Three-Qubit-Embedded Split Cayley Hexagon is Contextuality Sensitive

Frédéric Holweck1,2, Henri de Boutray3 and Metod Saniga4

1Laboratoire Interdisciplinaire Carnot de Bourgogne, ICB/UTBM, UMR 6303 CNRS,
Université de Bourgogne Franche-Comté, 90010 Belfort Cedex, France

2Department of Mathematics and Statistics, Auburn University, Auburn, AL, USA

3 ColibriTD, La Défense, Paris, France

4Astronomical Institute, Slovak Academy of Sciences, SK-05960 Tatranská
Lomnica, Slovak Republic

(29 January 2022)

Abstract: It is known that there are two non-equivalent embeddings of the
split Cayley hexagon of order two into $\mathcal{W}(5,2)$, the binary
symplectic polar space of rank three, called classical and skew. Labelling the
63 points of $\mathcal{W}(5,2)$ by the 63 canonical observables of the three-
qubit Pauli group subject to the symplectic polarity induced by the
(commutation relations between the elements of the) group, the two types of
embedding are found to be quantum contextuality sensitive. In particular, we
show that the complement of a classically-embedded hexagon is not contextual,
whereas that of a skewly-embedded one is.

Keywords: finite geometry, quantum contextuality, Pauli groups, generalized
polygons, Kochen-Specker Theorem, split Cayley hexagon of order two

##  1 Introduction

Quantum contextuality is one of the most counter-intuitive notions in quantum
physics that is assumed to be of main importance in quantum information [1, 2,
3, 4]. Roughly speaking, quantum contextuality rules out any Hidden Variables
Theory unless one assumes that the pre-existing values of a measurement depend
on the context, i.e. on the set of mutually commuting measurements in which a
given experience takes place. First proved mathematically by Kochen and
Specker [5], several alternative proofs [6, 7] and experimental validations
[8, 9] have been provided since then. For a recent comprehensive survey about
contextuality, see [10].

Among various formulations of quantum contextuality we will deal in this paper
with observable-based proofs of quantum contextuality as proposed by Mermin
[11] and Peres [12]. In this formulation, see Section 2, a proof of the
Kochen-Specker (KS) Theorem is provided by a configuration of sets of mutually
commuting observables, called contexts, such that all the constrains imposed
by the configuration on the eigenvalues of those observables cannot be
satisfied by a classical function unless that function is context-dependent.
We will call such a configuration contextual when it furnishes a proof of the
KS Theorem. These proofs of contextuality can also be tested experimentally
[13, 14, 15]. Interestingly, this type of contextual configurations, which
involves $N$-qubit Pauli observables, can be embedded as subgeometries of the
geometric realization of the $N$-qubit Pauli group known as the symplectic
polar space of rank $N$ and order two, usually denoted as
$\mathcal{W}(2N-1,2)$, see Section 2. This geometric perspective on $N$-qubit
observables was introduced some $15$ years ago [16, 17, 18] and led to several
surprising connections between physics and geometry [19] and also between
different branches of modern physics, like, for instance, the so-called black-
hole/qubit correspondence [20, 21].

In $\mathcal{W}(5,2)$ one can find a number of copies of a very special
configuration made of $63$ points and $63$ lines, called the split Cayley
hexagon of order two. This configuration is a generalized hexagon is the sense
that it is $m$-gon free for $m\leq 5$. The fact that it can be embedded into
the three-qubit polar space, $\mathcal{W}(5,2)$, was first employed in [22] to
establish connections between three-qubit observables and black-hole entropy
formulas. Later it was also proved that this configuration has, in fact, two
distinguished embeddings in $\mathcal{W}(5,2)$, one called classical, the
other skew [23]. In this article one shows that these two distinguished
embeddings behave differently in terms of contextuality. More precisely, if
one denotes the two embeddings by $\mathcal{H}_{C}$ and $\mathcal{H}_{S}$,
respectively, then one demonstrates that the configuration
$\overline{\mathcal{H}}_{C}=\mathcal{W}(5,2)\setminus\mathcal{H}_{C}$, i.e.
the configuration made of all the line-contexts that are in $\mathcal{W}(5,2)$
but not in $\mathcal{H}_{C}$, is not contextual, while
$\overline{\mathcal{H}}_{S}=\mathcal{W}(5,2)\setminus\mathcal{H}_{S}$ is
contextual.

The paper is organized as follows. In Section 2 we recall the basic notions on
observable-based proofs of contextuality as well as the notion of symplectic
polar spaces where corresponding configurations live. We also explain how we
prove that a given configuration is contextual by solving a particular linear
system. In Section 3 one details the differences between the two distinguished
symplectic embeddings of the split Cayley hexagon of order two and recall some
known results about these specific configurations. Finally in Section 4 we
collect our results and state precisely our hexagon-related contextuality
findings. Section 5 concludes this paper with general remarks and
perspectives.

##  2 Observable-based proofs of contextuality and the symplextic polar spaces
over the two-element field

Let us recall the principle of the Mermin and Peres [11, 12] operator-based
proofs of quantum contextuality. Consider the configuration of two-qubit
observables depicted in Figure 1 and known as a Peres-Mermin magic square.

Figure 1: The Peres-Mermin magic square: All two-qubit observables on the same
row/column mutually commute. The product of observables in each row/column is
equal to $\pm I_{4}$ as indicated by the signs. The measurement of each two-
qubit observable yields $\pm 1$. One observes that there is no deterministic
classical function that would predict the outcomes for each node and satisfy
all the signs constraints.

Each node of the square is labelled by a two-qubit observable with a
shorthanded notation $AB\equiv A\otimes B$. Here $A,B\in\\{I,X,Y,Z\\}$ with
$X,Y,Z$ being the usual Pauli matrices and $I$ the unit matrix. In each
row/column of the grid, two-qubit observables mutually commute and form what
we call a context, i. e. a set of compatible measurements whose product is, up
to a sign, the identity operator. In the Peres-Mermin square the product of
the observables on each context is $\pm I_{4}$ as indicated by the signs in
Figure 1. In each context, the product of the measurements, which are
eigenvalues of the observables, should be an eigenvalue of the product of the
observables. The eigenvalues of each node are $\pm 1$ and the constraint on
each context is $\pm 1$ according to the signs prescription on rows/columns.
Because there is an odd number of negative contexts in the square, it is clear
that there exists no classical function that can assign predefinite values to
each node and satisfy all the constrains imposed by the signs of the contexts
(rows/columns). If such a classical function existed, it would necessarily be
context-dependent, i. e. the predefined value assigned to a given node would
depend on the considered context. This elementary configuration of observables
furnishes a proof of the KS Theorem by showing that there is no Non-Contextual
Hidden Variable (NCHV) theory that can reproduce the outcomes of quantum
physics. Another elementary proof based on three-qubit observables was
proposed by Mermin in [11] and is reproduced in Figure 2.

Figure 2: The Mermin pentagram [11]. Each node is labelled by a three-qubit
Pauli observable. The lines of the pentagram are sets of mutually commuting
observables whose product is $+I_{8}$ (each thin line) or $-I_{8}$ (the double
line). The fact that there is only one (odd number) negative line allows us to
use the same argument as for the Peres-Mermin magic square. This configuration
therefore also furnishes a proof of the KS Theorem.

In [24] it was proved that Peres-Mermin magic squares and Mermin pentagrams
are the smallest possible proofs of the KS Theorem that can be achieved in
terms of the number of observables and contexts: $9$ observables and $6$
contexts for the Peres-Mermin magic square and $10$ observables and $5$
contexts for the pentagram.

If one considers two-qubit Pauli observables up to a global phase ($\pm 1$,
$\pm i$), there are $10$ copies of the magic square. One explains now how one
can naturally embed these $10$ grids into the symplectic polar space of rank
$N=2$ over the two-elements field $\mathbb{F}_{2}=\\{0,1\\}$. In full
generality let us consider the group of $N$-qubit Pauli observables, i.e.

$\mathcal{P}_{N}=\\{sA_{1}A_{2}\dots A_{N},s\in\\{\pm 1,\pm
i\\},A_{i}\in\\{I,X,Y,Z\\}\\},$ (1)

employing again the shorthand notation $A_{1}A_{2}\dots A_{N}\equiv
A_{1}\otimes A_{2}\otimes\dots\otimes A_{N}$. The center of $\mathcal{P}_{N}$
is $\mathcal{C}_{N}=\\{\pm I_{2^{N}},\pm iI_{2^{N}}\\}$. The abelian group
$\mathcal{P}_{N}/\mathcal{C}_{N}$ is isomorphic to the $2N$-dimensional vector
space $V_{N}=\mathbb{F}_{2}^{2N}$ as we now explain.

Up to a phase, each observable $A_{i}$ can be represented by a doublet
$(\mu_{i},\nu_{i})\in\mathbb{F}_{2}^{2}$ as follows

$A_{i}=Z^{\mu_{i}}\cdot X^{\nu_{i}}\text{ where }\cdot\text{ denotes the
ordinary product of matrices};$ (2)

more precisely, we have the following relations:

$I\leftrightarrow(0,0),X\leftrightarrow(0,1),Z\leftrightarrow(1,0),iY\leftrightarrow(1,1).$
(3)

Hence, to a given class $\overline{\mathcal{O}}=\\{A_{1}A_{2}\dots
A_{N},-A_{1}A_{2}\dots A_{N},iA_{1}A_{2}\dots A_{N},-iA_{1}A_{2}\dots
A_{N}\\}$ in $\mathcal{P}_{N}/\mathcal{C}_{N}$ one can associate a unique
$2N$-plet $(\mu_{1},\nu_{1},\dots,\mu_{N},\nu_{N})\in\mathbb{F}_{2}^{2N}$.
Moreover, the multiplicative structure of $\mathcal{P}_{N}/\mathcal{C}_{N}$ is
mapped to the additive structure of $V_{N}$. Considering now the projective
space associated with $V_{N}$, $PG(2N-1,2)$, one obtains a map between non-
trivial $N$-qubit observables, up to a global phase, and points of
$PG(2N-1,2)$

$\pi:\left\\{\begin{array}[]{ccc}(\mathcal{P}_{N}/\mathcal{C}_{N})\setminus
I_{2^{N}}&\to&PG(2N-1,2),\\\
\mathcal{O}&\mapsto&[\mu_{1}:\nu_{1}:\mu_{2}:\mu_{2}:\dots:\mu_{N}:\nu_{N}].\end{array}\right.$
(4)

This mapping represents (classes of) observables as points in $PG(2N-1,2)$,
but tells us nothing about the commutation relations in $\mathcal{P}_{N}$. In
order to recover this information, one introduces the following symplectic
form on $PG(2N-1,2)$:

$\langle p,q\rangle=\sum_{i=1}^{N}p_{i}q_{N+i}+p_{N+i}q_{i},$ (5)

where $p=[p_{1}:\dots:p_{2N}]$ and $q=[q_{1}:\dots:q_{2N}]\in PG(2N-1,2)$.
Then, if one considers two non-trivial classes $\overline{\mathcal{O}}_{p}$
and $\overline{\mathcal{O}_{q}}$ of $N$-qubit Pauli observables in
$\mathcal{P}_{N}/\mathcal{C}_{N}$ such that
$\pi(\overline{\mathcal{O}}_{p})=p$ and $\pi(\overline{\mathcal{O}}_{q})=q$,
then a straightforward calculation shows that

$\mathcal{O}_{p}\text{ and }\mathcal{O}_{q}\text{ commute
}\Leftrightarrow\langle p,q\rangle=0.$ (6)

Employing the above-defined symplectic form leads to the definition of
$\mathcal{W}(2N-1,2)$, the symplectic polar space of rank $N$ and order $2$.

######  Definition 1.

The space of totally isotropic subspaces111A linear space is said to be
totally isotropic if and only if the symplectic form vanishes identically on
the space. of $PG(2N-1,2)$ endowed with a nondegenerate symplectic form
$\langle,\rangle$ is called the symplectic polar space of rank $N$ and order
$2$, $\mathcal{W}(2N-1,2)$.

Note that because of the symplectic form, the group $\text{Sp}(2N,2)$ of
symplectic matrices acts transitively on $\mathcal{W}(2N-1,2)$. This group is
spanned by the transvections $T_{p}$ for all $p\in\mathcal{W}(2N-1,2)$:

$T_{p}:\left\\{\begin{array}[]{ccc}\mathcal{W}(2N-1,2)&\to&\mathcal{W}(2N-1,2),\\\
q&\mapsto&q+\langle p,q\rangle p.\end{array}\right.$ (7)

If we consider the action of transvections in terms of the labelling of
$\mathcal{W}(2N-1,2)$ by $N$-qubit Pauli observables, then for
$p=\pi(\overline{\mathcal{O}}_{p})$ and $q=\pi(\overline{\mathcal{O}}_{q})$ we
have

$T_{\mathcal{O}_{p}}(\overline{\mathcal{O}}_{q})=\left\\{\begin{array}[]{ccc}\overline{\mathcal{O}}_{q}&\text{
if and only if }&\overline{\mathcal{O}}_{q}\text{ and
}\overline{\mathcal{O}}_{q}\text{ commute},\\\
\overline{\mathcal{O}_{q}\cdot\mathcal{O}_{p}}&\text{ if and only if
}&\overline{\mathcal{O}}_{q}\text{ and }\overline{\mathcal{O}}_{q}\text{
anticommute}.\end{array}\right.$ (8)

The points of $\mathcal{W}(2N-1,2)$ are $N$-qubit observables, lines
correspond to triplet of mutually commuting elements whose product is $\pm
I_{N}$ and form contexts. Planes and higer dimensional linear subspaces of
$\mathcal{W}(2N-1,2)$ can also be of some use to generate contexts. Let us
illustrate this for $N=2$ and $N=3$ [25, 26].

###  2.1  $N=2$, the Doily $\mathcal{W}(3,2)$

The symplectic polar space of rank $2$ and order $2$, known as the doily,
encapsulates the commutation relations within the two-qubit Pauli group. This
space features $15$ points and $15$ lines and form a $15_{3}$ configuration,
i.e. a point-line incidence structure with $3$ points per line and $3$ lines
per point. Out of $245,342$ non isomorphic $15_{3}$-configurations, the doily
is the only one to be triangle free. Using the canonical representatives
($s=1$ in (1)), one gets a labelling of the doily by two-qubit Pauli
observables as given in Figure 3.

Figure 3: The symplectic polar space $\mathcal{W}(3,2)$ labelled by two-qubit
Pauli observables. Each line of the configuration forms a context. The three
negative contexts are indicated in blue.

Peres-Mermin magic squares are subgeometries of $\mathcal{W}(3,2)$. More
precisely they are geometric hyperplanes of $\mathcal{W}(3,2)$, i.e. sets of
points such that a line of $\mathcal{W}(3,2)$ is either fully contained in
such set or shares with it a single point. An example of such a set is given
in Figure 4. Starting with this canonical labeling one can form $9$ more
copies of Peres-Mermin square by acting by transvections. Each of them have an
odd number, namely $1$ or $3$, of negative contexts and is thus contextual.

Figure 4: A Peres-Mermin square as a subgeometry of the doily. The square is
the same as the one introduced in Figure 1. It is a geometric hyperplane in
the sense that a line of $\mathcal{W}(3,2)$ is either contained in it (six
such lines depicted in red) or has one point common with it (each of the
remaining nine lines).

To conclude this subsection it is worth mentioning that the space
$\mathcal{W}(3,2)$ of its own is a contextual configuration, as first proved
in [13]. In fact, Cabello shows that the configuration corresponding to points
and lines of $\mathcal{W}(2N-1,2)$ is contextual for any $N\geq 2$.

###  2.2  $N=3$, the symplectic polar space $\mathcal{W}(5,2)$

The symplectic polar space of rank $3$, $\mathcal{W}(5,2)$, comprises $63$
points, $315$ lines and $135$ Fano planes. An interesting study of geometric
hyperplanes of $\mathcal{W}(5,2)$ and their physical interpretations in terms
of representation theory and invariants can be found in [26]. Fano planes of
$\mathcal{W}(5,2)$ are totally isotropic $2$-dimensional spaces over
$\mathbb{F}_{2}$. An example of such a plane is given in Figure 5. The product
of the seven observables in a Fano plane yields $\pm I_{8}$. If one removes a
line, one gets a new type of context – an affine plane of order two – made of
four observables whose product is $\pm I_{8}$. For instance, in the negative
Fano plane of Figure 5, removing the line $ZZI-IZZ-ZIZ$, which is positive,
leads to the context $XXX-YYX-YXY-XYY$ that is exactly the negative line of
the Mermin pentagram of Figure 2. Using three-qubit Pauli operator contexts of
this type one can create altogether $12,096$ distinct Mermin’s pentagrams in
$\mathcal{W}(5,2)$, as it was first shown by computer calculations [27] and
later proved also rigorously by sole geometric arguments [28].

Figure 5: An example of the (negative) Fano plane in $\mathcal{W}(5,2)$; the
blue line is also negative.

In [29] it was demonstrated that geometric hyperplanes of $\mathcal{W}(5,2)$
defined by quadratic equations (hyperbolic or elliptic) are also contextual
configurations; contextual inequalities for the hyperbolic case were also
tested on a quantum computer [15].

In Section 3 we will introduce another prominent configuration, namely the
split Cayley hexagon of order two, which lives in $\mathcal{W}(5,2)$ in two
distinct modes featuring different contextuality behavior.

###  2.3 A simple method for checking contextuality of a configuration

In order to prove that a particular point-line configuration of observables,
i.e. a set of observables $\\{\mathcal{O}_{j}\\}_{j\in J}$ grouped into
contexts $\\{\mathcal{C}_{i}\\}_{i\in I}$, provides a KS proof, we reformulate
the question as a linear problem [29]. Indeed, the existence of an NCHV theory
that accommodates all contexts $\mathcal{C}_{i}$ of a configuration means that
there exists a classical function, $f$, such that $f(\mathcal{O}_{j})=\pm 1$
for all observables and for all contexts $\mathcal{C}_{i}$ of the
configuration. This means that $f$ satisfies the following system of
equations:

$\Pi_{\mathcal{O}_{j}\in\mathcal{C}_{i}}f(\mathcal{O}_{j})=\text{sign}(\mathcal{C}_{i}),\forall
i\in I,$ (9)

where $\text{sign}(\mathcal{C}_{i})$ is the sign of the context
$\mathcal{C}_{i}$.

Consider now the incidence matrix $A$ of the configuration, i. e. an $I\times
J$ matrix where $a_{ij}=1$ if and only if $\mathcal{O}_{j}\in\mathcal{C}_{i}$
and $a_{ij}=0$ otherwise. Let $b\in\mathbb{F}_{2}^{I}$ such that $b_{i}=0$ if
and only if $\text{sign}(\mathcal{C}_{i})=1$ and $b_{i}=1$ if and only if
$\text{sign}(\mathcal{C}_{i})=-1$. Then the existence of a classical function
$f$ that satifies the constrains of Eq. (9) boils down to finding a vector
$x\in\mathbb{F}_{2}^{J}$ which is the solution of the following linear system

$Ax=b.$ (10)

For instance, in the case of the Peres-Mermin configuration of Figure 1 the
linear system to be solved over the two-element field is

$\begin{pmatrix}1&1&1&0&0&0&0&0&0\\\ 0&0&0&1&1&1&0&0&0\\\ 0&0&0&0&0&0&1&1&1\\\
1&0&0&1&0&0&1&0&0\\\ 0&1&0&0&1&0&0&1&0\\\ 0&0&1&0&0&1&0&0&1\\\
\end{pmatrix}\cdot\begin{pmatrix}x_{1}\\\ x_{2}\\\ x_{3}\\\ x_{4}\\\ x_{5}\\\
x_{6}\\\ x_{7}\\\ x_{8}\\\ x_{9}\end{pmatrix}=\begin{pmatrix}0\\\ 0\\\ 0\\\
0\\\ 0\\\ 1\end{pmatrix}.$ (11)

One readily sees that this system has no solution as the sum of the first five
rows of $A$ equals the last row, while the sum of the first five coordinates
of $b$ does not equal the last coordinate. The lack of solution means that the
corresponding configuration is contextual, i. e. it furnishes a proof of the
KS Theorem. This linear formulation of the problem allows us to get a
relatively fast computer-based check of the contextual nature of larger sets
of contexts and observables. Note that the incidence matrix $A$ encodes the
geometry of the configuration, while the vector $b$ encodes the signs of the
contexts; the latter, of course, depends on the choice of labeling of the
points of the symplectic polar space by $N$-qubit observables.

##  3 The split Cayley hexagon of order two and its two non-equivalent
symplectic embeddings

A generalized $n$-gon $\mathcal{G}$ of order $(k,l)$ is an point-line
incidence structure such that every line contains $k+1$ points, every point is
contained in $l+1$ lines and $\mathcal{G}$ does not contain any ordinary
$m$-gons for $2\leq m<n$, but two points, two lines or a point and a line are
always contained in an $n$-gon [30]. When $k=l$ one says that the order of
$\mathcal{G}$ is $k$. The Fano plane, Figure 5, is the unique example of the
generalized triangle of order two and the doily $\mathcal{W}(3,2)$, Figure 3,
is the unique (self-dual) generalized quadrangle of order two. There is no
generalized $5$-gon of order two, but there exist two generalized hexagons of
order two, the split Cayley hexagon and its dual [30]. We will focus now on
the split Cayley hexagon, $\mathcal{H}$, which also lives in
$\mathcal{W}(5,2)$; this means that one can label the $63$ points of
$\mathcal{H}$ by the $63$ non trivial three-qubit observables in such a way
that the lines of the hexagon are lines of $\mathcal{W}(5,2)$, i. e. sets of
mutually commuting three-qubit observables.

As already stressed, there are two unequivalent embeddings of $\mathcal{H}$ in
$\mathcal{W}(5,2)$. The first one, called classical, was explicitly worked out
in [22] and in the context of quantum information further discussed, for
instance, in [20, 27, 32]; an example of such embedding is portrayed in Figure
6.

Figure 6: Three-qubit classical embedding of the split Cayley hexagon of order
two as described in [22]. The illustration of points and lines of the hexagon
follows closely that of Ref. [31]

.

In $\mathcal{W}(5,2)$ there are $120$ classically-embedded copies of
$\mathcal{H}$; each of them can be obtained from the copy shown in Figure 6 by
the action of the symplectic group $\text{Sp}(6,2)$. The labelling of Figure 6
was obtained from the observation that the split Cayley hexagon contains a
copy of the Heawood graph, the incidence graph of the Fano plane, and
following the procedure outlined in [33].

Another description of this embedding is the following one [30]. In $PG(6,2)$,
consider the following parabolic quadric $\mathcal{Q}$

$x_{1}x_{4}+x_{2}x_{5}+x_{3}x_{6}-x_{7}^{2}=0.$ (12)

This quadric contains $63$ points. Consider now the lines of $\mathcal{Q}$
that satisfy the following Plücker equations:

$\begin{array}[]{cccc}p_{62}=p_{17},&p_{13}=p_{72},&p_{24}=p_{37},&p_{35}=p_{74},\\\
p_{46}=p_{57},&p_{51}=p_{76},&p_{14}+p_{25}+p_{36}=0,&\end{array}$ (13)

where $p_{ij}=x_{i}x_{j}-x_{j}x_{i}$. There are $63$ such lines and the
obtained $63_{3}$ configuration is isomorphic to $\mathcal{H}$. This
configuration can be bijectively projected into $PG(5,2)$ and
$\mathcal{W}(5,2)$. Indeed the projection

$[x_{1}:\dots:x_{7}]\mapsto[x_{1}:\dots:x_{6}]$ (14)

is a bijection over $\mathbb{F}_{2}$ as
$x_{7}=x_{1}x_{2}+x_{3}x_{4}+x_{5}x_{6}$ and the lines of $\mathcal{Q}$ that
satisfy the Plücker relations (13) are mapped to totally isotropic lines of
$PG(5,2)$, i. e. the lines of $\mathcal{W}(5,2)$ for the sympletic form (5).
This embedding will be denoted by $\mathcal{H}_{C}$.

Interestingly, there exists another, non-equivalent embedding of $\mathcal{H}$
into $\mathcal{W}(5,2)$, as discovered by Coolsaet [23]. Coolsaet considers,
in $PG(6,2)$, the following coordinate map

$\epsilon:[x_{1}:\dots:x_{7}]\mapsto[x_{1}+x_{6}+f_{5}(x):x_{2}+x_{3}+f_{4}(x):x_{3}:x_{4}:x_{5}:x_{6}:x_{7}]$
(15)

with $f_{4}(x)=x_{3}x_{5}+x_{7}x_{4}$ and $f_{5}(x)=x_{4}x_{6}+x_{7}x_{5}$,
which indeed establishes a different type of embedding of $\mathcal{H}$ into
$\mathcal{Q}$, called skew; its projection to $\mathcal{W}(5,2)$ will be
denoted by $\mathcal{H}_{S}$. There are altogether $7560$ copies of
$\mathcal{H}_{S}$ in $\mathcal{W}(5,2)$; one of them is depicted in Figure 7.

Figure 7: A copy of the split Cayley hexagon of order two that is skew-
embedded into $\mathcal{W}(5,2)$.

We shall conclude this section by a brief description of the main difference
between the two embeddings. In $\mathcal{H}_{C}$, all the three lines passing
through any of its points are coplanar, i. e. lie in the same plane of
$\mathcal{W}(5,2)$. This is, however, not the case for $\mathcal{H}_{S}$,
where only 15 points exhibit this property; for each of the remaining 48
points, only two of the three lines are coplanar. To illustrate this, let us
take the point $XXX$ of the copy shown in Figure 7. The three lines passing
through it are $\\{XXX,XYY,IZZ\\}$, $\\{XXX,ZZI,YYX\\}$ and
$\\{XXX,IYZ,XZY\\}$; clearly, only the first two lie in the same plane. Next,
take any observable and all the 30 other observables that commute with it.
These 31 observables will form a geometric hyperplane in both
$\mathcal{H}_{C}$ and $\mathcal{H}_{S}$. But the two embeddings differ in that
while this hyperplane is of the same type for each point of $\mathcal{H}_{C}$,
an $\mathcal{H}_{S}$ features two different kinds of them.

##  4  $\overline{\mathcal{H}_{S}}$ is contextual, whereas
$\overline{\mathcal{H}}_{C}$ is not!

At this point an interesting question arises: Can the two embeddings be
ascribed some physical distinction? To answer this question (in affirmative),
we ran several different experiments on
$\mathcal{W}(5,2),\mathcal{H}_{C}^{\prime}s,\mathcal{H}_{S}^{\prime}s$ and
their complements. By the complement, $\overline{\mathcal{G}}$, of a
configuration $\mathcal{G}\subset\mathcal{W}(2N-1,2)$ we mean the set of all
line-contexts of $\mathcal{W}(2N-1,2)$ that are not in $\mathcal{G}$. We
tested by computer the contextual nature of all
$\mathcal{H}_{C},\mathcal{H}_{S},\overline{\mathcal{H}}_{C},\overline{\mathcal{H}}_{S}$
of $\mathcal{W}(5,2)$ and the results of our analysis are summarized in Table
1.

Geometry | Contextual | # of Copies
---|---|---
$\mathcal{H}_{C}$ | No | $120$
$\mathcal{H}_{S}$ | No | $7560$
$\overline{\mathcal{H}}_{C}$ | No | $120$
$\overline{\mathcal{H}}_{S}$ | Yes | $7560$
Table 1: Results on the contextuality properties of differently-embedded
hexagons and their complements. The difference between the two embeddings
reveals itself in terms of contextuality when one considers the complement of
the configuration.

Our calculations were performed with the software Magma and the computation
resources of the supercomputer of the Mésocentre de calcul de Franche-Comté.
All our codes and results are available on the QuantCert GitHub page222 URL:
https://quantcert.github.io/Magma-contextuality/. To implement the incidence
structures of $\mathcal{H}_{C}$ and $\mathcal{H}_{S}$ we first encoded the
description in coordinates of the two embeddings provided by Coolsaet [23]
(Eq. (12), Eq. (13) and Eq. (15)). Then we created all $120$ copies of
$\mathcal{H}_{C}$ and all $7560$ copies of $\mathcal{H}_{S}$ by repeatedly
acting by transvections. Their complements were easily obtained from the
implementation of $\mathcal{W}(5,2)$. Once all incidence structures were
created we checked their contextuality by using the function IsConsistent that
establishes if a given configuration is contextual by employing the procedure
described in Section 2.

The IsConsistent Magma intrinsic (compiled function) checks if the rank of the
augmented matrix $(A|b)$ is greater that the one of the coefficient matrix
$A$. This operation is in $O(|\mathcal{O}|+|\mathcal{C}|)$ operations; in
other words, its execution lasts a time proportional to the number of
observables and contexts of the geometry. Given the fact that this precise
system had $315-63=252$ contexts, the duration of this operation is quite
negligible, even repeated several thousands of times. Building the 7560 copies
of $\mathcal{H}_{S}$ was more intensive though: to reach all copies we started
with a single copy of $\mathcal{H}_{S}$ and ran every possible combination of
$4$ transvections, including the identity, for a total of
$64^{4}=2^{2}4\approx 17E6$ total operations. This computation lasted around
11 hours (10.87 to be more precise), which speaks for the efficiency of Magma.

##  5 Conclusion

We have found a very important physical property regarding the two non-
equivalent embeddings of a very special subconfiguration of the symplectic
polar space $\mathcal{W}(5,2)$, the split Cayley hexagon of order two. Using
the interpretation in terms of three-qubit Pauli observables we showed that
the complement of any skew-embedded hexagon is a contextual configuration, i.
e. provides a proof of the Kochen-Specker Theorem. We tested our findings on
all possible embeddings of the split Cayley hexagons and found out that only
skew embeddings enjoy this property. To extend this work one may try to
measure the degree of contextuality of $\overline{\mathcal{H}}_{S}$. The
degree of contextuality indicates how far is a given contextual configuration
to be satisfiable, i.e. non-contextual, if one changes the constrains imposed
by the vector $b$. In other words, if $|J|$ is the number of contexts and $P$
the maximum number of constraints that can be satisfied, then the degree of
contextuality is $d=|J|-P$. For a non-contextual configuration $d=0$. The
calculation of $d$ boils down to finding the Hamming distance between $b$ and
the image of $A$, the incidence matrix of the configuration. However, in the
case of the split Cayley hexagon a brute force calculation to compute $d$ is
out of reach of the supercomputer resources we have currently at our disposal.
It, therefore, requires a deeper understanding of the geometry to reduce the
calculation cost. The degree of contextuality $d$ is also necessary for
calculating the classical bound of the contextual inequalities given in [13]
as well as for a possible testing of these inequalities on a quantum computer
[15].

##  6 Acknowledgements

This work was supported by the Conseil Régional de Bourgogne Franche-Comté,
mobility grant GeoQuant, the Thomas Jefferson Fundation and by the Slovak VEGA
Grant Agency, Project $\\#$ 2/0004/20. Part of this work was done during a
visit of HdB and a one-year stay of FH at the Mathematics and Statistics
Department of Auburn University. Both authors wish to acknowledge the people
of the department that made our work here possible, in particular Dr. Luke
Oeding for his precious help and friendship as well as Drs. Gaetan Bakalli and
Mucyo Karemera for their hospitality and enthusiastic discussions about
mathematics and life.

## References

  * [1] Howard, M., Wallman, J., Veitch, V., & Emerson, J. (2014). Contextuality supplies the ‘magic’for quantum computation. Nature, 510(7505), 351-355.
  * [2] Bermejo-Vega, J., Delfosse, N., Browne, D. E., Okay, C., & Raussendorf, R. (2017). Contextuality as a resource for models of quantum computation with qubits. Physical review letters, 119(12), 120505.
  * [3] Lillystone, P., Wallman, J. J., & Emerson, J. (2019). Contextuality and the single-qubit stabilizer subtheory. Physical review letters, 122(14), 140405.
  * [4] Shahandeh, F. (2021). Quantum computational advantage implies contextuality. arXiv preprint arXiv:2112.00024.
  * [5] Kochen, S., & Specker, E. P. (1975). The problem of hidden variables in quantum mechanics. In The logico-algebraic approach to quantum mechanics (pp. 293-328). Springer, Dordrecht.
  * [6] Lisoněk, P., Badziag, P., Portillo, J. R., & Cabello, A. (2014). Kochen-Specker set with seven contexts. Physical Review A, 89(4), 042101.
  * [7] Cabello, A., Kleinmann, M., & Portillo, J. R. (2016). Quantum state-independent contextuality requires 13 rays. Journal of Physics A: Mathematical and Theoretical, 49(38), 38LT01.
  * [8] Bartosik, H., Klepp, J., Schmitzer, C., Sponar, S., Cabello, A., Rauch, H., & Hasegawa, Y. (2009). Experimental test of quantum contextuality in neutron interferometry. Physical review letters, 103(4), 040403.
  * [9] D’Ambrosio, V., Herbauts, I., Amselem, E., Nagali, E., Bourennane, M., Sciarrino, F., & Cabello, A. (2013). Experimental implementation of a Kochen-Specker set of quantum tests. Physical Review X, 3(1), 011012.
  * [10] Budroni, C., Cabello, A., Gühne, O., Kleinmann, M., & Larsson, J. Ǎ. (2021). Quantum contextuality. arXiv preprint arXiv:2102.13036.
  * [11] Mermin, N. D. (1993). Hidden variables and the two theorems of John Bell. Reviews of Modern Physics, 65(3), 803.
  * [12] Peres, A. (1990). Incompatible results of quantum measurements. Physics Letters A, 151(3-4), 107-108.
  * [13] Cabello, A. (2010). Proposed test of macroscopic quantum contextuality. Physical Review A, 82(3), 032110.
  * [14] Dikme, A., Reichel, N., Laghaout, A., & Björk, G. (2020). Measuring the Mermin-Peres magic square using an online quantum computer. arXiv preprint arXiv:2009.10751.
  * [15] Holweck, F. (2021). Testing quantum contextuality of binary symplectic polar spaces on a Noisy Intermediate Scale Quantum Computer. Quantum Information Processing 7/2021
  * [16] Saniga, M., & Planat, M. (2007). Multiple Qubits as Symplectic Polar Spaces of Order Two. Advanced Studies in Theoretical Physics, 1, 1-4.
  * [17] Thas, K. (2009). The geometry of generalized Pauli operators of N-qudit Hilbert space, and an application to MUBs. EPL (Europhysics Letters), 86(6), 60005.
  * [18] Havlicek, H., Odehnal, B., & Saniga, M. (2009). Factor-group-generated polar spaces and (multi-) qudits. SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, 5, 096.
  * [19] Saniga, M., Planat, M., & Rosu, H. (2004). Mutually unbiased bases and finite projective planes. Journal of Optics B: Quantum and Semiclassical Optics, 6(9), L19.
  * [20] Lévay, P., Saniga, M., Vrana, P., & Pracna, P. (2009). Black hole entropy and finite geometry. Physical Review D, 79(8), 084036.
  * [21] Lévay, P., & Holweck, F. (2019). Finite geometric toy model of spacetime as an error correcting code. Physical Review D, 99(8), 086015.
  * [22] Lévay, P., Saniga, M., & Vrana, P. (2008). Three-qubit operators, the split Cayley hexagon of order two, and black holes. Physical Review D, 78(12), 124022.
  * [23] Coolsaet, K. (2010). The smallest split Cayley hexagon has two symplectic embeddings. Finite Fields and Their Applications, 16(5), 380-384.
  * [24] Holweck, F., & Saniga, M. (2017). Contextuality with a small number of observables. International Journal of Quantum Information, 15(04), 1750026.
  * [25] Saniga, M., Planat, M., Pracna, P., & Havlicek, H. (2007). The Veldkamp space of two-qubits. SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, 3, 075.
  * [26] Lévay, P., Holweck, F., & Saniga, M. (2017). Magic three-qubit Veldkamp line: A finite geometric underpinning for form theories of gravity and black hole entropy. Physical Review D, 96(2), 026018.
  * [27] Planat, M., Saniga, M., & Holweck, F. (2013). Distinguished three-qubit ‘magicity’via automorphisms of the split Cayley hexagon. Quantum information processing, 12(7), 2535-2549.
  * [28] Lévay, P., Planat, M., & Saniga, M. (2013). Grassmannian connection between three-and four-qubit observables, Mermin’s contextuality and black holes. Journal of High Energy Physics, 2013(9), 1-35.
  * [29] de Boutray, H., Holweck, F., Giorgetti, A., & Masson, P. A. (2021). Automated synthesis of contextuality proofs from subspaces of symplectic polar spaces. arXiv preprint arXiv:2105.13798.
  * [30] Maldeghem, van H., Generalized Polygons, Monographs in Mathematics 93, Birkhäuser, Basel, 1998.
  * [31] Schroth, A. E. (1999). How to draw a hexagon. Discrete Math, 199, 161-171.
  * [32] Saniga, M., Planat, M., Pracna, P., & Lévay, P. (2012). ’Magic’Configurations of Three-Qubit Observables and Geometric Hyperplanes of the Smallest split Cayley hexagon. SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, 8, 083.
  * [33] Polster, B., Schroth, A. E., & Van Maldeghem, H. (2001). Generalized flatland. The Mathematical Intelligencer, 23(4), 33-47.