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\title{Varieties of contextuality based on probability and structural nonembeddability}
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\author{Karl Svozil}
\address{Institute for Theoretical Physics, TU Wien, Wiedner Hauptstrasse 8-10/136, 1040 Vienna,  Austria}

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\cortext[mycorrespondingauthor]{Karl Svozil}
\ead{svozil@tuwien.ac.at}

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\begin{abstract}
Different analytic notions of contextuality fall into two major groups: probabilistic and strong notions of contextuality. Kochen and Specker's Theorem~0~\cite{kochen1} is a demarcation criterion for differentiating between those groups. Whereas probabilistic contextuality still allows classical models, albeit with nonclassical probabilities, the logico-algebraic ``strong'' form of contextuality characterizes collections of quantum observables that have no faithfully embedding into (extended) Boolean algebras. Both forms indicate a classical in- or under-determination that can be termed ``value indefinite'' and formalized by partial functions of theoretical computer sciences.
\end{abstract}

\begin{keyword}
quantum contextuality \sep quantum randomness \sep Gleason theorem \sep Kochen-Specker theorem \sep Born rule \sep quantum logic \sep quantum entanglement \sep probability distributions \sep category formation
\end{keyword}


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\begin{document}

\title{Varieties of contextuality based on probability and structural nonembeddability}


\author{Karl Svozil}
\email{svozil@tuwien.ac.at}
\homepage{http://tph.tuwien.ac.at/~svozil}

\affiliation{Institute for Theoretical Physics,
TU Wien,
Wiedner Hauptstrasse 8-10/136,
1040 Vienna,  Austria}


\date{\today}

\begin{abstract}
Different analytic notions of contextuality fall into two major groups: probabilistic and strong notions of contextuality. Kochen and Specker's demarcation Theorem~0~\cite{kochen1} is a criterion for differentiating between those groups. Whereas probabilistic contextuality still allows classical models, albeit with nonclassical probabilities, the logico-algebraic ``strong'' form of contextuality characterizes collections of quantum observables that have no faithfully embedding into (extended) Boolean algebras. Both forms indicate a classical in- or under-determination that can be termed ``value indefinite'' and formalized by partial functions of theoretical computer sciences.
\end{abstract}

\keywords{Quantum contextuality, Gleason theorem, Kochen-Specker theorem, Born rule, quantum logic, probability distributions}
\pacs{03.65.Ca, 02.50.-r, 02.10.-v, 03.65.Aa, 03.67.Ac, 03.65.Ud}

\maketitle

\fi

\section{Types of quantum contextuality}

The main point of this paper is that there are at least two main types of contextuality: the first notion is based upon nonclassical phenomenology,
and in particular, on nonclassical probabilities contradicting Boole's conditions of physical experience~\cite{Boole-62}.
The second type goes beyond this,
and is based upon the absence of a classical logico-algebraic structure in terms of which the respective observables could be re-interpreted.
Formally this amounts to nonembaddability into Boolean algebras by classical means; that is, in formal terms,
by two-valued measures interpretable as classical truth assignments. The strongest form of this latter logico-algebraic contextuality
is the total absence of any such classical truth assignment~\cite{kochen1}.
Thereby, it is important to keep in mind that the mere existence of any such classical truth assignment
is necessary but not sufficient to ensure classical representability or embeddability:
indeed, even if there is an apparent ``abundance'' of classical truth assignments, these might
not provide sufficient means to classically embed a collection of quantum observables.

Early synthetic conceptions of contextuality emerged from
insights into the entangled complexion~\cite{schrodinger-gwsidqm2} of physical properties
retrieved from quantum measurements. As expressed by Bohr~\cite{bohr-1949}:
{``the impossibility of any sharp separation between the behavior of atomic
objects and the interaction with the measuring instruments which serve to define the conditions
under which the phenomena appear.''}
This yields a {``conditionality of phenomena''}~\cite{Khrennikov2017,Jaeger2019} relative to a
{``a complex of conditions under which the measurement is performed''}~\cite{Khrennikov2009a,Khrennikov2009b}.

In this line of thought, observable phenomena appear not as isolated properties of the object,
but as signals from the object-measurement apparatus composite.
(Entanglement may even extend to the observer~\cite{london-Bauer-1939,london-Bauer-1983}.)
Indeed, if entanglement is involved, these signals are about the relational properties of the combined entangled system.
It makes no sense to refer to a well-defined property of the individual object alone~\cite{schrodinger-gwsidqm2,zeil-99}.
Therefore, one should be careful interpreting a statement such as Bell's observation~\cite{bell-66} that
{``the result of an observation may reasonably depend $\ldots$ on the complete disposition of the apparatus''.}
In general there is no deterministic, one-to-one correspondence, association, or translation between relevant
(counterfactual) well-defined individual properties of the constituents (one imagined~\cite{hertz-94} as ``object'')
of an entangled quantized system on the one hand,
and the signal resulting from observation of this entangled state on the other hand. Entanglement evades such an association
because the constituents of an entangled quantum state have no well-defined individuality.


Subsequent attempts to specify and quantify contextuality
have presumed that individual objective properties nevertheless exist even for quantized systems,
and that these properties do not depend on any kind of disposition of the measurement apparatus;
in particular not on some compatible observables that are measured simultaneously.
The latter assumption is usually referred to as ``noncontextual''.
The former assumption of the general existence of counterfactual properties or observables can be called omni-existence.
Omni-existence is the ``totality'' assertion that, although due to complementarity not all observables can be measured simultaneously, they are nevertheless value definite;
that is, some of them have a definite counterfactual value that can by no physical means be measured~\cite{specker-60}.



Historical attempts to prove contextuality have assumed both omni-existence and noncontextuality
(thereby disregarding earlier synthetic concepts of contextuality by entanglement mentioned earlier), and
have concentrated on the differences between classical and quantum predictions.
Thus, given  omni-existence, any empirically (falsifiable) discrepancies between classical and quantum predictions are interpreted to signify contextuality.
However, one has to be careful and keep in mind that, just because omni-existence is often but not always~\cite{2015-AnalyticKS} assumed for the sake of contradiction,
a violation of noncontextuality does not imply or suggest the existence of any contextual hidden variable model.
Contemporary interpretations of contextuality indicate the non-existence of a noncontextual model~\cite{cabello2021contextuality}.

Most commonly, experimental violations of Boole-Bell-type inequalities are identified with quantum contextuality~\cite{cabello:210401,cabello2021contextuality}.
Other empirical signatures of quantum contextuality are the experimental violations of ad hoc configurations
whose classical interpretation
(i)
either merely assume the omni-existence of unrestricted classical noncontextual value assignments that do not depend on the
complete disposition of the apparatus~\cite{cabello:210401};
(ii)
or, on preselected input, predict classical functional output that is violated in quantized systems~\cite{2018-minimalYIYS}.

Theoretical arguments against omni-existent noncontextual value assignments consider finite configurations of
observables forming intertwining contexts that have no consistent classical value assignments~\cite{specker-60}.
Here the term ``intertwined'' is understood as introduced by Gleason~\cite{Gleason}:
in higher than two dimensions the orthonormal bases identified with contexts can ``share'' common elements---they need not be ``isolated'', that is, disjoint.
This is not the place for a historic review but the literature indicates that
what is now known as the Kochen-Specker theorem~\cite[Theorem~1]{kochen1} has been
discussed~\cite{ZirlSchl-65,kamber65} as a direct consequence of Gleason's theorem~\cite{Gleason}.

The variety of contextual signifiers has resulted in a great semantic spread of notions of contextuality that threatens to obscure subtle differences
concerning the quality of anomaly.
Because the same collection of observables, taken from quasi-classical or quantum experimental configurations alike,
may still allow very different types of probability distributions.
The resulting differences in the prediction may be taken as signatures for contextuality.
But this is incomparable to configurations of observables that do not, by any classical means, support such probability distributions,
either because the existing classical value assignments that do not depend on the complete disposition of the apparatus
are too scarce to resolve the logico-algebraic structure of observables at hand,
or because this structure does not allow any such classical value assignment at all.

In what follows I, therefore, suggest refining the notion of contextuality by differentiating between two cases,
depending on whether the collection of observables
\begin{itemize}
\item[(i)] violates some constraints on classical probabilities but still allows a faithful embedding into an extended Boolean subalgebra, or
\item[(ii)] does not, by any classical means, allow any faithful embedding into some extended Boolean subalgebra.
\end{itemize}

For the sake of this analysis note that maximal collections of (finitely many) mutually co-measurable observables can be ``wrapped up'' into
contexts (or blocks or maximal observables) which, from a probabilistic and structural point of view, intrinsically behave classically.
The respective probability distributions are in accord with Kolmogorov's axioms. In particular,
the probabilities of mutually exclusive events add up~\cite{Gleason}.

\section{Nomenclature}

For quantum mechanics we fix a positive integer $n\ge 2$.
Let $O$ be a nonempty set of one-dimensional projection observables on the Hilbert space $\mathbb{C}^n$~\cite{halmos-vs}.
A set $C\subset O$ is a {\em context} of $O$ if $C$ is an orthonormal basis of $\mathbb{C}^n$.
An equivalent definition is in terms of the spectrum of a maximal operator containing one-dimensional projection operators
onto the subspaces spanned by the vectors in $C$.
In more general empirical logic terms, a context can be conceptualized by a set of mutually exclusive observables whose disjunction is a tautology.


Quantum mechanics, as well as partition logics~\cite{svozil-2001-eua} from a generalized urn~\cite{wright} or finite automata models~\cite{e-f-moore,schaller-96} featuring complementarity, allow
two or more distinct contexts
which, for more than two mutually exclusive outcomes per context, may intertwine in some observable(s)~\cite{svozil-2018-b}.
The remaining nonintertwining observables,
taken from different contexts, exhibit complementarity.
(One convenient and compact graphical representation depicts contexts as smooth lines,
and mutually exclusive elementary observables by points on these lines.)

As has been mentioned earlier, despite exhibiting complementarity,
a collection of contexts may still allow some quasi-classical interpretation in terms of
extreme cases.
Such value assignments can be formalized by dispersionless {\em two-valued states} $v(x) \in \{0,1\}$ (or, logically interpreted, ``false'' and ``true'')
that are binary functions of the respective observables $x \in X$ forming the contexts
which are additive $v(x \vee y)= v(x) + v(y)$ for mutually exclusive observables $x\wedge y=\emptyset$, and add up to 1
for all mutually exclusive observables within any context.
A weak and general formalization for quantized systems that allows partial functions and thus value indefiniteness
is in terms of admissibility~\cite{2015-AnalyticKS}:
Let $O$ be a set of one-dimensional projection observables and let $v: O \to \{0,1\}$ be a value assignment function.
Then $v$ is {\em admissible} if the following two conditions hold for every context $C$ formed by $O$:
\begin{itemize}
\item[(i)] exclusivity: if there exists an $x\in C$ with $v(x)=1$, then $v(x')=0$ for all $x'\in C \setminus \{x\}$;
\item[(ii)] completeness: if there exists an $x\in C$ with $v(x')=0$ for all $x'\in C\setminus \{x\}$, then $v(x)=1$.
\end{itemize}


If the observables on which the aforementioned collections of contexts are based are quantum,
then there is no guarantee that ``sufficiently many'' classical value assignments corresponding to dispersionless two-valued $\{0,1\}$-states exist.
Indeed, there are finite configurations of observables with
no such classical value assignment, or ones that cannot support ``sufficiently many'' classical value assignments to allow
embeddings preserving the respective logico-algebraic structure.

Already Kochen and Specker discussed these issues and presented a demarcation criterion~\cite[Theorem~0]{kochen1} that
utilizes the (in)separability of the underlying binary elementary propositions by classical value assignments:
A set of observables  $X$ forming a collection of contexts
is faithfully embeddable into an extended Boolean algebra if and only if these observables in $X$ contained in the respective contexts
support or allow a separating set of two-valued states $V = \{v_1, \ldots ,v_n\}$ such that,
for any two observables $x , y \in X$,
there exists some $v_i \in V$ for which $v_i (x) \neq v_i (y)$.
(In its extreme form the collection of observables $X$ support no two-valued measure, and $V=\emptyset$.)

If the observables in $X$ support a separating set of two-valued states $V$ then $V$ facilitates three constructions:
\begin{itemize}
\item[(i)]
It yields all classical probability distributions
in the form of a convex combination
$
P(x) = \sum_{i=1}^n\lambda_i v_i(x)$,
such that
$
\sum_{i=1}^n\lambda_i=1
$
and
$
\lambda_j > 0
$
for all
$
j\in \left\{1,\ldots,n\right\}
$.

\item[(ii)]
The values of the value assignments formalized by dispersionless two-valued states on, say, $k$ observables can be arranged in $k$-tuples.
These tuples can
be interpreted as extreme points or vertices of a compact convex subset of $\mathbb{R}^k$, a convex polytope,
that has an equivalent representation in terms of its hull; that is, as a set of half-spaces which are (in)equalities.
In the quantum physical realm, these inequalities are often referred to as Boole-Bell-type inequalities~\cite{froissart-81,pitowsky-86}.
We shall later encounter such a hull computation in deriving the Suppes-Zanotti inequalities~(\ref{2020-ex-SZI}).

\item[(iii)]
A complete set of $n$ two-valued states allows a representation as a partition logic
that explicitly represents a classical embedding into an extended Boolean algebra $2^n$~\cite{svozil-2001-eua}.
\end{itemize}

Therefore, as disclosed earlier, it is suggested to adopt Kochen and Specker's demarcation criterion of embeddability for a refined definition of contextuality:
One could speak of {\em strong contextuality} if no classical representation of the respective observables, and also no classical probability distribution exists.
Strong contextuality always indicates
some ``essential'' scarcity of two-valued states associated with classical truth assignments;
a deficiency to supply sufficient ``structural information'' for a classical embedding of the respective quantum observables.
As will be discussed later such essential scarcity can be categorized and quantified by escalating levels of ``rarity'', a ``shortage''
of elements of the set of two-valued states that may become nonseparating, nonunital, or in its strongest form empty.

The weaker
{\em probabilistic contextuality}, while featuring complementarity (because more than one context is involved),
allows all kinds of classically embeddable collections of observables, as well as classical probability distributions---if
only the
probability distribution in some way differs from global classical Kolmogorovian probabilities that are not restricted to local contexts.
That is, the observables still allow classical probability
distributions---as well as ``hidden variables'' in terms of the ``larger'' extended Boolean algebra in which the observables can be homomorphically embedded---but
those probabilities are not realized by the  probabilistic contextual  systems at hand.

Why should one make such a distinction from a physical point of view?
Because if there do not exist separating sets of two-valued states and thus no faithful classical embeddability
then the respective structures do not any longer support classical probability distributions;
and also no ``hidden variables'' in terms of the ``larger'' extended Boolean algebra in which the observables can be homomorphically embedded.
This is quite different from any nonclassical probability distribution on an otherwise ``quasi-classical''  empirical logics~\cite{Foulis1976}
containing complementary observables that can still be imbedded into ``larger'' Boolean algebras.

For physical realizations we refer to Wright's generalized urn models~\cite{wright} or the initial state identification problem
for finite automata~\cite{e-f-moore,schaller-96}, amounting to partition logics~\cite{svozil-2001-eua}.
The resulting empirical logics~\cite{Foulis1976} are identical to a variety of quantum propositional structures~\cite{birkhoff-36}
These logics support both classical as well as quantum probabilities.
Which probability distribution needs to be chosen depends solely on the respective physical realization~\cite{svozil-2018-b}.
In particular, there is no structural logical distinction; therefore, classical ``hidden variables'' cannot be ruled out,
the difference is in the classical-versus-quantum performance: the respective quantum contextuality is probabilistic.


In what follows several quantum mechanical examples of probabilistic contextuality in $n$--dimensional Hilbert space will be enumerated.
To avoid unnecessary redundancies they are mentioned with a reference to the concrete computation.
Often two-valued states will be rewritten in terms of the
expectation values of dichotomic outcomes $E \in \{-1,1\}$
by affine transformations---a multiplication followed by a subtraction---from classical value assignments encoded by two-valued states
$v \in \{0,1\}$:
$
E = 2 v - 1
$, or conversely,
$
v =  \frac{1}{2}\left( E + 1 \right)
$.
In quantum mechanics and Hilbert spaces of dimension greater than one, $E$ generalizes to a unitary Householder transformation
$
\textsf{\textbf{E}}_{\bf x}
=
\mathbb{1}- 2   {\bf x}^\dagger {\bf x}
$,
where ${\bf x}$ is a unit vector and $\dagger$ represents the Hermitian adjoint (aka conjugate).
The resulting eigensystem of $\textsf{\textbf{E}}_{\bf x}$ has eigenvalues $\pm 1$:
\begin{itemize}
\item[$-1$:]
${\bf x}$ is an eigenvector of    $\textsf{\textbf{E}}_{\bf x}$ with eigenvalue $-1$.
\item[$+1$:]
The remaining $n-1$ mutually orthogonal eigenvectors span the $n-1$ dimensional subspace orthogonal to ${\bf x}$.
Every vector in that subspace has eigenvalue $+1$. (For $n>2$ the spectrum is degenerate.)
\end{itemize}
For any context represented by some orthonormal basis
$
%{\frak B}=
\{  {\bf e}_1  ,
  {\bf e}_2  , \ldots ,  {\bf e}_n  \}$,
the product of the respective unitary Householder transformations is minus the identity; that is,
$
\textsf{\textbf{E}}_{{\bf e}_1} \textsf{\textbf{E}}_{{\bf e}_2} \cdots \textsf{\textbf{E}}_{{\bf e}_n}
=
-\mathbb{1}
$.

All such approaches take ``a bag of'' observables from quantum mechanics---some of them complementary---and force a classical interpretation upon them.
This is done in terms of classical value assignments formalized by two-valued states or expectations of binary observables.
Note that there exist classical empirical models featuring complementarity,
such as the ones meantioned earlier: Moore automata~\cite{e-f-moore,schaller-96}, or generalized urn models~\cite{wright},
both yielding partition logics~\cite{svozil-2001-eua,svozil-2018-b}.


\section{Examples of probabilistic contextuality}

In what follows we shall concentrate on two subtypes of  probabilistic contextuality;
one based on Boole's  ``conditions of physical experience''~\cite{Boole-62},
and another one on the functional behaviour derived from terminal vertices of gadget graphs.

\subsection{Boole-Bell type signatures of contextuality by hull computations}


As mentioned earlier the hull computation of the convex polytope
formed by vertices that represent the encoded classical value assignments of a given selection of observables
yields inequalities that are identified
with optimal Boole-Bell type inequalities~\cite{froissart-81,pitowsky-86}.
Violations of these inequalities
by quantum probabilities are interpreted as signifying contextuality relative to the assumptions discussed earlier,
in particular, omni-existence and noncontextuality.


For the sake of concrete examples of historic configurations used for Boole-Bell type inequalities
consider configurations with
\begin{itemize}
\item[(i)] isolated contexts with no common observable, such as
\begin{itemize}
\item[(i.SZ)]
Suppes-Zanotti inequalities from three observables~\cite{Suppes-81}, as discussed later;
\item[(i.CHSH)]
Clauser-Horne-Shimony-Holt (CHSH) inequalities from four observables in two groups, and their respective tensor products forming four isolated contexts~\cite{Suppes-81};
\item[(i.TPB)]
two-party Bell inequalities from finitely many observables, and their respective tensor products forming isolated contexts~\cite{Avis2005};
\end{itemize}
\item[(ii)]
intertwining contexts from three-or higher dimensional Hilbert spaces with common observables, such as
\begin{itemize}
\item[(ii.SB)]
for the Specker bug configuration~\cite{kochen2} that serves as graph theoretic true-implies-false gadget~\cite{svozil-2001-cesena};
\item[(ii.KCBS)]
inequalities from five cyclically connected contexts~\cite{Klyachko-2008}.
(The Bub-Stairs inequality~\cite{Bub-2009} on the same configuration is ad hoc and does not follow from a hull
computation but from a classical probability assessment.)
\end{itemize}
\end{itemize}

Suppes and Zanotti's result~\cite{Suppes-81,svozil-2020-ex} as well as other bounds on classical probabilities for different configurations of observables
still allow {\em single} instances of $\{-1,1\}$-value assignments, as the respective set of two-valued states is not empty (as for Kochen-Specker configurations).
But the classical probabilistic analysis of such configurations reveals that the bounds on classical probabilities
are violated by the quantum probabilities of analogous quantized systems.
This has been, for instance, pointed out by tabulations of classical  $\{-1,1\}$-value assignments
for the CHSH configuration~\cite{chsh} by Asher Peres~\cite{peres222}.
Further quantitative investigations into the ``amount'', or a measure, of probabilistic contextuality
in terms of enumerations and tabellations of classical value assignments have, for instance,
been studied in References~\cite{svozil_2010-pc09,svozil-2011-enough,Dzhafarov-2015,Dzhafarov-2017,KujalaDzhafarov-2019,Dzhafarov-19PhysRevA}.

Let us, for the sake of an explicit example, enumerate the classical value assignments in the Suppes-Zanotti configuration
which consists of measurements of an Einstein-Podolski-Rosen~\cite{epr} type configuration of two observables on one ``side'',
and one observable on the other ``side'' (CHSH uses a symmetric 2-2 configuration).
It therefore involves $2+1=3$ binary observables $X,Y,Z\in \{-1,+1\}$ associated with $\{-1,1\}$-value assignments
that can be used to form the second-order distributions
from the three second-order expectations
$\textsf{\textbf{E}}(X,Y) = XY$,
$\textsf{\textbf{E}}(X,Z) = XZ$,
$\textsf{\textbf{E}}(Y,Z) = YZ$,
obtained by just multiplying the binary observables from their three possible pairs, respectively.
Classically this amounts to $2^3=8$ value assignments tabulated in Table~\ref{2021-context-table-sz}.
\begin{table}%[H] %add [H] placement to break table across pages
\centering
%\begin{ruledtabular}
\begin{tabular}{c|ccc|ccccccccccccccccccc}
\toprule
\textbf{\#} & $X$ & $Y$ & $Z$ & $XY$ & $XZ$ & $YZ$ \\
\midrule
$v_1$ & $+1$ & $+1$ & $+1$ & $+1$ & $+1$ & $+1$ \\
$v_2$ & $+1$ & $+1$ & $-1$ & $+1$ & $-1$ & $-1$ \\
$v_3$ & $+1$ & $-1$ & $+1$ & $-1$ & $+1$ & $-1$ \\
$v_4$ & $-1$ & $+1$ & $+1$ & $-1$ & $-1$ & $+1$ \\
$v_5$ & $+1$ & $-1$ & $-1$ & $-1$ & $-1$ & $+1$ \\
$v_6$ & $-1$ & $+1$ & $-1$ & $-1$ & $+1$ & $-1$ \\
$v_7$ & $-1$ & $-1$ & $+1$ & $+1$ & $-1$ & $-1$ \\
$v_8$ & $-1$ & $-1$ & $-1$ & $+1$ & $+1$ & $+1$ \\
\bottomrule
\end{tabular}
%\end{ruledtabular}
 \caption{\label{2021-context-table-sz}  The eight  $\{-1,1\}$-value assignments of the Suppes-Zanotti configuration.}
 \end{table}
These eight classical value assignments can be used to generate all classical higher-order distributions~\cite{Garg1984}
by convex summation of these value assignments; in particular, the probabilities
\begin{equation}
\begin{split}
p(x) = \lambda_1 v_1(x) +\cdots + \lambda_n v_n(x),
\\
{\rm with} \;
\lambda_1 +\cdots + \lambda_n=1
\text{ and }
 \lambda_j  \ge  0
, \;
j\in \left\{1,\ldots,n\right\}
.
\end{split}
\end{equation}
The second-order distributions can be geometrically characterized by a convex polytope~\cite{froissart-81,pitowsky-86}
formed by the ``classical vertices'' whose coordinates are arranged in three-tuples
$\begin{pmatrix}XY,XZ,YZ\end{pmatrix}^\intercal$ ($\intercal$ indicates transposition)
with respect to the Cartesian standard basis of $\mathbb{R}^3$  are identified
with  the respective last three row entries of Table~\ref{2021-context-table-sz}:
in this case the four vertex vectors (the other four vectors are duplicates) are
$\begin{pmatrix} 1,1,1\end{pmatrix}^\intercal$,
$\begin{pmatrix} 1,-1,-1\end{pmatrix}^\intercal$,
$\begin{pmatrix} -1,1,-1\end{pmatrix}^\intercal$,
and
$\begin{pmatrix} -1,-1,1\end{pmatrix}^\intercal$.
The resulting convex polytope has an equivalent representation in terms of its hull, formed by its half-spaces~\cite{ziegler,Schrijver,Fukuda-techrep}.
In the case of the Suppes-Zanotti configuration~\cite{Suppes-81,Brody-1989,Khrennikov2020} the hull equations are
the Suppes-Zanotti inequalities; in particular, the four half-spaces described by the inequalities~\cite{svozil-2020-ex}
\begin{equation}
- 1  \le  \pm \textsf{\textbf{E}}(X,Y) \pm \textsf{\textbf{E}}(X,Z) \pm \textsf{\textbf{E}}(Y,Z) \le 1
.
\label{2020-ex-SZI}
\end{equation}

Quantization of the Suppes-Zanotti configuration
with the associated operators~\cite{filipp-svo-04-qpoly-prl}
$
\textsf{\textbf{F}}(X,Y) \pm \textsf{\textbf{F}}(X,Z) \pm \textsf{\textbf{F}}(Y,Z)
$
with the quantum expectation $\textsf{\textbf{F}}$ yields the much larger quantum bounds
\begin{equation}
-3 <  \textsf{\textbf{F}}(X,Y) \pm \textsf{\textbf{F}}(X,Z) \pm \textsf{\textbf{F}}(Y,Z)   < 3
\label{2020-ex-SZIq}
\end{equation}
that allows a violation of the classical bounds~(\ref{2020-ex-SZI}),
which is a signature of probabilistic contextuality.





\subsection{Functional signatures of contextuality}

There exist configurations of observables that, interpreted classically, serve all kinds of (logical) functions.
(Graph theoretically they are gadgets.)
Usually, they have input and output terminals which, functionally interpreted, serve as arguments and functional values.
Two historic configurations realize either true-implies-false~\cite{kochen2,2018-minimalYIYS}
or true-implies-true functional relations~\cite{kochen1,svozil-2020-hardy}:
if a particular state is preselected on the input terminal then
classical value assignments (implementing omni-existence and noncontextuality)
enforce a particular dependent value assignment---either false or true, respectively---on
the output terminal.

Violations of these dependencies
by quantum probabilities are interpreted as signifying contextuality relative to the assumptions discussed earlier.
In particular, any classical ``hidden variable'' model cannot implement both omni-existence and have noncontextual admissible value assignments~\cite{peres222}.

For the sake of examples we refer to (extensions of)
the Specker bug~\cite{kochen2,2018-minimalYIYS}, or the examples in Refs.~\cite{Ramanathan-18,svozil-2018-whycontexts}
which use finite sets of quantum observables in three-dimensional Hilbert space,
as well as Hardy type configurations~\cite{kochen1,svozil-2020-hardy}
which use finite sets of quantum observables in four- or higher-dimensional~\cite{2018-minimalYIYS} Hilbert space.
All of these gadgets have a classical interpretation in terms of partition logic,
finite automaton models or generalized urn models.
Their respective logico-algebraic structure can be faithfully embedded into extended Boolean algebras;
for instance, $2^n$,
by identification with the union of elements of a partition obtained from analyzing a complete set of $n$ two-valued states~\cite{svozil-2001-eua}.

For the sake of an example we shall review a configuration of 13 observables in seven bi-intertwining contexts (maximal Boolean subalgebras $2^3$) introduced by Kochen and Specker~\cite{kochen2}.
It hypergraph representing contexts as smooth curves (lines) is depicted in Figure~\ref{2021-context-figure-sb}.
\ifx\revtex\undefined
\begin{figure}%[H] %add [H] placement to break table across pages
\else
\begin{figure*}%[H] %add [H] placement to break table across pages
\fi
\centering
\resizebox{0.8\textwidth}{!}{
\begin{tabular}{cccccc}
\begin{tikzpicture}  [scale=0.8]

\tikzstyle{every path}=[line width=1pt]

\newdimen\ms
\ms=0.1cm
\tikzstyle{s1}=[color=red,rectangle,inner sep=3.5]
\tikzstyle{c3}=[circle,inner sep={\ms/8},minimum size=5*\ms]
\tikzstyle{c2}=[circle,inner sep={\ms/8},minimum size=3*\ms]
\tikzstyle{c1}=[circle,inner sep={\ms/8},minimum size=2*\ms]
\tikzstyle{cs1}=[circle,inner sep={\ms/8},minimum size=1*\ms]

% Define positions of all observables


\coordinate (a7) at  (1,2);
\coordinate (a5) at (2,1);
\coordinate (a10) at (0.5,0.5);
\coordinate (a11) at (2,-1);
\coordinate (a1) at (1,-2);
\coordinate (a3) at (0,-1);
\coordinate (a4) at (1.5,0.5);
\coordinate (a9) at (0,1);



% draw contexts

\draw [color=orange] (a1) -- (a11)  coordinate[cs1,fill=white,draw=gray,pos=0.5,label=below right:{\color{black}$a_{12}$}] (a12);
\draw [color=blue] (a11) -- (a9);
\draw [color=red] (a9) -- (a7)  coordinate[cs1,fill=white,draw=gray,pos=0.5,label=above left:{\color{black}$a_8$}] (a8);
\draw [color=green] (a7) -- (a5)  coordinate[cs1,fill=white,draw=gray,pos=0.5,label=above right:{\color{black}$a_6$}] (a6);
\draw [color=gray] (a5) -- (a3);
\draw [color=magenta] (a1) -- (a3)  coordinate[cs1,fill=white,draw=gray,pos=0.5,label= below left:{\color{black}$a_2$}] (a2);
\draw [color=cyan] (a10) -- (a4)  coordinate[cs1,fill=white,draw=gray,pos=0.5,label=above:{\color{black}$a_{13}$}] (a13);




% draw atoms

\draw (a1) coordinate[c2,fill=orange,label=below:$a_1$];
\draw (a1) coordinate[c1,fill=magenta];


\draw (a3) coordinate[c2,fill=magenta,label=left:$a_3$];
\draw (a3) coordinate[c1,fill=gray];

\draw (a11) coordinate[c2,fill=blue,label=right:$a_{11}$];
\draw (a11) coordinate[c1,fill=orange];

\draw (a10) coordinate[c2,fill=cyan,label=below left:$a_{10}$];
\draw (a10) coordinate[c1,fill=blue];

\draw (a5) coordinate[c2,fill=gray,label=right:$a_{5}$];
\draw (a5) coordinate[c1,fill=cyan];

\draw (a9) coordinate[c2,fill=red,label=left:$a_9$];
\draw (a9) coordinate[c1,fill=blue];


\draw (a7) coordinate[c2,fill=gray,label=above:$a_7$];
\draw (a7) coordinate[c1,fill=green];

\draw (a4) coordinate[c2,fill=cyan,label=below right:$a_4$];
\draw (a4) coordinate[c1,fill=gray];


\end{tikzpicture}
&
%%%%%%%%%%%%%%%
&
\begin{tikzpicture}  [scale=0.8]

\tikzstyle{every path}=[line width=1pt]

\newdimen\ms
\ms=0.1cm
\tikzstyle{s1}=[color=red,rectangle,inner sep=3.5]
\tikzstyle{c3}=[circle,inner sep={\ms/8},minimum size=5*\ms]
\tikzstyle{c2}=[circle,inner sep={\ms/8},minimum size=3*\ms]
\tikzstyle{c1}=[circle,inner sep={\ms/8},minimum size=2*\ms]
\tikzstyle{cs1}=[circle,inner sep={\ms/8},minimum size=1*\ms]

% Define positions of all observables


\coordinate (a7) at  (1,2);
\coordinate (a5) at (2,1);
\coordinate (a10) at (0.5,0.5);
\coordinate (a11) at (2,-1);
\coordinate (a1) at (1,-2);
\coordinate (a3) at (0,-1);
\coordinate (a4) at (1.5,0.5);
\coordinate (a9) at (0,1);



% draw contexts

\draw [color=orange] (a1) -- (a11)  coordinate[cs1,fill=white,draw=gray,pos=0.5,label=below right:{\color{black}$p(a_{12})$}] (a12);
\draw [color=blue] (a11) -- (a9);
\draw [color=red] (a9) -- (a7)  coordinate[cs1,fill=white,draw=gray,pos=0.5,label=above left:{\color{black}$p(a_8)$}] (a8);
\draw [color=green] (a7) -- (a5)  coordinate[cs1,fill=white,draw=gray,pos=0.5,label=above right:{\color{black}$p(a_6$)}] (a6);
\draw [color=gray] (a5) -- (a3);
\draw [color=magenta] (a1) -- (a3)  coordinate[cs1,fill=white,draw=gray,pos=0.5,label= below left:{\color{black}$p(a_2)$}] (a2);
\draw [color=cyan] (a10) -- (a4)  coordinate[cs1,fill=white,draw=gray,pos=0.5,label=above:{\color{black}$p(a_{13})$}] (a13);




% draw atoms

\draw (a1) coordinate[c2,fill=orange,label=below:\colorbox{white}{\textcolor{blue}{$p(a_1)=\lambda_1+\lambda_2+\lambda_3$}}];
\draw (a1) coordinate[c1,fill=magenta];


\draw (a3) coordinate[c2,fill=magenta,label=left:$p(a_3)$];
\draw (a3) coordinate[c1,fill=gray];

\draw (a11) coordinate[c2,fill=blue,label=right:$p(a_{11})$];
\draw (a11) coordinate[c1,fill=orange];

\draw (a10) coordinate[c2,fill=cyan,label=below left:$p(a_{10})$];
\draw (a10) coordinate[c1,fill=blue];

\draw (a5) coordinate[c2,fill=gray,label=right:$p(a_{5})$];
\draw (a5) coordinate[c1,fill=cyan];

\draw (a9) coordinate[c2,fill=red,label=left:$p(a_9)$];
\draw (a9) coordinate[c1,fill=blue];


\draw (a7) coordinate[c2,fill=gray,label=above:\colorbox{white}{\textcolor{red}{$p(a_7)=\lambda_6+\lambda_{13}+\lambda_{14}$}}];
\draw (a7) coordinate[c1,fill=green];

\draw (a4) coordinate[c2,fill=cyan,label=below right:$p(a_4)$];
\draw (a4) coordinate[c1,fill=gray];


\end{tikzpicture}
&
%%%%%%%%%%%%%%%%%%%
&
\begin{tikzpicture}  [scale=0.8]

\tikzstyle{every path}=[line width=1pt]

\newdimen\ms
\ms=0.1cm
\tikzstyle{s1}=[color=red,rectangle,inner sep=3.5]
\tikzstyle{c3}=[circle,inner sep={\ms/8},minimum size=5*\ms]
\tikzstyle{c2}=[circle,inner sep={\ms/8},minimum size=3*\ms]
\tikzstyle{c1}=[circle,inner sep={\ms/8},minimum size=2*\ms]
\tikzstyle{cs1}=[circle,inner sep={\ms/8},minimum size=1*\ms]

% Define positions of all observables


\coordinate (a7) at  (1,2);
\coordinate (a5) at (2,1);
\coordinate (a10) at (0.5,0.5);
\coordinate (a11) at (2,-1);
\coordinate (a1) at (1,-2);
\coordinate (a3) at (0,-1);
\coordinate (a4) at (1.5,0.5);
\coordinate (a9) at (0,1);




% draw contexts

\draw [color=orange] (a1) -- (a11)  coordinate[cs1,fill=white,draw=gray,pos=0.5,label=below right:{\scriptsize \color{black}$\begin{pmatrix}0,1,\sqrt{3}\end{pmatrix}$}] (a12);
\draw [color=blue] (a11) -- (a9);
\draw [color=red] (a9) -- (a7)  coordinate[cs1,fill=white,draw=gray,pos=0.5,label=above left:{\scriptsize \color{black}$\begin{pmatrix}-2\sqrt{2},\sqrt{2},-3\sqrt{3}\end{pmatrix}$}] (a8);
\draw [color=green] (a7) -- (a5)  coordinate[cs1,fill=white,draw=gray,pos=0.5,label=above right:{\scriptsize \color{black}$\begin{pmatrix}2\sqrt{2},-1,-3\sqrt{3}\end{pmatrix}$}] (a6);
\draw [color=gray] (a5) -- (a3);
\draw [color=magenta] (a1) -- (a3)  coordinate[cs1,fill=white,draw=gray,pos=0.5,label= below left:{\scriptsize \color{black}$\begin{pmatrix}0, -1, \sqrt{3}]\end{pmatrix}$}] (a2);
\draw [color=cyan] (a10) -- (a4)  coordinate[cs1,fill=white,draw=gray,pos=0.5,label=above:{\tiny \color{black}$\begin{pmatrix}2,-2\sqrt{2},0\end{pmatrix}$}] (a13);




% draw atoms

 \draw (a1) coordinate[c2,fill=orange,label=below:{\colorbox{white}{\textcolor{blue}{$\begin{pmatrix}1,0,0\end{pmatrix}$}}}];
 \draw (a1) coordinate[c1,fill=magenta];


\draw (a3) coordinate[c2,fill=magenta,label=left:{$\begin{pmatrix}0,\sqrt{3},1\end{pmatrix}$}];
\draw (a3) coordinate[c1,fill=gray];

\draw (a11) coordinate[c2,fill=blue,label=right:{$\begin{pmatrix}0,\sqrt{3},-1\end{pmatrix}$}];
\draw (a11) coordinate[c1,fill=orange];

\draw (a10) coordinate[c2,fill=cyan,label=below left:{$\begin{pmatrix}\sqrt{2},1,\sqrt{3}\end{pmatrix}$}];
\draw (a10) coordinate[c1,fill=blue];

\draw (a5) coordinate[c2,fill=gray,label=right:{$\begin{pmatrix}-2\sqrt{2},1,-\sqrt{3}\end{pmatrix}$}];
\draw (a5) coordinate[c1,fill=cyan];

\draw (a9) coordinate[c2,fill=red,label=left:{$\begin{pmatrix}-2\sqrt{2},1,\sqrt{3}\end{pmatrix}$}];
\draw (a9) coordinate[c1,fill=blue];


\draw (a7) coordinate[c2,fill=gray,label=above:{\colorbox{white}{\textcolor{red}{$\begin{pmatrix}1,2\sqrt{2},0\end{pmatrix}$}}}];
\draw (a7) coordinate[c1,fill=green];

\draw (a4) coordinate[c2,fill=cyan,label=below right:{$\begin{pmatrix}\sqrt{2},1,-\sqrt{3}\end{pmatrix}$}];
\draw (a4) coordinate[c1,fill=gray];

\end{tikzpicture}
\\
(a)&&(b)&&(c)\\
\end{tabular}
}
\caption{\label{2021-context-figure-sb} (a) Configuration of 13 observables in seven bi-intertwining contexts serving as a symmetric (with respect to horizontal)
true-implies-false gadget; (b) the associated classical probability distributions obtained by the convex sum
$\lambda_1+\cdots +\lambda_{14}=1$, $\lambda_i\ge 0$, $1 \le i \le 14$;
(c) a quantum representation in terms of a faithful orthogonal vertex labeling
in terms of a vertex representation by vectors, preserving orthogonality of adjacent vertices~\cite{lovasz-79}
of the hypergraph that maximizes
the probability of $a_7$, given $a_1$.}
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\else
\end{figure*}%[H] %add [H] placement to break table across pages
\fi

\begin{table}%[H] %add [H] placement to break table across pages
\centering
%\begin{ruledtabular}
\begin{tabular}{c|cccccccccccccccccccccc}
\toprule
\textbf{\#} & $a_1$ & $a_2$ & $a_3$ & $a_4$ & $a_5$ & $a_6$ & $a_7$ & $a_8$ & $a_9$ & $a_{10}$ & $a_{11}$ & $a_{12}$ & $a_{13}$ \\
\midrule
$v_1$ & \colorbox{white}{\fbox{\textcolor{blue}{$1$}}}&0& 0& 1& 0& 1& 0& 0& 1& 0& 0& 0& 0\\
$v_2$ & \colorbox{white}{\fbox{\textcolor{blue}{$1$}}}&0& 0& 0& 1& 0& 0& 1& 0& 1& 0& 0& 0\\
$v_3$ & \colorbox{white}{\fbox{\textcolor{blue}{$1$}}}&0& 0& 0& 1& 0& 0& 0& 1& 0& 0& 0& 1\\
$v_4$ & 0&1& 0& 1& 0& 1& 0& 1& 0& 0& 1& 0& 0\\
$v_5$ & 0&1& 0& 1& 0& 1& 0& 0& 1& 0& 0& 1& 0\\
$v_6$ & 0&1& 0& 1& 0& 0& \colorbox{white}{\fbox{\textcolor{red}{$1$}}}& 0& 0& 0& 1& 0& 0\\
$v_7$ & 0&1& 0& 0& 1& 0& 0& 1& 0& 1& 0& 1& 0\\
$v_8$ & 0&1& 0& 0& 1& 0& 0& 1& 0& 0& 1& 0& 1\\
$v_9$ & 0&1& 0& 0& 1& 0& 0& 0& 1& 0& 0& 1& 1\\
$v_{10}$ & 0&0& 1& 0& 0& 1& 0& 1& 0& 1& 0& 1& 0\\
$v_{11}$ & 0&0& 1& 0& 0& 1& 0& 1& 0& 0& 1& 0& 1\\
$v_{12}$ & 0&0& 1& 0& 0& 1& 0& 0& 1& 0& 0& 1& 1\\
$v_{13}$ & 0&0& 1& 0& 0& 0& \colorbox{white}{\fbox{\textcolor{red}{$1$}}}& 0& 0& 1& 0& 1& 0\\
$v_{14}$ & 0&0& 1& 0& 0& 0& \colorbox{white}{\fbox{\textcolor{red}{$1$}}}& 0& 0& 0& 1& 0& 1\\
\bottomrule
\end{tabular}
%\end{ruledtabular}
 \caption{\label{2021-context-table-sb}  The 14  $\{0,1\}$-value assignments of the Specker bug configuration. Boxed values indicate nonvanishing contributions of the respective two-valued state to the probabilities of the terminal points $a_1$ and $a_7$ of the gadget.}
 \end{table}

This Specker bug (Specker's ``K\"afer'' configuration can be employed as a classical (noncontextual) true-implies-false (TIF) gadget, as
the two ``terminal points'' $a_1$ and $a_7$ cannot both be ``true'' (value 1) at the same time:
suppose $a_1$ and $a_7$ are both $1$ simultaneously.
Then admissibility demands that $a_3$, $a_5$, $a_9$ as well as $a_{11}$ must all be $0$ simultaneously.
Consequently  $a_4$ and $a_{10}$ must both be $1$ simultaneously---a complete contradiction since admissibility of two-valued $\{0,1\}$-states requires exactly one
element of a context to be 1, all other elements must be 0.
Nevertheless one can be true (have value 1) and the other one false (value 0).
Also they can both be false (value 0): their respective probabilities,
which can be obtained by the convex sum of its 14 two-valued states enumerated in Table~\ref{2021-context-table-sb}
and depicted in in Figure~\ref{2021-context-figure-sb}(b), are
\colorbox{white}{\textcolor{black}{$p(a_1)=\lambda_1+\lambda_2+\lambda_3$}}
and
\colorbox{white}{\textcolor{black}{$p(a_7)=\lambda_6+\lambda_{13}+\lambda_{14}$}}
are mutually exclusive.

Quantization in terms of faithful orthogonal vertex labeling---a
vertex representation by vectors, preserving orthogonality of adjacent vertices~\cite{lovasz-79}---as for instance,
depicted in Figure~\ref{2021-context-figure-sb}(c),
allows the simultaneous preparation by the pre-selection state $a_1$ and the detection of the post-selected state
$a_7$ with nonvanishing
probabilities $\vert     a_7  \cdot a_1  \vert^2 \le \frac{1}{9}$, thereby violating the classical predictions of zero chance that $a_7$ occurs if
$a_1$ occurred.
In this concrete realization with $a_1\equiv \begin{pmatrix}1,0,0\end{pmatrix}^\intercal$ and
$a_7\equiv \frac{1}{3} \begin{pmatrix}1,2\sqrt{2},0\end{pmatrix}^\intercal$
the violation with the classical prediction is maximal~\cite{Belinfante-73,redhead,cabello-1994}.

Also in this case the set of two-valued states allowing a classical co-representation even of complementary observables.
And yet, probabilistic contextuality manifests itself in the functional performance of the respective gadgets at its terminal points.


\subsection{Ad hoc signatures of contextuality}

We just mention without further discussion that there exist other ad hoc methods,
in particular, so-called ``state independent quantum contextuality''~\cite{cabello:210401}, to obtain probabilistic contextuality.
Often the observables of some Kochen-Specker configurations without any two-valued state are taken.
By relaxing the axioms of admissibility mentioned earlier, ``classical'' (relative to these relaxed assumptions) estimates and predictions are obtained
which are violated by experimentally testable quantum predictions~\cite{cabello2020converting}.


\section{Examples of strong contextuality}

In what follows we shall deal with (finite) configurations of observable whose classical interpretations in
terms of its two-valued states is insufficient for an embedding into any Boolean algebra.
Even though there still may exist ``many'' two-valued states associated with classical value assignments,
these assignments may not be able to resolve the structure of quantum observables supporting them.
There exist escalations of the ``smallness'' of the set of two-valued states in terms of
inseparability, nonunitality, or nonexistence that will be briefly reviewed next.
All these instances go beyond classical embeddability (and preservation of the logico-algebraic structure of the associated observables)
as they do not satisfy Kochen and Specker's demarcation criterion~\cite[Theorem~0]{kochen1} for separability.


\subsection{Nonseparability of classical value assignments}

As mentioned there exist finite sets of observables that do not allow separation by classical value assignments.
That is, in such circumstances, no
classical value assignment exists that
is capable to differentiate between, or separating, the individual constituents of some pair of distinct quantum observables.

For the sake of an example of contextuality based on inseparability,
take Kochen and Specker's combo~\cite[Graph~$\Gamma_3$]{kochen1} of intertwining true-implies-true gadgets~\cite[Graph~$\Gamma_1$]{kochen1}
that contains two pairs of observables that cannot be classically separated.
Its hypergraph is depicted in Figure~\ref{2021-context-figure-ksc}.
\ifx\revtex\undefined
\begin{figure}%[H] %add [H] placement to break table across pages
\else
\begin{figure*}%[H] %add [H] placement to break table across pages
\fi
\centering
\resizebox{0.6\textwidth}{!}{
\begin{tikzpicture}  [scale=1]

\tikzstyle{every path}=[line width=1pt]

\newdimen\ms
\ms=0.1cm
\tikzstyle{s1}=[color=red,rectangle,inner sep=3.5]
\tikzstyle{c3}=[circle,inner sep={\ms/8},minimum size=4*\ms]
\tikzstyle{c2}=[circle,inner sep={\ms/8},minimum size=3*\ms]
\tikzstyle{c1}=[circle,inner sep={\ms/8},minimum size=2*\ms]

% Define positions of all observables

% bug #1
\coordinate (a1) at  (1,2);
\coordinate (a2) at (1.5,{(1-(1-0.5)/2)*2});
\coordinate (a3) at (2,1);
\coordinate (a4) at (2,0);
\coordinate (a5) at (2,-1);
\coordinate (a6) at (1.5,{(-0.5-(1-0.5)/2)*2});
\coordinate (a7) at (1,-2);
\coordinate (a8) at (0.5,{(-0.5-(1-0.5)/2)*2});
\coordinate (a9) at (0,-1);
\coordinate (a10) at (0,0);
\coordinate (a11) at (0,1);
\coordinate (a12) at (0.5,{(1-(1-0.5)/2)*2});
\coordinate (a13) at (1,0);

\coordinate (a14) at (4,0);

% bug #2
\coordinate (a21) at  ({6+1},2);
\coordinate (a22) at ({6+1.5},{(1-(1-0.5)/2)*2});
\coordinate (a23) at ({6+2},1);
\coordinate (a24) at ({6+2},0);
\coordinate (a25) at ({6+2},-1);
\coordinate (a26) at ({6+1.5},{(-0.5-(1-0.5)/2)*2});
\coordinate (a27) at ({6+1},-2);
\coordinate (a28) at ({6+0.5},{(-0.5-(1-0.5)/2)*2});
\coordinate (a29) at ({6+0},-1);
\coordinate (a210) at ({6+0},0);
\coordinate (a211) at ({6+0},1);
\coordinate (a212) at ({6+0.5},{(1-(1-0.5)/2)*2});
\coordinate (a213) at ({6+1},0);


% draw contexts

% bug #1
\draw [color=orange] (a1) -- (a3);
\draw [color=blue] (a3) -- (a5);
\draw [color=red] (a5) -- (a7);
\draw [color=green] (a7) -- (a9);
\draw [color=gray] (a9) -- (a11);
\draw [color=magenta] (a11) -- (a1);
\draw [color=cyan] (a10) -- (a4);

\draw [color=lime] (a1) to [out=0,in={90+45}] (a14) to [out={180+90+45},in=180]  (a27);
\draw [color=olive] (a7) to [out=0,in={180+45}] (a14) to [out=45,in=180] (a21);

% bug #2
\draw [color=orange!40!white] (a21) -- (a23);
\draw [color=blue!40!white] (a23) -- (a25);
\draw [color=red!40!white] (a25) -- (a27);
\draw [color=green!40!white] (a27) -- (a29);
\draw [color=gray!40!white] (a29) -- (a211);
\draw [color=magenta!40!white] (a211) -- (a21);
\draw [color=cyan!40!white] (a210) -- (a24);



% draw atoms

% bug 1
\draw (a1) coordinate[c3,fill=lime,label=above:{\colorbox{white}{\textcolor{red}{$a_7$}}}];
\draw (a1) coordinate[c2,fill=orange];
\draw (a1) coordinate[c1,fill=gray];

\draw (a2) coordinate[c1,fill=orange,label=below left:$a_8$];

\draw (a3) coordinate[c2,fill=blue,label=right:$a_9$];
\draw (a3) coordinate[c1,fill=orange];

\draw (a4) coordinate[c2,fill=cyan,label=right:$a_{10}$];
\draw (a4) coordinate[c1,fill=blue];

\draw (a5) coordinate[c2,fill=red,label=right:$a_{11}$];
\draw (a5) coordinate[c1,fill=blue];

\draw (a6) coordinate[c1,fill=red,label=above left:$a_{12}$];

\draw (a7) coordinate[c3,fill=olive,label=below:{\colorbox{white}{\textcolor{blue}{$a_1$}}}];
\draw (a7) coordinate[c2,fill=green];
\draw (a7) coordinate[c1,fill=red];

\draw (a8) coordinate[c1,fill=green,label=below left:$a_2$];

\draw (a9) coordinate[c2,fill=gray,label=left:$a_3$];
\draw (a9) coordinate[c1,fill=green];

\draw (a10) coordinate[c2,fill=gray,label=left:$a_4$];
\draw (a10) coordinate[c1,fill=cyan];

\draw (a11) coordinate[c2,fill=magenta,label=left:$a_5$];
\draw (a11) coordinate[c1,fill=gray];

\draw (a12) coordinate[c1,fill=magenta,label=above left:$a_6$];

\draw (a13) coordinate[c1,fill=cyan,label=above:$a_{13}$];


\draw (a14) coordinate[c2,fill=lime,label=above:{$c$}];
\draw (a14) coordinate[c1,fill=olive];

\draw (a21) coordinate[c3,fill=olive,label=above:{\colorbox{white}{\textcolor{red}{$b_7$}}}];

\draw (a27) coordinate[c3,fill=lime,label=below:{\colorbox{white}{\textcolor{blue}{$b_1$}}}];

% bug 2
\draw (a21) coordinate[c2,fill=orange!40!white];
\draw (a21) coordinate[c1,fill=gray!40!white];

\draw (a22) coordinate[c1,fill=orange!40!white,label=above right:$b_8$];

\draw (a23) coordinate[c2,fill=blue!40!white,label=right:$b_9$];
\draw (a23) coordinate[c1,fill=orange!40!white];

\draw (a24) coordinate[c2,fill=cyan!40!white,label=right:$b_{10}$];
\draw (a24) coordinate[c1,fill=blue!40!white];

\draw (a25) coordinate[c2,fill=red!40!white,label=right:$b_{11}$];
\draw (a25) coordinate[c1,fill=blue!40!white];

\draw (a26) coordinate[c1,fill=red!40!white,label=below right:$b_{12}$];

\draw (a27) coordinate[c2,fill=green!40!white];
\draw (a27) coordinate[c1,fill=red!40!white];

\draw (a28) coordinate[c1,fill=green!40!white,label=above right:$b_2$];

\draw (a29) coordinate[c2,fill=gray!40!white,label=left:$b_3$];
\draw (a29) coordinate[c1,fill=green!40!white];

\draw (a210) coordinate[c2,fill=gray!40!white,label=left:$b_4$];
\draw (a210) coordinate[c1,fill=cyan!40!white];

\draw (a211) coordinate[c2,fill=magenta!40!white,label=left:$b_5$];
\draw (a211) coordinate[c1,fill=gray!40!white];

\draw (a212) coordinate[c1,fill=magenta!40!white,label=below right:$b_6$];

\draw (a213) coordinate[c1,fill=cyan!40!white,label=above:$b_{13}$];


\end{tikzpicture}
}
\caption{\label{2021-context-figure-ksc} The Kochen-Specker ``combo of Specker bugs'' whose set of classical truth assignments formalized by its
two-valued states cannot separate $a_1$ from $b_1$, as well as $a_7$ from $b_7$.}
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\end{figure*}%[H] %add [H] placement to break table across pages
\fi
If $a_1$ is true (has value 1) then $b_1$ has to be true (has value 1), and vice versa.
Likewise,
if $a_7$ is true (has value 1) then $b_7$ has to be true (has value 1), and vice versa.
Therefore,
$a_1$ cannot be classically separated from  $b_1$, and
$a_7$ cannot be classically separated from  $b_7$.
For a proof, note that,  if $a_1$ is assumed to be true, then the true-implies-false Specker bug gadget and
admissibility demands $a_7$ as well as $c$ to be false,
and thus $b_1$ to be true.
Likewise, whenever $b_1$ is  true,
$b_7$ as well as $c$ needs to be false,
and thus $a_1$ to be true. Therefore, $a_1$ cannot be
separated from $b_1$ by any classical means.
A symmetric argument (utilizing the symmetry of the true-implies-false Specker bug gadget)
yields nonseparability of $a_7$  from $b_7$.
The Kochen-Specker combo has no faithful embedding into any ``larger''
Boolean algebra, because any such faithful embedding would allow a classical resolution of the constituents of the pairs
$a_1$ and $b_1$, as well as $a_7$ and $b_7$.


A respective four-dimensional example inspired by Hardy's nonlocal configuration
can be found in Figure~5 of Reference~\cite{svozil-2020-hardy}.
Other explicit experimentally testable cases of inseparability
can be found in a configuration depicted in Figure~2 of Ref.~\cite{tkadlec-96},
as well as in Figure~24.2c analyzed in Table~24.1 of Ref.~\cite{Svozil-2018-p}, based on a configuration introduced in
Figure~2 of Ref.~\cite{2015-AnalyticKS}.

\subsection{Nonunitality of classical value assignments}

Another, even stronger (because it includes and extends inseparability)
form of logical contextuality are collections of observables with a unital set of two-valued states: if interpreted classically such structures
enforce the nonoccurrence (and occurrence) of certain observables.

Two explicit experimentally testable cases of inseparability
are the same as mentioned earlier in a configuration depicted in Figure~2 of Ref.~\cite{tkadlec-96},
as well as in Figure~24.2c analyzed in Table~24.1 of Ref.~\cite{Svozil-2018-p}, based on a configuration introduced in
Figure~2 of Ref.~\cite{2015-AnalyticKS}.

For the sake of demonstration we review this latter configuration depicted in Figure~\ref{2021-context-figure-ACS}.
Analysis of its set of eight two valued states enumerated in Table~\ref{2021-context-table-ACS}  reveals that eight observables, namely
 $a_2$, $a_{13}$, $a_{15}$, $a_{16}$, $a_{17}$, $a_{25}$, $a_{27}$, $a_{36}$ are always 0 (aka false, or nonoccurring) value because they are connected to $a_1$ which has to be 1
(aka true, or always occurring).
As a corollary, those eight observables with simultaneous values 0 cannot be separated from one another with classical means; that is, by two-valued states.

This configuration can, of course, be always subjected to a global rotation, such that its
faithful orthogonal representation,
as enumerated in Table~I of Ref.~\cite{2015-AnalyticKS}, matches $a_1$, or, alternatively, $a_2$.
Thereby, given any pure state, we can as a corollary construct a complete contradiction.
Because given any pure state ${\bf a}$ that can be represented as a unit vector of $\mathbb{R}^3$, we are free to choose
two faithful orthogonal vertex labelings of the hypergraph depicted in Figure~\ref{2021-context-figure-ACS}:
one that matches ${\bf a}$ with $a_1$,
and another one that matches ${\bf a}$ with $a_2$.
Hence ${\bf a}$ would need to be 1 and 0, aka true and false; a complete contradiction.
One could call this argument ``state independent'' because it applies to any pure state ${\bf a}$.



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\centering
\resizebox{0.85\textwidth}{!}{
\begin{tikzpicture}  [scale=0.5, rotate=0]
        %\tikzstyle{every path}=[line width=1.5pt]
%\tikzstyle{c1}=[circle,fill,inner sep=4]
%\tikzstyle{c2}=[circle,fill,inner sep=2.7]
       \tikzstyle{c1}=[color=gray,circle,inner sep=1.5]
       \tikzstyle{c2}=[color=blue,circle,inner sep=3]
        \tikzstyle{s1}=[color=red,rectangle,inner sep=4]
        \tikzstyle{l1}=[draw=none,circle,minimum size=6]

        % Define positions of all observables



\draw [color=orange]  (4,0)  coordinate[c1,fill,label=260:{\color{black}\footnotesize $a_{37}$}] (b) -- (13,0)    coordinate[c2,fill,label={[label distance=-1]270:{\footnotesize \color{black}  $a_2$}}] (2) -- (22,0)  coordinate[c1,fill,label={[label distance=-1]315:{\footnotesize \color{black}  $a_3$}}] (3);
\draw [color=blue] (3) -- (26,12)  coordinate[c1,fill,pos=0.8,label={[label distance=-1]0:{\footnotesize \color{black}  $a_{21}$}}] (21) coordinate[c1,fill,label={[label distance=-3]0:{\footnotesize \color{black}  $a_{23}$}}] (23);
\draw [color=green] (23) -- (22,18.5) coordinate[c1,fill,pos=0.4,label={[label distance=-1]0:{\footnotesize \color{black}  $a_{29}$}}] (29) coordinate[c1,fill,label={[label distance=-1]45:{\footnotesize \color{black}  $a_5$}}] (5);
\draw [color=magenta] (5)-- (13,18.5)coordinate[c2,fill,label=90:{\color{black}\footnotesize $a_{36}$}] (a) -- (4,18.5)  coordinate[c1,fill,label={[label distance=-1]135:{\footnotesize \color{black}  $a_4$}}] (4);
\draw [color=CadetBlue] (4) -- (0,12)   coordinate[c1,fill,pos=0.6,label={[label distance=-1]180:{\footnotesize \color{black}  $a_{10}$}}] (10)  coordinate[c1,fill,label={[label distance=-1]180:{\footnotesize \color{black}  $a_7$}}] (7);
\draw [color=brown] (7) -- (b)      coordinate[c1,fill,pos=0.2,label={[label distance=-1]180:{\footnotesize \color{black}  $a_6$}}] (6);






        \draw [color=gray] (a) -- (2) coordinate[s1,fill,pos=0.52,label={[label distance=-1, yshift=2]357.5:{\footnotesize \color{black}  $a_1$}}] (1);

        \draw [color=violet] (5) -- (22,6) coordinate[c1,fill,pos=0.4,label={[label distance=-1]0:{\footnotesize \color{black}  $a_{11}$}}] (11) coordinate[c1,fill,label={[label distance=-1]0:{\footnotesize \color{black}  $a_9$}}] (9);

\draw [color=Apricot] (9) -- (b) coordinate[c1,fill,pos=0.3,label={[label distance=-1]280:{\footnotesize \color{black}  $a_8$}}] (8);

\draw [color=TealBlue] (4) -- (4,6) coordinate[c1,fill,pos=0.4,label={[label distance=-1]180:{\footnotesize \color{black}  $a_{28}$}}] (28) coordinate[c1,fill,label={[label distance=-3]180:{\footnotesize \color{black}  $a_{22}$}}] (22);
\draw [color=YellowGreen] (22) -- (3) coordinate[c1,fill,pos=0.2,label={[label distance=-1]260:{\footnotesize \color{black}  $a_{19}$}}] (19);

        \coordinate (25) at ([xshift=-4cm]1);
        \coordinate (27) at ([xshift=4cm]1);

\draw [color=MidnightBlue]  (22) -- (25) coordinate[c1,fill,pos=0.5,label={[label distance=-1]180:{\footnotesize \color{black}  $a_{24}$}}] (24) coordinate[c2,fill,label={[label distance=-1]180:{\footnotesize \color{black}  $a_{25}$}}] (25);
\draw [color=Mulberry] (25) -- (9) coordinate[c1,fill,pos=0.8,label={[label distance=-1]90:{\footnotesize \color{black}  $a_{35}$}}] (35);

\draw [color=BrickRed]  (7) -- (27) coordinate[c1,fill,pos=0.5,label={[label distance=-3]90:{\footnotesize \color{black}  $a_{34}$}}] (34) coordinate[c2,fill,label={[label distance=-1]90:{\footnotesize \color{black}  $a_{27}$}}] (27);
\draw [color=Emerald] (27) -- (23) coordinate[c1,fill,pos=0.25,label={[label distance=-1]320:{\footnotesize \color{black}  $a_{26}$}}] (26);

\draw [color=BlueGreen]  (10) -- (15.5,17.5) coordinate[c1,fill,pos=0.5,label={[label distance=-1]90:{\footnotesize \color{black}  $a_{12}$}}] (12) coordinate[c2,fill,label={[label distance=-1,xshift=5]270:{\footnotesize \color{black}  $a_{13}$}}] (13);
\draw [color=Tan] (13) -- (29) coordinate[c1,fill,pos=0.4,label={[label distance=-1]90:{\footnotesize \color{black}  $a_{31}$}}] (31);

\draw [color=RawSienna]  (28) -- (10.5,15) coordinate[c1,fill,pos=0.5,label={[label distance=-3, yshift=-3]160:{\footnotesize \color{black}  $a_{30}$}}] (30) coordinate[c2,fill,label={[label distance=-5]45:{\footnotesize \color{black}  $a_{15}$}}] (15);
\draw [color=SpringGreen] (15) -- (11) coordinate[c1,fill,pos=0.6,label={[label distance=-1]90:{\footnotesize \color{black}  $a_{14}$}}] (14);

\draw [color=Salmon]  (15) -- (1) coordinate[c2,fill,pos=0.2,label={[label distance=-1, yshift=2]180:{\footnotesize \color{black}  $a_{17}$}}] (17);
\draw [color=Fuchsia] (1)-- (13) coordinate[c2,fill,pos=0.3,label={[label distance=-1]0:{\footnotesize \color{black}  $a_{16}$}}] (16);

\draw [color=CornflowerBlue]  (19) -- (16) coordinate[c1,fill,pos=0.3,label={[label distance=-1]180:{\footnotesize \color{black}  $a_{18}$}}] (18);
\draw [color=pink] (16) -- (8) coordinate[c1,fill,pos=0.7,label={[label distance=-1]180:{\footnotesize \color{black}  $a_{32}$}}] (32);

\draw [color=PineGreen]  (6) -- (17) coordinate[c1,fill,pos=0.7,label={[label distance=-1, yshift=2]180:{\footnotesize \color{black}  $a_{33}$}}] (33);
\draw [color=DarkOrchid] (17) -- (21) coordinate[c1,fill,pos=0.4,label={[label distance=-3]20:{\footnotesize \color{black}  $a_{20}$}}] (20);

\draw [color=black] (25) -- (1) -- (27);

\end{tikzpicture}
}
\caption{\label{2021-context-figure-ACS} A configuration of quantum observables with a nonunital set of classical two-valued states
in three-dimensional Hilbert space~\cite{Svozil-2018-p}. Admissibility demands that proposition $a_1$ must be true (value 1);
and the adjacent propositions $a_2$, $a_{13}$, $a_{15}$, $a_{16}$, $a_{17}$, $a_{25}$, $a_{27}$, $a_{36}$
sharing hyperedges with $a_1$ must be false (value 0).
All other observables are either 0 or 1, depending on the respective two-valued state enumerated in Table~\ref{2021-context-table-ACS}.}
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\begin{table}%[H] %add [H] placement to break table across pages
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\begin{table*}%[H] %add [H] placement to break table across pages
\fi
\centering
\setcounter{MaxMatrixCols}{40}
\resizebox{1\textwidth}{!}{
\begin{tabular}{c|cccccccccccccccccccccccccccccccccccccccccccccccccccccccccc}
\toprule
%\# & $a_{ 1}$ & $a_{ 2}$& $a_{ 3}$& $a_{ 4}$& $a_{ 5}$& $a_{ 6}$& $a_{ 7}$& $a_{ 8}$& $a_{ 9}$& $a_{10}$& $a_{11}$& $a_{12}$& $a_{13}$& $a_{14}$& $a_{15}$& $a_{16}$& $a_{17}$& $a_{18}$& $a_{19}$& $a_{20}$& $a_{21}$ & $a_{22}$&$a_{23}$& $a_{24}$& $a_{25}$& $a_{26}$& $a_{27}$& $a_{28}$& $a_{29}$& $a_{30}$& $a_{31}$& $a_{32}$& $a_{33}$& $a_{34}$& $a_{35}$& $a_{36}$ & $a_{37}$\\
%\midrule
%1&\colorbox{white}{\fbox{\textcolor{red}{1}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 1& 0& 0& 0& 0& 0& 0& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 1& 0& 1& 0& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 1& 1& 0& 1& 1& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\\
%2&\colorbox{white}{\fbox{\textcolor{red}{1}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 1& 0& 0& 0& 0& 0& 0& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 1& 1& 0& 0& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 0& 1& 1& 1& 1& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\\
%3&\colorbox{white}{\fbox{\textcolor{red}{1}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 0& 1& 0& 0& 0& 0& 1& 0& 0&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 1& 0& 1& 0& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 0& 0& 1& 1& 1& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\\
%4&\colorbox{white}{\fbox{\textcolor{red}{1}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 0& 1& 0& 0& 0& 0& 1& 0& 0&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 0& 0& 1& 1& 0& 0&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 0& 1& 1& 1& 1& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\\
%5&\colorbox{white}{\fbox{\textcolor{red}{1}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 1& 0& 1& 0& 1& 0& 0& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 0& 1& 0& 0& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 1& 1& 0& 0& 0& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0&\\
%6&\colorbox{white}{\fbox{\textcolor{red}{1}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 1& 0& 1& 0& 0& 1& 0& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 0& 1& 0& 0& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 1& 1& 0& 1& 0& 1& 0&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0&\\
%7&\colorbox{white}{\fbox{\textcolor{red}{1}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 0& 1& 1& 0& 1& 0& 1& 0& 0&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 0& 1& 0& 0& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 0& 0& 1& 0& 0& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0&\\
%8&\colorbox{white}{\fbox{\textcolor{red}{1}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 0& 1& 0& 1& 1& 0& 0& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 0& 1& 0& 0& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 0& 0& 1& 0& 1& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0&\\
\# & $a_{ 1}$ & $a_{ 2}$& $a_{ 3}$& $a_{ 4}$& $a_{ 5}$& $a_{ 6}$& $a_{ 7}$& $a_{ 8}$& $a_{ 9}$& $a_{10}$& $a_{11}$& $a_{12}$& $a_{13}$& $a_{14}$& $a_{15}$& $a_{16}$& $a_{17}$& $a_{18}$&$a_{19}$\\
\midrule
1&\colorbox{white}{\fbox{\textcolor{red}{1}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 1& 0& 0& 0& 0& 0& 0& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 1\\%& 0& 1& 0& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 1& 1& 0& 1& 1& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\\
2&\colorbox{white}{\fbox{\textcolor{red}{1}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 1& 0& 0& 0& 0& 0& 0& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 1\\%& 1& 0& 0& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 0& 1& 1& 1& 1& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\\
3&\colorbox{white}{\fbox{\textcolor{red}{1}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 0& 1& 0& 0& 0& 0& 1& 0& 0&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 1\\%& 0& 1& 0& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 0& 0& 1& 1& 1& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\\
4&\colorbox{white}{\fbox{\textcolor{red}{1}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 0& 1& 0& 0& 0& 0& 1& 0& 0&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 0\\%& 0& 1& 1& 0& 0&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 0& 1& 1& 1& 1& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\\
5&\colorbox{white}{\fbox{\textcolor{red}{1}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 1& 0& 1& 0& 1& 0& 0& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 0\\%& 1& 0& 0& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 1& 1& 0& 0& 0& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0&\\
6&\colorbox{white}{\fbox{\textcolor{red}{1}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 1& 0& 1& 0& 0& 1& 0& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 0\\%& 1& 0& 0& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 1& 1& 0& 1& 0& 1& 0&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0&\\
7&\colorbox{white}{\fbox{\textcolor{red}{1}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 0& 1& 1& 0& 1& 0& 1& 0& 0&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 0\\%& 1& 0& 0& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 0& 0& 1& 0& 0& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0&\\
8&\colorbox{white}{\fbox{\textcolor{red}{1}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 0& 1& 0& 1& 1& 0& 0& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 0\\%& 1& 0& 0& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 0& 0& 1& 0& 1& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0&\\
\midrule
\# &  $a_{20}$& $a_{21}$ & $a_{22}$&$a_{23}$& $a_{24}$& $a_{25}$& $a_{26}$& $a_{27}$& $a_{28}$& $a_{29}$& $a_{30}$& $a_{31}$& $a_{32}$& $a_{33}$& $a_{34}$& $a_{35}$& $a_{36}$ & $a_{37}$\\
\midrule
1&  0& 1& 0& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 1& 1& 0& 1& 1& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\\
2&  1& 0& 0& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 0& 1& 1& 1& 1& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\\
3&  0& 1& 0& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 0& 0& 1& 1& 1& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\\
4&  0& 1& 1& 0& 0&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 0& 1& 1& 1& 1& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\\
5&  1& 0& 0& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 1& 1& 0& 0& 0& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0&\\
6&  1& 0& 0& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0& 1& 1& 0& 1& 0& 1& 0&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0&\\
7&  1& 0& 0& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 0& 0& 1& 0& 0& 1& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0&\\
8&  1& 0& 0& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 1& 0& 0& 1& 0& 1& 0& 1&\colorbox{white}{\fbox{\textcolor{blue}{0}}}& 0&\\
\midrule
\bottomrule
\end{tabular}
}
\caption{\label{2021-context-table-ACS} The eight two-valued states on the configuration depicted in Figure~\ref{2021-context-figure-ACS}. Boxes indicate fixed values.}
\ifx\revtex\undefined
\end{table}%[H] %add [H] placement to break table across pages
\else
\end{table*}%[H] %add [H] placement to break table across pages
\fi



\subsection{Nonexistence of classical value assignments}

The most extreme form of strong contextuality occurs if the respective structure of observables allows no classical interpretation whatsoever.
This result had already been announced by Specker in 1960~\cite{specker-60},
and is nowadays called the Kochen-Specker theorem~\cite[Graph~$\Gamma_2$]{kochen1}.
It has been perceived~\cite{ZirlSchl-65,kamber65} as a direct consequence of Gleason's theorem~\cite{Gleason}.


One may, in a certain sense and relative to the mathematical means employed,
extend these results by proving that there exist finite configurations of observables
that do not allow any classical value definite existence beyond a single classical value assignment,
and the (continuity of) contexts containing this extreme case.
Proofs relative to global and total classical value assignments are in Refs.~\cite{pitowsky:218,hru-pit-2003}.
Similar results are obtained with weaker assumptions allowing partial value assignments in
Refs.~\cite{2012-incomput-proofsCJ,PhysRevA.89.032109,2015-AnalyticKS}

\section{Further observations}

%Let me add some pertinent issues:
%First, I would like to raise some concerns about
%a method of ``collapsing'' probabilities, also called the ``support'' of a standard probabilistic Hardy model~\cite{Abramsky2011},
%such that any non-zero probability is mapped onto one,
%which serves as the basis of various ``logic'' arguments%~\cite{Abramsky2012,Abramsky2015,Abramsky2014,Abramsky-2017,Abramsky2018}.
%This ``logical'' method works well for particular gadgets such as true-implies-false gadgets because
%they are symmetrical with respect to their ``terminals'': true-implies-false gadgets
%also work ``the other way round'' by inverting the terminal observables,
%but this is no longer the case for true-implies-true gadgets gadgets.
%In general by effectively ``collapsing'' all two-valued states, one might lose important relational information.
%
%Second, to pronounce that contextuality ``supplies the magic for quantum computation''
%suggests that contextuality might be some computational capacity or resource that raises high expectations
%based on a rather vague notion. One may even argue that contextuality might indicate restrictions rather than extensions
%of classical computations.

\subsection{Omni-existence}

The use of the term ``contextual'' might imply or implicitly suggest, without direct empirical evidence,
a form of omni-existence. But omni-existence is a metaphysical concept because it lacks any direct operational test. Those arguments involve counterfactuals~\cite{specker-60}.

A generalized Jaynes' principle is called ``plausible reasoning'':
one should not introduce unnecessary epistemic bias, superficial information, and individual ontologic projections into empirical evidence
but rather stick to the ``knowable'' facts.
In Jaynes' words~\cite[Section~10.11, p.~331]{jaynes}, {``the onus is always on the user $\ldots$~that the full extent of his ignorance is also
properly represented''}.

Therefore, it might be more appropriate to talk about ``quantum indeterminacy'' as Pitowsky did~\cite{pitowsky:218,hru-pit-2003},
and to allow partial functions and value indefiniteness.
Partial functions have been first conceptualized~\cite{Kleene1936} in theoretical computer science
to cope with and formalize computability; in particular, with the recursive unsolvability of the halting problem.
They are essential in the theory of recursive functions
and indicate lack of capacities that go beyond certain limits of consistent formal expressibility.
It thus might me more appropriate to use the terms ``partial functions'' and ``value indefinite''
instead of ``contextuality''~\cite{2012-incomput-proofsCJ,Abbott:2010uq,PhysRevA.89.032109,2015-AnalyticKS}.

We conjecture that because of Pitowsky's principle of indeterminism~\cite{pitowsky:218,hru-pit-2003}
and newer theorems allowing partial functions as value assignments~\cite{2015-AnalyticKS},
the ``message'' of the quantum is straightforward: quantum systems are defined in their frame of preparation, and undefined in directions
other than perpendicular or collinear.
%Insistence in the assumption of omni-definiteness yields to ``contextual'' as well as stochastic value assignments.
%The respective measurement outcomes do not directly reflect any objective property prior to measurement,
%but on the complexion of the entire entangled state including the measurement apparatus.


\subsection{Is contextuality haunted?}

It should be kept in mind that there exists only indirect empirical tests of contextuality invoking counterfactuals.
Indeed, any experimentally verifiable contextuality remains indirect (or ``haunted'') and not direct, as
quantum mechanics predicts the absence of
direct verifications~\cite{svozil-1999-haunted-qc,svozil:040102,Griffiths2017,Griffiths2019}:
if say, two contexts $\{a,b,c\}$ and $\{a,d,e\}$ intertwining at observable $a$ are considered,
quantum mechanics is not ambiguous about the outcome corresponding to $a$, regardless of the context measured.
This can be directly experimentally tested on a single quantized particle, or on entangled particle pairs.
A remaining ``haunted'' context-dependence might manifest itself in a hidden and uncontrollable outcome dependence of the remaining
complementary observable pairs $\{b,c\}$ and $\{d,e\}$.
In four or higher dimensions this applies also to all quantum observables common to different contexts.
For instance, for a configuration $\{a,b,c,d\}$ and $\{a,b,e,f\}$,
quantum mechanics predicts that the observables $a$ and $b$ are noncontextual, whereas $\{c,d\}$ and $\{e,f\}$
might show (hidden and haunted) outcome dependence.

\subsection{Historic aspects regarding the importance of embeddability for Specker}


Let me add some afterthoughts on Kochen and Specker's demarcation criterion~\cite[Theorem~0]{kochen1}
for structure-preserving (non)embeddability mentioned earlier
as a decisive benchmark or measure for strong contextuality or value indefiniteness.
I consider it not entirely unreasonable to speculate that Specker meant the lack of embeddability by announcing~\cite{specker-60}:
{``An elementary geometrical argument shows that such an assignment is
impossible and that therefore (aside from the exceptions noted above) no
consistent prediction concerning a quantum mechanical system is possible.''}
Indeed, the `Comment' section of the respective article in Specker's `Selecta'~\cite[p.~385]{specker-ges}
explicitly notes (the references are updated to match the current ones used here):
{``The impossibility to embed the lattice of subspaces of $\mathbb{R}^3$
into a Boolean algebra, mentioned at the end of~\cite{specker-60}, is proved in~\cite{kochen1}
(theorem 1 and subsequent remarks).''}

%It might well be that,
%as Ad\'an Cabello has suggested~\cite{cabello-2021-03-10},
%these results had been ``in Specker's mind'' already in the late 1940s, possibly in Specker's {\it annus mirabilis} 1948.
%In his brief autobiography, written under the pseudonym ``Jonas Meon''~\cite[p~90]{Engeler-2012},
%Specker writes~\cite[pp.~XV,XXI]{specker-ges} (again my references, one corrected):
%{\em ``Another
%seminar which greatly influenced Ernst's later work was the seminar of
%Gonseth and Pauli on the foundations of quantum theory. It was essentially based on von Neumann's book, but there were also talks on the
%Birkhoff-von Neumann paper and the work of Destouches.
%$\ldots$~The basic theorem of the paper [\cite{specker-60},~\cite[pp. 135-140]{hooker}
%in English translation] was proved shortly afterward, but published
%much later. When I once asked Ernst why he had postponed publication
%for so many years, he told me of certain results William Craig had already
%proved in Zurich and only published much later. And he added: Some
%people like to be told to publish.''}
%The time Specker mentioned can be fixed by referring to Pauli's activities after his return to the ETH~\cite[Part~III, p.~189]{enz-Pauli}:
%{\em ``In the summer semester of 1948 he [[Pauli]] conducted a colloquium on the foundations of quantum mechanics together with the mathematics
% professors Paul Bernays and Ferdinand Gonseth; Pauli also occasionally contributed to their journal ``Dialectica''.}
%(German original: {\em ``Im Sommersemester 1948 f\"uhrte er [[Pauli]] gemeinsam mit den Mathematik-Professoren Paul Bernays und Ferdinand Gonseth ein Kolloquium \"uber
%die Grundlagen der Quantenmechanik durch;
%auch an deren Zeitschrift ``Dialectica'' wirkte Pauli gelegentlich mit.''}).
%So young Specker might have had proven the main theorems, announce 12 years afterward and proven almost 20 years afterward by Kochen and Specker,
%shortly after the summer semester of 1948 that ended on July 17th, 1948.
%Thus one reading of Specker's Selecta is consistent with claims that Specker proved the main theorems of the Kochen-Specker paper
%shortly after mid-July 1948 -- Specker's {\it annus mirabilis}.


\section{Summary}

Numerous notions of contextuality exist in the literature on quantum foundations.
The concept of ``contextuality'' is often used in ambiguous or contradictory ways.
This paper has taken a logic-based approach to clarify, identify, and categorize these different notions.
In particular, we propose to categorize different notions of contextuality into two major groups:
probabilistic and strong notions of contextuality.
We suggest using Kochen and Specker's demarcation Theorem~0~\cite{kochen1} as a criterion to differentiate between those groups.

The following review summarises and concentrates the issues raised.
\begin{itemize}
\item[(i)]
The supposition of a well-defined physical operationalization of the properties associated with quantum observables, and,
in particular, omni-existence lies at the basis of current so-called empirical tests of contextuality.
This does not take into account the entanglement between the object under observation and the measurement apparatus.
However, such a conception of measurement entails that the constituents of the entangled object-apparatus state are in no definite individual state.

\item[(ii)]
As long as the explicit functional context-dependence of quantum observables common to different contexts is directly tested
it is absent in quantized systems. Therefore, it might be ``haunted',' as such dependence may only occur indirectly, and
without direct experimental testability.

\item[(iii)]
It might be prudent to differentiate between the logico-algebraic structure formed by the observables via
complementarity and the probability distributions such logics can or cannot support.

\item[(iv)]
Kochen and Specker gave a ``demarcation criterion''
for nonembeddability in terms of (in)separability:
if the set of two-valued states on the logic can discriminate between every pair of atomic observables
(aka elementary propositions in the sense of Birkhoff and von Neumann)
then it can support classical models (and also quantum ones if there exist faithful orthogonal representations).
If no classical value assignment aka two-valued state can separate between two observables, then
embeddability in some presumably extended Boolean algebra breaks down.
Such situations can be termed strong contextuality.

\item[(v)]
Current empirical corroborations of contextuality associated with Boole-Bell-type inequalities,
as long as they are based on hull computations of classical value assignments on suitable ensembles of quantum observables
with separating sets of two-valued states encoding these value assignments,
are merely about probabilistic contextuality.
The same is true for separable ensembles of quantum observables with functional relations on endpoints.


%\item[(vi)]
%In the spirit of theoretical computer science and of Itamar Pitowsky I suggest~\cite{Svozil-2018-p} to call these properties ``partial value assignments'',
%``value indefinite', or ``indeterminate''.

\item[(vi)]
Strong contextuality implies probabilistic contextuality because the former always indicates
some ``essential'' scarcity of two-valued states associated with classical truth assignments.
Because, by convex summation, two-valued states form the basis for classical probability distributions,
any ``essential'' lack thereof indicates limits to classical physical phenomenology that provide distinctions from quantized systems.
Essential here stands for nonseparating, nonunital, or in its strongest form not existing.
\end{itemize}

These matters are pertinent not only to foundational questions but also to the computational capacities of quantized systems.





\ifx\revtex\undefined

\section*{Acknowledgments}

This research was funded in whole, or in part, by the Austrian Science Fund (FWF), Project No. I 4579-N. For the purpose of open access, the author has applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission.

%The author acknowledge TU Wien Bibliothek for financial support through its Open Access Funding Programme.

%I kindly thank the IFISC (Institute for Cross-Disciplinary Physics and Complex Systems),
%a joint research institute of the University of the Balearic Islands (UIB) and the Spanish National Research Council (CSIC)
%for hospitality during my stay in February 2021;
%and Adan Cabello as well as Jose Portillo of the University of Sevilla for our ongoing cooperation and during this stay.


%I kindly acknowledge discussions with and suggestions by
%Cristian Calude, Kelly James Clark, Silvia Jonas, Jeffrey Koperski, Irem Kurtsal, Emil Salim, Mohammad Hadi Shekarriz, and Noson S. Yanofsky.

I thank Ad\'an Cabello for sharing his thoughts and questions on the time of origin of what is nowadays called~\cite{Stigler1980} Kochen-Specker theorem.

All misconceptions and errors are mine.

The author declares no conflict of interest.

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%\bibliography{svozil}


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\end{thebibliography}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\else


\begin{acknowledgments}

This research was funded in whole, or in part, by the Austrian Science Fund (FWF), Project No. I 4579-N. For the purpose of open access, the author has applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission.

The authors acknowledge TU Wien Bibliothek for financial support through its Open Access Funding Programme.

%I kindly thank the IFISC (Institute for Cross-Disciplinary Physics and Complex Systems),
%a joint research institute of the University of the Balearic Islands (UIB) and the Spanish National Research Council (CSIC)
%for hospitality during my stay in February 2021;
%and Adan Cabello as well as Jose Portillo of the University of Sevilla for our ongoing cooperation and during this stay.


%I kindly acknowledge discussions with and suggestions by
%Cristian Calude, Kelly James Clark, Silvia Jonas, Jeffrey Koperski, Irem Kurtsal, Emil Salim, Mohammad Hadi Shekarriz, and Noson S. Yanofsky.

I thank Ad\'an Cabello for sharing his thoughts and questions on the time of origin of what is nowadays called~\cite{Stigler1980} Kochen-Specker theorem.

All misconceptions and errors are mine.

The author declares no conflict of interest.
\end{acknowledgments}


\bibliography{svozil}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\fi

\end{document}

Dear Editor,

In the spirit of suggestions of both Referees, and also to give the reader a better impression of the contents, I have changed the title of the article to

"Varieties of contextuality based on complexion, quantum probabilities, and structural nonembeddability"

This alludes to the synthetic notion of contextuality Bohr, as mentioned by the second Referee, and emphasizes the two main analytical aspects, mentioned by the first Referee: non-classical probabilisty distributions and structural, that is, logico-algebraic, properties such as nonembeddibily into a classical mold.

According to a suggestion of the first Referee I have attempted to define the relevant terms (two-valued state as classical value assignment, expectation value, probabilities from convex sums over extreme states) in the article, and have thereby been able to drop and reduce a great deal of references.

Furthermore, following the suggestion of the first Referee, I have explicitly referred to explicit cases of experimentally testable probabilistic contextuality.

I also streamlined and clarified references to explicit cases in the existing literature, so that it is easier to follow up on them.

The abstract has be rewritten accordingly to emphasize and reflect upon these issues:

"Contextuality can either be synthetically defined in terms of outcome conditionality on the measurement conditions, or in terms of non-classical probability distributions. Another logico-algebraic ``strong'' form of contextuality characterizes collections of quantumobservables that have no faithfully embedding into (extended) Boolean algebras. Any of these forms indicate a classical in- or underdetermination that can be termed ``value indefinite'', and formalized by partial functions of theoretical computer sciences."

I have followed the suggestion of the second referee and mentioned conceptualisations related to Bohr's original observation at the beginning of the text as follows:

"Early synthetic conceptions of contextuality emerged from
insights into the entangled complexion~\cite{schrodinger-gwsidqm2} of physical properties
retrieved from quantum measurements. As expressed by Bohr~\cite{bohr-1949}:
{``the impossibility of any sharp separation between the behavior of atomic
objects and the interaction with the measuring instruments which serve to define the conditions
under which the phenomena appear.''}
This yields a {``conditionality of phenomena''}~\cite{Khrennikov2017,Jaeger2019}
relative to a {``a complex of conditions under which the measurement is performed''}~\cite{Khrennikov2009a,Khrennikov2009b}."

I have also mentioned this approach to contextuality in the discussion at the end of the paper.

Overall I have rewritten the arguments, and made them more explicit. These revisions should improve comprehensibility and clarity.

I hope that, with these revisions, the paper can be published.

Please do not hesitate to contact me should further revisions be necessary.

Thank you and the Referees for your kind attention and for all your efforts!

With kind regards,

Karl Svozil

~~~~~~~~~~~~~~ 2nd round of RR


Dear Editors,

thank you and the Referees for the care and attention dedicated to the manuscript.

I have taken some (hopefully most) of the concerns of the first Referee, and (hopefully) all the suggestions of the 4th Referee into account as follows:

Response to Reviewer 1:

The Reviewer kindly repeats my changes that, I have hoped, would improve readability and would make the argument more transparent.

Insofar the Reviewer disagrees with what I have stated I have tried to clarify the semantics of what I have in mind.

More explicitly, this disagreement is mentioned in the Reviewer's paragraph starting with

"Instead of an operational definition, the author then offers
Sec.II "EXAMPLES OF PROBABILISTIC CONTEXTUALITY"..."

In this paragraph the Reviewer argues as follows:

"... I did not look
into all that references, but those I have looked into do not
support any "probabilistic contextuality." E.g., Suppes and
Zanotti [25] in 1981 considered variable bounds for particular
combinations of probabilities that observables would have values
from {-1,1} but that does mean that any "probabilistic
contextuality" is involved, i.e., that "classically embeddable
observables, as well as classical probability distributions,
are allowed" for assignments from {-1,1}. The point is that in
a contextual set such an assignment is not possible and Suppes
and Zanotti's result does not enable any such classical
"probabilistic contextuality." ..."

I am afraid that, with all due respect, I have to disagree with the Reviewer in this aspect; indeed the existence of a two-valued measure (indeed, multiple such measures) and the existence of an associated {-1,1}-value assignment is the exact point of my argument; in particular, also for the Suppes and Zanotti's result.

My recent analysis of Suppes and Zanotti's result (DOI 10.1007/s10773-021-04850-9) still allows *single* instances of {-1,1}-value assignments (a subject never denied by Suppes and Zanotti). But the classical probabilistic analysis of their interesting configuration reveals the bounds derived by Suppes and Zanotti, that are violated by the quantum probabilities of analogous quantized systems. These arguments have been brought forward for the Clauser-Horne-Shimony-Holt configuration by Asher Peres (DOI: 10.1119/1.11393).

Assuming that many Readers might raise similar concerns and objections as the Referee, I have added the following passage to the manuscript at the end of this section:

"Suppes and Zanotti's result~\cite{Suppes-81,svozil-2020-ex} as well as other bounds on classical probabilities for different configurations of observables
still allow {\em single} instances of $\{-1,1\}$-value assignments, as the respective set of two-valued states is not empty (as for Kochen-Specker configurations).
But the classical probabilistic analysis of such configurations reveals that the bounds on classical probabilities
are violated by the quantum probabilities of analogous quantized systems.
This has been explicitly and constructively demonstrated  by tabulations of classical  $\{-1,1\}$-value assignments
for the Clauser-Horne-Shimony-Holt configuration~\cite{chsh} by Asher Peres~\cite{peres222}.
Further quantitative investigations into the ``amount'', or a measure, of probabilistic contextuality
in terms of enumerations and tabellations of classical value assignments have, for instance,
been studied in References~\cite{svozil_2010-pc09,svozil-2011-enough,Dzhafarov-2015,Dzhafarov-2017,KujalaDzhafarov-2019,Dzhafarov-19PhysRevA}."

I have also greatly expanded the definition section on "Notation", as well as included some concrete examples that should improve comprehension.


Response to Reviewer 4:

In the beginning of the Report, the Referee so nicely summarises the paper that I have used a (slightly modified) version of this summary and included it at the ent of the paper:

"\section{Summary}

Numerous notions of contextuality exist in the literature on quantum foundations.
The concept of ``contextuality'' is often used in ambiguous or contradictory ways.
This paper has taken a logic-based approach to clarify, identify, and categorize these different notions.
In particular, we propose to categorize different notions of contextuality into two major groups:
probabilistic and strong notions of contextuality.
We suggest using Kochen and Specker's demarcation Theorem~0~\cite{kochen1} as a criterion to differentiate between those groups."

As the Referee agrees with Referee 1 that there is a lack of clarity of the argument; and actual concrete examples are lacking from the paper's exposition, I have included the following examples now:

(i) A concrete tabellation and discussion of classical simultaneous {-1,1} values for the Suppes-Zanotti configuration;

(ii) A concrete enumeration of a functional contextuality, the "Specker bug" gadget whose hypergraph representation of contexts is depicted in Figure 1;

(iii) A concrete enumeration of strong contextuality by the kochen and Specker "combo" whose hypergraph representation of contexts is depicted in Figure 2.

These examples wil hopefully will contribute towards a clarification of some of the issues discussed.

Here is my response to the seven issues that the 4th Referee considered in need to be seriously addressed, one-by-one:

~~~~~~~~~~~~~

1) The main suggestion of the paper is to categorise contextuality into two cases, that are
termed probabilistic and strong contextuality. However, it is very difficult to extract a
formal understanding of what these notions are, as they are defined very implicitly and
rather informally. Given their central role, I think these need to be clearly and formally
defined. It is also important to clarify the relation between these notions (does one imply
the other?)

Response to 1) I have, on many occations tried to stree state the central criterion id embeddability of the set of quantum observables into supposedly ``larger'' boolean algebras, while maintaining the logico-algebraic structure.

I have added parts of a sentence in the abstract: "... in terms of non-classical probability distributions, while still maintaining classical representability of the quantum observables."

I have also added an introductory paragraph as follows:

"The main point of this paper will be to argue that there are at least two main types of contextuality: the first notion is based upon nonclassical phenomenology,
and in particular, on nonclassical probabilities contradicting Boole's conditions of physical experience~\cite{Boole-62}.
The second type goes beyond this,
and is based upon the absence of a classical logico-algebraic structure in terms of which the respective observables could be re-interpreted.
Formally this amounts to nonembaddability into Boolean algebras by classical means; that is, in formal terms,
by two-valued measures interpretable as classical truth assignments. The strongest form of this latter logico-algebraic contextuality
is the total absence of any such classical truth assignment~\cite{kochen1}.
Thereby, it is important to keep in mind that the mere existence of any such classical truth assignment
is necessary but not sufficient to ensure classical representability or embeddability:
indeed, even if there is an apparent ``abundance'' of classical truth assignments, these might
not provide sufficient means to classically embed a collection of quantum observables."

I have added a sentence explaining omni-existence:

"Omni-existence is the insistance that, although due to complementarity not all observables can be measured simultaneously, they are nevertheless value definite;
that is, some of them must have a definite counterfactual value that can by no physical means be measuremed~\cite{specker-60}."

~~~~~~~~~~~

2) The decision to make this distinction seems to be based on the logical difference
between the notions. This is an interesting approach, but I would like to see it justified
further from a physical point of view. In particular, it is not so clear to me why, from a
physical point of view, it is relevant that some observables "allow classical
probabilities, but those probabilities are not realized by the [...] systems at hand." In
either case (probabilistic or strong contextuality), the physical objects don't admit of a
classical non-contextual model, so why does it matter what kind of models they *don't*
allow?

Response to 2)

I have modified one paragraph and added another one:

"Why should one make such a distinction from a physical point of view?
Because if there do not exist separating sets of two-valued states and thus no faithful classical embeddability
then the respective structures do not any longer support classical probability distributions;
and also no ``hidden variables'' in terms of the ``larger'' extended Boolean algebra in which the observables can be homomorphically embedded.
This is quite different from any non-classical probability distribution on an otherwise ``quasi-classical''  empirical logics~\cite{Foulis1976}
containing complementary observables that can still be imbedded into ``larger'' extended Boolean algebras.

For physical realizations we refer to Wright's generalized urn models~\cite{wright} or the initial state identification problem
for finite automata~\cite{e-f-moore,schaller-96}, amounting to partition logics~\cite{svozil-2001-eua}.
The resulting empirical logics~\cite{Foulis1976} are identical to the quantum propositional structures~\cite{birkhoff-36}
we shall discuss in those instances of ``weak'' quantum contextuality that we term probabilistic.
These logics support both classical as well as quantum probabilities.
Which probability distribution needs to be chosen depends solely on the respective physical realization~\cite{svozil-2018-b}.
In particular, there is no structural logicaldistinction; the difference is in their performance.
In such cases classical ``hidden variables'' cannot be ruled out."

With regard to the Reviewer's question: "In
either case (probabilistic or strong contextuality), the physical objects don't admit of a
classical non-contextual model, so why does it matter what kind of models they *don't*
allow?"  --- one needs to emphasise that in the case of probabilistic contextuality classical non-contextual model cannot be ruled out.
Indeed, this is one of my concerns with this type of contextuality.

~~~~~~~~~~~~~~~~~~~~

3) In the literature, a distinction is often made between state dependent and state
independent contextuality. What is the relation of this to the types of contextuality
described in the present article? It also seems to me that the distinction between strong
and probabilistic contextuality has already been made in the literature (e.g.,
https://arxiv.org/pdf/1911.03521.pdf, Eq. (4.1)). How does this relate to what this paper
is defining? If the notions are not the same, then why should these terms be defined in one
way rather than the other? More generally, the paper should probably be better embedded in
the modern contextuality literature. Many papers are cited, but the majority of these are
rather old (there are very few references from the last 10 years, excepting those involving
the author and some philosophical references).

Response to 3)

I hesitate to include both "state independent contextuality" as well as contextuality discussed by Prof. Samson Abramsky because the article is already in danger of taking too many side steps. I have, nevertheless, quoted Cabello's early paper on this in the introduction by saying "(i) either merely assume the omni-existence of unrestricted classical noncontextual value assignments that do not depend on the complete disposition of the apparatus [15];"

The first approach of "state independent contextuality" is not based on two-valued measures, but insted ventures into non-admissible yet non-contextual value assignments. I agree that this might have some relevance, but it would not contribute to the demarcation criterium.

I also agree that Prof. Samson Abramsky's approach is fascinating. Yet it is in many ways a reformulation of the findings in quantum logic, such as the theorems by Specker, and Kochen & Specker.

Therefore I hope that the Reviewer will forgive my not mentioning these interesting approaches.

~~~~~~~~~~~~~~~~~~~~~

4) The paper has two sections on "examples of [probabilistic/strong] contextuality", but
these sections appear rather to be literature surveys rather than examples. To understand
the notions presented, it would help a lot to have some concrete examples actually
presented in the paper, rather than simply cited. It might help to rework the structure of
the paper a little. For example, by (for each notion of contextuality) first defining it
formally, then giving some concrete examples, then discussing how it fits with the
literature.

Resonse to 4)

I have added now two respective examples, as suggested by the Referee.

~~~~~~~~~~~~~~~~~~~~~

5) The claim that, in the literature, omni-existence is often taken for granted is, I
believe, somewhat misleading. Just because it is assumed for the sake of contradiction
doesn't necessarily mean contextuality is generally interpreted as the existence of a
contextual hidden variable model, as seems to be claimed in the introduction. Indeed, these
days, contextuality is primarily taken, at least formally, to mean the non-existence of a
noncontextual model (see, e.g., the recent review https://arxiv.org/abs/2102.13036).
Indeed, the way it is interpreted is often somewhat inconsistent, leading to many of the
issues this paper tries to address, but I would avoid imposing interpretations on others as
much as possible, as this compounds, rather than helps, the problem.

Response to 5) Here I am a little bit of a loss. I knwo Cabello's work quite well (I even co-authored a paper with him), but I believe that he would agree with me.

Take, for instance, his "state independent contextuality" paper DOI 10.1103/PhysRevLett.101.210401 which he begins with "Because of the lack of spacelike separation between one
observer’s choice and the other observer’s outcome, the immense majority of the experimental violations of Bell inequalities does not prove quantum nonlocality, but just
quantum contextuality." --- which basically boils down to probabilistic contextuality.

Abbott, Calude (earlier Conder) and myself have developed a notion of contextuality that does not require total omni-existence of value assignments but allows for
partial functions in the sense of Kleene; and I have discussed this in some depth.

~~~~~~~~~~~~~~~

6) Some of the claims in the discussion need to be backed up further. For example, it is
not really clear to me what the "concerns about a method of collapsing probabilities" is
(or at least, why they're problematic), and the comment on contextuality supplying the
magic in quantum computation appears unmerited (at the least the claims should be cited;
the papers I imagine the author has in mind are, from a formal point of view, clear). In
Sec. IVA, the author argues against one particular interpretation of the KS contradiction
as being unjustified, before suggesting an alternative interpretation that (by the very
same reasons) appears no more justified to me. It is alost unclear for me what the term
"haunted" refers to in IVB.

Responsae to 6)

I have eliminated the paragraph discussing "collapsing probabilities" as it may have been distractive.

"Haunted contextuality" is again the idea that there exists no direct, only indirect empirical tests invoking counterfactuals of this concept.
I have mentionedit because I have been asked by Professor Khrennikov and others to include this issue in a discussion recently.
I have added the sentence:

"It should be kept in mind that there exists no direct, only indirect empirical tests invoking counterfactuals of this concept."

~~~~~~~~~~~~~~~~~~~~~

7) Finally, the paper users many terms that are not standard in the literature, and often
phrases things in particularly complicated ways. An effort should be made to simplify the
language to improve clarity throughout.

Response to 7) I have carefully revised and polished some passages in order to improve readability, and streamline the subject.

I have, for instance, changed the title to "Varieties of contextuality based on probability and structural nonembeddability"
to emphasize this important point from the onset.  Also, the abstract has been rewritten with this in mind.

In a new "nomenclature" section the notion of context and two-valued measures are formally defined by using "admissibility", a notion that has been put forward by Professor Calude in whose honour this special volume of TCS is dedicated.

I hope with these revisions the paper can be accepted.

Thank you and the Referees for their kind attention and efforts dedicated to this manuscript.

With kind regards,

Karl Svozil


~~~~~~~~~~~~~~ 3rd round of RR

Dear Editor,

Thank you and Reviewer #4 for all the attention and work dedicated to the manuscript! I have tried to fully comply with the criticism and suggestions of Reviewer #4, and have made the following revisions, as outlined below:


1) At the end of the paragraph in the introduction beginning with "Historical attempts...", it is stated that any discrepancies "are interpreted to signify noncontextuality". Surely this should read contextuality. On the same point, I find the authors response to point 5 of my previous report unconvincing. Simply because A. Cabello may (or may not) agree with the authors view does not provide support for how a KS type contradiction is often interpreted. Indeed, the review I cited makes this clear.

Response to 1)

(i) I thank the Referee for pointing out that the passage should not read "are interpreted to signify noncontextuality" but instead "are interpreted to signify contextuality". I have changed this text (thereby eliminating "non").

(ii) I have also followed the Referee's suggestion to not confuse the axioms entering the Kochen-Specker argument---a proof by contradiction---suggesting "omni-existence". Indeed, after some thought, I believe that the Referee is right on this point, and my wording may have been potentially misleading.

I, therefore, deleted the sentence "Thereby, omni-existence is usually implicitly taken for granted." and instead added

"However, one has to be careful and keep in mind that, just because omni-existence is often but not always~\cite{2015-AnalyticKS} assumed for the sake of contradiction,
a violation of noncontextuality does not imply or suggest the existence of any contextual hidden variable model.
Contemporary interpretations of contextuality indicate the non-existence of a noncontextual model~\cite{cabello2021contextuality}."

2) The notations of intertwining and isolated contexts are referred to several times but never properly explained or defined. It is claimed that in the CHSH scenario there are four isolated contexts, but from a KS perspective it seems that the relevant observables are the tensor products of observables for each party, which do not form isolated contexts. This should be clarified. (A similar remark concerns the Suppes-Zanotti example, where it is claimed one multiplies the binary observables of the parties; instead, one multiplies the results of the measurements - the observables are tensor products of the given observables.)

Response to 2)

(i) I have added a sentence explaining the notion of intertwining and isolated contexts:

"Here the term ``intertwined'' is understood as introduced by Gleason~\cite{Gleason}:
in higher than two dimensions the orthonormal bases identified with contexts can ``share'' common elements---they need not be ``isolated'', that is, disjoint."

(ii) I have simplified the Suppes-Zanotti inequality case and specified the CHSH and other cases by stating:

"\item[(i.SZ)]
Suppes-Zanotti inequalities from three observables~\cite{Suppes-81}, as discussed later;
\item[(i.CHSH)]
Clauser-Horne-Shimony-Holt (CHSH) inequalities from four observables in two groups, and their respective tensor products forming four isolated contexts~\cite{Suppes-81};
\item[(i.TPB)]
two-party Bell inequalities from finitely many observables, and their respective tensor products forming isolated contexts~\cite{Avis2005};"


3) I still find the claim at the end of V.A rather difficult to justify. It is a very speculative comment. The goal of the paper is to remove or clarify some of the confusion around contextuality, but in Sec. V several speculative interpretations or comments are given, which goes completely against this aim.

Response to 3)

I agree with the Referee that this comment is speculative and might add confusion to the discussion. Therefore I eliminated the following passage at the end of V.A entirely:

Eliminated "Insistence in the assumption of omni-definiteness yields to ``contextual'' as well as stochastic value assignments.
The respective measurement outcomes do not directly reflect any objective property prior to measurement,
but on the complexion of the entire entangled state including the measurement apparatus."

4) Point vi) of Section VI refers to "these properties", but it is not clear what properties are being referred to. Again this comment seems very speculative and against the spirit of the paper.

Response to 4)

Again I agree with the Referee that this comment might add confusion to the discussion. Therefore I eliminated the entire point vi):

Eliminated "\item[(vi)]
In the spirit of theoretical computer science and of Itamar Pitowsky I suggest~\cite{Svozil-2018-p} to call these properties ``partial value assignments'',
``value indefinite', or ``indeterminate''."

5) The comment vii) of Section VI surely should comme earlier in the discussion, as it seems to be a crucial statement about the relation between the two types of contextuality the paper discusses, rather than a comment tacked on at the end?

Response to 5)

Thank you for this suggestion; I have added a respective passage as motivation and outlook right after the introduction of strong contextuality:

"Strong contextuality always indicates
some ``essential'' scarcity of two-valued states associated with classical truth assignments;
a deficiency to supply sufficient ``structural information'' for a classical embedding of the respective quantum observables.
As will be discussed later such essential scarcity can be categorized and quantified by escalating levels of ``rarity'', a ``shortage''
of elements of the set of two-valued states that may become nonseparating, nonunital, or in its strongest form empty."

There are also a few grammatical or stylistic points that could be corrected or improved:

6) In the introduction: "omni-realisim or emni-existence". Why not stick to just the first time, rather than introducing further unused terminology?

Response to 6)

I eliminated the term "omni-realism" for "omni-existence".

7) Before point (i) at the end of page 2 a sentence is not clear: "V then signifiers" facilitates three tasks". This sentence makes no sense to me.

Response to 7)

I kindly thank the Referee for pointing this out; I have corrected the sentence:

"If the observables in $X$ support a separating set of two-valued states $V$ then $V$ facilitates three constructions:"

8) In the intro to Section III, the term "weak" contextuality is introduced, but unused elsewhere. Again, it would be less confusing to avoid introducing further unneeded terminology.

Response to 8)

According to the suggestion of the Referee I have eliminated the term "weak contextuality". I also clarified the paragraph by adding:

"In particular, there is no structural logical distinction; therefore, classical ``hidden variables'' cannot be ruled out,
the difference is in the classical-versus-quantum performance: the respective quantum contextuality is probabilistic."


I hope with these revisions the paper can be accepted.

My sincerest thanks to the Editor and the Referee for their kind attention and efforts dedicated to this manuscript.

With kind regards,

Karl Svozil
