But hyperedges that connect two vertices that do not both belong to the
aforementioned 6 vertices, e.g., 15 or 1Q, are not taken into account in this
calculation at all. Let us see how this can be amended.

The string of the 30-108 MMP given in Appendix LABEL:app2 offers us the
following probabilities. Vertex 1 is in the hyperedge 1234 and has the
probability of $\frac{1}{4}$ of being detected. But it is also in the
following 7 hyperedges 15, 18, 1F, 1G, 1I, 1K, and 1Q within each of which it
has the probability $\frac{1}{2}$ of being detected. The arithmetic mean of
these probabilities is $(7\frac{1}{2}+\frac{1}{4})/8=\frac{15}{32}$. There are
16 such blocks. The rest are organized in 7-hyperedges blocks: four
$(5\frac{1}{2}+2\frac{1}{4})/7=\frac{3}{7}$, four
$(6\frac{1}{2}+\frac{1}{4})/7=\frac{13}{28}$, and six
$7\frac{1}{2}/7=\frac{1}{2}$. The total sum of probabilities probabilities is
$\frac{197}{14}=14.07=\alpha_{r}^{*}$, which differs from
$\alpha^{*}(\Gamma)=9$ [magic-14, Proof of Th. 1]

$\displaystyle
8=\alpha(\Gamma)<14.07=\alpha_{r}^{*}(\Gamma)\neq\alpha^{*}(\Gamma)=9.$ (31)

The question arises whether $\alpha^{*}(\Gamma)=\alpha_{r}^{*}(\Gamma)$ under
another approach. We discus this below. In any case it does not seem correct
to assign the probabilities $\frac{1}{2}$ and $\frac{1}{4}$ to the
aforementioned 6 and 24 vertices based on their containment in the 2-vertex-
and 4-vertex-hyperedges, respectively, and ignore their containment in 86
hyperedges that connect vertices in the 2-vertex-hyperedges with those in the
4-vertex-hyperedges or two vertices in different 4-vertex-hyperedges via
2-vertex-hyperedges. For the time being, let us elaborate on discarding two
states from a tensor product of states of two qubits.

Any two mutually orthogonal vertices from two-vertex-hyperedges in the 30-108
MMP belong to an edge and therefore to two qubits. The two-vertex-hyperedge
only means that two of four states are discarded. Thus 202 of 232 vertices are
excluded. Each qubit has a coordinatization and a complete measurement of each
edge must involve all four vertex states, i.e., only complete states (both
vertices from each hyperedges) can build up tensor products of two qubits. The
quantum systems must pass out-ports no matter whether we take them into
account in a later elaboration of our data or not. Related to that, the claim
that $\alpha=8$ should be clarified, because, e.g., in Fig. 16 we see that the
vertices 7,9,D,K,Q,N,S,U,V,n, etc., are independent and yield the independence
number $\alpha\geq 101$ that violates the inequalities $\alpha<\alpha^{*}$ and
$\alpha<\alpha_{r}^{*}$. It remains to be explained how come that vertices
with $m=1$ do not contribute to the independence set when they actually build
up a KS MMP hypergraph 232-108. If it were a result of the construction of
$\Gamma$, then the role of $\alpha=p^{3}=8$ should also be explained, because
there are contextual critical non-binary MMP hypergraphs with higher $\alpha$
which violate Eq. (24) (see Table 4).

When we take into account all vertices of all qubits we get $\Gamma$(filled)
232-108 KS MMP, whose string is given in Appendix LABEL:app2. Its parameters
are given in Table 5. It is not critical, and the only critical set contained
in it, that we obtained, is the 152-71 critical KS MMP shown in Fig. 16(d)
(where we dropped $m=1$ vertices) whose hypergraph string is also given in the
Appendix LABEL:app2. It satisfies
$64\leq\alpha(152$-$71)>\alpha_{r}^{*}(152$-$71)=38$. The string of 152-71
with dropped $m=1$ vertices—24-71 MMP—is given in the Appendix LABEL:app2. It
has 5 independent vertices: 4,8,K ($m=7$) and Q,M ($m=6$). Their parameters
are also given in Table 5. We obtain
$5=\alpha(24$-$71)<\alpha_{r}^{*}(24$-$71)=7.12$.

Table 5: Terms for the inequalities of 4-dim contextual non-binary $\Gamma$ MMP hypergraphs from Fig. 16: $\alpha_{r}^{*}$-inequality: $\alpha\leq\alpha_{r}^{*}$ (violated for 232-108 and 152-71), v-inequality: $HI^{m}_{cM}<l$, and eMax-inequality: $l_{cM}<l$; $m_{M}$ is the maximal $m$; computer search for vectors formed from simple components ($0,\pm 1,\pm i,\pm\omega,\pm 2,\pm\sqrt{2},\pm 3,\pm 5$) failed; finding of $HI_{cm},l_{cM},l_{cm}$ for 30-180 and 24-71 requires tweaking of the program One so as to provide us with the parameters when some hyperedges (from an $n$-dim space) contain less than $n$ vertices, what we have not done as of yet; note that non-critical 30-108, 232-108, and 24-71 MMPs generate thousands of smaller non-binary $\overline{\rm subhypergraphs}$. dim | KS hypergraphs | $HI_{cM}$ | $HI_{cm}$ | $l_{cM}$ | $l_{cm}$ | $l$ | $m_{M}$ | $\alpha_{r}^{*}$ | crit. | vector
---|---|---|---|---|---|---|---|---|---|---
$\leftrightarrow\alpha$ | components
$\Gamma$ MMPs | 30-108 | 8 | - | - | - | 108 | 8 | 14.07 | no | ?
232-108 | 101 | 59 | 107 | 101 | 108 | 8 | 58 | no | ?
| 24-71 | 5 | - | - | - | 70 | 7 | 7.12 | no | ?
| 152-71 | 64 | 41 | 70 | 64 | 71 | 7 | 38 | yes | ?

While deriving their fractional independence number inequality (Eq. (24)) for
their $\Gamma$ graphs, Howard, Wallman, Veitech, and Emerson [magic-14] refer
to the paper of Cabello, Severini and Winter [cabello-severini-winter-14] who
claim that the GLS inequality is a noncontextuality inequality. In contrast,
Theorem 4.20 shows that for quantum YES-NO measurements carried out on MMP
hypergraphs for which the raw data statistics 4.13(a) is formed, the
$\alpha^{*}$-inequality (27) should reduce to the $\alpha^{*}_{r}$-inequality
(24) and therefore cannot be considered a noncontextuality inequality since
arbitrarily many contextual and noncontextual MMP hypergraphs violate it, as
exemplified in Figs. 6(a-e), 9(a,h), 12(d), and 13(b).

On the other hand, in order to deal with the 30-108 MMP hypergraph we first
have to implement 232-108, measure all 232 vertices within their hyperedges
and only then postselect 30 vertices to form 30-108 MMP hypergraph and prove
the contextuality. But the 232-108 KS MMP has far too intricate a
coordinatization for an implementation. We tried to generate it from simple
vector components but did not get anything within months of running our
programs on a supercomputer.

Besides, there are practically arbitrary many simpler non-binary contextual
MMP hypergraphs that can be automatically generated and whose contextuality
can be automatically verified via existing algorithms and programs which then
satisfy the inequalities (19) or (20). Here, one should only answer the
question of what such an inequality for an MMP hypergraph offers to a quantum
computer once it has already been verified that it is contextual.

###  5.5 Peres-Mermin non-binary MMP hypergraphs and the smallest MMP
hypergraph that exists revisited

In Sec. 4.1, Eq. (4), we referred to an operator-based inequality for the
4-dim KS 18-9 MMP hypergraph and in Sec. 4.3, Eqs. (28)-(29), we consider an
analogous operator-based inequality for a general critical MMP hypergraph and
for the 21-11 MMP, in particular. In both cases the operators are defined via
vectors/states/vertices of a given MMP hypergraph. In contrast, the so-called
Peres-Mermin square is defined via operators alone, i.e., without a vector-
defined set underlying the operator set. The operator set is defined by means
of the following nine operators [mermin90]:

$\displaystyle\Sigma_{1}=\sigma_{z}^{(1)}\\!\otimes\\!I^{(2)},\
\Sigma_{2}=I^{(1)}\\!\otimes\\!\sigma_{z}^{(2)},\
\Sigma_{3}=\sigma_{z}^{(1)}\\!\otimes\\!\sigma_{z}^{(2)},$
$\displaystyle\Sigma_{4}=I^{(1)}\\!\otimes\\!\sigma_{x}^{(2)},\
\Sigma_{5}=\sigma_{x}^{(1)}\\!\otimes\\!I^{(2)},\
\Sigma_{6}=\sigma_{x}^{(1)}\\!\otimes\\!\sigma_{x}^{(2)},$
$\displaystyle\Sigma_{7}=\sigma_{z}^{(1)}\\!\otimes\\!\sigma_{x}^{(2)},\
\Sigma_{8}=\sigma_{x}^{(1)}\\!\otimes\\!\sigma_{z}^{(2)},\
\Sigma_{9}=\sigma_{y}^{(1)}\\!\otimes\\!\sigma_{y}^{(2)}.$ (32)

The Peres-Mermin square schematic is shown in the figure (a) below. The square
has 9 dots and 6 lines and it is claimed that the Peres-Mermin square which is
“convert[ible] … to KS vectors” [budroni-cabello-rmp-22, p. 8] “exhibits SIC
[state independent contextuality]” [asadian-15].

Why, then, do Cabello, Kleinmann and Portillo claim that “according to quantum
theory, no SIC set with less than 13 rays exists” [cabello-klein-port-prl-14]?

Do they have some particular vector-based set which one can derive from the
operator-based Peres-Mermin square in mind?

Because, it might be argued that the Peres-Mermin square is not a SIC set and
even not a consistent contextual set in the following sense. Eqs. (9) and (32)
show that there are three operators in each row and/or column which multiply
so as to give $\pm I$. But there is no common eigenstate or a combination of
eigenstates $|\psi\rangle$ of operators $\Sigma_{j}$ which would
counterfactually enable $\Sigma_{j}|\psi\rangle=\pm|\psi\rangle$. Hence, their
classical counterparts $S_{j}$ (Eq. (10)) cannot be assigned values $\pm 1$,
either. In other words, since such counterfactually assumed clicks of
nondestructive measurements carried out via each of operators $\Sigma_{j}$
cannot occur, assignments of $\pm 1$ to classical counterparts of these
assumed measurements is ungrounded.

This statement is at odds with the overwhelming acceptance and acclaim of the
Peres-Mermin square as a contextual set and a KS proof in the literature. Let
us dissect it.

######  Peres-Mermin contradiction. 5.3.

  1. (i)

We have

$\displaystyle\Sigma_{1}\Sigma_{2}\Sigma_{3}=\Sigma_{4}\Sigma_{5}\Sigma_{6}=\Sigma_{7}\Sigma_{8}\Sigma_{9}=\Sigma_{1}\Sigma_{4}\Sigma_{7}=\Sigma_{2}\Sigma_{5}\Sigma_{8}=I,\qquad\Sigma_{3}\Sigma_{6}\Sigma_{9}=-I;$
(33)

  2. (ii)

all operator/projector-based or hypergraph-based contextual or KS sets assume
YES-NO measurements (counterfactual or actual) carried out on systems emerging
from prepared gates; upon leaving a gate determined by operators/projectors or
vectors/vertices the systems are projected to or detected in a particular
state or not; so, a quantum system in state $|\Psi\rangle$ which enters
sequences of three gates, whose actions are described by operators
$\Sigma_{j}$ from Eq. (32), should either counterfactually or actually emerge
from each of the gates either in the state $|\Psi\rangle$ or in the state
$-|\Psi\rangle$;

  3. (iii)

a classical set of states of a classical system which would be a counterpart
of the quantum system described in (ii) should experience predetermined
actions of classical gates described by observables $S_{j}$ which would assign
either ‘$1$’ or ‘$-1$’ to each state;

  4. (iv)

there is no state $|\Psi\rangle$ for which we would have

$\displaystyle\Sigma_{i}|\Psi\rangle=|\Psi\rangle\qquad{\rm
and}\qquad\Sigma_{j}|\Psi\rangle=-|\Psi\rangle\qquad$ (34)

for $i,j\in\\{1,\dots,9\\};\ i\neq j$;

  5. (v)

statements (ii) and (iv) contradict each other, so the statement (iii) cannot
hold either.

To be more specific, let us consider the following case. It is generally
assumed that the Peres-Mermin contradiction is state independent and Eq. (33)
is offered as a support for the claim. Our point is that a quantum system in
state $|\Psi\rangle$ has to pass three gates in a succession; say first
through $\Sigma_{1}$, then through $\Sigma_{2}$, and finally through
$\Sigma_{3}$ so as to emerge in states $\pm|\Psi\rangle$. But to which
counterfactual (not performed) quantum measurements (“clicks”) assumed
valuations of classical counterparts $S_{i}$ might correspond? According to
point $(iv)$ eigenvalues of $\Sigma_{i}$ cannot play such a role. For an
arbitrary state, say the triplet
$|\Psi^{+}\rangle=|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle$, we
obtain:
$\Sigma_{1}\Psi^{+}\rangle=|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle$
and
$\Sigma_{2}|\Psi^{+}\rangle=|\uparrow\uparrow\rangle+|\downarrow\downarrow\rangle$
The observables change the triplet state into other states (singlet and
another triplet). So, the operators do not act on the same state and this is
the meaning of point $(ii)$ above.

Our conclusion is that it is inconsistent to assume the existence of a
classical observable $S_{j}$ which would assign $\pm 1$ to the states of a
system because there is no quantum state $|\Psi\rangle$ of a system which
$\Sigma_{j}$ would project to states $\pm|\Psi\rangle$. Since the
noncontextuality cannot be formulated we cannot talk about Peres-Mermin square
contextuality either.

On the other hand, Budroni, Cabello, Gühne, Kleinmann and Larsson claim
[budroni-cabello-rmp-22, p. 8]: “The [Peres-Mermin] magic square can be
converted into a standard proof of the KS theorem with vectors [from]
[peres].” This is, however, incorrect. The Peres-Mermin square cannot be
converted to the vectors given in [peres, Table 2]. When properly organized in
hyperedges (not explicitly carried out in [peres]) those hyperedges yield one
of 1,233 KS MMP hypergraphs [pmm-2-09] contained in the master MMP hypergraph
24-24. The latter set is obtained by adding further hyperedges (not provided
in [peres]) to the former ones [mpglasgow04-arXiv-0] “although [Peres] has
most probably never tried to identify all 24 tetrads for his 24 vectors.” But
there exists no conversion of the Peres-Mermin square given on p. L179 of
[peres] to or from any of 1,233 KS MMP hypergraphs generated by vectors from
Table 2 on p. L177 of [peres]. They are simply independently given in the same
paper.

Let us therefore see whether we can arrive at vectors for which some plausible
linkage with the Peres-Mermin operators would be possible. The idea is to
establish a correspondence with the Pauli operators $\Sigma$ structure, as a
contextual “square of orthogonalities” which would support a postselection of
9 measurements in a $3\times 3$ arrangement of YES-NO measurements shown in
Fig. 17:

Figure 17: (a) Peres-Mermin operator schematics; (b,c) filled MMPs
(noncontextual); (d) filled MMP with extended orthogonalities and gray $m=1$
vertices (noncontextual); (e) MMP with extended orthogonalities and $m=1$
vertices dropped—contextual, but not a KS set and not a critical set; (f)-(i)
critical subsets of (e).

  * •

(a) a direct translation 12,23,45,56,78,89,14,47,25,58,36,69. does not work
(hyperedges connect vertices pairwise and consecutively); e.g., 1 is not
orthogonal to 3 but it should be because $\Sigma_{i}$, $i=1,2,3$ mutually
commute;

  * •

(b) 15-6: 1A23,4B56,7C89,1D47,2E58,3F69. is noncontextual; hyperedges go
through all three vertices; e.g., 1 is orthogonal to 2 and 3 in the same
hyperedge; 9-6 (15-6 with gray vertices A,B,…,F dropped
123,456,789,147,258,369.) is also noncontextual;

  * •

(c) has 45 vertices/vectors and 18 hyperedges and its string is given in
Appendix LABEL:app3: hyperedges connect vertices pairwise but exhaustively;
e.g., 1 is orthogonal to 2 via one hyperedge and to 3 via another); it is not
contextual; some of its properties are: $HI_{cm}=9$ and $HI_{cM}=18$ and they
violate the v-inequality:

$\displaystyle HI_{cM}=18=HI_{q}=18.$ (35)

$l_{cm}=l_{cM}=18$ and they violate the e-inequalities:

$\displaystyle l_{cM}=18=l=18$ (36)

  * •

(d) dropping all gray vertices with $m=1$ from (c) yields a contextual 9-18
non-binary MMP hypergraph
12,23,13,45,56,46,78,89,79,14,47,17,25,58,28,36,69,39.All the remaining
vertices have $m=4$; $HI_{cM}=3$. The $\alpha_{r}^{*}$-inequality reads

$\displaystyle 3=HI_{cM}=\alpha<\alpha_{r}^{*}=\frac{9}{2}=4.5.$ (37)

The eMax-inequality reads:

$\displaystyle l_{cM}=12<l=18,$ (38)

and its span shows us that it is not critical. It might be implemented by
port-detections at each hyperedge/gate, but not via letting systems fly
through triple consecutive gates as in Peres-Mermin square measurements in the
literature.

The non-binary 9-18 MMP hypergraph contains the critical subhypergraphs 3-3,
5-5, 7-7, and 9-9 for whose we have $l_{cM}=l-1$. These criticals are shown in
Fig. 17(e-h). They all have $l_{cM}=(l-1)/2$ and satisfy the e-inequalities.
Their filled versions 9-3, 15-5, 21-7, and 27-9 all have $l_{cM}=l$ and
therefore they violate the e-inequalities.

We can get the coordinatization of these critical sets by reading it off from
Appendix LABEL:app3 for the corresponding vertices.

The obtained smallest quantum contextual MMP hypergraph 3-3, which differs
from the one obtained in Sec. 5.3 (Def. 5.1) only by its coordinatization, is
not a SIC set in the standard sense of the word, because it is not an
operator-based set. It is state independent in the sense that its
contextuality is based on its hypergraph structure, meaning that it holds for
any set of states that can support it, i.e., build its coordinatization.

###  5.6 5-dim KS MMP hypergraphs/sets

The 5-dim hypergraph spaces are defined by spin-2 systems and cannot include
qubits. So far, to our knowledge, only several MMP hypergraphs were obtained
in [cabell-est-05],[waeg-aravind-pra-17, Supp. Material] “by hand.” By our
automated generation we obtained up to 30 millions critical non-isomorphic MMP
hypergraphs from the vector components $\\{0,\pm 1\\}$. Several smallest ones
are shown in Fig. LABEL:fig:5dim.

Their structural property parameters are shown in Table 5.6.

Table 6: Structural properties of KS subhypergraphs of the 5-dim KS master set
105-136 generated from $\\{0,\pm 1\\}$ components. All MMPs violate the
$\alpha^{*}_{r}$-inequality.

Dim | KS MMPs | $HI_{cM}$ | $HI_{cm}$ | $l_{cM}$ | $l_{cm}$ | crit | vec. comp.
---|---|---|---|---|---|---|---
Real 5-dim MMP hypergraphs | 29-16 | 7 | 3 | 15 | 10 | yes | $\\{0,\pm 1\\}$
30-16 | 8 | 3 | 15 | 11 | yes | $\\{0,\pm 1\\}$
| 30-17 | 8 | 3 | 16 | 10 | yes | $\\{0,\pm 1\\}$
| 34-17 | 8 | 3 | 16 | 11 | yes | $\\{0,\pm 1\\}$
| 58-40 | 15 | 7 | 39 | 22 | yes | $\\{0,\pm 1\\}$
| 65-40 | 17 | 8 | 39 | 25 | yes | $\\{0,\pm 1\\}$
|  105-136  (master) | 23 | 13 | 122 | 80 | no | $\\{0,\pm 1\\}$

In Appendix LABEL:app5string we give the ASCII strings and coordinatization
for all MMP hypergraphs from Table 5.6 which include the four hypergraphs
given in Fig. LABEL:fig:5dim.

###  5.7 6-dim KS MMP hypergraphs/sets

In 2014 a star-like 6D 21-7 KS set was found [lisonek-14] and implemented
[canas-cabello-14]. The coordinatization used was defined by the vector
components from the set $\\{0,1,\omega,\omega^{2}\\}$. In [pavicic-pra-17] it
was shown that the set can be given a simpler coordinatization based only on
the components from $\\{0,1,\omega\\}$ components and that the star-like
graphical representation is isomorphic to the triangular representation given
in [pavicic-pra-17, Fig. 11]. In the same reference, a polytope-based class
236-1216 of 6-dim KS hypergraph was generated but it did not contain the 21-7
star/triangle-like set; its vectors had components from the following set:
$\\{0,\pm\frac{1}{2},\pm\frac{1}{\sqrt{3}},\pm\frac{1}{\sqrt{2}},1\\}$. Also,
based on many other failed attempts to generate real coordinatization of 21-7,
we conjecture that it might have only complex ones.

Figure 19: (a-f) the smallest critical 6-dim KS MMP hypergraphs obtained from $\\{0,1,\omega,\omega^{2}\\}$ components; see text; (g,h) smallest critical 6-dim KS MMP hypergraphs obtained from $\\{0,\pm 1\\}$ components; see text. Table 7: Structural properties of the KS subhypergraphs of the 6-dim KS master set/hypergraph 81-162. Only 20-5 and 31-11 violate the $\alpha_{r}^{*}$-inequality; $5>3.\dot{3}$ and $6>5.17$, respectively. Dim | KS MMPs | $HI_{cM}$ | $HI_{cm}$ | $l_{cM}$ | $l_{cm}$ | crit. | vec. comp.
---|---|---|---|---|---|---|---
Complex 6-dim MMP hypergraphs | 20-5 Fig. 4(d) | 5 | 4 | 5 | 5 | non-KS | $\\{0,\pm 1,2\\}$
21-7 [lisonek-14] | 3 | 3 | 6 | 6 | yes | (0,1,$\omega$)
| 27-9 [pm-entropy18, pwma-19] | 4 | 4 | 8 | 8 | yes | (0,1,$\omega$)
| 31-11 [pm-entropy18, pwma-19] | 6 | 4 | 10 | 9 | yes | (0,1,$\omega$,$\omega^{2}$)
| 32-11 [pwma-19] | 5 | 4 | 10 | 8 | yes | (0,1,$\omega$,$\omega^{2}$)
| 33-11 [pwma-19] | 5 | 3 | 10 | 8 | yes | (0,1,$\omega$)
| 36-13 [pwma-19] | 6 | 4 | 12 | 10 | yes | (0,1,$\omega$,$\omega^{2}$)
| 39-13 [pwma-19] | 6 | 4 | 12 | 8 | yes | (0,1,$\omega$,$\omega^{2}$)
| 81-162 [pwma-19] (sub-master) | 11 | 7 | 132 | 84 | no | (0,1,$\omega$,$\omega^{2}$)
Table 8: Structural properties of KS subhypergraphs of the 6-dim KS master set 236-1216. All the MMPs violate the $\alpha^{*}_{r}$-inequality. Dim | KS MMPs | $HI_{cM}$ | $HI_{cm}$ | $l_{cM}$ | $l_{cm}$ | crit | vec. comp.
---|---|---|---|---|---|---|---
Real 6-dim MMP hypergraphs | 34-16 [pavicic-pra-17] | 7 | 3 | 15 | 10 | yes | $\\{0,\pm 1\\}$
35-16 [pavicic-pra-17] | 7 | 3 | 15 | 10 | yes | $\\{0,\pm 1\\}$
| 37-16 [pavicic-pra-17] | 7 | 3 | 15 | 10 | yes | $\\{0,\pm 1\\}$
| 37-17 [pavicic-pra-17] | 8 | 3 | 16 | 11 | yes | $\\{0,\pm 1\\}$
| 37-18 [pavicic-pra-17] | 8 | 3 | 17 | 11 | yes | $\\{0,\pm 1\\}$
| 38-18 [pavicic-pra-17] | 8 | 4 | 17 | 10 | yes | $\\{0,\pm 1\\}$
| 236-1216 [pavicic-pra-17] (sub-master) | 66 | 36 | 105 | 78 | no | $\\{0,\pm 1\\}$

In [pm-entropy18] we obtain a master KS set 216-153 from $\\{0,1,\omega\\}$
components. It contains just three critical sets 21-7, 27-9, and 33-11 (the
last one has 8 non-isomorphic instances); Fig. 19(a,b,e). The 21-7 and 33-11
strings are given in [pwma-19, Supp. Material]. In [pwma-19] we also generate
two master sets 591-1123 and 81-162 and the corresponding classes (25 million
non-isomorphic critical KS sets) from $\\{0,1,\omega,\omega^{2}\\}$
components. The 31-11 string is given in [pwma-19, Supp. Material]. See Table
7.

In this paper, we generate the 332-1408 master from $\\{0,\pm 1\\}$
components. It contains two unconnected sub-masters: a non-KS 96-192 one and a
236-1216 KS one, which turns out to be isomorphic with the aforementioned
polytope based 236-1216 one.

The structure of the smaller 6-dim MMP hypergraphs from the 81-162 class is
different from the 4-dim ones because the $\delta$ feature allows the 6-dim
hypergraphs to be more interwoven than the 4-dim ones, as presented in [pm-
entropy18, Fig. A2] and [pwma-19, Fig. 4] and because of the coordinatization
with complex vectors. As a consequence, fewer vertices need to be assigned
value 1 to satisfy the KS conditions (i) and (ii) of the KS theorem.

The KS MMP hypergraphs from the 236-1216 class [pavicic-pra-17] obtained by
means of real vector components have much larger smallest hypergraphs, as
shown in Table 8. The $\\{0,\pm 1\\}$ components yield the 332-1408 MMP master
which consists of two unconnected sub-masters: the KS 236-1216 and a non-KS
(noncontextual) 96-192.

Both MMP classes, 81-162 and 236-1216 exhibit the $\delta$ feature and the
distinguishers which determine the sizes of minimal MMPs are complex vs. real
vectors.

###  5.8 7- and 8-dim KS MMP hypergraphs/sets

Here we present a few examples from the 7- and 8-dim spaces. The distribution
of the 7-dim KS MMP class is provided in [pavicic-pra-22] and of the 8-dim one
in [pavicic-pra-17].

Figure 20: (a) The smallest critical 7-dim KS MMP hypergraphs we obtained from
the 805-9936 master generated by $\\{0,\pm 1\\}$ components in [pavicic-
pra-22, Fig. 3]; (b-d) smallest critical 8-dim KS MMP hypergraphs obtained we
obtained from the 3280-1361376 master (more precisely its 2768-1346016 sub-
master) generated by $\\{0,\pm 1\\}$ components; (c) and (d) are
isomorphic—they also have a triangular representation (given in [pavicic-
pra-17, Fig. 14]) as the 6-dim 21-7 in Fig. 19 does; as for (d), cf. 4-dim
Fig. 15(c); (e) the smallest MMP with 52 vertices; cf. 52-16 in [pavicic-
pra-17, Fig. 14].

We give the structural properties of the examples in Table 9 and their ASCII
strings and coordinatizations in Appendix LABEL:app78string. The 8-dim MMP
hypergraphs given in Fig. 20 are highly symmetrical, so the reader might
easily assign ASCII symbols to vertices in the figure.

Table 9: Structural properties of KS MMP subhypergraphs of the 7-dim KS master set 805-9936 and 8-dim KS (sub-)master 2768-1346016, both obtained from $\\{0,\pm 1\\}$ components. The 7-dim 34-14 and 202-97 and 8-dim 37-11 and 52-14 violate the $\alpha^{*}_{r}$-inequality, while 8-dim 34-9 and 36-9 satisfy it. |  |  |  |  |  |  | Vector
---|---|---|---|---|---|---|---
dim | KS MMPHs | $HI_{cM}$ | $HI_{cm}$ | $l_{cM}$ | $l_{cm}$ | critical | components
7-dim | 34-14 | 7 | 3 | 13 | 8 | yes | $\\{0,\pm 1\\}$
202-97 | 44 | 21 | 94 | 68 | yes | $\\{0,\pm 1\\}$
8-dim | 34-9 | 4 | 3 | 8 | 8 | yes | $\\{0,\pm 1\\}$
36-9 | 4 | 4 | 8 | 8 | yes | $\\{0,\pm 1\\}$
37-11 | 5 | 3 | 11 | 8 | yes | $\\{0,\pm 1\\}$
52-13 | 8 | 6 | 12 | 10 | yes | $\\{0,\pm 1\\}$
120-2024 (sub-master) | 8 | 3 | 1080 | 404 | no | $\\{0,\pm 1\\}$

In [pavicic-pra-17] we obtained the 36-8(c) by hand and 34-9 and 36-9(d) from
the master 120-2024 generated by the Lie algebra E8. Coordinatizations of 34-9
and 36-9(d) in [pavicic-pra-17] therefore differ from the one obtained here
via our program Vecfind, i.e., from $\\{0,\pm 1\\}$ components. Since any
8-dim MMP hypergraphs with the latter vector components is a subhypergraph of
the 8-dim KS master MMP hypergraph 3280-1361376 generated from $\\{0,\pm 1\\}$
components, the 34-9 and 36-9 are subhypergraphs of both the 120-2024 and the
3280-1361376 master sets and have coordinatizations from both $\\{0,\pm 1\\}$
and Lie E8 components. Actually, the 120-2024 itself, as proven in [pm-
entropy18], can have both coordinatizations and is therefore a sub-
hypergraph/sub-master of the 3280-1361376 master. The latter master consists
of two larger unconnected sub-masters: a KS 2768-1346016 one and a binary
(noncontextual) 512-15360 one. This means that all 6,925,540 MMP critical
hypergraphs [pavicic-pra-17, Fig. 12] obtained from the Lie E8 components of
the 120-2024 master are also subhypergraphs of the 2768-1346016 master with
$\\{0,\pm 1\\}$-based coordinatizations.

In Table 9 we obtain $l_{cM}=94$ instead of expected 96 (for a critical MMP
hypergraph). This might be due to the algorithm and program One imperfections:
for such highly interwoven hyperedges some assignments of 1s to vertices might
have failed. Recall a similar outcome for the original Kochen-Specker MMP
hypergraph discussed below Table 2. An independent algorithm and program which
would check on these discrepancies is needed.

##  6 Implementation

The experiments cited in Sec. 1 were all focused on proving that contextual
sets really are contextual. In particular, they mostly carried out repeated
measurements with operators acting on different states to prove their state-
independence (SIC). However, we entered the realm of generation of arbitrary
many contextual sets of any structure in any dimension via automated
algorithms and programs and the next stage of their application should be a
direct implementation of the sets from the data base we obtained. Instead of
proving that contextual sets really are contextual, we should simply accept
that they are and start using them in quantum computation and quantum
communication.

For instance, generalised Stern-Gerlach experiment which makes use of both
magnetic and electric fields [anti-shimony] can generate any quantum state in
any dimension and therefore provide us with implementation of any hypergraph
which would yield these states.

On the other hand, there is a universal two-qubit quantum gate by means of
which one can build any quantum network/gates in a $2^{n}$-dim space that can
be realized by QED, nuclear spins, quantum dots, trapped ions, or photon-
photon coupling in an all-optical realization [weinf95].

In a projector formulation, all KS MMP hypergraph with a coordinatization are
state-independent and for non-KS non-binary MMP hypergraphs one still has to
find a general automated approach. However, in addition to numerous already
known small non-KS non-binary MMP hypergraphs [pavicic-entropy-19] we do have
an abundance of KS MMP hypergraphs of any size and structure in odd [pavicic-
pra-22] and even [pavicic-pra-17, pm-entropy18, pwma-19] dimensional spaces.

Taken together, future research in the field should be focused on finding
general automated algorithms and programs for implementation of arbitrary
contextual sets in quantum gate networks.

##  7 Discussion

In this paper we elaborate on particular approaches to features of quantum
contextual sets that determine their generation, usage, applications,
implementations, and perspectives of future research.

###  7.1 Operator-based vs. MMP-hypergraph-based contextuality

In Secs. 2, 4.1, and 4.3 we compare operator-based and MMP-hypergraph-based
approaches to contextual sets and show that the former one relies on the
latter. In the literature only a handful of smallest operator-based contextual
set have been analyzed while there are billions of contextual MMP hypergraphs
[bdm-ndm-mp-fresl-jmp-10, pmm-2-09, mp-7oa, pavicic-pra-17, pm-entropy18,
pavicic-entropy-19, pwma-19].

MMP hypergraph language is introduced in Sec. 2 and contrasted with obsolete
and/or inappropriate graph and general hypergraph language throughout the
paper. The latter approaches are vividly graphically presented in Figs. 1, 5,
15, 16, and 17 and their disadvantages discussed in the text surrounding them.

In Sec. 3.1 we consider several extensions of contextual Kochen-Specker (KS)
vector sets. The most important is a non-binary contextual MMP hypergraph
extension given by Def. 3.5 in which we dispense with vectors
(coordinatization), i.e., states, and rely only on the very structure of MMP
hypergraphs. In this sense they are state-independent. To make use of
states/vectors or to define operators we attach a compatible coordinatization
to them using simple vector components.

In Sec. 4.1 we consider three approaches to obtain operator-based contextual
sets, two of which (hyperedge (i) and vertex (ii) ones) generate operators
directly from MMP hypergraphs $k$-$l$, mainly via projectors
$P=|v_{i}\rangle\langle v_{i}|$, $i=1,\dots,k$, where $k$ is the number of
vertices and $l$ the number of hyperedges. We conjecture that the following
rule universally holds.

######  Operator $\boldsymbol{\leftrightarrow}$ MMP Rule. 7.1.

Every MMP hypergraph which might serve for a construction of an operator-based
contextual set via its states/vectors is itself a non-binary contextual MMP
hypergraph.

We give a number of examples to this rule in Secs. 4.1, 4.2, and 4.3. We would
like to single out Yu-Oh’s 13-16 set shown in Fig. 8; see the text above it.

An important notion we introduce in order to generate MMP hypergraphs is the
multiplicity of vertices, $m$, given by Def. 4.3 that tells us how many
hyperedges each vertex shares. In relation to it, we can induce/generate
contextual MMP hypergraphs from both non-binary (contextual) and binary
(noncontextual) MMP hypergraphs in two ways:

  * •

by dropping data obtained by measuring states of systems related to vertices
with $m=1$ as well as vertices themselves from the MMP hypergraphs; this works
for non-binary MMPs hypergraphs (e.g., for 3-dim ones shown in fig.
LABEL:fig:3dup and extensively elaborated on in [pavicic-entropy-19]) as well
as for the binary ones (e.g., Peres-Mermin’s 45-18 MMP shown in Fig. 17);
thousands of such ${\rm\overline{subhypergraphs}}$ which serve the purpose are
generated in [pavicic-entropy-19, Sec. II.D];

  * •

by finding smaller contextual non-binary MMP ${\rm\overline{subhypergraphs}}$
(Def. 4.2) contained in binary MMP hypergraphs with $m\neq 1$, as carried out
on two examples in [cabello-severini-winter-14].

###  7.2 Raw- and postselected data statistics and their inequalities; the
Grötschel-Lovász-Schrijver (GLS) $\alpha^{*}$-inequality is not a
noncontextuality inequality

The role of the multiplicity $m$ (Def. 4.3) with which vertices share
hyperedges in every hypergraph is characterized by Eq. (17) and Lemma 4.10. As
we explain in Secs. 4.2 and 4.3 it enables us to distinguish the standard
Hypergraph Statistics based on raw measurement data (4.13(a)) from the one
based on postprocessed measurement data (4.13(b)). It also determines the
structure of the hypergraphs as shown in Sec. 5.1, Fig. 9, and Table 1.

A standard tool for discriminating contextual from noncontextual sets has
lately been claimed to be noncontextuality inequalities (Def. 2), in
particular the operator/projector-based ones [klyachko-08, cabello-08, badz-
cabel-09, beng-blan-cab-pla12, kurz-kasz-12, kurz-cabello-14, ram-hor-14,
cabell-klein-budr-prl-14, cabello-severini-winter-14, yu-tong-14, xu-chen-su-
pla15, yu-tong-15]. We review them in Secs. 4.1 and 4.2. They are mostly
defined by states/vectors of contextual MMP hypergraphs what means that the
MMP hypergraph structure together with its coordinatization serves us to build
operator/projector structure.

Vertex multiplicity enables us to introduce a new kind of hypergraph-based
vertex v-inequalities (Def. 4.11) and relate the operator-based inequalities
with the hypergraph-based (hyper)edge e-inequalities (Defs. 4.14, 4.15). We
also consider the $\alpha^{*}$\- (Eq. (27)), and $\alpha_{r}^{*}$\- (Eq. (24),
and $\alpha_{p}^{*}$\- (Eq. (26)) inequalities. The $\alpha$ in them is the
maximum number of pairwise non-adjacent vertices (Def. 4.7), i.e., the maximum
number of vertices to which one can assign ‘1’ (Lemma 4.8); it is called the
independence number and also the stability number. Table 10 provides us with a
list of inequalities.

  * •

The v-inequality, relates the maximal classical multiplexed vertex indices
$HI_{cm}$, $HI_{cM}$, $HI^{m}_{cM}$, Def. 4.5 (the total number of 1s we can
assign to vertices each multiplied by its multiplicity or not) to quantum
hypergraph index $HI_{q}$, Def. 4.9 (the sum of the probabilities of getting
quantum measurement clicks within hyperedges:$HI_{cm}\leq HI_{cM}\leq
HI^{m}_{cM}<HI_{q}$, Def. 4.11); see Tables 2, 4, 7, and 8.

  * •

The eMax\- and emin-inequalities quantifies the KS theorem generalisation
(Def. 3.5), according to which we cannot assign 1 to all hyperedges of a non-
binary MMP hypergraph, i.e., they simply relate the maximum and minimum
number, respectively, of hyperedges which can contain 1 ($l_{cM},l_{cm}$; Def.
4.6) with the actual number of hyperedges of a considered MMP hypergraph
$k$-$l$ containing $k$ vertices and $l$ hyperedges. The eMax-inequality (20)
corresponds to Badziág, Bengtsson, Cabello, and Pitowsky’s $\beta$-inequality
[badz-cabel-09], given in Eq. (8), and $l_{cM}$ corresponds to $\beta$; eMax-
inequality has a trivial form $l_{cM}+1=l$ for critical KS MMP hypergraphs
(except perhaps for the original KS one—see Sec. 5.2). They become relevant
($l_{cM}$ becomes significantly smaller than $l$) for non-critical and master
KS MMP hypergraphs as shown in Tables 4, 7, and 8. The emin-inequality (21) is
more suitable for an implementation because of its greater span between its
terms. We conjecture that $l_{cm}$ is the “rank of contextuality” recently
introduced by Horodecki at al. [horod-22].

  * •

The $\alpha_{r}^{*}$\- (24) and $\alpha_{p}^{*}$\- (26) inequalities, with
constant/fixed probabilities of detecting (within quantum YES-NO measurements)
a quantum particle in a particular state (assigned to vertices within each
hyperedge), are special cases of the GLS $\alpha^{*}$-inequality which, in
contrast to the former ones, rely to variable/free probabilities (Defs. 4.17,
4.18, and Eq. (27)): $\alpha\leq\alpha^{*}$ (27). The latter inequality is
valid for any graph or hypergraph, contextual or not, with variable/free
probabilities assigned to vertices within each hyperedge. Linear programming
or any other algorithms for solving linear optimization problems then
determines which values must the probabilities have within each hyperedge
(where their sum must be $\leq 1$). The $\alpha_{r}^{*}$-inequality is based
on the Raw data statistics 4.13(1) and it is not a noncontextuality
inequality. The $\alpha_{p}^{*}$-inequality is based on the Postprocessed data
statistics 4.13(2); it is just another another form of the v-inequality (19)
and is therefore a noncontextuality inequality. For example, a spin-1 particle
passing through a Stern-Gerlach gate/hyperedge has the probability of
$\frac{1}{3}$ to exit from any of its ports along any of its 3 vertices/ports.
But then, for arbitrary many contextual non-binary MMP hypergraphs, the
$\alpha_{r}^{*}$-inequality fails (Cf. Figs. 6(a-d), 12(d), 13(b)).

Table 10: List of the inequalities elaborated on in the paper. Inequality |  Eq. | Noncontextuality inequality (1)
---|---|---
v | $HI_{cm}\leq HI_{cM}\leq HI^{m}_{cM}<l$ | (19) | yes
eMax | $l_{cM}<l$ | (20) | yes
emin | $l_{cm}<l$ | (21) | yes
$\alpha^{*}_{r}$ | $\alpha\leq\alpha^{*}_{r}=\frac{k}{n}$ | (24) | no
$\alpha^{*}_{p}$ | $\alpha<\alpha^{*}_{p}=l$ | (26) | yes
GLS | $\alpha\leq\alpha^{*}$ | (27) | no

Hypergraph v-, e-, $\alpha_{r}^{*}$, and $\alpha_{p}^{*}$-inequalities can be
generated via automated procedures directly from non-binary MMP hypergraphs.
The hypergraphs themselves can be generated in automated ways from simple
vector components such as $\\{0,\pm 1,i\\}$ (Tables 4,8) or
$\\{0,1,\pm\omega\\}$ (Table 7), etc. The operator approach is suitable for
forming quantum gates which can be applied to arbitrary states to generate
inequalities for evaluating the contextual measurements, while the latter
automated hypergraph approach is suitable for testing a level of contextuality
of hypergraph states by postprocessing measurements carried out at out-ports
of gates determined by hyperedges as well as for verifying contextual
properties of a chosen hypergraph.

The v-, e-, and $\alpha_{p}$ inequalities are the only genuine
noncontextuality inequalities (Def. 4.1).

###  7.3 Structure and features of particular MMP hypergraphs

The MMP hypergraph language applied to several well-known contextual sets
yields the following results.

####  7.3.1 MMP vertex multiplicity

Throughout the paper we show that the multiplicity of vertices plays
significant roles in determining the features of $n$-dim MMP hypergraphs
$k$-$l$. In particular, by the Vertex-Hyperedge Lemma 4.10 we show that the
sum of multiplicities is equal to $nl$; in Sec. 5.1 we show that MMP
hypergraphs with odd number of hyperedges predominantly have vertices with
even multiplicities (see Fig. 9 and the figure in Appendix LABEL:app4) and in
Table 1 that the multiplicities of vertices uniquely characterize master MMP
hypergraphs we use to generate all known MMP hypergraphs classes from.

####  7.3.2 3-dim MMP hypergraphs; Graph vs. MMP hypergraph representations

In Sec. 5.2 we discuss (see Fig. LABEL:fig:3dup) the four previously known
3-dim KS MMP hypergraphs: Bub’s 49-36, Conway-Kochen’s 51-37, Peres’ 57-40,
and Kochen-Specker’s 192-118 and point out that they are critical, i.e., that
none of them contains any smaller KS sets. By removing vertices with $m=1$
from these KS MMP hypergraphs, we obtain, via the method presented in Sec.
7.1, the non-binary contextual 33-36, 31-37, 33-40, and 117-118 sets,
respectively, but they are not KS sets, contrary to what [budroni-cabello-
rmp-22, Table 1, p. 21] might mislead the reader in.

In Fig. LABEL:fig:3dup1 we give five new 3-dim critical MMP hypergraphs
obtained among thousands of such new 3-dim ones in [pavicic-pra-22].

In Sec. 5.2 we show that Kochen and Specker’s original presentation of their
192-118 set or Budroni, Cabello, Gühne, Kleinmann and Larsson’s [budroni-
cabello-rmp-22, Fig. 1] “simplification” of that set is neither a graph, nor a
general hypergraph, nor an MMP hypergraph.

Note that no definite vectors for the KS set 192-118 have been given prior to
the ones provided in [pavicic-pra-22].

None of the 3-dim KS sets satisfies the $\alpha_{r}^{*}$-inequality, but most
of their smaller $\overline{\rm subhyper}$\- $\overline{\rm graphs}$ do, in
particular the critical ones [pavicic-entropy-19]. Some are shown in Fig. 12
and Table 3.

####  7.3.3 The smallest non-binary MMP hypergraph that exists and other small
4-dim MMPs

In Sec. 5.3 we review several chosen small 4-dim MMP hypergraphs and give
their parameters in Table 4 and their figures in Fig. 13. Binary 18-9, non-
binary critical 22-13, and binary 22-13 shown in Figs. 13(a,b,c),
respectively, violate the $\alpha_{r}^{*}$-inequality. When the $m=1$ vertices
are dropped from the binary 18-9 one obtains a non-binary MMP $\overline{\rm
subhypergraph}$ 10-9 shown in Fig. 14(a). One of its critical $\overline{\rm
subhypergraphs}$ is the non-binary 3-3 MMP hypergraph shown in Fig. 14(d)—the
smallest 4-dim MMP hypergraph with coordinatization that exists. Since the
3-dim 3-3 MMP hypergraph does not have a coordinatization, the obtained 4-dim
3-3 is the smallest contextual set that exists in any dimension. This is the
Result 5.1.

Another result obtained in that section is the Result 5.2.

####  7.3.4 4-dim: Graph vs. MMP hypergraph case

In Sec. 5.4 we analyze a recent experiment [d-ambrosio-cabello-13] and show
how and why graph representation of contextual sets lead to wrong experimental
and theoretical results.

In particular they claim that all 18 vectors they consider contribute with an
equal weight and that therefore their implementation is a proper KS set. We
show their graph in Fig. 15(e). Their red and green edges contain only two
vertices, though. For example, vertices 1,2,5,10,15,18, i.e., edges 1-10, 2-15
and 5-18, the probabilities $p_{1,10}$, $p_{2,15}$, and $p_{5,18}$ from
measurement data were obtained. We show their set in the MMP hypergraph
representation in Fig. 15(g). It is an MMP 18-18, but it should have a
coordinatization, i.e., $m=1$ vertices should be added while performing the
experiment and they are shown in Fig. 15(f) as gray dots—we end up with an MMP
with 36 vertices and 18 hyperedges—and that leads us into a contradiction:
36-18 MMP has no coordinatization. So, not only that the set is not a KS set,
but the measurement data themselves are inconsistent.

There are two possible remedies for the contradiction:

  * •

merge triples of gray vertices at the intersections of red and green
hyperedges as shown in Fig. 15(h); new measurements should be carried out for
the additional six vertices of the new 24-18 MMP hypergraph; it is one of 1233
KS MMP hypergraphs [pmm-2-09] contained in Peres’ 24-24 master set;

  * •

abandon green and red hyperedges altogether and reduce the implementation to
the 18-9 KS MMP hypergraph (Fig. 15(c));

####  7.3.5 $\alpha^{*}$-inequality vs. quantum computation and quantum
indeterminacy

As presented in details in Sec. 5.4 Howard, Wallman, Veitech, and Emerson have
shown that stabilizer operations with quantum bits initialized superposition
of states (“magic states”), can be used to purify quantum gates provided they
exhibit the contextuality [magic-14]. As a proof that considered sets are
contextual the authors make use of the GLS inequality [gro-lovasz-schr-81, p.
192] by invoking Ref. [cabello-severini-winter-14].

In the latter reference, two simple examples are given for inducing small
contextual non-binary MMP hypergraphs from bigger noncontextual binary ones.
This is essentially the second procedure we referred to at the end of Sec.
7.1. A generalization of the procedure is offered, which would consist in a
recognition of the GLS inequality [gro-lovasz-schr-81] as a noncontextuality
inequality (Def. 4.1). A similar approach is carried out by Cabello
[cabello-21, Supp. Material, Sec. IV]. However, as we show in Theorem 4.20,
the assumption of variable probabilities of detecting outputs from hyperedges
would clash with the postulate of quantum indeterminacy 4.19 and therefore the
$\alpha_{r}^{*}$-inequality given by Eq. (24) should be used, instead. The
theorem then states that $\alpha_{r}^{*}$-inequality is not a noncontextuality
inequality since arbitrary many contextual and noncontextual MMP hypergraphs
violate it.

In Ref. [magic-14] the same problem emerges. The exclusivity graph—“a source
of quantum computer’s power” [bartlett-nature-14, magic-14]—is a non-binary
30-108 MMP hypergraph and a $\overline{\rm subhypergraph}$ of a non-critical
KS 232-108 MMP hypergraph as analyzed in Sec. 5.4. In Fig. 16(b) we see that
we can extend the original 30-108 MMP hypergraph to the KS 232-108 one by
adding $m=1$ vertices to the former one. These added vertices enable us to
identify additional independent vertices in addition to the original 8 thus
exceeding the upper bound. We obtain $101\leq\alpha>\alpha_{r}^{*}=58$ (Table
5), and it is an open problem to prove that that is not relevant for the proof
that the appropriate inequality is $\alpha=2^{3}=8<\alpha^{*}=2^{3}+1=9$ as
given in Ref. [bartlett-nature-14]. On the other hand, for the 30-108 MMP
hypergraph itself, a calculation which takes into account all its edges and
their 30 vertices yields $\alpha_{r}^{*}=14.07$—see Eq. (31).

Taken together, if the only reason for invoking the GLS inequality was to
prove that exclusivity MMP hypergraphs suitable for quantum computation are
contextual, then the more efficient approach would be to directly check the
obtained measurement data on contextuality, e.g., via e-inequalities.

####  7.3.6 Peres-Mermin square: operators vs. MMP hypergraphs

The Peres-Mermin square has received a great deal of attention both
theoretically and experimentally [peres90, mermin90, mermin93, cabello-08,
k-cabello-blatt-09, pavicic-book-13, hofer-20]. So far it has been formulated
only via operators (the claim that it “can be converted into a standard proof
of the KS theorem with vectors” [budroni-cabello-rmp-22, p. 8] is incorrect as
explained in Sec. 5.5), and in this paper we arrive at an MMP hypergraph
representation of the basic features of the Peres-Mermin operators so as to
examine possible candidates for such a representation. We find that a non-KS
45-18 MMP hypergraph, shown in Fig. 17(d), serves the purpose. This is because
it contains contextual 9-18 non-binary MMP hypergraph (Fig. 17(e)) which we
obtain by dropping all vertices with multiplicity $m=1$ from the 45-18 MMP
(Fig. 17(d)). Notice that there are no states which would satisfy the
conditions given by Eqs. (33) and (34), so there is no MMP contradiction which
would correspond to the operator Peres-Mermin contradiction. Notice also that
vertices belonging to rows and columns of the original operator-based
schematics (Fig. 17(d)) are mutually orthogonal in correspondence to mutual
commutations of operators in the same rows and columns.

The 9-18 MMP hypergraph is not critical. It contains 3-3, 5-5, 7-7, and 9-9
criticals as shown in Figs. 17(e-h). Their possible experimental
implementations might use Peres-Mermin operators but not in the square
arrangements in which systems run through triple gates as depicted in Fig.
17(a). Instead, one of the Fig. 17(c-h) arrangements might be used.

As for the critical MMP subhypergraph 5-5, it is isomorphic to the pentagon
from Ref. [klyachko-08] but they are not equivalent since they live in two
different spaces, 4-dim and 3-dim, respectively. That is why the
coordinatization of complete, filled, MMP hypergraphs are so essential. For
example, while 5-5 can be represented in both 3-dim and 4-dim spaces, 3-3 or
4-4 cannot, because coordinatizations for a filled 3-3 (e.g., 1A2,2B3,3C1.) or
a filled 4-4 (e.g., 1A2,2B3,3C4,4D1.) do not exist in a 3-dim space.

To sum up, the original operator formulation of the Peres-Mermin square is
inconsistent because classical observables $S_{j}$ which would assign $\pm 1$
to the states have no quantum counterparts since there is no quantum state
$|\Psi\rangle$ of a system which $\Sigma_{j}$ might project to states
$\pm|\Psi\rangle$ (eigenstates with $\pm 1$ eigenvalues). The fact that a
correlated noncontextuality cannot be formulated and that therefore the Peres-
Mermin square contextuality is void of its meaning we presented as the Peres-
Mermin contradiction 5.3.

####  7.3.7 The pentagon case

In Sec. 4.2 we make use of different coordinatizations to compare hypergraph
inequalities with the operator ones, on the example of Klyachko, Can,
Binicioğlu, and Shumovsky’s 3-dim pentagon. They consider particular
coordinatization with vectors/states which an operator projects to a chosen
state $\Psi$ so as to give the maximum quantum mean value $\sqrt{5}$ as in Eq.
(13) and the operator inequality $2<\sqrt{5}$, i.e., the inequality is state-
dependent. The hypergraph approach, on the other hand, gives the quantum
hypergraph index $HI_{q}=5$, the e- and v-inequalities $4<5$ (Eq. (15)), and
the $\alpha_{r}^{*}$ inequality $2=\alpha<\alpha_{r}^{*}=\frac{5}{2}$. The
inequalities arise from the structure of the MMP hypergraph alone
independently of the states that build its coordinatization and that makes the
MMP hypergraph state independent in the sense that its contextuality holds for
any set of states that can build its coordinatization and even when there are
no such states. (Note that no projections of or to any states are involved.)
Still, the coordinatization plays a role in the geometric representation of
MMP hypergraphs; e.g., a 3-dim pentagon can never be planar even when all five
of its $m>1$ vectors span a plane.

###  7.4 Higher dimensions

In Secs. 5.6, 5.7 and 5.8 we give give figures and structural properties of
the smallest MMP hypergraphs from 6- to 8-dim spaces, respectively.

###  7.5 Implementation and research perspectives

In Sec. 6 we presented existing general implementation schemes for
implementation of arbitrary MMP hypergraphs and a broad class of a qubit gate
network. This opens up a research road to possible universal automated
preparation algorithms for implementation of arbitrary contextual sets in
contrast to existing approach in the literature which usually limits itself to
experiments with the smallest contextual sets and theoretical constructs based
on them which however have not been exemplified on arbitrarily chosen
contextual sets. Actually, such small-set-approach often yields inconsistent
results, some of which we discussed in the paper in an attempt to purify the
handling of the contextual sets and find their general features. An abundance
of available contextual sets would support achieving this goal and therefore
we also generated billions of them in odd (3 to 9; [pavicic-pra-22, pavicic-
entropy-19]) and even (4 to 32; [pavicic-pra-17, pm-entropy18, pwma-19])
dimensional spaces and stored them in a freely available data base wherefrom
one can download sets with required structure and complexity for any
application.

The programs are available at http://puh.srce.hr/s/Qegixzz2BdjYwFL