and 20
tricentric triads.

The $N$-qubit observables we will be dealing with belong to the set

${\cal S}_{N}=\\{G_{1}\otimes G_{2}\otimes\cdots\otimes
G_{N}:~{}G_{j}\in\\{I,X,Y,Z\\},~{}j\in\\{1,2,\ldots,N\\}\\}\backslash\\{{\cal
I}_{N}\\}$ (10)

where ${\cal I}_{N}\equiv I_{(1)}\otimes I_{(2)}\otimes\ldots\otimes I_{(N)}$,

$X=\left(\begin{array}[]{rr}0&1\\\ 1&0\\\
\end{array}\right),~{}~{}Y=\left(\begin{array}[]{rr}0&-i\\\ i&0\\\
\end{array}\right),~{}~{}{\rm and}~{}~{}Z=\left(\begin{array}[]{rr}1&0\\\
0&-1\\\ \end{array}\right)$ (11)

are the Pauli matrices, $I$ is the identity matrix and ‘$\otimes$’ stands for
the tensor product of matrices. ${\cal S}_{N}$, whose elements are simply
those of the $N$-qubit Pauli group if the global phase is disregarded,
features two kinds of observables, namely symmetric (i.e., observables
featuring an even number of $Y$’s) and skew-symmetric; the number of symmetric
observables is $(2^{N-1}+1)(2^{N}-1)$. We shall further employ a finer
classification where an observable having $N-1$, $N-2$, $N-3$, …$I$’s will be,
respectively, of type $A$, $B$, $C$, …; also, whenever it is clear from the
context, $G_{1}\otimes G_{2}\otimes\cdots\otimes G_{N}$ will be short-handed
to $G_{1}G_{2}\cdots G_{N}$.

For a particular value of $N$, the $4^{N}-1$ elements of ${\cal S}_{N}$ can be
bijectively identified with the same number of points of $W(2N-1,2)$ in such a
way that the images of two commuting elements lie on the same line of this
polar space, and generators of $W(2N-1,2)$ correspond to maximal sets of
mutually commuting elements. If we take the symplectic form defined by eq.
(1), then this bijection acquires the form

$G_{j}\leftrightarrow(x_{j},x_{j+N}),~{}j\in\\{1,2,\dots,N\\},$ (12)

assuming that

$I\leftrightarrow(0,0),~{}X\leftrightarrow(0,1),~{}Y\leftrightarrow(1,1),~{}{\rm
and}~{}Z\leftrightarrow(1,0).$ (13)

Employing the above-introduced bijection (for more details see, e. g., [12]),
it can be shown that given an observable $O$, the set of symmetric observables
commuting with $O$ together with the set of skew-symmetric observables not
commuting with $O$ will lie on a certain non-degenerate quadric of
$W(2N-1,2)$, this quadric being hyperbolic (resp. elliptic) if $O$ is
symmetric (resp. skew-symmetric). We can express this important property by
making, whenever appropriate, this associated observable explicit in a
subscript, $\mathcal{Q}^{\pm}_{(O)}(2N-1,2)$, noting that there exists a
particular hyperbolic quadric associated with ${\cal I}$:

$\begin{split}\mathcal{Q}^{+}_{({\cal
I})}(2N-1,2):=\\{(x_{1},x_{2},\dots,x_{2N})\in
W(2N-1,2)~{}|~{}x_{1}x_{N+1}+x_{2}x_{N+2}+\\\
\ldots+x_{N}x_{2N}=0\\}.\end{split}$ (14)

Given a point-line incidence geometry $\Gamma(P,L)$, a geometric hyperplane of
$\Gamma(P,L)$ is a subset of its point set such that a line of the geometry is
either fully contained in the subset or has with it just a single point in
common. The Veldkamp space $\mathcal{V}(\Gamma)$ of $\Gamma(P,L)$ is the space
in which [16]: (i) a point is a geometric hyperplane of $\Gamma$ and (ii) a
line is the collection, denoted $H^{\prime}H^{\prime\prime}$, of all geometric
hyperplanes $H$ of $\Gamma$ such that $H^{\prime}\cap
H^{\prime\prime}=H^{\prime}\cap H=H^{\prime\prime}\cap H$ or
$H=H^{\prime},H^{\prime\prime}$, where $H^{\prime}$ and $H^{\prime\prime}$ are
distinct points of $\mathcal{V}(\Gamma)$. For a $\Gamma(P,L)$ with three
points on a line, all Veldkamp lines are of the form
$\\{H^{\prime},H^{\prime\prime},\overline{H^{\prime}\Delta
H^{\prime\prime}}\\}$ where $\overline{H^{\prime}\Delta H^{\prime\prime}}$ is
the complement of symmetric difference of $H^{\prime}$ and $H^{\prime\prime}$,
i. e. they form a vector space over GF$(2)$. As demonstrated in [10],
$\mathcal{V}(W(2N-1,2))\cong$ PG$(2N,2)$. Its points are both hyperbolic and
elliptic quadrics of $W(2N-1,2)$, as well as its perp-sets. Given a point $x$
of $W(2N-1,2)$, the perp-set $\widehat{\mathcal{Q}}_{(x)}(2N-1,2)$ of $x$
consists of all the points collinear with it,

$\widehat{\mathcal{Q}}_{(x)}(2N-1,2):=\\{y\in
W(2N-1,2)~{}|~{}\sigma(x,y)=0\\};$ (15)

the point $x$ being referred to as the nucleus of
$\widehat{\mathcal{Q}}_{(x)}(2N-1,2)$.

Figure 1: The three kinds of geometric hyperplanes of $W(3,2)$. The 15 points of the doily are represented by small circles and its 15 lines are illustrated by the straight segments as well as by the segments of circles; note that not every intersection of two segments counts for a point of the doily. The upper panel shows grids (red bullets), the middle panel perp-sets (yellow bullets) and the bottom panel ovoids (blue bullets). Each picture $-$ except that located in the bottom right-hand corner $-$ stands for five different hyperplanes, the four other being obtained from it by its successive rotations through 72 degrees around the center of the pentagon. Table 1: An overview of the properties of the five different types of lines of $\mathcal{V}(W(3,2))$ in terms of the core (i. e., the set of points common to all the three hyperplanes forming the line) and the types of geometric hyperplanes featured by a generic line of a given type. The last column gives the total number of lines per each type. Type | Core | Perps | Ovoids | Grids | $\\#$
---|---|---|---|---|---
I | Two Secant Lines | 1 | 0 | 2 | 45
II | Single Line | 3 | 0 | 0 | 15
III | Tricentric Triad | 3 | 0 | 0 | 20
IV | Unicentric Triad | 1 | 1 | 1 | 60
V | Single Point | 1 | 2 | 0 | 15

We shall briefly recall basic properties of the Veldkamp space of the doily,
$\mathcal{V}(W(3,2))\simeq$ PG(4, 2), whose in-depth description can be found
in [11]. The 31 points of $\mathcal{V}(W(3,2))$ comprise 15 perp-sets, ten
grids and six ovoids $-$ as also illustrated in Figure 1. The 155 lines of
$\mathcal{V}(W(3,2))$ split into five distinct types as summarized in Table 1
and depicted in Figure 2. (Table 1, as well as Figure 1 and Figure 2, were
taken from [17].)

In what follows we will mainly be focused on $W(2N-3,2)$’s that are located in
$W(2N-1,2)$. These are, in general, of two different kinds [10]. A $W(2N-3,2)$
of the first kind, to be called linear, is isomorphic to the intersection of
two perp-sets with non-collinear nuclei and their number in $W(2N-1,2)$ is

$|W|_{W_{l}}=\frac{1}{3}4^{N-1}(4^{N}-1).$ (16)

A $W(2N-3,2)$ of the second kind, to be called quadratic, is isomorphic to the
intersection of a hyperbolic quadric and an elliptic quadric and $W(2N-1,2)$
features

$|W|_{W_{q}}=4^{N-1}(4^{N}-1)$ (17)

of them. By way of example, in $W(3,2)$ a linear (resp. quadratic) $W(1,2)$
corresponds to a tricentric (resp. unicentric) triad.

Figure 2: An illustrative portrayal of representatives (rows) of the five
(numbered consecutively from top to bottom) different types of lines of
$\mathcal{V}(W(3,2))$, each being uniquely determined by the properties of its
core (black bullets).

In the sequel, when referring to $W(2N-1,2)$ and its subspaces we will always
have in mind the $W(2N-1,2)$ and its subspaces whose points are labelled by
$N$-qubit observables from the set ${\cal S}_{N}$ as expressed by eqs. (12)
and (13). Moreover, a linear subspace of such $W(2N-1,2)$ will be called
positive or negative according as the (ordinary) product of the observables
located in it is $+{\cal I}_{N}$ or $-{\cal I}_{N}$, respectively. Let us
illustrate this point taking again the $N=2$ case. Up to isomorphism, there is
just one type of the two-qubit doily. Its six observables of type $A$ are
$IX$, $XI$, $IY$, $YI$, $IZ$, and $ZI$ and its nine ones of type $B$ are $XX$,
$XY$, $XZ$, $YX$, $YY$, $YZ$, $ZX$, $ZY$ and $ZZ$, the latter lying on a
particular hyperbolic quadric, $\mathcal{Q}_{(YY)}^{+}(3,2)$. Among the 15
lines only the three lines $\\{XX,YY,ZZ\\}$, $\\{XY,YZ,ZX\\}$ and
$\\{XZ,YX,ZY\\}$ are negative, forming also one system of generators of
$\mathcal{Q}_{(YY)}^{+}(3,2)$.

##  3 $\boldsymbol{W(3,2)}$ and its Two-Qubit $\boldsymbol{W(1,2)}$’s

This is a rather trivial case. As already mentioned in Sec. 2, the doily
contains three negative lines, of the same ($B-B-B$) type. Among its
$W(1,2)$’s, we find two types of linear ones and three types of quadratic
ones, whose properties are summarized in Table 2.

Table 2: Classification of $W(1,2)$’s living in $W(3,2)$. Column one ($T$)
shows the type, columns two and three ($O_{A}$ and $O_{B}$) indicate the
number of observables of corresponding types, and columns four ($W_{l}$) and
five ($W_{q}$) yield, respectively, the number of ‘linear’ and ‘quadratic’
$W(1,2)$’s of a given type.

$T$ | $O_{A}$ | $O_{B}$ | $W_{l}$ | $W_{q}$
---|---|---|---|---
1 | 0 | 3 | $-$ | 6
2 | 1 | 2 | $-$ | 36
3 | 1 | 2 | 18 | $-$
4 | 2 | 1 | $-$ | 18
5 | 3 | 0 | 2 | $-$

It is worth noticing that the six quadratic $W(1,2)$’s (i.e., unicentric
triads) of Type 1 lie on the distinguished quadric
$\mathcal{Q}_{(YY)}^{+}(3,2)$, being in fact its six ovoids.

##  4 $\boldsymbol{W(5,2)}$ and its Three-Qubit Doilies

The space $W(5,2)$ contains 63 points, 315 lines and 135 generators, the
latter being all Fano planes. Among the 63 canonical three-qubit observables
associated to the points, nine are of type $A$, 27 are type $B$ and 27 are of
type $C$. Through an observable of type $C$ there pass six negative lines, all
being of type $C-C-B$; the total number of negative lines of this type thus is
$\frac{27\times 6}{2}=81.$ Through an observable of type $B$ there pass four
negative lines. Of them, three are of the above-mentioned type and the fourth
one is of type $B-B-B$; the total number of negative lines of the latter type
is $\frac{27\times 1}{3}=9.$ As no negative line features an observable of
type $A$, one finds that the $W(5,2)$ accommodates as many as $(81+9=)~{}90$
negative lines.

When we pass to $W(3,2)$’s, we find a (much) richer structure, because
alongside the types of observables we can employ one more parameter, namely
the number of negative lines a given $W(3,2)$ contains. In fact, we find that
the 336 linear doilies (see eq. 16) fall into six different types and the 1008
quadratic ones (see eq. 17) into seven types; we note in passing that Type 9
splits further into two subtypes depending on whether the two observables of
type $A$ do (Type $9A$, 162 members) or do not (Type $9B$, 54 members)
commute. This classification is summarized in Table 3 and also pictorially
illustrated in Figure 3. It is worth noticing here that there are two
different types of doilies (Type 3 and Type 6) exhibiting an even number of
negative lines.

Table 3: Classification of doilies living in $W(5,2)$. Column one ($T$) shows
the type, column two ($C^{-}$) the number of negative lines in a doily of the
given type, columns three to five ($O_{A}$ to $O_{C}$) indicate the number of
observables of corresponding types, and columns six ($D_{l}$) and seven
($D_{q}$) yield, respectively, the number of ‘linear’ and ‘quadratic’ doilies
of a given type.

$T$ | $C^{-}$ | $O_{A}$ | $O_{B}$ | $O_{C}$ | $D_{l}$ | $D_{q}$
---|---|---|---|---|---|---
1 | 7 | 0 | 7 | 8 | $-$ | 81
2 | 7 | 0 | 9 | 6 | 27 | $-$
3 | 6 | 1 | 5 | 9 | $-$ | 108
4 | 5 | 2 | 5 | 8 | 162 | $-$
5 | 5 | 2 | 7 | 6 | $-$ | 162
6 | 4 | 3 | 5 | 7 | $-$ | 324
7 | 3 | 0 | 9 | 6 | 9 | $-$
8 | 3 | 0 | 15 | 0 | $-$ | 36
9 | 3 | 2 | 7 | 6 | $-$ | 216
10 | 3 | 2 | 9 | 4 | 81 | $-$
11 | 3 | 4 | 5 | 6 | 54 | $-$
12 | 3 | 4 | 7 | 4 | $-$ | 81
13 | 3 | 6 | 9 | 0 | 3 | $-$

The 27 observables of type $B$ lie on an elliptic quadric of $W(5,2)$, which
can be defined as follows:

$\mathcal{Q}^{-}_{(YYY)}(5,2):=x_{1}^{2}+x_{1}x_{4}+x_{4}^{2}+x_{2}^{2}+x_{2}x_{5}+x_{5}^{2}+x_{3}^{2}+x_{3}x_{6}+x_{6}^{2}=0.$
(18)

Here, we took a coordinate basis of $W(5,2)$ in which the symplectic form
$\sigma(x,y)$ is given by eq. (1),

$\displaystyle\sigma(x,y)=(x_{1}y_{4}-x_{4}y_{1})+(x_{2}y_{5}-x_{5}y_{2})+(x_{3}y_{6}-x_{6}y_{3}),$

so that the correspondence between the 63 three-qubit observables (see eq.
(10))

$\displaystyle{\cal S}_{3}=\\{G_{1}\otimes G_{2}\otimes
G_{3}:~{}G_{j}\in\\{I,X,Y,Z\\},~{}j\in\\{1,2,3\\}\\}\backslash{\cal I}_{3}$

and the 63 points of $W(5,2)$ is of the form (see eq. (12))

$\displaystyle G_{j}\leftrightarrow(x_{j},x_{j+3}),~{}j\in\\{1,2,3\\},$

taking also into account eqs. (13).

Figure 3: Representatives – numbered consecutively from left to right, top to
bottom – of the 13 different types of three-qubit doilies; Type 1 is top left,
Type 13 bottom middle; we also distinguish between Type $9A$ (3rd row, right)
and Type $9B$ (4th row, left). The three different types of observables are
distinguished by different colors and the negative lines are drawn heavy.

This special quadric $\mathcal{Q}^{-}_{(YYY)}(5,2)$, like any non-degenerate
quadric, is a geometric hyperplane of $W(5,2)$. As a doily is also a
subgeometry of $W(5,2)$, it either lies fully in
$\mathcal{Q}^{-}_{(YYY)}(5,2)$ (Type 8), or shares with
$\mathcal{Q}^{-}_{(YYY)}(5,2)$ a set of points that form a geometric
hyperplane; an ovoid (Types 3, 4, 6 and 11), a perp-set (Types 1, 5, 9 and 12)
and a grid (Types 2, 7, 10 and 13). One also observes that no quadratic doily
shares a grid with $\mathcal{Q}^{-}_{(YYY)}(5,2)$.

In addition to the distinguished elliptic quadric, there are also three
distinguished hyperbolic quadrics in $W(5,2)$, namely: the quadric whose 35
observables feature either two $X^{\prime}$s or no $X$,

$\mathcal{Q}^{+}_{(ZZZ)}(5,2):=x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{1}x_{4}+x_{2}x_{5}+x_{3}x_{6}=0,$
(19)

the one whose 35 observables feature either two $Y^{\prime}$s or no $Y$ (see
eq. (14)),

$\mathcal{Q}^{+}_{(III)}(5,2):=x_{1}x_{4}+x_{2}x_{5}+x_{3}x_{6}=0,$ (20)

and the one whose 35 observables feature either two $Z^{\prime}$s or no $Z$,

$\mathcal{Q}^{+}_{(XXX)}(5,2):=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{1}x_{4}+x_{2}x_{5}+x_{3}x_{6}=0.$
(21)

Accordingly, there are three distinguished doilies of Type 8, namely the ones
the quadric $\mathcal{Q}^{-}_{(YYY)}(5,2)$ shares with these three hyperbolic
quadrics.

Take the two-qubit doily. Add formally to each observable, at the same
position, the same mark from the set $\\{X,Y,Z\\}$. Pick up a geometric
hyperplane in this three-qubit labeled doily, and replace by $I$ the added
mark in each observable that belongs to this geometric hyperplane. One
obviously gets a three-qubit doily. Now, there are 31 geometric hyperplanes in
the doily, three possibilities $(X,Y,Z)$ to pick up a mark, and three
possibilities (left, middle, right) where to insert the mark; so there will be
$31\times 3\times 3=279$ doilies created this way. In particular, out of the
$15\times 9=135$ doilies ‘induced’ by perp-sets, 81 are of Type 10 and 54 of
Type 11; out of the $10\times 9=90$ doilies ‘generated’ by grids, 81 are of
Type 12 and 9 of Type 8; finally, the $6\times 9=54$ doilies stemming from
ovoids are all of the same type $9B$. So, if we look at Table 3, all doilies
of Types 1 to 7, 27 doilies of type 8 and all doilies of type $9A$ can be
regarded as ‘genuine’ three-qubit guys, 9 doilies of type 8 that originate
from grids (henceforth referred to as Type $8^{\prime}$) and all doilies of
types $9B$ to 13 can be viewed as ‘built from the two-qubit guy; with Type 13
doilies being even more two-qubit-like.

Figure 4: A formal decomposition of a three-qubit doily (top) into three
‘single-qubit residuals’ (bottom). In each doily of the bottom row, the three
geometric hyperplanes forming a Veldkamp line are distinguished by different
color, with their common points being drawn black; also, the nuclei of perp-
sets are represented by double circles.

This stratification of three-qubit doilies can also be spotted in a different
way. Take a representative doily of a particular type, for example that of
Type 3 depicted in Figure 4, top. From its three-qubit labels, keep first only
the left mark (bottom left figure), then the middle mark (bottom middle
figure) and, finally, the right mark (bottom right figure). In each of these
three ‘residual’ doilies it is easy to see that if you take the points
featuring a given non-trivial mark (i.e., $X$, $Y$ or $Z$) together with the
points featuring $I$, these always form a geometric hyperplane, and the whole
set form a Veldkamp line of the doily where the points featuring $I$ represent
its core! Employing Table 1 we readily see that this Veldkamp line is of type
V (the core is a single point) for the left residual doily, type III (the core
is a tricentric triad) for the middle doily and of type IV (the core is a
unicentric triad) for the right one. To account this way for the 13 types of
three-qubit doilies, we also need the concept of a trivial Veldkamp line of
the doily, i. e. a line consisting of a geometric hyperplane counted twice and
the full doily, which exactly accounts for those doilies ‘generated’ by the
two-qubit doily! This classification is summarized in Table 4. Here, columns
two to six give the number of ordinary Veldkamp lines of a given type, columns
seven to nine show the same for trivial Veldkamp lines and the last column
corresponds to the degenerate case when all the points of a residual doily
bear the label $I$. Note that all doilies stemming from the two-qubit doily
(i. e., Types $8^{\prime}$ to 13) feature ordinary Veldkamp lines of the same
type.

Table 4: A refined classification of doilies living in $W(5,2)$. We use the
following abbreviations for the cores of Veldkamp lines: $2cl$ – two
concurrent lines, $le$ – line, $ttr$ – tricentric triad, $utr$ – unicentric
triad, $pt$ – point, $ov$ – ovoid, $ps$ – perp-set, $gr$ – grid and $fl$
stands for the full doily.

$T$ | 2cl | le | ttr | utr | pt | ov | ps | gr | fl
---|---|---|---|---|---|---|---|---|---
1 | 1 | $-$ | $-$ | $-$ | 2 | $-$ | $-$ | $-$ | $-$
2 | $-$ | 3 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$
3 | $-$ | $-$ | 1 | 1 | 1 | $-$ | $-$ | $-$ | $-$
4 | $-$ | 1 | 2 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$
5 | 1 | $-$ | $-$ | 2 | $-$ | $-$ | $-$ | $-$ | $-$
6 | 1 | $-$ | 1 | 1 | $-$ | $-$ | $-$ | $-$ | $-$
7 | $-$ | 3 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$
8 | 3 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$
$9A$ | 1 | $-$ | $-$ | 2 | $-$ | $-$ | $-$ | $-$ | $-$
$8^{\prime}$ | $-$ | $-$ | 2 | $-$ | $-$ | $-$ | $-$ | 1 | $-$
$9B$ | $-$ | $-$ | 2 | $-$ | $-$ | 1 | $-$ | $-$ | $-$
10 | $-$ | $-$ | 2 | $-$ | $-$ | $-$ | 1 | $-$ | $-$
11 | $-$ | $-$ | 2 | $-$ | $-$ | $-$ | 1 | $-$ | $-$
12 | $-$ | $-$ | 2 | $-$ | $-$ | $-$ | $-$ | 1 | $-$
13 | $-$ | $-$ | 2 | $-$ | $-$ | $-$ | $-$ | $-$ | 1

Using computer, we have also found out a very interesting property that given
a doily and any geometric hyperplane in it, there are three other doilies
having the same geometric hyperplane. Figure 5 serves as a visualisation of
this fact when the common geometric hyperplane is an ovoid. The four doilies
sharing a geometric hyperplane, however, do not stand on the same footing.
This is quite easy to spot from our example depicted in Figure 5. A point of
the doily is collinear with three distinct points of an ovoid, the three
points forming a unicentric triad. Let us pick up such a triad, say
$\\{ZYI,XYI,YYI\\}$ and look for its centers in each of the four doilies.
These are $IYI$ (top doily), $IIX$ (left doily), $IIY$ (right doily) and $IYZ$
(bottom one). We see that the last three observables are mutually
anticommuting, whereas the first observable commutes with each of them. This
property is found to hold for each of ${5\choose 3}=10$ triads contained in an
ovoid. Hence, the top doily of Figure 5 has indeed a different footing than
the remaining three. A similar $3+1$ split up is also observed in any
quadruple of doilies having a grid in common because a point of the doily is
also collinear with three points of a grid that form a unicentric triad.
However, when the shared hyperplane is a perp-set, one gets a different,
namely $2+2$ split, because in this case the corresponding triple of points
forms a tricentric triad.

Figure 5: An illustration of the case when four different doilies share an
ovoid (boldfaced). The top doily is of Type 11, the bottom one of Type 8, and
both the left and right doilies are of Type 3.

A tricentric triad of a linear resp. quadratic doily of $W(5,2)$ defines a
line resp. plane in the ambient PG$(5,2)$. The latter type of a triad is found
to be shared by four quadratic doilies. Given the three observables of such a
triad, there are seven observables commuting with each of them, the
corresponding seven points lying in a Fano plane (namely in the polar plane to
the plane defined by the triad) in the ambient PG$(5,2)$. One of the seven
observables has a distinguished footing as it commutes with each of the
remaining six ones, these six observables forming three commuting pairs. Out
of the six observables one can form just four tricentric triads of which each
is complementary to the triad we started with and thus defines with the latter
a unique quadratic doily. These properties are also illustrated in Figure 6.

Figure 6: Four three-qubit doilies on a ‘planar’ tricentric triad (represented
by hexagons). The seven observables commuting with the three hexagonal ones
are, for better illustration, colored differently. The three red lines of the
Fano plane that meet at the distinguished observable (gray) are totally
isotropic, whilst the remaining four (depicted green) are not. The four
complementary triads (of observables) are illustrated by a full black circle
and three half-circles.

Among the 13 different types of three-qubit doilies, there is one type, namely
Type 3, which has two remarkable properties. The first property is that there
is one point (to be called a deep point) such that all three lines passing
through it are negative. Let’s take a representative doily of such a type
shown in Figure 3, 1st row right. The deep point is $ZIZ$. Then one sees that
there are just two points (to be called zero-points) such that neither of them
lies on a negative line; one is $IIY$ and the other is $XIZ$. These two points
and the deep point form in the doily a tricentric triad, hence a copy of
‘linear’ $W(1,2)$! The second property is related to the fact that through
each observable of type $B$ there pass four negative lines. Three of them are
such that each features one observable of type $B$ and two observables of type
$C$, whereas the remaining one consists of all observables of type $B$.
Written vertically, the four negative lines passing through our deep point
$ZIZ$ are:

$ZIZ$ | $ZIZ$ | $ZIZ$ | $ZIZ$
---|---|---|---
$XXX$ | $XYX$ | $XZX$ | $XIX$
$YXY$ | $YYY$ | $YZY$ | $YIY$

We see that the three lines that are located in the doily are of the same
type, viz. $B-C-C$. If we include also the fourth negative line, viz. the
$B-B-B$ one, we get what we can call a ‘doily with a tail.’ Taking into
account the above-mentioned four-doilies-per-hyperplane property, we see that
there are altogether 12 doilies, four per each observable, having the same
tail and all being of Type 3.

##  5 ‘Conwell’ Heptads of Doilies in $\boldsymbol{W(5,2)}$

Let’s recall a famous Sylvester’s construction of $W(3,2)$ [18]. Given a six-
element set $M_{6}\equiv\\{1,2,3,4,5,6\\}$, a duad is an unordered pair
$(ij)\in M_{6}$, $i\neq j$, and a syntheme is a set of three pairwise disjoint
duads, i. e. a set $\\{(ij),(kl),(mn)\\}$ where $i,j,k,l,m,n\in M_{6}$ are all
distinct. The point-line incidence structure whose points are duads and whose
lines are synthemes, with incidence being inclusion, is isomorphic to
$W(3,2)$, as also illustrated in Figure 7.

Figure 7: A duad-syntheme model of $W(3,2)$.

Next, take a seven-element set, $M_{7}\equiv\\{1,2,3,4,5,6,7\\}$. One can form
from it ${7\choose 3}=35$ unordered triples $(ijk)$, $i\neq j\neq k\neq i$.
From each set of fifteen triples having the same element in common we can
create a doily using the duad-syntheme construction on that six-element subset
of $M_{7}$ where the common element is omitted. So, we get seven different
doilies, one per each element, as depicted in Figure 8. Any two of them have
an ovoid in common; because each ovoid is characterized by two elements, say
$a$ and $b$, and it is of the form $\\{(abc),(abd),(abe),(abf),(abg)\\}$,
where $a,b,c,d,e,f,g\in M_{7}$ are all different, hence it belongs to both the
$a$-doily and the $b$-doily. Also, any triple is shared by three doilies.

Figure 8: An abstract heptad of doilies on a seven-element set.

A remarkable fact is that this abstract heptad of doilies has a neat
realization in our three-qubit $W(5,2)$. To see this, we have to introduce the
notion of a Conwell heptad of PG$(5,2)$. Given a $\mathcal{Q}^{+}(5,2)$ of
PG$(5,2)$, a Conwell heptad [19] (in the modern language [20] also known as a
maximal exterior set) with respect to $\mathcal{Q}^{+}(5,2)$ is a set of seven
off-quadric points such that each line joining two distinct points of the
heptad is skew to the $\mathcal{Q}^{+}(5,2)$. There are exactly 8 heptads with
respect to $\mathcal{Q}^{+}(5,2)$. Any two of them have exactly one point in
common and any point off $\mathcal{Q}^{+}(5,2)$ is exactly in two heptads;
also any six points of a heptad are linearly independent in PG$(5,2)$. Next
[21], let $P$ be a point on $\mathcal{Q}^{+}(5,2)$. The tangent hyperplane of
$\mathcal{Q}^{+}(5,2)$ at $P$ intersects a heptad $C$ in exactly three points
$P_{1},P_{2}$ and $P_{3}$ such that the points $P,P_{1},P_{2}$ and $P_{3}$ are
coplanar and $P_{1},P_{2}$ and $P_{3}$ are not collinear; that is, the points
$P_{1},P_{2}$ and $P_{3}$ represent a conic in the plane and the point $P$ is
its knot (the common intersection of its tangents). Hence, there exists a
bijection from the set of the 35 points of $\mathcal{Q}^{+}(5,2)$ onto the set
of the 35 triples of points of $C$.

Now, let us take a $\mathcal{Q}^{+}(5,2)$ that belongs to $W(5,2)$, for
example $\mathcal{Q}_{(III)}^{+}(5,2)$ (see eq. 20) that accommodates all
symmetric observables from ${\cal S}_{3}$. The eight Conwell heptads with
respect to this distinguished hyperbolic quadric, expressed in terms of three-
qubit observables, are:

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
---|---|---|---|---|---|---|---
$ZYX$ | $YZI$ | $YIZ$ | $YZI$ | $YIZ$ | $YXI$ | $XYI$ | $YII$
$YIX$ | $YXZ$ | $YZX$ | $YXI$ | $YIX$ | $YZZ$ | $ZYZ$ | $ZYI$
$YZZ$ | $YXX$ | $YXX$ | $IYZ$ | $XYI$ | $YZX$ | $ZYX$ | $XYZ$
$XYX$ | $IYI$ | $IYX$ | $IYX$ | $IZY$ | $IYI$ | $ZIY$ | $XYX$
$IYZ$ | $IXY$ | $ZYZ$ | $ZIY$ | $IXY$ | $IZY$ | $XZY$ | $XIY$
$YXZ$ | $XZY$ | $IIY$ | $YYY$ | $YYY$ | $XXY$ | $XXY$ | $ZZY$
$IIY$ | $ZZY$ | $XYZ$ | $XIY$ | ZYI | $ZXY$ | $YII$ | $ZXY$

We see that each Conwell heptad entails seven pairwise anticommuting
observables and so, in fact, corresponds to a set of generators of a seven-
dimensional Clifford algebra [22]. Let us pick up one of them, say the heptad
number 1, and associate its observables with the elements of $M_{7}$ as
follows:

$1\leftrightarrow ZYX,2\leftrightarrow YIX,3\leftrightarrow
YZZ,4\leftrightarrow XYX,5\leftrightarrow IYZ,6\leftrightarrow
YXZ,7\leftrightarrow IIY.$

From the above-described relation between tangent hyperplanes to a hyperbolic
quadric and a Conwell heptad it follows that any unordered triple $(ijk)$,
$i,j,k\in M_{7}$, will be associated with a particular point on
$\mathcal{Q}_{(III)}^{+}(5,2)$ and its associated observable is the (ordinary)
product of the observables associated with elements/points $i$, $j$, and $k$;
for example, $146\leftrightarrow ZYX.XYX.YXZ=IXZ$. Hence, all seven doilies of
the heptad lie fully in $\mathcal{Q}_{(III)}^{+}(5,2)$ and, since no two of
them share a line, they partition the set of 105 lines of
$\mathcal{Q}_{(III)}^{+}(5,2)$. Figure 9 serves as a visualization of this
particular ‘Conwell’ heptad of doilies. As $W(5,2)$ contains 36 hyperbolic
quadrics (see eq. 6), it features altogether $36\times 8=288$ such heptads of
doilies.

Figure 9: A ‘Conwell’ heptad of doilies in the three-qubit $W(5,2)$. Following
our convention, different types of observables are distinguished by different
colors and negative lines are shown in bold.

##  6 $\boldsymbol{W(7,2)}$ and its Four-Qubit $\boldsymbol{W(5,2)}$’s

The space $W(7,2)$ possesses 255 points, 5355 lines, 11475 planes and 2295
generators, the latter being all PG(3,2)’s. Among the 255 canonical four-qubit
observables associated to the points, 12 are of type $A$, 54 of type $B$, 108
of type $C$ and 81 of type $D$. Through an observable of type $D$ there pass:
four negative lines of type $D-D-D$, totaling to $\frac{81\times 4}{3}=108$;
12 negative lines of type $D-D-B$, totaling to $\frac{81\times 12}{2}=486$;
and 12 negative lines of type $D-C-C$, totaling to $81\times 12=972$. Through
an observable of type $C$ there pass, apart from the above-mentioned lines of
type $D-C-C$, six negative lines of type $C-C-B$, totaling to $\frac{108\times
6}{2}=324$. Through an observable of type $B$ there passes, apart from the
already discussed two types of lines, a single negative line of type $B-B-B$,
the total number of such lines being $\frac{54\times 1}{3}=18$. Since no
negative line can contain an observable of type $A$, the four-qubit $W(7,2)$
thus exhibits five distinct types of negative lines whose total number is
$(108+486+972+324+18=)~{}1908$.

Table 5: Classification of $W(5,2)$’s living in $W(7,2)$. Column one ($T$)
shows the type, column two ($C^{-}$) the number of negative lines in a
$W(5,2)$ of the given type, columns three to six ($O_{A}$ to $O_{D}$) indicate
the number of observables featuring three $I$’s, two $I$’s, one $I$ or no $I$,
respectively, columns seven ($W_{l}$) and eight ($W_{q}$) yield, respectively,
the number of ‘linear’ and ‘quadratic’ $W(5,2)$’s of a given type, the last
but one column depicts the type of intersection of a representative $W(5,2)$
with the distinguished hyperbolic quadric and the last column indicates the
type of geometric hyperplane featuring the trivial mark ($I$) for composite
$W(5,2)$’s.

$T$ | $C^{-}$ | $O_{A}$ | $O_{B}$ | $O_{C}$ | $O_{D}$ | $W_{l}$ | $W_{q}$ | Int | GH
---|---|---|---|---|---|---|---|---|---
1 | 130 | 3 | 9 | 33 | 18 | 108 | $-$ | ell | $---$
2 | 126 | 0 | 24 | 0 | 39 | $-$ | 108 | full | $---$
3 | 126 | 1 | 13 | 27 | 22 | $-$ | 1944 | hyp | $---$
4 | 126 | 2 | 10 | 30 | 21 | $-$ | 1620 | perp | $---$
5 | 122 | 1 | 15 | 27 | 20 | 972 | $-$ | hyp | $---$
6 | 122 | 2 | 10 | 30 | 21 | $-$ | 648 | perp | $---$
7 | 118 | 0 | 16 | 32 | 15 | $-$ | 324 | perp | $---$
8 | 118 | 3 | 9 | 33 | 18 | 648 | $-$ | ell | $---$
9 | 118 | 3 | 11 | 25 | 24 | $-$ | 1296 | hyp | $---$
10 | 114 | 1 | 15 | 27 | 20 | 324 | $-$ | hyp | $---$
11 | 114 | 1 | 17 | 27 | 18 | $-$ | 216 | hyp | $---$
12 | 114 | 3 | 13 | 25 | 22 | 1944 | $-$ | hyp | $---$
13 | 114 | 4 | 12 | 28 | 19 | $-$ | 1944 | perp | $---$
14 | 110 | 3 | 15 | 25 | 20 | $-$ | 1944 | hyp | $---$
15 | 110 | 5 | 11 | 23 | 24 | 648 | $-$ | hyp | $---$
16 | 106 | 5 | 13 | 23 | 22 | $-$ | 1944 | hyp | $---$
17 | 102 | 1 | 21 | 27 | 14 | $-$ | 648 | hyp | $---$
18 | 102 | 2 | 18 | 30 | 13 | $-$ | 324 | perp | $---$
19 | 102 | 3 | 15 | 25 | 20 | $-$ | 648 | hyp | $---$
20 | 102 | 4 | 12 | 28 | 19 | $-$ | 1944 | perp | $---$
21 | 90 | 0 | 36 | 0 | 27 | $-$ | 12 | full | ell: $O=YYY$
22 | 90 | 2 | 22 | 30 | 9 | $-$ | 108 | perp | hyp: all 9 $O$’s featuring two $Y$’s
23 | 90 | 3 | 9 | 33 | 18 | 36 | $-$ | ell | $---$
24 | 90 | 3 | 21 | 25 | 14 | 324 | $-$ | hyp | perp: all 27 $O$’s of type $C$
25 | 90 | 4 | 16 | 28 | 15 | $-$ | 324 | perp | ell: all 27 $O$’s featuring one $Y$
26 | 90 | 5 | 15 | 31 | 12 | 324 | $-$ | ell | perp: all 27 $O$’s of type $B$
27 | 90 | 6 | 18 | 26 | 13 | $-$ | 324 | perp | hyp: 26 $O$’s having no $Y$ \+ $III$
28 | 90 | 7 | 17 | 21 | 18 | 108 | $-$ | hyp | perp: all 9 $O$’s of type $A$
29 | 90 | 9 | 27 | 27 | 0 | 4 | $-$ | ell | full $W(5,2)$

When it comes to $W(5,2)$’s, we find 11 types among their 5440 linear members
and as many as 18 types among their 16320 quadratic cousins $-$ as summarized
in Table 5. It represents no difficulty to check that 54 observables of type
$B$ and 81 ones of type $D$ lie on a particular hyperbolic quadric in
$W(7,2)$, to be referred to as the distinguished hyperbolic quadric
$\mathcal{Q}^{+}_{(YYYY)}(7,2)$, which is also a geometric hyperplane in the
latter space. A $W(5,2)$ either lies fully in this quadric (Types 2 and 21) or
shares with it a set of points that forms a geometric hyperplane. Hence, the
sum of $O_{B}$ and $O_{D}$ in each row of Table 5 must be one of the following
numbers: 27 (when the hyperplane of $W(5,2)$ is an elliptic quadric), 31 (a
perp-set) and/or 35 (a hyperbolic quadric); for the reader’s convenience, the
type of such geometric hyperplane is explicitly listed in column 9 of Table 5.
One sees that no linear $W(5,2)$ shares with $\mathcal{Q}^{+}_{(YYYY)}(7,2)$ a
perp-set and no quadratic $W(5,2)$ cuts this distinguished quadric in an
elliptic quadric. Comparing Table 5 with Table 3 one readily discerns that
whereas $W(3,2)$’s in $W(5,2)$ are endowed with both an even and odd number of
negative lines, for $W(5,2)$’s in $W(7,2)$ this number is always even; in
addition, the difference in $C^{-}$ for any two distinct types of four-qubit
$W(5,2)$’s is a multiple of four.

Let us have a closer look at $W(5,2)$’s featuring 90 (i.e., the smallest
possible number of) negative lines. We can easily show that almost all of them
originate from the three-qubit $W(5,2)$. First, by adding $I$ to each three-
qubit observable at the same position we get the four trivial four-qubit
$W(5,2)$’s of Type 29. Next, adding to each observable at the same position a
mark from the set $\\{X,Y,Z\\}$, picking up a geometric hyperplane in this
four-qubit labeled $W(5,2)$ and replacing by $I$ the added mark of each
observable in the geometric hyperplane one gets a four-qubit $W(5,2)$ with 90
negative lines. Now, there are 28 ($\\#$ of elliptic quadrics) + 36 ($\\#$ of
hyperbolic quadrics) + 63 ($\\#$ of perp-sets) = 127 geometric hyperplanes in
the $W(5,2)$, three possibilities $(X,Y,Z)$ to pick up a mark, and four
possibilities (left, middle-left, middle-right, right) where to insert the
mark. So, there will be $127\times 3\times 4=1524$ four-qubit $W(5,2)$’s
created this way, which only falls short by 36 the total number of $W(5,2)$’s
endowed with 90 negative lines (the four guys of Type 29 being, of course,
disregarded). A concise summary is given in the last column of Table 5, where
the type of geometric hyperplane is further specified by the character/type of
the associated (three-qubit) observable. One observes that Type 23 is the only
irreducible type of $W(5,2)$’s having 90 negative lines.

We shall illustrate this process by a couple of examples. Let us start with
the perp-set of the three-qubit $W(5,2)$ whose nucleus is an observable of
type $A$, say $XII$. Out of 31 observables commuting with this observable
there are 7 of type $A$ ($XII$, $IXI$, $IIX$, $IYI$, $IIY$, $IZI$ and $IIZ$),
15 of type $B$ ($IXX$, $IXY$, $IXZ$, $XXI$, $XIX$, $IYX$, $IYY$, $IYZ$, $XYI$,
$XIY$, $IZX$, $IZY$, $IZZ$, $XZI$, and $XIZ$) and 9 of type $C$ ($XXX$, $XXY$,
$XXZ$, $XYX$, $XYY$, $XYZ$, $XZX$, $XZY$, and $XZZ$). Hence, out of 32
observables off the perp, there will be $9-7=2$ of type $A$, $27-15=12$ of
type $B$ and $27-9=18$ of type $C$:

$\widehat{\mathcal{Q}}_{(XII)}$ | $O_{A}$ | $O_{B}$ | $O_{C}$
---|---|---|---
on | 7 | 15 | 9
off | 2 | 12 | 18

.

Next, each observable of the perp-set acquires a trivial mark $I$ and hence
goes into the four-qubit observable of the same type. However, an observable
lying off the perp-set gets a non-trivial label $X$, $Y$ or $Z$ and so yields
the four-qubit observable of the subsequent type; that is, $O_{A}^{(3)}\to
O_{B}^{(4)}$, $O_{B}^{(3)}\to O_{C}^{(4)}$ and $O_{C}^{(3)}\to O_{D}^{(4)}$.
Hence, in our case we get:

($\widehat{\mathcal{Q}}_{(XII)}$) | $O_{A}$ | $O_{B}$ | $O_{C}$ | $O_{D}$
---|---|---|---|---
(on – type intact) | 7 | 15 | 9 | 0
(off – type shifted) | 0 | 2 | 12 | 18
Total | 7 | 17 | 21 | 18

.

Comparing with Table 5 we see that this is a four-qubit $W(5,2)$ of Type 28.

As the second example we shall take the case when the geometric hyperplane of
$W(5,2)$ is an elliptic quadric generated by an antisymmetric observable of
type $B$, say $YXI$. This quadric, $\mathcal{Q}_{(YXI)}^{-}(5,2)$, consists of
all symmetric observables that commute with $YXI$ and all antisymmetric
observables that anticommute with $YXI$. In particular, it contains 4
observables of type $A$ ($IXI$, $IIX$, $IIZ$ and $IYI$), 11 observables of
type $B$ ($XZI$, $ZZI$, $YIY$, $IXX$, $IXZ$, $YZI$, $IYX$, $IYZ$, $XIY$, $ZIY$
and $IZY$) and 12 observables of type $C$ ($XZX$, $ZZX$, $XZZ$, $ZZZ$, $YXY$,
$XYY$, $ZYY$, $YZX$, $YZZ$, $XXY$, $ZXY$ and $YYY$). So, out of 36 observables
off the quadric, there will be 5, 16 and 15 of type $A$, $B$ and $C$,
respectively. In a succinct form,

$\mathcal{Q}_{(YXI)}^{-}(5,2)$ | $O_{A}$ | $O_{B}$ | $O_{C}$
---|---|---|---
on | 4 | 11 | 12
off | 5 | 16 | 15

.

From this it follows that the corresponding four-qubit $W(5,2)$ is of Type 25:

($\mathcal{Q}_{(YXI)}^{-}(5,2)$) | $O_{A}$ | $O_{B}$ | $O_{C}$ | $O_{D}$
---|---|---|---|---
(on – type intact) | 4 | 11 | 12 | 0
(off – type shifted) | 0 | 5 | 16 | 15
Total | 4 | 16 | 28 | 15

.

##  7 Conclusion

We have introduced a remarkable observable-based taxonomy of subspaces of
$W(2N-1,2)$, $2\leq N\leq 4$, whose rank is just one less than that of the
ambient space. Alongside the distribution of various types of observables, an
important parameter of the classification was the number of negative lines
contained in a subspace. As already mentioned in the introduction, this latter
parameter is essential in checking whether a given finite geometric
configuration is contextual or not. For example, our preliminary analysis
shows that all three-qubit and four-qubit doilies are, like their two-qubit
sibling, contextual. In a separate paper we plan to address this question in
more detail, employing also the degree of contextuality, for a variety of
other symplectic subspaces. However, when approaching this way subspaces of
higher rank, it would be natural to include as parameters the number of
negative linear subspaces of every viable dimension from $1$ to $N-2$, i. e.
consider negative lines, negative planes, …, negative generators; so, already
in the case of $N=4$ we can add one more parameter, the number of negative
planes a four-qubit $W(5,2)$ is endowed with, to get an interesting refinement
of our Table 5. As the three-qubit $W(5,2)$ features 54 negative planes [23],
each composite four-qubit $W(5,2)$ must have the same number of negative
planes; in connection with this fact it would be interesting to check whether
also each irreducible four-qubit $W(5,2)$ having 90 lines (Type 23) enjoys
this property.

Another interesting extension/variation of our taxonomy would be to take into
account the number of negative lines passing through a point of the subspace.
Let us call this number the order of a point and for each subspace $W(2s-1,2)$
define the following string of parameters
$[p_{0},p_{1},p_{2},\ldots,p_{4^{s-1}-1}]$, where $p_{k}$, $0\leq k\leq
4^{s-1}-1$, stands for the number of points of order $k$ the subspace
contains. Applying this to three-qubit doilies ($s=2$), we find the following
five patterns (as readily discerned from Figure 3): $[0,9,6,0]$ (Types 1 and
2), $[2,9,3,1]$ (Type 3), $[5,5,5,0]$ (Types 4 and 5), $[6,6,3,0]$ (Type 6)
and $[6,9,0,0]$ (Types 7 to 13).

A slightly different possibility of employing our strategy is to analyse other
distinguished subgeometries of $W(2N-1,2)$ like, for example, the split Cayley
hexagon of order two [24]. This generalized polygon can be embedded into
$W(5,2)$, and in two different ways at that [25], called classical and skew.
We have already discerned two distinct kinds of the former, and as many as 13
different types of the latter. Yet, a full understanding of the case requires
a more rigorous computer-assisted approach and will, therefore, be treated in
a separate paper.

## Acknowledgments

This work was supported by the Slovak VEGA Grant Agency, Project $\\#$
2/0004/20, the French “Investissements d’Avenir” programme, project ISITE-BFC
(contract ANR-15-IDEX-03) and by the EIPHI Graduate School (contract
ANR-17-EURE-0002). The computations have been performed on the supercomputer
facilities of the Mésocentre de calcul de Franche-Comté; the code is freely
available at https://quantcert.github.io/Magma-contextuality/. We are also
indebted to Zsolt Szabó and Petr Pracna for their help with the figures and
thank Péter Vrana for providing us with a list of doilies of $W(5,2)$.

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every starting memory mode ${\sf m}^{\prime}$ and starting time $t^{\prime}\in
T_{i_{0}}$ that the probability of visiting $c_{0}$ before revisiting $u$ is
upper-bounded, i.e.,

$\displaystyle{\mathcal{P}}_{{\mathcal{G}}_{k+1},c_{i_{0}},\sigma[{\sf
m}^{\prime}](t^{\prime}),\pi}({{c_{0}}\ \mathsf{before}\ u})$
$\displaystyle\leq\prod_{i=1}^{i_{0}}(1-\min_{{\sf m}}\inf_{t\in
T_{i}}\alpha(i,{\sf m},t,\pi))$

Since $\sum_{i}\alpha(i)$ diverges, it follows from (13) that

$\sum_{i=1}^{\infty}\min_{{\sf m}}\inf_{t\in T_{i}}\alpha(i,{\sf
m},t,\pi)\geq\left(\frac{1}{k+1}\sum_{i}\alpha(i)\right)-\frac{1}{2k+2}$

also diverges. From Proposition A.2 we obtain

$\prod_{i=1}^{\infty}(1-\min_{{\sf m}}\inf_{t\in T_{i}}\alpha(i,{\sf
m},t,\pi))=0$

and hence

$\lim_{i_{0}\to\infty}\prod_{i=1}^{i_{0}}(1-\min_{{\sf m}}\inf_{t\in
T_{i}}\alpha(i,{\sf m},t,\pi))=0.$

Thus, for every $\varepsilon>0$, there exists a sufficiently large $i_{0}$
such that for all $t^{\prime}\in T_{i_{0}}$ and ${\sf m}^{\prime}$

${\mathcal{P}}_{{\mathcal{G}}_{k+1},c_{i_{0}},\sigma[{\sf
m}^{\prime}](t^{\prime}),\pi}({{c_{0}}\ \mathsf{before}\ u})\leq\varepsilon.$

Hence, for all $t\in\mathbb{N}$ and all ${\sf m}$ we have

${\mathcal{P}}_{{\mathcal{G}}_{k+1},u,\sigma[{\sf m}](t),\pi}({{c_{0}}\
\mathsf{before}\ u})\leq\varepsilon.$

Analogously to 5.1, we can pick $i_{0}^{l}$ and $\varepsilon^{l}=\delta\cdot
2^{-l}$ after the $l$-th visit to $u$ such that for all $t\in\mathbb{N}$ and
all ${\sf m}$ we have

${\mathcal{P}}_{{\mathcal{G}}_{k+1},u,\sigma[{\sf
m}](t),\pi}(\mathtt{Reach}(\\{c_{0}\\}))\leq\sum_{l=1}^{\infty}\varepsilon_{l}=\delta$

as required.

Now we consider the second case where $\sum_{i}\alpha(i)$ converges. We
construct a Minimizer strategy $\pi$ in ${\mathcal{G}}_{k+1}$ that plays as
follows. First $\pi$ picks the transition $u{\longrightarrow}b_{i_{0}}$ for a
sufficiently high $i_{0}\in\mathbb{N}$, to be determined. At every state
$d_{i,j}$, outside of the subgames, $\pi$ always plays action “1”, i.e.,
$d_{i,j}{\longrightarrow}r_{i,j}^{1}$. This implies that each state $c_{i}$
with $i\geq i_{0}$ is visited at most once, and states $c_{i}$ with $i<i_{0}$
are never visited. (If Maximizer plays action “1” then he wins and if he plays
“0” then the games goes up to $c_{i+1}$.)

By Eq. 11, for every $i\geq 1$ there exists a time $t_{i}\in\mathbb{N}$ such
that

(14) $\forall t\geq t_{i}.\ \alpha(i,t)\leq\alpha(i)+\frac{1}{4}2^{-i}.$

Let $T_{i}=\\{t\in\mathbb{N}\mid t\geq t_{i}\\}$ be the set of the times $t$
that satisfy (14). At each state $b_{i}$ Minimizer’s strategy $\pi$ delays
sufficiently long such that $c_{i}$ is reached at a time $t\in T_{i}$. This is
possible for every arrival time $t^{\prime}$ at $b_{i}$.

Consider a state $s$ in the subgame ${\mathcal{G}}_{k}^{i}$ that is reached at
some time $t$ when Maximizer’s strategy is in some memory mode ${\sf m}$, and
let $\pi^{\prime}$ be a Minimizer strategy. Let $\beta(s,i,{\sf
m},t,\pi^{\prime})$ be the probability that Maximizer will play action “1”
(wrt. the encoded concurrent game) in the next round in ${\mathcal{G}}_{k+1}$
after winning the subgame ${\mathcal{G}}_{k}^{i}$ (i.e., after reaching
$\mathrm{win}^{k,i}$), or loses the subgame (never reaches
$\mathrm{win}^{k,i}$). So $\beta(s,i,{\sf m},t,\pi^{\prime})$ is the
probability, from state $s$, of losing the subgame ${\mathcal{G}}_{k}^{i}$
plus $\sum_{j}(1/j)\cdot p_{j}$, where $p_{j}$ is the probability of winning
the subgame and then directly going to $d_{i,j}$ (without visiting any other
state $c_{i}$ in between). To define this formally, let
$C_{-i}\stackrel{{\scriptstyle\text{{\tiny{def}}}}}{{=}}\\{c_{j}\mid j\neq
i\\}$. We let

$\displaystyle\beta(s,i,{\sf m},t,\pi^{\prime})$
$\displaystyle\stackrel{{\scriptstyle\text{{\tiny{def}}}}}{{=}}{\mathcal{P}}_{{\mathcal{G}}_{k+1},s,\sigma[{\sf
m}](t),\pi^{\prime}}(\neg\mathtt{Reach}(\\{\mathrm{win}^{k,i}\\}))$
$\displaystyle+\sum_{j}(1/j){\mathcal{P}}_{{\mathcal{G}}_{k+1},s,\sigma[{\sf
m}](t),\pi^{\prime}}(d_{i,j}\ \mathsf{before}\ C_{-i})$

Let $\beta(s,i,{\sf
m},t)\stackrel{{\scriptstyle\text{{\tiny{def}}}}}{{=}}\sup_{\pi^{\prime}}\beta(s,i,{\sf
m},t,\pi^{\prime})$ be the supremum over all Minimizer strategies. Let

$\beta(s,i,t)\stackrel{{\scriptstyle\text{{\tiny{def}}}}}{{=}}\min_{{\sf
m}\in\\{0,\dots,k\\}}\beta(s,i,{\sf m},t)$

be the minimum over all memory modes.

###### Claim E.1.

For all states $s$ in ${\mathcal{G}}_{k}^{i}$ and times $t$ we have
$\beta(s,i,t+1)\leq(k+1)\alpha(i,t)$.

###### Proof.

Consider the situation where we are at state $c_{i}$ at time $t$ and the
Maximizer strategy $\sigma$ is in some memory mode ${\sf m}$ (unknown to
Minimizer). Then some particular Minimizer strategy $\hat{\pi}$ could play
$c_{i}{\longrightarrow}s$ to arrive at $s$ in one step at time $t+1$.
Meanwhile, $\sigma$ can update its memory to some other mode ${\sf
m}^{\prime}$ (or a distribution over memory modes), still unknown to
Minimizer. Then $\hat{\pi}$ hedges her bets by guessing Maximizer’s memory
mode ${\sf m}^{\prime}$. For each of the $k+1$ possible modes ${\sf
m}^{\prime}$, the Minimizer strategy $\hat{\pi}$ plays, with probability
$\frac{1}{k+1}$, an $\varepsilon$-optimal strategy to maximize the probability
that Maximizer plays action “1” after winning ${\mathcal{G}}_{k}^{i}$ (or
loses the subgame). Thus

$\displaystyle\alpha(i,t)$ $\displaystyle=\min_{\sf
m}\sup_{\pi^{\prime}}\alpha(i,{\sf m},t,\pi^{\prime})$
$\displaystyle\geq\min_{\sf m}\alpha(i,{\sf m},t,\hat{\pi})$
$\displaystyle\geq\min_{{\sf m}^{\prime}}\frac{1}{k+1}\beta(s,i,{\sf
m}^{\prime},t+1,\hat{\pi})$ $\displaystyle\geq\frac{1}{k+1}(\min_{{\sf
m}^{\prime}}\sup_{\pi^{\prime}}\beta(s,i,{\sf
m}^{\prime},t+1,\pi^{\prime})-\varepsilon)$
$\displaystyle=\frac{1}{k+1}(\beta(s,i,t+1)-\varepsilon).$

Since this holds for every $\varepsilon>0$, the claim follows. ∎

By E.1 for every $s$ in ${\mathcal{G}}_{k}^{i}$ and time $t+1$ there exists at
least one memory mode ${\sf m}(s,t+1)$ (a mode ${\sf m}$ where the minimum
$\beta(s,i,t+1)=\min_{{\sf m}\in\\{0,\dots,k\\}}\beta(s,i,{\sf m},t+1)$ is
realized) such that if $\sigma$ enters $s$ at time $t+1$ in mode ${\sf
m}(s,t+1)$ then after winning ${\mathcal{G}}_{k}^{i}$ (if at all) Maximizer
plays action “1” with a “small” probability $\leq(k+1)\alpha(i,t)$. Crucially,
this property holds for the $\sup$ over the Minimizer strategies and thus for
_every_ Minimizer strategy inside the subgame ${\mathcal{G}}_{k}^{i}$. In
particular it holds for the Minimizer strategy $\pi$ that we will construct.
We call ${\sf m}(s,t+1)$ the _forbidden_ memory mode for state $s$ at time
$t+1$.

Above we have defined our Minimizer strategy $\pi$ so that it adds sufficient
delays in the states $b_{i}$ such that $c_{i}$ is only visited at times $t\geq
t_{i}$. This implies that states $s$ in ${\mathcal{G}}_{k}^{i}$ are only
visited at times $t+1$ where $t\geq t_{i}$. Since for these times Eq. 14 is
satisfied, we obtain

(15) $\forall t\geq t_{i}.\ \beta(s,i,{\sf
m}(s,t+1),t+1)\leq(k+1)(\alpha(i)+\frac{1}{4}2^{-i}).$

Let $\sigma^{\prime}$ be a restriction of $\sigma$ that, inside the subgame
${\mathcal{G}}_{k}^{i}$, is never in the forbidden memory mode ${\sf
m}(s,t+1)$ at state $s$ at time $t+1$, or else concedes defeat.

###### Claim E.2.

Consider a step counter plus $(k+1)$ mode Maximizer strategy $\sigma^{\prime}$
in ${\mathcal{G}}_{k}^{i}$ that is never in the forbidden memory mode ${\sf
m}(s,t+1)$ at state $s$ at time $t+1$. Then there exists a step counter plus
$k$ mode Maximizer strategy $\sigma^{\prime\prime}$ in ${\mathcal{G}}_{k}^{i}$
that performs equally well as $\sigma^{\prime}$ against any Minimizer
strategy.

###### Proof.

While $\sigma^{\prime}$ has memory modes $\\{0,\dots,k\\}$, the strategy
$\sigma^{\prime\prime}$ only has memory modes $\\{0,\dots,k-1\\}$ (in addition
to the step counter). However, $\sigma^{\prime\prime}$ can directly imitate
the behavior of $\sigma^{\prime}$ as follows. Whenever $\sigma^{\prime}$
enters memory mode $k$ at some state $s$ and time $t+1$ then
$\sigma^{\prime\prime}$ enters memory mode ${\sf m}(s,t+1)$ instead. Whenever,
$\sigma^{\prime\prime}$ is in ${\sf m}(s,t+1)$ at time state $s$ and time
$t+1$ then it plays like $\sigma^{\prime}$ at state $s$ in memory mode $k$. By
the condition on the behavior of $\sigma^{\prime}$ there is no confusion. ∎

By E.2, the Maximizer strategy $\sigma^{\prime}$ is equivalent to a strategy
with just a step counter and $k$ memory modes. By the induction hypothesis
(9), for this restricted Maximizer strategy $\sigma^{\prime}$ there exists a
Minimizer strategy $\pi_{i}$ in ${\mathcal{G}}_{k}^{i}$ such that

(16) $\forall{\sf m}\ \forall t\
{\mathcal{P}}_{{\mathcal{G}}_{k}^{i},u^{k,i},\sigma^{\prime}[{\sf
m}](t),\pi_{i}}(\mathtt{Reach}(\\{c_{0}^{k,i}\\}))\leq\delta\cdot 2^{-(i+1)}.$

We are now ready to construct Minimizer’s strategy $\pi$ in
${\mathcal{G}}_{k+1}$. At every state $d_{i,j}$, outside of the subgames,
$\pi$ always plays action “1”, i.e., $d_{i,j}{\longrightarrow}r_{i,j}^{1}$.
This implies that each state $c_{i}$ is visited at most once. At the states
$b_{i}$, Minimizer chooses the delays such that $c_{i}$ is reached at a time
$t\geq t_{i}$, as described above. From each $c_{i}$ Minimizer goes to state
$u^{k,i}$ of the subgame ${\mathcal{G}}_{k}^{i}$. Inside each subgame
${\mathcal{G}}_{k}^{i}$ Minimizer plays like $\pi_{i}$. By (16), $\pi_{i}$
performs well in ${\mathcal{G}}_{k}^{i}$ (regardless of the initial memory
mode and time) if Maximizer limits himself to $\sigma^{\prime}$.

Now we show that $\pi$ performs well in ${\mathcal{G}}_{k+1}$. Since $\pi$
first picks the transition $u{\longrightarrow}b_{i_{0}}$ and then always plays
action “1” outside of the subgames, it follows that each subgame
${\mathcal{G}}_{k}^{i}$ with $i\geq i_{0}$ is played at most once, and
subgames ${\mathcal{G}}_{k}^{i}$ with $i<i_{0}$ are never played. For each
subgame ${\mathcal{G}}_{k}^{i}$, let $\mathrm{Forb}_{i}$ be the set of plays
where Maximizer enters a forbidden memory mode (for the current state and
time) at least once. 222Strictly speaking, $\mathrm{Forb}_{i}$ is not an event
in ${\mathcal{G}}_{k}^{i}$, since it refers to the memory mode of Maximizer’s
strategy $\sigma$. However, since we fix $\sigma$ first, we can consider the
MDP that is induced by fixing $\sigma$ in ${\mathcal{G}}_{k}^{i}$. Then
Maximizer’s memory mode ${\sf m}$ and the step counter $t$ are encoded into
the states, which are of the form $(s,{\sf m},t)$. In this MDP,
$\mathrm{Forb}_{i}$ is a measurable event, actually an open set. However,
since Maximizer’s memory is private, Minimizer has only partial observation in
this MDP, i.e., she cannot distinguish between states $(s,{\sf m},t)$ and
$(s,{\sf m}^{\prime},t)$. Indeed the Minimizer strategy that we construct does
not assume any knowledge of the memory mode, but instead hedges her bets.

From Eq. 16 we obtain that Maximizer loses ${\mathcal{G}}_{k}^{i}$ (and thus
${\mathcal{G}}_{k+1}$) with high probability if he _never_ enters a forbidden
memory mode.

(17) $\displaystyle\max_{{\sf m}}\sup_{t}$
$\displaystyle\;{\mathcal{P}}_{{\mathcal{G}}_{k+1},c_{i},\sigma[{\sf
m}](t),\pi}(\mathtt{Reach}(\\{c_{0}\\})\cap\overline{\mathrm{Forb}_{i}})$
$\displaystyle\leq\max_{{\sf
m}}\sup_{t}{\mathcal{P}}_{{\mathcal{G}}_{k}^{i},u^{k,i},\sigma^{\prime}[{\sf
m}](t),\pi_{i}}(\mathtt{Reach}(\\{c_{0}^{k,i}\\}))$
$\displaystyle\leq\delta\cdot 2^{-(i+1)}.$

On the other hand, we can show that if Maximizer does enter a forbidden memory
mode (for the current state and time) in ${\mathcal{G}}_{k}^{i}$ then his
chance of playing action “1” (and thus winning ${\mathcal{G}}_{k+1}$ in that
round) after (and if) winning the subgame ${\mathcal{G}}_{k}^{i}$ is small.
This holds for _every_ Minimizer’s strategy inside ${\mathcal{G}}_{k}^{i}$ and
thus in particular this holds for our chosen Minimizer strategy $\pi_{i}$.

Recall that $\pi$ ensures that states in ${\mathcal{G}}_{k}^{i}$ are only
reached at times $t+1$ where $t\geq t_{i}$, and thus Eq. 15 applies. Hence, at
$c_{i}$, Maximizer’s chance of satisfying $\mathrm{Forb}_{i}$ and still
winning the game in this round (without going to $c_{i+1}$ and the next
subgame $G_{k}^{i+1}$) are upper bounded by
$(k+1)(\alpha(i)+\frac{1}{4}2^{-i})$. For all ${\sf m}$ and all $t\geq t_{i}$
we have

(18) $\displaystyle{\mathcal{P}}_{{\mathcal{G}}_{k+1},c_{i},\sigma[{\sf
m}](t),\pi}(\mathtt{Reach}(\\{c_{0}\\})\cap\mathrm{Forb}_{i}\cap\neg\mathtt{Reach}(\\{c_{i+1}\\}))$
$\displaystyle\leq(k+1)(\alpha(i)+\frac{1}{4}2^{-i})$

Since in our current case $\sum_{i}\alpha(i)$ converges, it follows that
$\sum_{i}(k+1)(\alpha(i)+\frac{1}{4}2^{-i})$ also converges, and thus there
exists a sufficiently large $i_{0}\in\mathbb{N}$ such that

(19) $\sum_{i\geq i_{0}}(k+1)(\alpha(i)+\frac{1}{4}2^{-i})\leq\delta/2$

Let

${\it nFo}(i,{\sf
m}^{\prime},t^{\prime})\stackrel{{\scriptstyle\text{{\tiny{def}}}}}{{=}}{\mathcal{P}}_{{\mathcal{G}}_{k+1},c_{i},\sigma[{\sf
m}^{\prime}](t^{\prime}),\pi}(\mathtt{Reach}(\\{c_{0}\\})\cap\overline{\mathrm{Forb}_{i}})$

and

$\begin{array}[]{l}{\it Fo}(i,{\sf
m}^{\prime},t^{\prime})\stackrel{{\scriptstyle\text{{\tiny{def}}}}}{{=}}\\\
{\mathcal{P}}_{{\mathcal{G}}_{k+1},c_{i},\sigma[{\sf
m}^{\prime}](t^{\prime}),\pi}(\mathtt{Reach}(\\{c_{0}\\})\cap\mathrm{Forb}_{i}\cap\neg\mathtt{Reach}(\\{c_{i+1}\\}))\end{array}$

Then from (17), (18) and (19) we obtain that for every initial memory mode
${\sf m}$ and time $t$

$\begin{array}[]{llll}{\mathcal{P}}_{{\mathcal{G}}_{k+1},u,\sigma[{\sf
m}](t),\pi}(\mathtt{Reach}(\\{c_{0}\\})\\\ \leq\sum_{i\geq i_{0}}\max_{{\sf
m}^{\prime}}\sup_{t^{\prime}\geq t_{i}}{\it nFo}(i,{\sf
m}^{\prime},t^{\prime})\\\ +\sum_{i\geq i_{0}}\max_{{\sf
m}^{\prime}}\sup_{t^{\prime}\geq t_{i}}{\it Fo}(i,{\sf
m}^{\prime},t^{\prime})\\\ \leq\sum_{i\geq i_{0}}\delta\cdot
2^{-(i+1)}+\sum_{i\geq i_{0}}(k+1)(\alpha(i)+\frac{1}{4}2^{-i})\\\
\leq\delta/2+\delta/2=\delta\end{array}$

∎