for all $e,e^{\prime}\in E(\Sigma_{G})$, i.e.,
$\forall[m|M]:\sum_{s_{e}}p(m,s_{e}|M,S_{e})=\sum_{s_{e^{\prime}}}p(m,s_{e^{\prime}}|M,S_{e^{\prime}})$,
for all $e,e^{\prime}\in E(\Sigma_{G})$.

An example of such a source events hypergraph was considered in Ref. [12],
albeit without the additional source labelled by $e_{*}$ here [16]. We
illustrate it here in Fig. 6 for the KCBS scenario.

Figure 6: The source events hypergraph with the operational equivalences
between the source settings separately specified.

##  4 A key hypergraph invariant: the weighted max-predictability

We now define a hypergraph invariant that will be relevant for our noise-
robust noncontextuality inequalities:

$\beta(\Gamma_{G},q)\equiv\max_{p\in\mathcal{G}(\Gamma_{G})|_{\rm
ind}}\sum_{e\in E(\Gamma_{G})}q_{e}\zeta(M_{e},p),$ (45)

where $q_{e}\geq 0$ for all $e\in E(\Gamma_{G})$, $\sum_{e\in
E(\Gamma_{G})}q_{e}=1$, and

$\zeta(M_{e},p)\equiv\max_{v\in e}p(v)$

is the maximum probability assigned to a vertex in $e\in E(\Gamma_{G})$ by an
extremal indeterministic probabilistic model
$p\in\mathcal{G}(\Gamma_{G})|_{\rm ind}$.303030An extremal indeterministic
probabilistic model refers to those extremal $p\in\mathcal{G}(\Gamma_{G})$ for
which $\zeta(M_{e},p)<1$ for some $e\in E(\Gamma_{G})$.

We call $\beta(\Gamma_{G},q)$ the weighted max-predictability of the
measurement settings (i.e., hyperedges) in $\Gamma_{G}$, where the hyperedges
$e\in E(\Gamma_{G})$ are weighted according to the probability distribution
$q\equiv\\{q_{e}\\}_{e\in E(\Gamma_{G})}$.

We now outline how this quantity is related to properties of an operational
theory $\mathbb{T}$ admitting a measurement noncontextual ontological model.
$\Gamma_{G}$ represents a particular configuration of operational equivalences
that a set of measurement events in $\mathbb{T}$ may realize. The
probabilistic models on $\Gamma_{G}$ that can be realized by $\mathbb{T}$ are,
as earlier, denoted by $\mathcal{T}(\Gamma_{G})$. Since $\mathbb{T}$ admits a
measurement noncontextual ontological model,313131This will always be the case
for any operational theory we consider: the assumption of measurement
noncontextuality on its own can always be satisfied by a trivial ontological
model of the type we outlined in Section 2.5. Indeed, quantum theory satisfies
it, the Beltrametti-Bugajski model [52] that was discussed in Ref. [18] being
an example of a measurement noncontextual ontological model of quantum theory.
It is only when this assumption is supplemented with something else – outcome
determinism in the case of KS-noncontextuality and preparation
noncontextuality in the case of generalized noncontextuality [18] – that it
can produce a contradiction with the predictions of an operational theory. its
predictions for the specific case of $\Gamma_{G}$ can be reproduced by such a
model. But since, in keeping with the CSW approach [22], we will look at
witnesses of contextuality tailored to particular experiments ($\Gamma_{G}$
representing features of one such experiment), we do not need an ontological
model for the full theory $\mathbb{T}$ to reproduce its predictions for a
particular experiment. Indeed, to construct a measurement noncontextual
ontological model for the set of probabilistic models
$\mathcal{T}(\Gamma_{G})$, it suffices to assume (without loss of generality)
that the extremal probabilistic models on $\Gamma_{G}$ – given by
$\mathcal{G}(\Gamma_{G})|_{\rm det}\sqcup\mathcal{G}(\Gamma_{G})|_{\rm ind}$ –
are in bijective correspondence with the ontic states ($\Lambda$) of the
physical system on which the measurements are carried out. This is because,
firstly, any probabilistic model in $\mathcal{G}(\Gamma_{G})$ can be expressed
as a convex mixture of extremal probabilistic models in
$\mathcal{G}(\Gamma_{G})|_{\rm det}\sqcup\mathcal{G}(\Gamma_{G})|_{\rm ind}$,
and, secondly, associating each ontic state in the ontological model with an
extremal probabilistic model323232Representing response functions for the
ontic state, i.e., $p(v)=\xi(v|\lambda),\forall v\in V(\Gamma_{G})$ in
$\mathcal{G}(\Gamma_{G})|_{\rm det}\sqcup\mathcal{G}(\Gamma_{G})|_{\rm ind}$
means that any probabilistic model in $\mathcal{G}(\Gamma_{G})$ corresponding
to predictions of an operational theory (in particular, any
$p\in\mathcal{T}(\Gamma_{G})\subseteq\mathcal{G}(\Gamma_{G})$) can be obtained
by an appropriate probability distribution over this set of ontic states.
Denoting the set of ontic states corresponding to
$\mathcal{G}(\Gamma_{G})|_{\rm det}$ by $\Lambda_{\rm det}$ and the set of
ontic states corresponding to $\mathcal{G}(\Gamma_{G})|_{\rm ind}$ by
$\Lambda_{\rm ind}$, we have that the measurement noncontextual ontological
model given by $\Lambda\equiv\Lambda_{\rm det}\sqcup\Lambda_{\rm ind}$
reproduces the predictions $\mathcal{T}(\Gamma_{G})$ of any operational theory
$\mathbb{T}$ that admits a measurement noncontextual ontological model: that
is, for every $p\in\mathcal{G}(\Gamma_{G})$ (and therefore also
$p\in\mathcal{T}(\Gamma_{G})$),

$p(v)=\sum_{\lambda\in\Lambda}\xi(v|\lambda)\mu(\lambda)$

for all $v\in V(\Gamma_{G})$, for some probability distribution
$\mu:\Lambda\rightarrow[0,1]$ such that
$\sum_{\lambda\in\Lambda}\mu(\lambda)=1$.333333As a corollary, note that as
long as the polytope $\mathcal{G}(\Gamma_{G})$ has a finite number of extreme
points, we can take the ontic state space to consist of a finite number of
ontic states (as we have done) without any loss of generality. The hypergraphs
$\Gamma_{G}$ we study – representing the measurement events of interest in a
contextuality experiment – have this property because of their finiteness. We
can also then rewrite $\beta(\Gamma_{G},q)$ as

$\beta(\Gamma_{G},q)=\max_{\lambda\in\Lambda_{\rm ind}}\sum_{e\in
E(\Gamma_{G})}q_{e}\zeta(M_{e},\lambda),$ (46)

where $\zeta(M_{e},\lambda)\equiv\max_{m_{e}}\xi(m_{e}|M_{e},\lambda)$.

##  5 Noise-robust noncontextuality inequalities

We will now proceed to obtain our noise-robust noncontextuality inequalities
following the ideas outlined in Ref. [16].

###  5.1 Key notions from CSW

We first recall some key notions from the CSW framework [22] before obtaining
our inequalities.

Consider the positive linear combination of the probabilities of measurement
events,

$R([s|S])\equiv\sum_{v\in V(G)}w_{v}p(v|S,s),$ (47)

where $w_{v}>0$ for all $v\in V(G)$.

The fundamental result of CSW is that this quantity is bounded for different
sets of correlations — KS-noncontextual, those realizable by projective
quantum measurements, and those satisfying consistent exclusivity — by graph-
theoretic invariants as follows:

$\forall[s|S]:R([s|S])\overset{\rm KS}{\leq}\alpha(G,w)\overset{\rm
Q}{\leq}\theta(G,w)\overset{{\rm CE}^{1}}{\leq}\alpha^{*}(G,w),$ (48)

where ${\rm KS}$ denotes operational theories that admit KS-noncontextual
ontological models and thus realize probabilistic models on $\Gamma_{G}$ that
fall in the set $\mathcal{C}(\Gamma_{G})$, ${\rm Q}$ denotes quantum theory
with projective measurements which assigns probabilistic models on
$\Gamma_{G}$ denoted by $\mathcal{Q}(\Gamma_{G})$, and ${\rm CE}^{1}$ denotes
operational theories satisfying consistent exclusivity and thus realizing the
set of probabilistic models $\mathcal{CE}^{1}(\Gamma_{G})$ on $\Gamma_{G}$.
The graph invariants of the weighted graph $(G,w)$, namely, $\alpha(G,w)$,
$\theta(G,w)$, and $\alpha^{*}(G,w)$ are defined as follows:

  1. 1.

Independence number $\alpha(G,w)$:

$\alpha(G,w)\equiv\max_{\mathcal{I}}\sum_{v\in\mathcal{I}}w_{v},$ (49)

where $\mathcal{I}\subseteq V(G)$ is an independent set of vertices of $G$,
i.e., a set of nonadjacent vertices of $G$, so that none of the vertices in
this set shares an edge with any other vertex in the set.

  2. 2.

Lovasz theta number $\theta(G,w)$:

$\theta(G,w)\equiv\max_{\\{|u_{v}\rangle\\}_{v\in
V(G)},|\psi\rangle}\sum_{v\in V(G)}w_{v}|\langle\psi|u_{v}\rangle|^{2},$ (50)

where $\\{|u_{v}\rangle\\}_{v\in V(G)}=\\{|u_{v}\rangle\\}_{v\in V(\bar{G})}$
(each $|u_{v}\rangle$ a unit vector in $\mathbb{R}^{d}$) is an orthonormal
representation (OR) of the complement of $G$, namely, $\bar{G}$, and the unit
vector $|\psi\rangle\in\mathbb{R}^{d}$ is called a handle.

Here $V(\bar{G})\equiv V(G)$ and
$E(\bar{G})\equiv\\{(v,v^{\prime})|v,v^{\prime}\in V(G),(v,v^{\prime})\notin
E(G)\\}$, and we have in an orthonormal representation that $\langle
u_{v^{\prime\prime}}|u_{v^{\prime\prime\prime}}\rangle=0$ for all pairs of
nonadjacent vertices, $(v^{\prime\prime},v^{\prime\prime\prime})$, in
$\bar{G}$, or equivalently, for all
$(v^{\prime\prime},v^{\prime\prime\prime})\in E(G)$.

  3. 3.

Fractional packing number $\alpha^{*}(G,w)$:

$\alpha^{*}(G,w)\equiv\max_{\\{p_{v}\\}_{v\in V(G)}}\sum_{v\in
V(G)}w_{v}p_{v},$ (51)

where $\\{p_{v}\\}_{v\in V(G)}$ is such that $p_{v}\geq 0$ for all $v\in V(G)$
and $\sum_{v\in c}p_{v}\leq 1$ for all cliques $c$ in $G$.

Note that since we are always considering $\Gamma_{G}$ such that
$\mathcal{CE}^{1}(\Gamma_{G})=\mathcal{G}(\Gamma_{G})$, we, in fact, have the
bounds

$\forall[s|S]:R([s|S])\overset{\rm KS}{\leq}\alpha(G,w)\overset{\rm
Q}{\leq}\theta(G,w)\overset{{\rm GPT}}{\leq}\alpha^{*}(G,w),$ (52)

where “GPT” denotes the full set of probabilistic models on $\Gamma_{G}$,
i.e., $\mathcal{G}(\Gamma_{G})$.

In terms of the notation we have already introduced, where $R([s|S])\leq
R_{\rm KS}$ was a Bell-KS inequality, we now have — from CSW [22] — that
$R_{\rm KS}=\alpha(G,w)$.

###  5.2 Key notion not from CSW:
source-measurement correlation, ${\rm Corr}$

We need to define a new quantity not in the CSW framework, namely,

${\rm Corr}\equiv\sum_{e\in
E(\Gamma_{G})}q_{e}\sum_{m_{e},s_{e}}\delta_{m_{e},s_{e}}p(m_{e},s_{e}|M_{e},S_{e}),$
(53)

where $\\{q_{e}\\}_{e\in E(\Gamma_{G})}$ is a probability distribution, i.e.,
$q_{e}\geq 0$ for all $e\in E(\Gamma_{G})$ and $\sum_{e\in
E(\Gamma_{G})}q_{e}=1$, such that $\beta(\Gamma_{G},q)<1$ holds.343434Indeed,
for the strongest possible constraint on ${\rm Corr}$, one must pick $q$ such
that $\beta(\Gamma_{G},q)$ is minimized. In previous work [12, 16], we have
taken $q$ to be the uniform distribution $q_{e}=\frac{1}{|E(\Gamma_{G})|}$,
but the derivation of the noncontextuality inequalities is independent of that
choice (as we’ll see here). Also, note that we have chosen the following
labelling convention for outcomes of source setting $S_{e}$ (namely, $s_{e}$)
and measurement setting $M_{e}$ (namely, $m_{e}$): the source outcomes $s_{e}$
for source setting $S_{e}$ take values in the same set as measurement outcomes
$m_{e}$ for measurement setting $M_{e}$, i.e., $V_{S_{e}}=V_{M_{e}}$
(recalling notation from Section 2). In particular, outcomes corresponding to
the measurement event $[v|e]$ (representing $[m_{e}|M_{e}]$) and its
corresponding source event $v_{e}$ (representing $[s_{e}|S_{e}]$) are both
denoted by the same label, so that $m_{e}=s_{e}$ for them. An example of this
from Figs. 5 and 6 would be to, say, denote the outcomes of a particular $e\in
E(\Gamma_{G})$ (measurement setting $M_{e}$) by $m_{e}\in
V_{M_{e}}\equiv\\{0,1,2\\}$ and corresponding outcomes of $e\in E(\Sigma_{G})$
(source setting $S_{e}$) by $s_{e}\in V_{S_{e}}\equiv\\{0,1,2\\}$; so if
$[v|e]$ denotes $[m_{e}=0|M_{e}]$, then $v_{e}$ will denote $[s_{e}=0|S_{e}]$,
etc.

###  5.3 Obtaining the noise-robust noncontextuality inequalities

####  5.3.1 Expressing operational quantities in ontological terms

We begin with expressing the operational quantities of interest in terms of a
noncontextual ontological model. In an ontological model, $R([s|S])$ is given
by

$R([s|S])=\sum_{\lambda\in\Lambda}\sum_{v\in
V(G)}w_{v}p(v|\lambda)\mu(\lambda|S,s).$ (54)

Defining $R(\lambda)\equiv\sum_{v\in V(G)}w_{v}p(v|\lambda)$, we have that

$R([s|S])=\sum_{\lambda\in\Lambda}R(\lambda)\mu(\lambda|S,s).$ (55)

Similarly, ${\rm Corr}$ is given by

$\displaystyle{\rm Corr}$ $\displaystyle=$
$\displaystyle\sum_{\lambda\in\Lambda}\sum_{e\in
E(\Gamma_{G})}q_{e}\sum_{m_{e},s_{e}}\delta_{m_{e},s_{e}}\xi(m_{e}|M_{e},\lambda)\mu(\lambda,s_{e}|S_{e})$
$\displaystyle=$ $\displaystyle\sum_{\lambda\in\Lambda}\sum_{e\in
E(\Gamma_{G})}q_{e}$
$\displaystyle\sum_{m_{e},s_{e}}\delta_{m_{e},s_{e}}\xi(m_{e}|M_{e},\lambda)\mu(s_{e}|S_{e},\lambda)\mu(\lambda|S_{e}).$

Here, we have used the fact that

$\mu(\lambda,s_{e}|S_{e})=\mu(s_{e}|S_{e},\lambda)\mu(\lambda|S_{e})$

to express ${\rm Corr}$ in a way that treats sources and measurements
similarly.

Using preparation noncontextuality (cf. Eq. (2.5)), we have that

$\displaystyle\forall e,e^{\prime}$ $\displaystyle\in$ $\displaystyle
E(\Sigma_{G}):[\top|S_{e_{\top}}]\simeq[\top|S_{{e^{\prime}}_{\top}}]$
$\displaystyle\Rightarrow\mu(\lambda|S_{e})$ $\displaystyle=$
$\displaystyle\mu(\lambda|S_{e^{\prime}})\equiv\nu(\lambda),\forall\lambda\in\Lambda.$
(57)

Then we can rewrite ${\rm Corr}$ as

$\displaystyle{\rm Corr}$ $\displaystyle=$
$\displaystyle\sum_{\lambda\in\Lambda}\sum_{e\in
E(\Gamma_{G})}q_{e}\sum_{m_{e},s_{e}}\delta_{m_{e},s_{e}}\xi(m_{e}|M_{e},\lambda)\mu(s_{e}|S_{e},\lambda)\nu(\lambda).$

Note that the only $\lambda$ that contribute to ${\rm Corr}$ are those for
which $\nu(\lambda)>0$. Also, $\mu(s_{e}|S_{e},\lambda)$ and
$\mu(\lambda|S_{e},s_{e})$ satisfy the condition
$\mu(s_{e}|S_{e},\lambda)\nu(\lambda)=\mu(\lambda|S_{e},s_{e})p(s_{e}|S_{e})$,
so that $\mu(s_{e}|S_{e},\lambda)$ is well-defined whenever $\nu(\lambda)>0$.

Defining

${\rm Corr}(\lambda)\equiv\sum_{e\in
E(\Gamma_{G})}q_{e}\sum_{m_{e},s_{e}}\delta_{m_{e},s_{e}}\xi(m_{e}|M_{e},\lambda)\mu(s_{e}|S_{e},\lambda),$
(59)

we have that

${\rm Corr}=\sum_{\lambda\in\Lambda}{\rm Corr}(\lambda)\nu(\lambda),$ (60)

Recalling that $\zeta(M_{e},\lambda)=\max_{m_{e}}\xi(m_{e}|M_{e},\lambda)$,
note that ${\rm Corr}(\lambda)$ is upper bounded as follows (for any
$\lambda\in\Lambda$):

$\displaystyle{\rm Corr}(\lambda)$ (61) $\displaystyle\equiv$
$\displaystyle\sum_{e\in
E(\Gamma_{G})}q_{e}\sum_{m_{e},s_{e}}\delta_{m_{e},s_{e}}\xi(m_{e}|M_{e},\lambda)\mu(s_{e}|S_{e},\lambda)$
$\displaystyle\leq$ $\displaystyle\sum_{e\in
E(\Gamma_{G})}q_{e}\zeta(M_{e},\lambda)\sum_{s_{e}}\mu(s_{e}|S_{e},\lambda)$
$\displaystyle=$ $\displaystyle\sum_{e\in
E(\Gamma_{G})}q_{e}\zeta(M_{e},\lambda).$

If $\lambda\in\Lambda_{\rm det}$, then this upper bound is trivial, i.e.,
${\rm Corr}(\lambda)\leq 1$, since every measurement has deterministic
response functions. On the other hand, for all $\lambda\in\Lambda_{\rm ind}$,
we have (from Eq. (46))

${\rm Corr}(\lambda)\leq\beta(\Gamma_{G},q).$ (62)

Similarly, for $\lambda\in\Lambda_{\rm det}$ we have
$R(\lambda)\leq\alpha(G,w)$, while for $\lambda\in\Lambda_{\rm ind}$ we have
$R(\lambda)\leq\alpha^{*}(G,w)$.

Using the fact that

$\nu(\lambda)=\mu(\lambda|S)=\sum_{s}\mu(\lambda|S,s)p(s|S),$

for any $S\equiv S_{e},e\in E(\Sigma_{G})$, we have

$\displaystyle{\rm Corr}$ (63) $\displaystyle=$
$\displaystyle\sum_{s}\left(\sum_{\lambda}{\rm
Corr}(\lambda)\mu(\lambda|S,s)\right)p(s|S)$ $\displaystyle=$
$\displaystyle\sum_{s}{\rm Corr}_{s}p(s|S).$

where we have defined ${\rm Corr}_{s}\equiv\sum_{\lambda}{\rm
Corr}(\lambda)\mu(\lambda|S,s)$.

####  5.3.2 Derivation of the noncontextual tradeoff for any graph $G$

We are now in a position to express our general noise-robust noncontextuality
inequality as a tradeoff between three operational quantities: ${\rm Corr}$,
$R([s_{e_{*}}=0|S_{e_{*}}])$, and $p(s_{e_{*}}=0|S_{e_{*}})$.

First, note that KS-contextuality is witnessed when for some choice of
$[s|S]$, here given by $[s_{e_{*}}=0|S_{e_{*}}]$, we have

$R([s_{e_{*}}=0|S_{e_{*}}])>\alpha(G,w).$

This means that for some set of ontic states in the support of
$[s_{e_{*}}=0|S_{e_{*}}]$, i.e.,

$\displaystyle\lambda$ $\displaystyle\in{\rm
Supp}\\{\mu(.|S_{e_{*}},s_{e_{*}}=0)\\}$
$\displaystyle\equiv\\{\lambda\in\Lambda:\mu(\lambda|S_{e_{*}},s_{e_{*}}=0)>0\\},$
(64)

we have $R(\lambda)>\alpha(G,w)$. For such a set of ontic states one must then
have ${\rm Corr}(\lambda)<1$ (because these $\lambda\in\Lambda_{\rm ind}$ and
we have Eq. (62)), which in turn implies that ${\rm Corr}_{s_{e_{*}}=0}<1$. On
the other hand, for $s_{e_{*}}=1$, we have no constraints: ${\rm
Corr}_{s_{e_{*}}=1}\leq 1$. Thus,

$\displaystyle{\rm Corr}$ (65) $\displaystyle=$ $\displaystyle{\rm
Corr}_{s_{e_{*}}=0}p(s_{e_{*}}=0|S_{e_{*}})+{\rm
Corr}_{s_{e_{*}}=1}p(s_{e_{*}}=1|S_{e_{*}})$ $\displaystyle\leq$
$\displaystyle p_{0}{\rm Corr}_{s_{e_{*}}=0}+1-p_{0},$

where $p_{0}\equiv p(s_{e_{*}}=0|S_{e_{*}})$.

Defining $\mu_{\rm det}\equiv\sum_{\lambda\in\Lambda_{\rm
det}}\mu(\lambda|S_{e_{*}},s_{e_{*}}=0)$ and $\mu_{\rm
ind}\equiv\sum_{\lambda\in\Lambda_{\rm
ind}}\mu(\lambda|S_{e_{*}},s_{e_{*}}=0)$, we now have

$\mu_{\rm det}+\mu_{\rm ind}=1,$ (66) ${\rm Corr}_{s_{e_{*}}=0}\leq\mu_{\rm
det}+\beta(\Gamma_{G},q)\mu_{\rm ind},$ (67) $R\leq\alpha(G,w)\mu_{\rm
det}+\alpha^{*}(G,w)\mu_{\rm ind}.$ (68)

Note that assuming $\mu_{\rm det}=1$ would reduce these constraints to a
standard Bell-KS inequality, $R\leq\alpha(G,w)$. However, since we are not
assuming this, simply eliminating $\mu_{\rm det}$ and $\mu_{\rm ind}$ from
these constraints leads us to353535To see this explicitly, just use Eq. (66)
to make the substitution $\mu_{\rm ind}=1-\mu_{\rm det}$ in Eqs. (67) and
(68), then eliminate $\mu_{\rm det}$ from Eq. (67) by using the upper bound on
it from Eq. (68).

$\displaystyle{\rm Corr}_{s_{e_{*}}=0}$ $\displaystyle\leq$ $\displaystyle
1-(1-\beta(\Gamma_{G},q))\frac{R-\alpha(G,w)}{\alpha^{*}(G,w)-\alpha(G,w)}$

where the upper bound is nontrivial if and only if $\beta(\Gamma_{G},q)<1$ and
$R-\alpha(G,w)>0$.

If we are given that $\beta(\Gamma_{G},q)<1$, then we have a trivial upper
bound on ${\rm Corr}_{s_{e_{*}}=0}$ for the remaining cases: the upper bound
is $1$ for $R=\alpha(G,w)$ and greater than $1$ for $R<\alpha(G,w)$.

Thus, our noise-robust noncontextuality inequality now reads:

${\rm Corr}\leq
1-p_{0}(1-\beta(\Gamma_{G},q))\frac{R-\alpha(G,w)}{\alpha^{*}(G,w)-\alpha(G,w)},$
(70)

which can be rewritten as

$R\leq\alpha(G,w)+\frac{\alpha^{*}(G,w)-\alpha(G,w)}{p_{0}}\frac{1-{\rm
Corr}}{1-\beta(\Gamma_{G},q)}.$ (71)

Note that Eq. (70) expresses the constraint from noncontextuality as an upper
bound on the source-measurement correlations ${\rm Corr}$, reminiscent of the
noise-robust noncontextuality inequality first derived in Ref. [12] (and later
treated in hypergraph-theoretic terms in Ref. [34]), except here the upper
bound on ${\rm Corr}$ depends not only on the hypergraph invariant
$\beta(\Gamma_{G},q)$ but also two of the graph invariants from the CSW
framework [22], namely, $\alpha(G,w)$ and $\alpha^{*}(G,w)$, besides also the
operational quantity $R$, which is the figure-of-merit for KS-contextuality
($R>\alpha(G,w)$ witnesses KS-contextuality) in the CSW framework. Eq. (70)
indicates that the source-measurement correlations would fail to be perfect
(i.e., ${\rm Corr}<1$) in an operational theory admitting a noncontextual
ontological model if and only if $R>\alpha(G,w)$ and $\beta(\Gamma_{G},q)<1$.
Contextuality would be witnessed when the source-measurement correlations are
stronger than the constraint from Eq. (70). For $R\leq\alpha(G,w)$, in
particular, there is no constraint from noncontextuality on ${\rm Corr}$.

On the other hand, rewriting the constraint from noncontextuality as Eq. (71),
one is reminded of the CSW framework [22], where $R$ is taken to be the
quantity that is upper bounded by KS-noncontextuality. Here, instead, we have
that $R$ is upper bounded by a term that includes the source-measurement
correlations ${\rm Corr}$ that can be achieved for the measurements and thus
penalizes for measurements that cannot be made highly predictable with respect
to some preparations, i.e., ${\rm Corr}<1$ makes it harder to violate the
upper bound on $R$. When the upper bound reaches $\alpha^{*}(G,w)$, it becomes
trivial and $R$ is no longer constrained by noncontextuality on account of
noise in the measurements. Indeed, trivial POVMs (cf. Appendices A.1.2 and C)
never violate such a noncontextuality inequality because of the penalty
incurred via ${\rm Corr}$, as we later show in Section 6.3.

####  5.3.3 When is the noncontextual tradeoff violated?

The inequality of Eq. (71) can be rewritten as the following tradeoff between
${\rm Corr}$, $p_{0}$, and $R$:

${\rm
Corr}+p_{0}(1-\beta(\Gamma_{G},q))\frac{R-\alpha(G,w)}{\alpha^{*}(G,w)-\alpha(G,w)}\leq
1.$ (72)

Writing the constraint from noncontextuality in the form of Eq. (72) (in
contrast to Eqs. (70) and (71)) makes it more even-handed in its treatment of
the two operational quantities $R$ (which is key in the CSW framework [22])
and ${\rm Corr}$ (which is key in noise-robust noncontextuality inequalities
inspired by logical proofs of the KS theorem [12, 34]) and emphasizes that
noise-robust noncontextuality inequalities inspired by statistical proofs of
the KS theorem [16] are tradeoffs between $R$ (which is about the strength of
correlations between measurements) and ${\rm Corr}$ (which is about the
predictability of measurements) that must be satisfied by any operational
theory admitting a noncontextual ontological model. Roughly speaking, a high
degree of predictability for measurements (e.g., ${\rm Corr}=1$) cannot
coexist with very strong correlations between the measurements (e.g.,
$R=\alpha^{*}(G,w)$) when the operational theory admits a noncontextual
ontological model.

For a nontrivial constraint – and hence, the possibility of witnessing
contextuality via violation of this inequality (Eq. (72)) – the upper bound on
${\rm Corr}$ (the right-hand-side of Eq. (70)) should be strictly bounded
above by 1, and the upper bound on $R$ (the right-hand-side of Eq. (71))
should be strictly bounded above by $\alpha^{*}(G,w)$ (the algebraic upper
bound on $R$), that is

$\displaystyle p_{0}$ $\displaystyle>$ $\displaystyle 0\textrm{ and
}\beta(\Gamma_{G},q)<1,$ $\displaystyle R$ $\displaystyle>$
$\displaystyle\alpha(G,w),$ $\displaystyle{\rm Corr}$ $\displaystyle>$
$\displaystyle 1-p_{0}(1-\beta(\Gamma_{G},q)).$ (73)

These are the minimal benchmarks necessary — besides the requirement of
tomographic completeness of a finite set of procedures and the possibility of
inferring secondary procedures with exact operational equivalences using
convexity of the operational theory [13] — to witness contextuality in a
Kochen-Specker type experiment adapted to our framework following Spekkens
[18].

Suppose one achieves, by some means, a value of $R=\theta(G,w)$, the upper
bound on the quantum value with projective measurements. When would this value
be an evidence of contextuality? For this to be the case, we must have:

${\rm
Corr}>1-p_{0}(1-\beta(\Gamma_{G},q))\frac{\theta(G,w)-\alpha(G,w)}{\alpha^{*}(G,w)-\alpha(G,w)}.$
(74)

Now, for the ideal quantum realization where measurement events are
projectors, and the corresponding source events are eigenstates, it is always
the case that ${\rm Corr}=1$, hence contextuality is witnessed. However, it’s
possible to witness contextuality even if ${\rm Corr}<1$, as long as it
exceeds the lower bound specified above. In a sense, for quantum theory, this
allows for a quantitative accounting of the effect of nonprojectiveness in the
measurements (or mixedness in preparations) on the possibility of witnessing
contextuality, a feature that is absent in traditional Kochen-Specker
approaches [21, 22, 23, 25]. Indeed, as long as one achieves any value of
$R>\alpha(G,w)$, it is possible to witness contextuality for a sufficiently
high value of ${\rm Corr}$ (see Eq. (70)).

###  5.4 Example: KCBS scenario

We will now illustrate our hypergraph framework by applying it to the KCBS
scenario to make differences with respect to the CSW graph-theoretic framework
[22] explicit.

The graph $G$ for the KCBS scenario is given in Fig. 3, the measurement events
hypergraph $\Gamma_{G}$ is given in Fig. 5, and the source events hypergraph
$\Sigma_{G}$ is given in Fig. 6. We then have

$R([s|S])=\sum_{v\in V(G)}p(v|S,s),$ (75)

where the (vertex) weights $w_{v}=1$ for all $v\in V(G)$, i.e., it’s an
unweighted graph and we will use $\alpha(G)$ and $\alpha^{*}(G)$ to denote its
independence number and the fractional packing number, respectively. These are
given by

$\alpha(G)=2\quad{\rm and}\quad\alpha^{*}(G)=5/2.$ (76)

The source-measurement correlation term is given by

${\rm Corr}=\sum_{e\in
E(\Gamma_{G})}q_{e}\sum_{m_{e},s_{e}}\delta_{m_{e},s_{e}}p(m_{e},s_{e}|M_{e},S_{e})$
(77)

for any choice of probability distribution $q\equiv\\{q_{e}\\}_{e\in
E(\Gamma_{G})}$. For simplicity, we will just take this probability
distribution to be uniform, i.e., $q_{e}=\frac{1}{5}$ for all $e\in
E(\Gamma_{G})$. Note that the only extremal probabilistic model on
$\Gamma_{G}$ corresponding to an indeterministic assignment (in $\Lambda_{\rm
ind}$) assigns $\xi(v|\lambda)=\frac{1}{2}$ for all $v\in V(G)$. This means

$\beta(\Gamma_{G},q)=\frac{1}{2}\quad\forall q.$ (78)

The noncontextuality inequality of Eq. (71)

$R\leq\alpha(G,w)+\frac{\alpha^{*}(G,w)-\alpha(G,w)}{p_{0}}\frac{1-{\rm
Corr}}{1-\beta(\Gamma_{G},q)}$ (79)

then becomes (in the KCBS scenario)

$R\leq 2+\frac{1/2}{p_{0}}\frac{1-{\rm Corr}}{1/2},$ (80)

or

$R\leq 2+\frac{1-{\rm Corr}}{p_{0}}.$ (81)

Recall that the KCBS inequality [47, 22] reads $R\leq 2$ and it would be a
valid noncontextuality inequality in our framework if and only if one can find
measurements and preparations such that ${\rm Corr}=1$. In the standard KCBS
construction [47] that violates the inequality $R\leq 2$, we have the five
vertices in $G$ (say $v_{i}$, $i\in\\{1,2,3,4,5\\}$, labelled cyclically)
associated with five projectors $\Pi_{i}=|l_{i}\rangle\langle l_{i}|$,
$i\in\\{1,2,3,4,5\\},$ on a qutrit Hilbert space, given by the vectors
$|l_{i}\rangle=(\sin\theta\cos\phi_{i},\sin\theta\sin\phi_{i},\cos\theta)$,
$\phi_{i}=\frac{4\pi i}{5}$, and $\cos\theta=\frac{1}{\sqrt[4]{5}}$. The
special source event $[s_{e_{*}}=0|S_{e_{*}}]$ is associated with the quantum
state $|\psi\rangle=(0,0,1)$, so that

$R=\sum_{i=1}^{5}|\langle l_{i}|\psi\rangle|^{2}=\sqrt{5}>2.$ (82)

Figure 7: Geometric configuration of the vectors appearing in the KCBS
construction [47].

See Fig. 7 for a depiction of the geometric configuration of these vectors.

To turn this KCBS construction into an argument against noncontextuality in
our approach, we need additional ingredients beyond the graph $G$. Firstly,
for both the measurement events hypergraph $\Gamma_{G}$ and the source events
hypergraph $\Sigma_{G}$, we denote the hyperedges by $e_{i}$,
$i\in\\{1,2,3,4,5\\}$. In $\Gamma_{G}$, the measurement events for the setting
$M_{e_{i}}$ are given by $\\{[m_{e_{i}}=0|M_{e_{i}}]=|l_{i}\rangle\langle
l_{i}|,[m_{e_{i}}=1|M_{e_{i}}]=\mathbb{I}-|l_{i}\rangle\langle
l_{i}|-|l_{i+1}\rangle\langle
l_{i+1}|,[m_{e_{i}}=2|M_{e_{i}}]=|l_{i+1}\rangle\langle l_{i+1}|\\}$, where
for $i=5$, $i+1=1$ (addition modulo 5). Similarly, in $\Sigma_{G}$, the source
events corresponding to source setting $S_{e_{i}}$ are given by
$\\{[s_{e_{i}}=0|S_{e_{i}}]=|l_{i}\rangle\langle l_{i}|$,
$[s_{e_{i}}=2|S_{e_{i}}]=|l_{i+1}\rangle\langle l_{i+1}|$, and
$[s_{e_{i}}=1|S_{e_{i}}]=\mathbb{I}-|l_{i}\rangle\langle
l_{i}|-|l_{i+1}\rangle\langle l_{i+1}|\\}$, where
$p(s_{e_{i}}=b|S_{e_{i}})=\frac{1}{3}$ for all $b\in\\{0,1,2\\}$. The special
source setting $S_{e_{*}}$ consists of source events
$\\{[s_{e_{*}}=0|S_{e_{*}}]=|\psi\rangle\langle\psi|,[s_{e_{*}}=1|S_{e_{*}}]=\frac{\mathbb{I}-|\psi\rangle\langle\psi|}{2}\\}$,
where $p(s_{e_{*}}=0|S_{e_{*}})=\frac{1}{3}$ and
$p(s_{e_{*}}=1|S_{e_{*}})=\frac{2}{3}$. We thus have the operational
equivalences we need between the source settings:

$[\top|S_{e_{\top}}]\simeq[\top|S_{{e^{\prime}}_{\top}}]=\frac{\mathbb{I}}{3},\quad\forall
e,e^{\prime}\in E(\Sigma_{G}).$ (83)

This choice of representation for $\Gamma_{G}$ and $\Sigma_{G}$ yields
$p_{0}=\frac{1}{3}$, ${\rm Corr}=1$, and
$R([s_{e_{*}}=0|S_{e_{*}}])=\sqrt{5}$, so that the inequality

$R\leq 2+\frac{1-{\rm Corr}}{p_{0}}$ (84)

is violated. However, note that this is an idealization (under which ${\rm
Corr}=1$) and, typically, the source events and measurement events will not be
perfectly correlated (${\rm Corr}<1$) and the operational equivalences between
the source settings need not correspond to the maximally mixed state. All that
is required for a test of noncontextuality using this inequality is that the
operational equivalences hold for some choice of preparations and measurements
which need not be the same as that in the ideal KCBS construction.

To illustrate what happens when ${\rm Corr}<1$, we consider the effect of a
depolarizing channel on the states and measurements in the ideal KCBS
construction. The channel is given by

$\displaystyle\mathcal{D}_{r}(\cdot)=rI(\cdot)\mathbb{I}+(1-r)\frac{1}{3}\mathbb{I}\text{Tr}(\cdot),r\in[0,1].$
(85)

The action of this channel – with parameter $r_{1}\in[0,1]$, say – on the pure
states $\\{\\{|l_{i}\rangle\langle
l_{i}|\\}_{i=1}^{5},|\psi\rangle\langle\psi|\\}$ yields the noisy states given
by

$\displaystyle\mathcal{D}_{r_{1}}(|l_{i}\rangle\langle
l_{i}|)={r_{1}}|l_{i}\rangle\langle
l_{i}|+(1-{r_{1}})\frac{\mathbb{I}}{3},\forall i\in[5],$ (86)
$\displaystyle\mathcal{D}_{r_{1}}(|\psi\rangle\langle\psi|)={r_{1}}|\psi\rangle\langle\psi|+(1-{r_{1}})\frac{\mathbb{I}}{3},$
(87)

and the action of its adjoint – with parameter $r_{2}\in[0,1]$, say – on the
ideal projectors, $\\{|l_{i}\rangle\langle l_{i}|\\}_{i=1}^{5}$, involved in
the measurements correspondingly yields the POVM elements given by

$\displaystyle\mathcal{D}^{\dagger}_{r_{2}}(|l_{i}\rangle\langle
l_{i}|)={r_{2}}|l_{i}\rangle\langle
l_{i}|+(1-{r_{2}})\frac{\mathbb{I}}{3},\forall i\in[5].$ (88)

Hence, we are imagining a situation where the preparation procedures are
affected by depolarizing noise with parameter $r_{1}$ and measurement
procedures are affected by depolarizing noise with parameter $r_{2}$, similar
to the situation considered previously in Section II of the Supplemental
material of Ref. [12]. The operational equivalences required for our argument
from preparation and measurement noncontextuality are satisfied by these noisy
preparations and measurements. That is, in $\Gamma_{G}$, we can represent the
measurement events for the setting $M_{e_{i}}$ (where $i\in[5]$, $i+1=1$,
i.e., addition modulo 5) by

$\displaystyle[m_{e_{i}}=0|M_{e_{i}}]=\mathcal{D}^{\dagger}_{r_{2}}(|l_{i}\rangle\langle
l_{i}|),$ (89)
$\displaystyle[m_{e_{i}}=1|M_{e_{i}}]=\mathcal{D}^{\dagger}_{r_{2}}(\mathbb{I}-|l_{i}\rangle\langle
l_{i}|-|l_{i+1}\rangle\langle l_{i+1}|),$ (90)
$\displaystyle[m_{e_{i}}=2|M_{e_{i}}]=\mathcal{D}^{\dagger}_{r_{2}}(|l_{i+1}\rangle\langle
l_{i+1}|).$ (91)

It is easy to verify that these form elements of a valid POVM denoted by the
measurement setting $M_{e_{i}}$ and that the operational equivalences between
the measurement events (represented by $\Gamma_{G}$) are indeed respected. On
the other hand, in $\Sigma_{G}$, the source events corresponding to source
setting $S_{e_{i}}$ can be represented by

$\displaystyle[s_{e_{i}}=0|S_{e_{i}}]=\mathcal{D}_{r_{1}}(|l_{i}\rangle\langle
l_{i}|),$ (92)
$\displaystyle[s_{e_{i}}=1|S_{e_{i}}]=\mathcal{D}_{r_{1}}(\mathbb{I}-|l_{i}\rangle\langle
l_{i}|-|l_{i+1}\rangle\langle l_{i+1}|\\})$ (93)
$\displaystyle[s_{e_{i}}=2|S_{e_{i}}]=\mathcal{D}_{r_{1}}(|l_{i+1}\rangle\langle
l_{i+1}|),$ (94)

where $p(s_{e_{i}}=b|S_{e_{i}})=\frac{1}{3}$ for all $b\in\\{0,1,2\\}$, while
the source events for source setting $S_{e_{*}}$ can be represented by

$\displaystyle[s_{e_{*}}=0|S_{e_{*}}]=\mathcal{D}_{r_{1}}(|\psi\rangle\langle\psi|),$
(95)
$\displaystyle[s_{e_{*}}=1|S_{e_{*}}]=\mathcal{D}_{r_{1}}\left(\frac{\mathbb{I}-|\psi\rangle\langle\psi|}{2}\right),$
(96)

where $p(s_{e_{*}}=0|S_{e_{*}})=\frac{1}{3}$ and
$p(s_{e_{*}}=1|S_{e_{*}})=\frac{2}{3}$. These satisfy the operational
equivalences

$[\top|S_{e_{\top}}]\simeq[\top|S_{{e^{\prime}}_{\top}}]=\frac{\mathbb{I}}{3},\quad\forall
e,e^{\prime}\in E(\Sigma_{G}).$ (97)

We then have

$\displaystyle{\rm Corr}=\frac{1}{5}\sum_{e\in
E(\Gamma_{G})}\frac{1}{3}\sum_{b\in\\{0,1,2\\}}p(m_{e}=b|M_{e},S_{e},s_{e}=b).$
(98)

Noting that for any qutrit pure state $|\phi\rangle$ and its corresponding
projector $|\phi\rangle\langle\phi|$, each affected by depolarizing noise with
parameters $r_{1}$ and $r_{2}$, respectively, we have

$\displaystyle\text{Tr}(\mathcal{D}_{r_{1}}(|\phi\rangle\langle\phi|)\mathcal{D}^{\dagger}_{r_{2}}(|\phi\rangle\langle\phi|))$
$\displaystyle=$ $\displaystyle\frac{1}{3}+\frac{2}{3}r_{1}r_{2}.$ (99)

Now, each term in the summation defining ${\rm Corr}$, namely,
$p(m_{e}=b|M_{e},S_{e},s_{e}=b)$, is obtained from a calculation of the type
in Eq. (5.4). Hence, we have for each such term,

$\displaystyle p(m_{e}=b|M_{e},S_{e},s_{e}=b)$ $\displaystyle=$
$\displaystyle\frac{1}{3}+\frac{2}{3}r_{1}r_{2},$ (100)

so that

$\displaystyle{\rm Corr}=\frac{1}{3}+\frac{2}{3}r_{1}r_{2}.$ (101)

In the noiseless regime, i.e., $r_{1}=r_{2}=1$, this reduces to the ideal KCBS
scenario. On the other hand, we have

$\displaystyle R([s_{e_{*}}=0|S_{e_{*}}])$ $\displaystyle=$
$\displaystyle\sum_{v\in V(G)}p(v|S_{e_{*}},s_{e_{*}}=0)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{5}\text{Tr}\left(\mathcal{D}^{\dagger}_{r_{2}}(|l_{i}\rangle\langle
l_{i}|)\mathcal{D}_{r_{1}}(|\psi\rangle\langle\psi|)\right)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{5}\left(r_{1}r_{2}|\langle
l_{i}|\psi\rangle|^{2}+\frac{r_{2}(1-r_{1})}{3}+\frac{r_{1}(1-r_{2})}{3}+\frac{(1-r_{1})(1-r_{2})}{3}\right)$
$\displaystyle=$ $\displaystyle r_{1}r_{2}\sum_{i=1}^{5}|\langle
l_{i}|\psi\rangle|^{2}+\frac{5}{3}(1-r_{1}r_{2}).$ (102)

Recall that violation of the noncontextuality inequality requires that

$R>2+\frac{1-{\rm Corr}}{p_{0}}.$ (103)

That is,

$\displaystyle r_{1}r_{2}\sum_{i=1}^{5}|\langle
l_{i}|\psi\rangle|^{2}+\frac{5}{3}(1-r_{1}r_{2})$ $\displaystyle>$
$\displaystyle
2+3\left(1-\left(\frac{1}{3}+\frac{2}{3}r_{1}r_{2}\right)\right).$ (104)

Given that $\sum_{i=1}^{5}|\langle l_{i}|\psi\rangle|^{2}=\sqrt{5}$, this
becomes

$\displaystyle r_{1}r_{2}\sqrt{5}+\frac{5}{3}(1-r_{1}r_{2})$ $\displaystyle>$
$\displaystyle 2+2(1-r_{1}r_{2}).$ (105)

Rewriting this, we obtain

$\displaystyle r_{1}r_{2}>1-\frac{\sqrt{5}-2}{\sqrt{5}+\frac{1}{3}}\approx
0.908,$ (106)

that is, the noncontextuality inequality can be violated only when the
depolarizing noise is below a certain threshold given by $r_{1}r_{2}>0.908$.
In terms of ${\rm Corr}$, this requires ${\rm Corr}>0.939$. The noiseless case
$r_{1}=r_{2}=1$ takes us back to the ${\rm Corr}=1$ regime that we previously
discussed.

##  6 Discussion

###  6.1 Measurement-measurement correlations vs. source-measurement
correlations

Note that the usual Kochen-Specker experiment, as conceptualized in Refs. [21,
22, 23, 25], for example, involves only the quantity $R([s|S])$, representing
correlations between various measurement events when all the measurements are
implemented on a system prepared according to the same preparation procedure,
denoted by the source event $[s|S]$. Thus, $R$ represents measurement-
measurement correlations on a system prepared according to a fixed choice of
preparation procedure.

On the other hand, the experiment we have conceptualized in this paper
involves, besides the quantity $R$, a quantity ${\rm Corr}$ representing
source-measurement correlations, characterizing the quality of the
measurements in terms of their response to corresponding preparations.

Our noncontextuality inequalities represent a trade-off relation that must
hold between $R$ and ${\rm Corr}$ in an operational theory that admits a
noncontextual ontological model. Here we note that the first example of such a
tradeoff relation, albeit only for the case of operational quantum theory with
unsharp measurements, appeared in Ref. [39] as the Liang-Spekkens-Wiseman
(LSW) inequality [40] which has been shown to be experimentally violated in
Ref. [53].363636This experiment, however, is not in a position to make claims
about contextuality without presuming the operational theory is quantum theory
simply because the LSW inequality presumes operational quantum theory. The
noncontextuality inequalities in this paper do not require the operational
theory to be quantum theory and can therefore be experimentally tested using
techniques from Refs. [13, 54, 37]. And, indeed, the developments reported in
Ref. [16] and the present paper have their origins in the idea of such a
trade-off relation that first appeared in Ref. [39].

###  6.2 Can our noise-robust noncontextuality inequalities be saturated by a
noncontextual ontological model?

A natural question concerns the tightness of these noncontextuality
inequalities, i.e., can Eq. (72) be saturated by a noncontextual ontological
model? This requires one to specify a noncontextual ontological model
reproducing the operational equivalences between the measurement events and
between the source settings, such that

${\rm
Corr}+p_{0}(1-\beta(\Gamma_{G},q))\frac{R-\alpha(G,w)}{\alpha^{*}(G,w)-\alpha(G,w)}=1.$
(107)

The assumption of measurement noncontextuality is already implicit in our
characterization of the response functions $\xi(m_{e}|M_{e},\lambda)$, and for
this reason it is, indeed, trivial to satisfy measurement noncontextuality
while saturating these noncontextuality inequalities. Measurement
noncontextuality, alone, in fact even allows a violation of the inequality
(when no preparation noncontextuality is imposed), the extreme case being
$R=\alpha^{*}(G,w)$ and $1\geq{\rm Corr}>1-p_{0}(1-\beta(\Gamma_{G},q))$. It’s
the assumption of preparation noncontextuality that is nontrivial to satisfy
and we do not know if there exists a general construction of a noncontextual
ontological model saturating our noncontextuality inequalities. We outline the
general situation below.

####  6.2.1 The special case of facet-defining Bell-KS inequalities: Corr=1

If outcome determinism is presumed (as in traditional Bell-KS type
treatments), then we know that there exists a necessary and sufficient set of
Bell-KS inequalities (each corresponding to a particular choice of $R([s|S])$)
that are satisfied by any operational theory admitting a KS-noncontextual
ontological model. In particular, each such (facet) Bell-KS inequality can be
saturated by KS-noncontextual ontological models that yield probabilities
(from $\mathcal{G}(\Gamma_{G})$) corresponding to the facet-defining Bell-KS
inequality, i.e., which satisfy $R([s|S])=\alpha(G,w)$ for such a Bell-KS
inequality. Indeed, our noise-robust noncontextuality inequalities
corresponding to these choices of $R([s|S])$ (i.e., facet-defining Bell-KS
inequalities of the Bell-KS polytope which is given by the convex hull of
points in $\mathcal{G}(\Gamma_{G})|_{\rm det}$) can always be saturated when
${\rm Corr}=1$, because in that case outcome determinism is justified by
preparation noncontextuality (cf. Ref. [16]) and our inequalities are
identical to the Bell-KS inequalities (saturated by $R=\alpha(G,w)$).

####  6.2.2 The general case: ${\rm Corr}<1$

Since we do not want to assume outcome determinism, nor necessarily the
idealization of ${\rm Corr}=1$, what is at stake here is the assumption of
preparation noncontextuality. This assumption must be satisfied while
saturating the noise-robust noncontextuality inequality in order for a
measurement noncontextual ontological model to be universally noncontextual.
Constructing such a noncontextual ontological model amounts to specifying the
distributions $\mu(s_{e}|S_{e},\lambda)$ and $\nu(\lambda)$ such that

$\forall\lambda\in\Lambda:\mu(\lambda|S_{e})=\nu(\lambda),\forall e\in
E(\Sigma_{G}),$ (108)

i.e., preparation noncontextuality holds, and we have (rewriting the
saturation condition from Eq. (107))

$\displaystyle(\alpha^{*}(G,w)-\alpha(G,w)){\rm
Corr}+p_{0}(1-\beta(\Gamma_{G},q))R$ $\displaystyle=$
$\displaystyle(\alpha^{*}(G,w)-\alpha(G,w))+p_{0}\alpha(G,w)(1-\beta(\Gamma_{G},q)),$

where

${\rm Corr}=\sum_{s_{e_{*}}}p(s_{e_{*}}|S_{e_{*}}){\rm Corr}_{s_{e_{*}}},$
(110) ${\rm Corr}_{s_{e_{*}}}=\sum_{\lambda\in\Lambda}{\rm
Corr}(\lambda)\mu(\lambda|S_{e_{*}},s_{e_{*}}),$ (111) $\displaystyle{\rm
Corr}(\lambda)$ $\displaystyle\equiv$ $\displaystyle\sum_{e\in
E(\Gamma_{G})}q_{e}\sum_{m_{e},s_{e}}\delta_{m_{e},s_{e}}\xi(m_{e}|M_{e},\lambda)\mu(s_{e}|S_{e},\lambda),$
(112)

and

$R=\sum_{\lambda\in\Lambda}R(\lambda)\mu(\lambda|S_{e_{*}},s_{e_{*}}=0).$
(113)

Unfortunately, we do not have a general construction that can show this to be
possible for any noise-robust noncontextuality inequality obtained according
to the approach we have outlined. We therefore leave it as an open question
whether such an inequality can (always?) be saturated by a noncontextual
ontological model.

###  6.3 Can trivial POVMs ever violate these noncontextuality inequalities?

No.

Recall that a trivial POVM is defined as an assignment of positive operators
$p(v)\mathbb{I}$ to the vertices of $\Gamma_{G}$, where $\mathbb{I}$ is the
identity operator on some Hilbert space and $p:V(\Gamma_{G})\rightarrow[0,1]$,
such that $\sum_{v\in e}p(v)=1$ for all $e\in E(\Gamma_{G})$, is a
probabilistic model on $\Gamma_{G}$.

####  6.3.1 The case $p\in\mathcal{C}(\Gamma_{G})$

Consider trivial POVMs corresponding to any KS-noncontextual probabilistic
model, i..e., $p\in\mathcal{C}(\Gamma_{G})$ is a convex mixture of
deterministic vertices, $\mathcal{G}(\Gamma_{G})|_{\rm det}$, or equivalently,
of ontic states in $\Lambda_{\rm det}$. In other words,
$\mathcal{C}(\Gamma_{G})\equiv{\rm ConvHull}(\mathcal{G}(\Gamma_{G})|_{\rm
det})$, the convex hull of points in $\mathcal{G}(\Gamma_{G})|_{\rm det}$. The
largest value ${\rm Corr}$ can take in this case is less than or equal to 1.
This means that the upper bound on $R$ from our noncontextuality inequality,
Eq. (71), will be greater than or equal to $\alpha(G,w)$, whereas we know that
for a KS-noncontextual probabilistic model, $R\leq\alpha(G,w)$. Hence, there
is no violation of our noncontextuality inequality for such trivial POVMs.

####  6.3.2 The case $p\in{\rm ConvHull}(\mathcal{G}(\Gamma_{G})|_{\rm ind})$

Now consider trivial POVMs that correspond to the indeterministic vertices,
$\mathcal{G}(\Gamma_{G})_{\rm ind}$ (correspondingly, $\Lambda_{\rm ind}$), or
their convex mixtures. We know that for these trivial POVMs, ${\rm
Corr}\leq\beta(\Gamma_{G},q)$. For any $R\leq\alpha^{*}(G,w)$ that is achieved
by these trival POVMs, our noncontextuality inequality reads

${\rm Corr}\leq
1-p_{0}(1-\beta(\Gamma_{G},q))\frac{R-\alpha(G,w)}{\alpha^{*}(G,w)-\alpha(G,w)},$
(114)

A sufficient condition for this inequality to be satisfied is that

$\beta(\Gamma_{G},q)\leq
1-p_{0}(1-\beta(\Gamma_{G},q))\frac{R-\alpha(G,w)}{\alpha^{*}(G,w)-\alpha(G,w)},$
(115)

which reduces, for $R>\alpha(G,w)$, to

$p_{0}\leq\frac{\alpha^{*}(G,w)-\alpha(G,w)}{R-\alpha(G,w)},$ (116)

where the upper bound is greater than or equal to 1, since
$\alpha(G,w)<R\leq\alpha^{*}(G,w)$. This is trivially satisfied since
$p_{0}\leq 1$.

For $R<\alpha(G,w)$, the sufficient condition of Eq. (115) is again trivially
satisfied since it reduces to

$p_{0}\geq-\frac{\alpha^{*}(G,w)-\alpha(G,w)}{\alpha(G,w)-R},$ (117)

and we must anyway have $p_{0}\geq 0$.

For $R=\alpha(G,w)$, the sufficient condition reduces to
$\beta(\Gamma_{G},q)\leq 1$, which is again trivially satisfied since
$\beta(\Gamma_{G},q)<1$ by definition.

####  6.3.3 The general case $p\in\mathcal{G}(\Gamma_{G})$

In general, a probabilistic model achieved by trivial POVMs can be in the
convex hull of both deterministic ($\Lambda_{\rm det}$) and indeterministic
($\Lambda_{\rm ind}$) ontic states, with the total weight on deterministic
ontic states denoted by ${\rm Pr}(\Lambda_{\rm det})$ and that on
indeterministic ontic states by ${\rm Pr}(\Lambda_{\rm ind})$, so that ${\rm
Pr}(\Lambda_{\rm det})+{\rm Pr}(\Lambda_{\rm ind})=1$. We then have

$\displaystyle{\rm Corr}$ $\displaystyle\leq$ $\displaystyle{\rm
Pr}(\Lambda_{\rm det})+{\rm Pr}(\Lambda_{\rm ind})\beta(\Gamma_{G},q),$
$\displaystyle R$ $\displaystyle\leq$ $\displaystyle{\rm Pr}(\Lambda_{\rm
det})\alpha(G,w)+{\rm Pr}(\Lambda_{\rm ind})\alpha^{*}(G,w).$

A sufficient condition for satisfaction of the noncontextuality inequality is
then

$\displaystyle 1-{\rm Pr}(\Lambda_{\rm ind})(1-\beta(\Gamma_{G},q))$
$\displaystyle\leq$ $\displaystyle
1-p_{0}(1-\beta(\Gamma_{G},q))\frac{R-\alpha(G,w)}{\alpha^{*}(G,w)-\alpha(G,w)},$

which becomes

$p_{0}\leq\frac{\alpha^{*}(G,w)-\alpha(G,w)}{R-\alpha(G,w)}{\rm
Pr}(\Lambda_{\rm ind})$ (120)

when $R>\alpha(G,w)$. Noting that

$R\leq\alpha(G,w)+{\rm Pr}(\Lambda_{\rm ind})(\alpha^{*}(G,w)-\alpha(G,w)),$

we have

${\rm Pr}(\Lambda_{\rm
ind})\geq\frac{R-\alpha(G,w)}{\alpha^{*}(G,w)-\alpha(G,w)},$ (121)

so that the sufficient condition for satisfaction of the noncontextuality
inequality becomes $p_{0}\leq 1$, which is trivially satisfied.

When $R=\alpha(G,w)$, the sufficient condition becomes
$\beta(\Gamma_{G},q)\leq 1$, which is again trivially satisfied.

Finally, when $R<\alpha(G,w)$, the sufficient condition becomes

$p_{0}\geq-\frac{\alpha^{*}(G,w)-\alpha(G,w)}{\alpha(G,w)-R}{\rm
Pr}(\Lambda_{\rm ind}),$ (122)

which is again trivially satisfied since $p_{0}\geq 0$.

Hence trivial POVMs cannot yield a violation of our noncontextuality
inequalities. This is the sense in which trivial POVMs cannot lead to
nonclassicality in our approach, unlike the case of traditional Kochen-Specker
approaches [21, 22, 23, 25] applied to the case of POVMs [73]. To violate our
noncontextuality inequalities, the POVMs must necessarily have some nontrivial
projective component (that is not the identity operator or zero) but they need
not be projectors. Further, we do not rely on restricting the notion of joint
measurability [44] (cf. Section 2.4) to commutativity for POVMs. Taking joint
measurability to be just commutativity is the approach adopted in, for
example, Ref. [25]. We refer to Appendix A and Appendix C for more discussion
on these issues, in particular Appendix C for the role of commutativity vs.
joint measurability.

##  7 Conclusions

We have obtained a hypergraph framework for obtaining noise-robust
noncontextuality inequalities corresponding to KS-colourable scenarios,
suitably augmented with preparation procedures in the spirit of Spekkens
contextuality [18]. The inequalities take the form of a noncontextual tradeoff
between the three operational quantities ${\rm Corr}$, $R$, and $p_{0}$, cf.
Eq. (72). This framework leverages the graph invariants from the graph-
theoretic framework of CSW for doing this, in addition to a new hypergraph
invariant (Eq. (45)) that we call the weighted max-predictability. Our
approach is general enough to be applicable to any situation involving noisy
preparations and measurements that arises from a KS-colourable contextuality
scenario.

We conclude with a list of open questions raised in this paper and other
directions for future research:

  1. 1.

Characterizing structural Specker’s principle from probabilistic models on a
hypergraph $\Gamma$:

Given that $\mathcal{CE}^{1}(\Gamma)=\mathcal{G}(\Gamma)$ for some $\Gamma$,
is it the case that $\Gamma$ must then necessarily satisfy structural
Specker’s principle, namely, that every clique in $O(\Gamma)$ is a subset of
some hyperedge in $\Gamma$? Or is it the case that there exists a hypergraph
$\Gamma^{\prime}$ for which
$\mathcal{CE}^{1}(\Gamma^{\prime})=\mathcal{G}(\Gamma^{\prime})$ but
structural Specker’s principle fails?

More generally, is there any characterization of a hypergraph satisfying
structural Specker’s principle entirely in terms of the probabilistic models
on it?

As already pointed out earlier, this open question relates to the open Problem
7.2.3 of Ref. [23] of characterizing $\Gamma$ for which
$\mathcal{CE}^{1}(\Gamma)=\mathcal{G}(\Gamma)$. It is known that $\Gamma$
representing bipartite Bell scenarios [55] satisfy the property
$\mathcal{CE}^{1}(\Gamma)=\mathcal{G}(\Gamma)$ and we have provided a generic
recipe for converting any $\Gamma$ that does not satisfy structural Specker’s
principle to a $\Gamma^{\prime}$ that does satisfy it so that
$\mathcal{CE}^{1}(\Gamma^{\prime})=\mathcal{G}(\Gamma^{\prime})$. The question
is if there are any other $\Gamma$ that also satisfy
$\mathcal{CE}^{1}(\Gamma)=\mathcal{G}(\Gamma)$.

  2. 2.

Almost quantum theory: We know that an almost quantum theory cannot satisfy
Specker’s principle [49] but it satisfies statistical Specker’s principle (or
consistent exclusivity). An open question that remains is:

Can an almost quantum theory satisfy structural Specker’s principle?

If not, this would render the satisfaction of consistent exclusivity by an
almost quantum theory unexplained by a natural structural feature of
measurements in the theory, namely, the satisfaction of structural Specker’s
principle, i.e., almost quantum theory would not fall in the category of
operational theories envisaged in Ref. [30].

  3. 3.

Conditions for saturating the noise-robust noncontextuality inequalities:

As mentioned in Section 6.2, it is an open question whether the noise-robust
noncontextuality inequalities of Eq. (72) based on our generalization of the
CSW framework [22] can be saturated by a noncontextual ontological model.

More generally, the status of these noise-robust noncontextuality inequalities
vis-à-vis the algorithmic approach of Ref. [17] for finding necessary and
sufficient conditions for noncontextuality in a general prepare-and-measure
scenario remains to be explored. One would suspect that the algorithmic
approach of Ref. [17] when adapted to the kind of situation considered in this
paper would yield nontrivial noncontextuality inequalities that aren’t merely
generalizations of the ones obtained in the CSW framework [22]. It would be
interesting to investigate the full structure of this set of inequalities and
compare it with the facet-defining Bell-KS inequalities of the CSW framework.

  4. 4.

Properties of the weighted max-predictability, $\beta(\Gamma_{G},q)$:

Since the crucial new hypergraph-theoretic ingredient in our inequalities is
the weighted max-predictability, it would be interesting to understand
properties of this hypergraph invariant on both counts: as a new mathematical
object in its own right, one we haven’t been able to find a reference to in
the hypergraph theory literature, as well as an important parameter of a
hypergraph relevant for noise-robustness of a noise-robust noncontextuality
inequality. Indeed, as we point out in Footnote 34, identifying a distribution
$q$ (in the definition of ${\rm Corr}$, Eq. (53)) that minimizes
$\beta(\Gamma_{G},q)$ for a given $\Gamma_{G}$ would lead to better noise-
robustness in the inequalities of Eqs. (70) or (71).

  5. 5.

Noise-robust applications of quantum protocols based on KS-contextuality:

A general research direction is to construct noise-robust versions of
applications that have previously been suggested for KS-contextuality. Our
approach provides a recipe for doing this for any Bell-KS inequality appearing
in such applications. Besides serving as a witness for strong nonclassicality
[56] (i.e., Spekkens contextuality),373737As opposed to weak nonclassicality
that can arise in epistemically restricted classical theories [57]. See also
the talk at Ref. [56], 41:43 minutes, for a short discussion. noise-robust
versions of these applications can help benchmark the experiments in terms of
the noise that can be tolerated while still witnessing nonclassicality.
Examples of such applications include those from Refs. [58, 59, 60, 61, 62,
63].

## Acknowledgments

I would like to thank Andreas Winter for his comments on an earlier version of
some of these ideas, Tobias Fritz for the ping-pong and the sing-song in which
we often talked about hypergraphs, Rob Spekkens for the often argumentative –
but always productive – conversations over lunch, and participants at the
Contextuality conference (CCIOSA) at Perimeter Institute, during July 24 - 28,
2017, for very stimulating discussions that fed into the narrative of this
paper. I would also like to thank David Schmid, Ana Belén Sainz, Elie Wolfe,
and Tomáš Gonda for helping me better articulate the difference between
structural vs. statistical readings of Specker’s principle, and Eric
Cavalcanti for comments on the manuscript. Theorem 1 owes its origin to a
discussion with Tomáš Gonda. I would also like to thank anonymous referees for
suggestions that immensely improved the presentation of these results.
Research at Perimeter Institute is supported by the Government of Canada
through the Department of Innovation, Science and Economic Development Canada,
and by the Province of Ontario through the Ministry of Research, Innovation
and Science.

##  Appendix A Status of KS-contextuality as an experimentally testable notion
of nonclassicality for POVMs in quantum theory

The purpose of this section is to emphasize how the progression from KS-
contextuality to Spekkens contextuality for KS-type contextuality experiments
is a natural one rather than an ad hoc move from one framework to another.
That is, Spekkens contextuality is not just another notion of nonclassicality
that is incomparable with KS-contextuality, but is indeed intimately connected
in its motivations to the limitations of KS-contextuality [18]. In particular,
we will focus on the role of KS-contextuality with respect to POVMs and why
allowing arbitrary POVMs poses a difficulty for KS-contextuality as a notion
of nonclassicality that is experimentally testable, i.e., a notion that
applies to noisy measurements (POVMs) typically implemented in a laboratory
experiment.383838And how a rather compelling way to arrive at a notion that is
experimentally testable is Spekkens contextuality. While one may be tempted to
reject this premise for assessing the suitability of KS-contextuality as a
notion of nonclassicality – claiming instead that KS-contextuality was never
meant for POVMs and applies only to “purified” experiments (namely, ones with
only PVMs and pure states) – the reasons for doing so are rooted in the
literature on KS-contextuality where POVMs have indeed been considered and (at
least) two kinds of conclusions drawn: one, that there exists a Kochen-Specker
contradiction for POVMs, even on a qubit, so KS-contextuality for POVMs is
interesting [64] and two, that allowing arbitrary POVMs in assessing
nonclassicality would make the research program of identifying device-
independent principles for quantum correlations in KS-contextuality
experiments ill-defined, so quantum correlations allowing arbitrary POVMs are
“pathological” [73]. We will look at these arguments in turn and use the
latter, in particular, to segue into our motivations for the framework
proposed in this paper.

###  A.1 Limitations of KS-contextuality vis-à-vis POVMs

####  A.1.1 KS-contextuality for POVMs in the literature

The first paper that applied KS-contextuality to the case of POVMs was by
Cabello [64] where a KS-uncolourability argument for POVMs on a single qubit
was proposed. This was motivated by the Gleason-type derivation of the Born
rule starting with the structure of POVMs due to Busch [66] and Caves et al.
[67], analogous to the case of the Kochen-Specker theorem [19] which can be
seen as motivated by Gleason’s theorem [68]. Insofar as there exists a
Gleason-type theorem for POVMs [66, 67], one could motivate KS-contextuality
as a reasonable notion of nonclassicality for POVMs, as was presumably the
case in Ref. [64]. The role of this notion of nonclassicality is then just to
argue – using a finite set of POVM elements – that no KS-noncontextual
assignment of outcomes is possible for certain finite sets of POVMs in quantum
theory. Should we, however, assume that it is reasonable to demand
deterministic assignment of outcomes to POVM elements in an ontological model,
just as we do for PVM elements? The argument of Ref. [64] was later criticized
on various counts [18, 65, 28] and we refer the reader to Ref. [28] for
criticisms pertinent to this paper, namely, that outcome determinism for all
unsharp measurements (ODUM in Ref. [28]) in quantum theory is
untenable.393939Ref. [28] is also a good resource for a detailed analysis of
arguments concerning dilations of POVMs, which we will not get into here.
Besides, it also provides a principled recipe for assigning response functions
to POVMs. Other works in the literature where KS-contextuality for POVMs has
been explored include Refs. [69, 70, 71, 72].

Besides, doubts about the experimental testability of the KS theorem were
raised in the late ‘90s in a series of papers by Meyer, Clifton, and Kent [74,
75, 76]. A review can be found in Ref. [77]. These doubts were premised on the
idea that the set of KS-colourable projectors (or PVMs) on any given Hilbert
space is dense in the set of all projectors (or PVMs) on that Hilbert space.
That is, for any given set of PVMs yielding a KS contradiction, it is always
possible to find PVMs which are arbitrarily “close” to the PVMs required for a
KS contradiction (for any finite precision) but which do not themselves lead
to a KS contradiction. The property of denseness of KS-colourable sets of
measurements in the set of all measurements in fact extends to even the most
general case when the measurements are POVMs on any Hilbert space. So, even a
KS contradiction for POVMs (such as the one in Ref. [64]) falls prey to the
Meyer-Clifton-Kent argument [77]. As Ref. [77] notes:

> Dealing with projective measurements is arguably not enough. One quite
> popular view of quantum theory holds that a correct version of the
> measurement rules would take POV measurements as fundamental, with
> projective measurements either as special cases or as idealisations which
> are never precisely realised in practice. In order to define an NCHV theory
> catering for this line of thought, Kent constructed a KS-colourable dense
> set of positive operators in a complex Hilbert space of arbitrary dimension,
> with the feature that it gives rise to a dense set of POV decompositions of
> the identity (Kent, 1999). Clifton and Kent constructed a dense set of
> positive operators in complex Hilbert space of arbitrary dimension with the
> special feature that no positive operator in the set belongs to more than
> one decomposition of the identity (Clifton & Kent, 2000). Again, the
> resulting set of POV decompositions is dense, and the special feature
> ensures that one can average over hidden states to recover quantum
> predictions.

Hence, in any finite precision experiment it would be impossible to test the
Kochen-Specker theorem, i.e., such an experimental test would require an
infinitely precise measurement and measurements in a real-world laboratory are
never infinitely precise. Although there was a lively debate along these lines
(see the references in [77]), the resolutions that were proposed all involved
modifying the notion of KS-noncontextuality by adding auxiliary assumptions
that seek to exclude the Meyer-Clifton-Kent type arguments. A recent attempt
in this direction can be found in Ref. [78] where a notion of “ontological
faithfulness” is proposed. As such, it was already recognized – for reasons
independent of Spekkens contextuality [18] – that the notion of KS-
noncontextuality needs to be revised if one is to make it experimentally
testable.404040Of course, this takes nothing away from the importance of the
Kochen-Specker theorem [19] as a no-go theorem concerning the logical
structure of quantum theory and the constraints it places on the ontological
models possible for the theory. What Spekkens brought to the fore [18],
besides generalizing the notion of contextuality to all experimental
procedures rather than measurements alone, was the idea that an experimental
test of noncontextuality should not rely on inequalities that presume outcome
determinism, just as a test of local causality does not require the assumption
of outcome determinism. Indeed, the assumption of outcome determinism for
sharp measurements in quantum theory is derived in the Spekkens framework from
the assumption of preparation noncontextuality rather than being assumed
independently.

We will now consider the more modern approach to KS-contextuality along the
lines of the frameworks in Refs.[22, 23, 25] to segue into our framework for
Spekkens contextuality which we develop in this paper.

####  A.1.2 Classifying probabilistic models: restriction of quantum models to
PVMs

Research on KS-contextuality took a different turn with the advent of the
graph-theoretic framework of Cabello, Severini and Winter in 2010 [21]
(revised slightly in 2014 [22]), the sheaf-theoretic framework of Abramsky and
Brandenburger in 2011 [25], and the hypergraph based formalism of Acìn, Fritz,
Leverrier, and Sainz in 2012 [23]. The unifying theme of these contributions
was that they took the key mathematical idea underlying KS-noncontextuality
and Bell-locality — namely, that both are instances of the classical marginal
problem [26, 32, 33] — and built frameworks that sought to distinguish between
classical theories (namely, those admitting KS-noncontextual ontological
models), quantum theory, and post-quantum general probabilistic theories by
classifying their empirical predictions relative to a Kochen-Specker
experiment into these categories. All these frameworks, motivated by the
device-independence paradigm, eschewed the erstwhile restriction of the notion
of KS-noncontextuality to quantum theory and sought to make their analysis
theory-independent, relying only on empirical predictions relative to a KS
experiment to classify theories. They separated the assumption of KS-
noncontextuality from the operational theory – namely, quantum theory – to
which it was originally meant to apply, allowing arbitrary operational
theories in their analysis. However, there was a key distinction between Bell
scenarios and KS-contextuality scenarios that was lost in this formal
unification: namely, that while the definition of a quantum probabilistic
model in a Bell scenario need not be restricted to (local) PVMs (and arbitrary
local POVMs can be allowed without changing the set of quantum models), the
same is not true of a KS-contextuality scenario. Indeed, as Henson and Sainz
note in their work [73],414141Proposing a principle bounding the KS-
contextuality possible in quantum theory, namely, “Macroscopic
Noncontextuality”. reflecting on the question of allowing arbitrary POVMs in
the definition of a quantum probabilistic model:

> …if we allow general POVMs rather than projective measurements then no
> principle that places a non-trivial restriction on correlations will be
> respected. Thus, this kind of “quantum model” is clearly pathological.

One way to motivate the present work is as a response to the pathology that
Henson and Sainz allude to: that trivial POVMs can realize any probabilistic
model, hence allowing arbitrary POVMs makes the problem of finding principles
to identify quantum correlations in KS-contextuality scenarios trivial, i.e.,
all probabilistic models are quantum and there is nothing to be learnt about
post-quantum probabilistic models. This is because any set of probabilities
satisfying the “no-disturbance” or “no-signalling” condition (of which the
$E1$ correlations of CSW [22] are a subset, in general) can be achieved by
(trivial) POVMs by simply multiplying an identity operator with every
probability in such an assignment of probabilities.424242Trivial POVMs are,
therefore, trivial resolutions of the identity, where every POVM element is
proportional to identity, i.e., $\\{a\mathbb{I}\\}_{a}$, such that $a\in[0,1]$
and $\sum_{a}a=1$. By the lights of KS-noncontextuality as one’s notion of
classicality, then, trivial POVMs saturating the general probabilistic bound
on the correlations would seem to be maximally nonclassical (i.e., maximally
KS-contextual). To avoid such “pathological” quantum models, they restrict the
definition of a quantum model to allow only projective measurements. Indeed,
with recent work on a sensible notion of “sharp” measurement in a general
probabilistic theory [30, 31], an appeal to the “fundamental sharpness” of all
measurements (see, e.g., [29]) is made to restrict attention to sharp
measurements in both quantum theory and general probabilistic theories.

On the other hand, the approach in this paper is different. In particular, we
want our approach to capture the intuition that trivial POVMs are “classical”
(and not pathological), so we must go beyond KS-noncontextuality. A simple
operational sense in which trivial POVMs are “classical” is that they reveal
nothing about the quantum state on which they are measured, being incapable of
distinguishing any pair of states whatsoever.434343Indeed, any trivial POVM
can be realized in the following operational manner: take the quantum system
prepared in some state, throw it in the garbage, and then sample from the
classical probability distribution corresponding to the trivial POVM. The
correlations (denoted by $R([s|S])$) usually examined in a KS-contextuality
experiment do not allow such experiments to witness the “triviality” of
trivial POVMs, i.e., the fact that they correspond to a fixed probability
distribution that doesn’t vary even as the choice of preparation is varied.
Moreover, since all nonprojective measurements are excluded by fiat in
traditional Kochen-Specker type approaches [22, 23] for reasons alluded to by
Henson and Sainz [73], one loses out on the potential to explore the
possibilities that nontrivial and nonprojective measurements offer with
respect to contextuality.444444All trivial POVMs are nonprojective, but not
all nonprojective POVMs are trivial. Indeed, see Refs. [39, 40, 41, 42] for
examples of generalized contextuality [18] with nonprojective measurements,
albeit assuming operational quantum theory. Our approach, therefore, is to
allow arbitrary POVMs when considering probabilistic models arising from
quantum theory (and not restricting to any notion of “sharp measurements” in
general probabilistic theories) but examine more quantities than are examined
in traditional approaches, i.e., besides the quantity $R$ typical in a KS-
contextuality scenario, we invoke the quantity ${\rm Corr}$ to account for
noise in the measurements.

If one restricts attention to operational theories that can always achieve
${\rm Corr}=1$ for any KS-contextuality scenario, then the usual
classification of probabilistic models following Refs. [22, 23] holds (Eq.
(72)). What is of interest in our framework, however, is the tradeoff between
$R$ and ${\rm Corr}$: how large can both $R$ and ${\rm Corr}$ be in an
operational theory? (See Eq. (72).)

###  A.2 Robustness of Bell nonlocality vis-à-vis POVMs

Note that whenever we refer to “Bell-KS” functionals or inequalities for
Kochen-Specker type experiments, we are not thinking of experiments that are
Bell experiments [4, 5, 7, 9, 10, 11], which have spacelike separation between
multiple parties, each performing local measurements on a shared multipartite
preparation. For the case of Bell experiments, trivial local POVMs assigned to
each party in a Bell experiment do not lead to Bell violations for a simple
reason: the trivial POVMs for each party are all compatible with each other,
thereby admitting a joint probability distribution over their outcomes for
each party; taking a product of these local joint probability distributions
(one for each party) results in a joint distribution over all measurements of
all parties, hence satisfying Bell inequalities. The fact that the POVMs are
trivial ensures that the Bell inequalities are satisfied regardless of the
choice of shared quantum state. On the other hand, forgetting the constraint
of local POVMs, there always exist global trivial POVMs that can violate Bell
inequalities: e.g., just take the Popescu-Rohrlich (PR) box distribution [43],
and multiply an identity operator (on the joint Hilbert space of Alice and
Bob) with each probability in the PR-box; this results in four trivial POVMs,
defined over the joint Hilbert space, that together violate the CHSH
inequality maximally. But, of course, this violation is uninteresting because
it doesn’t obey the locality constraint on the measurements in a Bell
experiment. This is mathematically reflected in the fact that the PR-box
distribution cannot be written as a convex mixture of product distributions,
one for each party, hence the corresponding trivial POVM cannot be understood
in terms of trivial local POVMs. Hence, it is the locality of the trivial
POVMs in a Bell experiment that prevents them from violating a Bell inequality
and renders them non-pathological, unlike in the case of KS-contextuality. The
fact that they are “trivial” in the sense of being unable to distinguish two
quantum states plays a role in the sense that, regardless of the shared
quantum state, these POVMs yield fixed distributions over the measurement
outcomes, thus always allowing the construction of a fixed (that is,
independent of the quantum state) global joint probability distribution over
all measurements in a Bell scenario. Since there are no such locality
constraints on the form of the POVM elements in a Kochen-Specker experiment,
they can easily violate any KS-noncontextuality inequality, e.g., the two-
party CHSH experiment considered as a Kochen-Specker experiment with four
observables in a 4-cycle where adjacent pairs are jointly measurable allows
for trivial POVMs (like the PR-box trivial POVM above) violating the CHSH-type
Bell-KS inequality in this scenario maximally. By the lights of KS-
noncontextuality, this violation would indicate the maximum possible KS-
contextuality with respect to this CHSH-type inequality.454545See Appendix C
for more discussion. For all these reasons, our discussion of KS-
noncontextuality as a notion of classicality — in an experiment with no
locality constraints on the measurements — does not extend to the case of
Bell-locality (or local causality) as a notion of classicality in a Bell
experiment, where the experiment must respect locality constraints on the
measurements for a Bell inequality violation to be meaningful.

The unification of Bell nonlocality and KS-contextuality à la Refs. [22, 23,
25] forces a certain dichotomy in these approaches: while in Bell scenarios,
one need not restrict to any notion of a “sharp” measurement in the definition
of probabilistic models (and thus claim “theory independence”), in Kochen-
Specker scenarios, one must make some statement about the nature of the
measurements (concerning their presumed sharpness [29], or that their joint
measurability [44, 42] is restricted to commutativity [25]), rendering any
putative “theory independence” claim (on a level at par with Bell nonlocality)
unfounded.464646See Ref. [27] for how this lack of locality of measurements in
a Kochen-Specker type experiment translates, at the ontological level, to the
unreasonableness of assuming factorizability in the ontological model; this
factorizability (or the stronger condition of outcome determinism) is invoked
to justify the resulting derivation of Bell-KS inequalities as constraints
from a classical marginal problem.

##  Appendix B Ontological models without respecting coarse-graining relations

Here we will construct explicit examples where the coarse-graining relations
are not respected in an ontological model, in contrast to the requirement on
the representation of coarse-grainings that we invoked in Section II.C of the
main text. The goal is to emphasize that the requirement of Section II.C is
necessary not only for the treatment of Spekkens contextuality but also for
Kochen-Specker contextuality. Below, we first demonstrate how a “KS-
noncontextual” model can be constructed for any scenario that proves the KS
theorem by using the example of the KCBS setup [47]. We then proceed to
demonstrate how a “preparation and measurement noncontextual” model can be
constructed in a similar way whenc considering generalized noncontextuality
[18].

###  B.1 How to construct a “KS-noncontextual” ontological model of the KCBS
experiment [47] without coarse-graining relations

Here we have that $\mathbb{M}$ contains at least the following measurement
settings: $\\{M_{i}\\}_{i=1}^{5}$, each with three possible outcomes,
$m_{i}\in\\{0,1,2\\}$. The measurement events for each measurement setting
$M_{i}$ can be coarse-grained into two different ways, defining new
measurement settings $M^{\prime}_{i}$ (with outcomes
$m^{\prime}_{i}\in\\{0,\bar{0}\\}$) and $M^{\prime\prime}_{i}$ (with outcomes
$m^{\prime\prime}_{i}\in\\{2,\bar{2}\\}$), where the coarse-graining relations
are given by

$\displaystyle[0|M^{\prime}_{i}]\equiv[0|M_{i}],$ (123)
$\displaystyle[\bar{0}|M^{\prime}_{i}]\equiv[1|M_{i}]+[2|M_{i}],$ (124)
$\displaystyle[2|M^{\prime\prime}_{i}]\equiv[2|M_{i}],$ (125)
$\displaystyle[\bar{2}|M^{\prime}_{i}]\equiv[0|M_{i}]+[1|M_{i}].$ (126)

In the operational theory, these coarse-graining relations are respected,
i.e., for all $[s|S]$, $s\in V_{S},S\in\mathbb{S}$,

$\displaystyle p(0,s|M^{\prime}_{i},S)\equiv p(0,s|M_{i},S),$ (127)
$\displaystyle p(\bar{0},s|M^{\prime}_{i},S)\equiv
p(1,s|M_{i},S)+p(2,s|M_{i},S),$ (128) $\displaystyle
p(2,s|M^{\prime\prime}_{i},S)\equiv p(2,s|M_{i},S),$ (129) $\displaystyle
p(\bar{2},s|M^{\prime\prime}_{i},S)\equiv p(0,s|M_{i},S)+p(1,s|M_{i},S).$
(130)

However, we do not require that these relations be respected in an ontological
model. Now, the KCBS argument requires the following operational equivalences,

$[2|M^{\prime\prime}_{i}]\simeq[0|M^{\prime}_{i+1}],$ (131)

for all $i\in\\{1,2,3,4,5\\}$, where addition is modulo 5, so that $i+1=1$ for
$i=5$. A KS-noncontextual ontological model for this experiment requires that

$\xi(2|M^{\prime\prime}_{i},\lambda)=\xi(0|M^{\prime}_{i+1},\lambda)\in\\{0,1\\},\quad\forall\lambda\in\Lambda.$
(132)

Constructing such a model requires one to specify response functions for the
measurements $\\{M_{i},M^{\prime}_{i},M^{\prime\prime}_{i}\\}_{i=1}^{5}$.
However, since there are no constraints from coarse-graining relations on
these response functions, there is no obstruction to the construction of a
“KS-noncontextual model” of this type for any set of operational statistics.
In particular, since we do not require that
$\forall\lambda\in\Lambda:\xi(0|M^{\prime}_{i+1},\lambda)\equiv\xi(0|M_{i+1},\lambda)$,
nor that
$\forall\lambda\in\Lambda:\xi(2|M^{\prime\prime}_{i},\lambda)\equiv\xi(2|M_{i},\lambda)$,
we can assign arbitrary response functions to
$\\{M^{\prime}_{i},M^{\prime\prime}_{i}\\}_{i=1}^{5}$, subject only to the
condition from KS-noncontextuality that
$\forall\lambda\in\Lambda:\xi(2|M^{\prime\prime}_{i},\lambda)=\xi(0|M^{\prime}_{i+1},\lambda)\in\\{0,1\\}$.474747This
“KS-noncontextual” ontological model will thus reproduce operational
equivalences of the type $[2|M^{\prime\prime}_{i}]\simeq[0|M^{\prime}_{i+1}]$
(cf. Eq. (131)). Note that, because coarse-graining relations are not
respected, this does not imply that
$\forall\lambda\in\Lambda:\xi(2|M_{i},\lambda)=\xi(0|M_{i+1},\lambda)\in\\{0,1\\}$,
which is the usual constraint we would have presumed from KS-noncontextuality
when coarse-graining relations are respected in the ontological model. In the
absence of any such constraints on the response functions for
$\\{M_{i}\\}_{i=1}^{5}$, one can always reproduce their operational
statistics, in particular the operational equivalences of the type
$[2|M_{i}]\simeq[0|M_{i+1}]$, which follow from Eqs. (123),(125), and (131).

###  B.2 How to construct a “preparation and measurement noncontextual”
ontological model without coarse-graining relations

Just as for measurements in the case of KS-noncontextuality, abandoning the
coarse-graining relations for preparations in the case of generalized
noncontextuality [18] makes possible the existence of a “preparation and
measurement noncontextual” ontological model for any set of operational
statistics. For the kinds of proofs of contextuality relevant to this article,
the relevant notion of coarse-graining is that of complete coarse-graining:
that is, consider two source settings $S$ and $S^{\prime}$ with (respective)
source events $\\{[s|S]\\}_{s\in V_{S}}$ and
$\\{[s^{\prime}|S^{\prime}]\\}_{s^{\prime}\in S^{\prime}}$, that can be
completely coarse-grained to yield the operational equivalence
$[\top|S_{\top}]\simeq[\top|S^{\prime}_{\top}]$, cf. Eq. (18). In the
operational description, where we assume the coarse-graining relation is
respected, this is represented by

$\displaystyle\forall[m|M],m\in V_{M},M\in\mathbb{M}:$
$\displaystyle\sum_{s}p(m,s|M,S)=\sum_{s^{\prime}}p(m,s^{\prime}|M,S^{\prime}).$
(133)

In the ontological description, however, we do not impose the coarse-graining
relations $\mu(\lambda,\top|S_{\top})\equiv\sum_{s}\mu(\lambda,s|S)$ and
$\mu(\lambda,\top|S^{\prime}_{\top})\equiv\sum_{s^{\prime}}\mu(\lambda,s^{\prime}|S^{\prime})$,
which makes it trivial to write down probability distributions
$\mu(\lambda,\top|S_{\top})$ and $\mu(\lambda,\top|S^{\prime}_{\top})$ such
that $\mu(\lambda,\top|S_{\top})=\mu(\lambda,\top|S^{\prime}_{\top})$ (as
required by preparation noncontextuality applied to
$[\top|S_{\top}]\simeq[\top|S^{\prime}_{\top}]$) but where we do not require
that $\sum_{s}\mu(\lambda,s|S)=\sum_{s}\mu(\lambda,s|S)$ (which is not
required by preparation noncontextuality). Note how the refusal to respect the
coarse-graining relations, i.e., identifying $\mu(\lambda,\top|S_{\top})$ with
$\sum_{s}\mu(\lambda,s|S)$ and $\mu(\lambda,\top|S^{\prime}_{\top})$ with
$\sum_{s^{\prime}}\mu(\lambda,s^{\prime}|S^{\prime})$, lifts the constraint
from preparation noncontextuality that would have been in place if the coarse-
graining relations were respected. The same refusal for the case of
measurements lifts any constraints (just as in the case of KS-noncontextuality
above) from measurement noncontextuality on the ontological model. It thus
becomes trivial to construct a “preparation and measurement noncontextual”
ontological model without coarse-graining relations.

##  Appendix C Trivial POVMs

###  C.1 Bell-CHSH scenario

We have the Hilbert space $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ for Alice
($\mathcal{H}_{A}$) and Bob ($\mathcal{H}_{B}$). Consider four binary-outcome
POVMs, $\\{A^{(0)},A^{(1)},B^{(0)},B^{(1)}\\}$, where

$\displaystyle A^{(0)}\equiv\\{A^{(0)}_{0},A^{(0)}_{1}\\},$ $\displaystyle
A^{(1)}\equiv\\{A^{(1)}_{0},A^{(1)}_{1}\\},$ $\displaystyle
B^{(0)}\equiv\\{B^{(0)}_{0},B^{(0)}_{1}\\},$ $\displaystyle
B^{(0)}\equiv\\{B^{(1)}_{0},B^{(1)}_{1}\\},$ (134)

$0\leq A^{(0)}_{0},A^{(1)}_{0}\leq\mathbb{I}_{\mathcal{H}_{A}}$, $0\leq
B^{(0)}_{0},B^{(1)}_{0}\leq\mathbb{I}_{\mathcal{H}_{B}}$,
$A^{(0)}_{0}+A^{(0)}_{1}=A^{(1)}_{0}+A^{(1)}_{1}=\mathbb{I}_{\mathcal{H}_{A}}$,
and
$B^{(0)}_{0}+B^{(0)}_{1}=B^{(1)}_{0}+B^{(1)}_{1}=\mathbb{I}_{\mathcal{H}_{B}}$.
The quantum probability, given a shared quantum state $\rho_{AB}$ defined on
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, is given by

$p(a,b|x,y)=\text{Tr}(\rho_{AB}A^{(x)}_{a}\otimes B^{(y)}_{b}),$ (135)

for $a,b,x,y\in\\{0,1\\}$. Here $A^{(x)}\otimes\mathbb{I}_{\mathcal{H}_{B}}$
is jointly measurable with $\mathbb{I}_{\mathcal{H}_{A}}\otimes B^{(y)}$, just
because of the commutativity of their respective POVM elements. The joint
observable being measured is $A^{(x)}\otimes B^{(y)}$. Now, consider the case
when all the POVM elements are trivial, i.e.,
$A^{(x)}_{a}=q^{(x)}_{a}\mathbb{I}_{\mathcal{H}_{A}}$ and
$B^{(y)}_{b}=r^{(y)}_{b}\mathbb{I}_{\mathcal{H}_{B}}$, for some
$q^{(x)}_{a},r^{(y)}_{b}\in[0,1]$ for all $a,b,x,y\in\\{0,1\\}$. We then have

$p(a,b|x,y)=q^{(x)}_{a}r^{(y)}_{b},\forall a,b,x,y\in\\{0,1\\}.$ (136)

A global joint probability distribution which reproduces the above as
marginals is simply given by their product:

$p(a^{(0)},a^{(1)},b^{(0)},b^{(1)})\equiv
q^{(0)}_{a^{(0)}}q^{(1)}_{a^{(1)}}r^{(0)}_{b^{(0)}}r^{(1)}_{b^{(1)}}.$ (137)

Hence, trivial POVMs never violate any Bell-CHSH inequality for this scenario.

###  C.2 CHSH-type contextuality scenario: 4-cycle

We now consider the Bell-CHSH scenario without the constraint of spacelike
separation. What the lack of spacelike separation means from the quantum
perspective is that one no longer needs to model this spacelike separation by
requiring a tensor product structure, or (more generally) by requiring the
commutativity of the observables that are jointly measured [79, 80, 25]. That
is, there is no physical justification for imposing the tensor product
structure or the commutativity of jointly measured observables.484848On the
other hand, what this lack of spacelike separation means from the perspective
of an ontological model is that one no longer has a justification for assuming
factorizability [25] and, consequently, the generalization of Fine’s theorem
[26] fails to prove that there is no loss of generality in assuming outcome
determinism in discussions of KS-contextuality (unlike the case of Bell
scenarios, where factorizability is justified by spacelike separation); there
is a definite loss of generality, in that measurement noncontextual and
outcome-indeterministic ontological models that are non-factorizable are not
empirically equivalent to measurement noncontextual and outcome-deterministic
(or KS-noncontextual) ontological models. See Ref. [27] for a discussion of
this aspect.

Thus, we have the Hilbert space $\mathcal{H}$ and we consider four binary-
outcome POVMs, $\\{A^{(0)},A^{(1)},B^{(0)},B^{(1)}\\}$, on $\mathcal{H}$,
where

$\displaystyle A^{(0)}\equiv\\{A^{(0)}_{0},A^{(0)}_{1}\\},$ $\displaystyle
A^{(1)}\equiv\\{A^{(1)}_{0},A^{(1)}_{1}\\},$ $\displaystyle
B^{(0)}\equiv\\{B^{(0)}_{0},B^{(0)}_{1}\\},$ $\displaystyle
B^{(0)}\equiv\\{B^{(1)}_{0},B^{(1)}_{1}\\},$ (138)

$0\leq
A^{(0)}_{0},A^{(1)}_{0},B^{(0)}_{0},B^{(1)}_{0}\leq\mathbb{I}_{\mathcal{H}}$,
$A^{(0)}_{0}+A^{(0)}_{1}=A^{(1)}_{0}+A^{(1)}_{1}=B^{(0)}_{0}+B^{(0)}_{1}=B^{(1)}_{0}+B^{(1)}_{1}=\mathbb{I}_{\mathcal{H}}$.
Further, the following sets of POVMs are jointly measurable:
$\\{A^{(0)},B^{(0)}\\}$, $\\{A^{(0)},B^{(1)}\\}$, $\\{A^{(1)},B^{(0)}\\}$,
$\\{A^{(1)},B^{(1)}\\}$. The most general joint observable for a pair of
compatible POVMs $\\{A^{(x)},B^{(y)}\\}$ is given by a POVM
$G^{(xy)}\equiv\\{G^{(xy)}_{00},G^{(xy)}_{01},G^{(xy)}_{10},G^{(xy)}_{11}\\}$
(that isn’t necessarily unique [42]) such that:
$G^{(xy)}_{00}+G^{(xy)}_{01}=A^{(x)}_{0},G^{(xy)}_{10}+G^{(xy)}_{11}=A^{(x)}_{1},G^{(xy)}_{00}+G^{(xy)}_{10}=B^{(y)}_{0},G^{(xy)}_{01}+G^{(xy)}_{11}=B^{(y)}_{1}$.
In particular, if (and only if) the POVMs $A^{(x)}$ and $B^{(y)}$ commute, we
can construct the joint POVM as a product:
$G^{(xy)}_{ab}=A^{(x)}_{a}B^{(y)}_{b}$ for all $a,b,x,y\in\\{0,1\\}$. In the
absence of such commutativity, the joint POVM cannot be written as a product.

The quantum probability, given a quantum state $\rho$ on $\mathcal{H}$, is
given by

$p(a,b|x,y)=\text{Tr}(\rho G^{(xy)}_{ab}),$ (139)

for $a,b,x,y\in\\{0,1\\}$. Note that this probability depends on the joint
measurement $G^{(xy)}$ implementing $A^{(x)}$ and $B^{(y)}$ together, and
that, in general, there may be multiple choices of $G^{(xy)}$ possible. This
is easy to see since there is one undetermined positive operator in the joint
measurement that is not fixed by $A^{(x)}$ or $B^{(y)}$, i.e., we can write
the POVM elements of $G^{(xy)}$ as: $G^{(xy)}_{01}=A^{(x)}_{0}-G^{(xy)}_{00}$,
$G^{(xy)}_{10}=B^{(y)}_{0}-G^{(xy)}_{00}$,
$G^{(xy)}_{11}=\mathbb{I}-A^{(x)}_{0}-B^{(y)}_{0}+G^{(xy)}_{00}$, where
$G^{(xy)}_{00}$ is a positive semidefinite operator satisfying
$A^{(x)}_{0}+B^{(y)}_{0}-\mathbb{I}_{\mathcal{H}}\leq G^{(xy)}_{00}\leq
A^{(x)}_{0},B^{(y)}_{0}$. Here $G^{(xy)}_{00}$ represents the freedom in the
choice of how the joint measurement might be implemented within quantum
theory. This freedom reflects the fact that since the jointly measured
observables are no longer spacelike separated, it is possible to introduce
correlations between them that are stronger than what is allowed in the
corresponding Bell scenario in quantum theory. The strength of these
correlations is only limited by the constraints on $G^{(xy)}_{00}$ imposed by
the marginal observables $A^{(x)}$ and $B^{(y)}$. This is in contrast to the
case where $A^{(x)}$ and $B^{(y)}$ are spacelike separated observables and the
only choice of joint POVM consistent with spacelike separation is fixed by
$G^{(xy)}_{00}=A^{(x)}_{0}B^{(y)}_{0}$, i.e., the strength of correlations
between $A^{(x)}$ and $B^{(y)}$ is fixed entirely by them and there is no
freedom in choosing $G^{(xy)}$.

Thus, we have that $A^{(x)}$ is jointly measurable with $B^{(y)}$ and
$G^{(xy)}$ denotes a joint POVM of $A^{(x)}$ and $B^{(y)}$. Now, consider the
case when all the POVM elements are trivial, i.e.,
$A^{(x)}_{a}=q^{(x)}_{a}\mathbb{I}_{\mathcal{H}}$ and
$B^{(y)}_{b}=r^{(y)}_{b}\mathbb{I}_{\mathcal{H}}$, for some
$q^{(x)}_{a},r^{(y)}_{b}\in[0,1]$ for all $a,b,x,y\in\\{0,1\\}$.

In particular, consider the case where $q^{(x)}_{a}=r^{(y)}_{b}=\frac{1}{2}$
for all $a,b,x,y\in\\{0,1\\}$. A possible joint POVM for these trivial POVMs
is then the product POVM:

$G^{(xy)}_{ab}=A^{(x)}_{a}B^{(y)}_{b}=\frac{1}{4}\mathbb{I}_{\mathcal{H}}.$
(140)

If one restricted joint measurability of $A^{(x)}$ and $B^{(y)}$ to just
commutativity — a sufficient but not necessary condition for joint
measurability494949Particularly in the absence of spacelike separation. It is
the need to model spacelike separation in a quantum Bell experiment that makes
commutativity a necessary (and sufficient) condition for joint measurability
of spacelike separated observables in a Bell scenario [44] — we would take the
above choice of the product POVM as a “natural” one. Being a product of
trivial POVMs, this choice will never lead to a violation of the CHSH-type
inequality for this scenario. Indeed, the structure of a Bell scenario —
requiring the decomposition of the Hilbert space as
$\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ (tensor product paradigm),
or more generally, imposing the commutativity requirement
$[A^{(x)}_{a},B^{y}_{b}]=0$ (commutativity paradigm) — is such that the only
possible choice of joint measurement that can be implemented by spacelike
separated parties is the one that corresponds to the product POVM, given by
operators $G^{(xy)}_{ab}=A^{(x)}_{a}B^{(y)}_{b}$.

However, this is not the only allowed joint measurement for these trivial
POVMs, particularly when there is no locality constraint on the measurements
from spacelike separation.505050To incorporate such a constraint, spacelike
separation needs to be modelled via either the tensor product paradigm or the
commutativity paradigm. Both these ways of modelling spacelike separation lead
to the same set of quantum correlations for any finite-dimensional Hilbert
space $\mathcal{H}$ [79]. The question of whether the two paradigms lead to
the same set of correlations in the case of infinite dimensional Hilbert
spaces is the subject of Tsirelson’s problem [79, 80]. Most studies of Bell-
nonlocality are primarily concerned with finite dimensional Hilbert spaces;
should one encounter infinite dimensional Hilbert spaces, the commutativity
paradigm is the proper way to model spacelike separation. An extreme choice of
joint POVM is the following:

$G^{PR(xy)}_{ab}=\frac{\mathbb{I}_{\mathcal{H}}}{2}\delta_{a\oplus b,xy},$
(141)

which leads to the probability distribution
$p(a,b|x,y)=\frac{1}{2}\delta_{a\oplus b,xy}$ for any choice of quantum state.
Hence, this joint POVM $G^{PR(xy)}$ always yields statistics corresponding to
the PR-box, maximally violating the CHSH-type inequality for this scenario,
namely,

$\sum_{a,b,x,y\big{|}a\oplus b=xy}\frac{1}{4}p(a,b|x,y)\leq\frac{3}{4}.$ (142)

Physically, it’s possible to implement this (without requiring any quantum
resources) by providing a box that always produces these correlations between
measurement settings denoted by $(xy)\in\\{0,1\\}^{2}$, regardless of the
input state. Such a black-box would maximally violate the CHSH-type inequality
(viewed as a Bell-KS inequality witnessing KS-contextuality), but that
shouldn’t be surprising in the absence of spacelike separation. Also, the
trivial PR-box joint POVM $G^{PR(xy)}_{ab}$ is a perfectly valid way to
implement the joint measurement of trivial POVMs $A^{(x)}$ and $B^{(y)}$
within the standard paradigm of operational quantum theory.515151Note that the
point of this demonstration is to show how, in the absence of spacelike
separation justifying commutativity or a promise that the measurements are
sharp, arbitrary correlations are achievable in quantum theory if unsharp
measurements are allowed. All trivial POVMs are unsharp, but the converse is
not true. That is, one can consider nontrivial POVMs that don’t violate the
CHSH-type inequality maximally, but which violate it (arbitrarily) more than
is allowed by sharp measurements in quantum theory. One could construct them,
for example, by just taking a convex combination of the PR-box trivial POVM
with some sharp (and thus product) joint POVM.

To summarize, we note the following:

  * •

Within the traditional framework of KS-noncontextuality, if one wants to go
beyond projective measurements to arbitrary POVMs in a contextuality scenario,
then one must – in order to avoid the pathology of trivial POVMs violating the
Bell-KS inequalities maximally – restrict by fiat the notion of joint
measurability to merely commutativity. This is, for example, the attitude
adopted in Ref. [25].

  * •

However, if one is going beyond projective measurements, we know that
commutativity is only a sufficient condition for joint measurability, not a
necessary one [44].

  * •

This brings us to our observation that the traditional notion of KS-
noncontextuality is pathological once the most general situation in quantum
theory is considered: arbitrary POVMs with the general notion of joint
measurability (see, e.g., Ref. [44] for this notion and its relation to
commutativity). In particular, in the absence of spacelike separation, there
is no physical justification to restrict the notion of joint measurability to
merely commutativity.

  * •

A similar consideration applies at the level of a KS-noncontextual ontological
model: there, factorizability is not justified in the absence of spacelike
separation. So, on those grounds alone, one should go beyond KS-
noncontextuality as one’s notion of classicality; particularly, if one wants a
notion of classicality that does not presume outcome determinism, just as
local causality doesn’t presume it. This was argued in Ref. [27]: imagine an
adversarial setting where because of the absence of spacelike separation in a
KS-contextuality experiment, two measurement settings on the same system can
exhibit correlations that are independent of those induced by the system on
which the measurements are being implemented, thus allowing them to exhibit
stronger correlations than are possible in a KS-noncontextual model. We use
trivial POVMs only to drive home that this can be done arbitrarily well
(achieving PR-box type correlations, in fact) if there is no constraint on the
strength of correlations the measurement settings can exhibit. The way such
constraints on the correlations between the measurement settings show up in
our analysis within the Spekkens framework is in terms of the quantity ${\rm
Corr}$: if ${\rm Corr}$ is really high, the measurements in a noncontextual
ontological model cannot be arbitrarily strongly correlated, i.e., $R$ cannot
be arbitrarily high (cf. Eq. (72)).

##  Appendix D The KS-uncolourable hypergraph $\Gamma_{18}$

It is instructive to consider the KS-uncolourable hypergraph $\Gamma_{18}$,
originally appearing in Ref. [51], and studied in the light of Spekkens
contextuality in Ref. [12]. This hypergraph fails both criteria for the
hypergraphs $\Gamma$ considered in this paper, namely,
$\mathcal{C}(\Gamma)\neq\varnothing$ (KS-colourability) and
$\mathcal{CE}^{1}(\Gamma)=\mathcal{G}(\Gamma)$.

For probabilistic models on $\Gamma_{18}$, the following hold:
$\mathcal{C}(\Gamma_{18})=\varnothing\subsetneq\mathcal{CE}^{1}(\Gamma_{18})\subsetneq\mathcal{G}(\Gamma_{18})$.
This was considered in Ref. [12], where $\mathcal{CE}^{1}(\Gamma_{18})$
excludes the extremal probabilistic model in $\mathcal{G}(\Gamma_{18})$ that
corresponds to the upper bound on the noise-robust noncontextuality inequality
of Ref. [12]. As argued in Ref. [12], this noise-robust noncontextuality
inequality is the appropriate operational generalization (to possibly noisy
measurements) of the Kochen-Specker contradiction first demonstrated in Ref.
[51]; this generalization cannot be accommodated in our generalization of the
CSW framework [22].

If one extends the KS-uncolourable $\Gamma_{18}$ to a KS-colourable hypergraph
$\Gamma_{27}$ with 9 “no-detection” events, one for each hyperedge, then we
have $\mathcal{C}(\Gamma_{27})\neq\varnothing$, but it’s still the case that
$\mathcal{C}(\Gamma_{27})\subsetneq\mathcal{CE}^{1}(\Gamma_{27})\subsetneq\mathcal{G}(\Gamma_{27})$
for this hypergraph.525252This follows from noting that extremal probabilistic
models on $\Gamma_{18}$ are still extremal probabilistic models on
$\Gamma_{27}$: ones where the no-detection events are assigned zero
probabilities. See Theorem 2.5.3 of Ref. [23]. Hence, $\Gamma_{27}$ cannot be
understood in our generalization of the CSW framework either.535353Note that
adding these no-detection events is equivalent to allowing subnormalized
probabilities (i.e., sum of probabilities assigned to measurement events in a
hyperedge can be less than 1) on $\Gamma_{18}$. Hence, even allowing for
subnormalization on $\Gamma_{18}$, which means that one is looking at
probabilistic models on the hypergraph $\Gamma_{27}$, does not eliminate the
gap between CE1 probabilistic models and general probabilistic models, so that
any upper bound on a Bell-KS expression given by probabilistic models in
$\mathcal{CE}^{1}(\Gamma_{27})$ is not always the same as the general
probabilistic upper bound from probabilistic models in
$\mathcal{G}(\Gamma_{27})$. The CSW framework only considers the upper bound
given by $\mathcal{CE}^{1}(\Gamma_{27})$ probabilistic models.

Indeed, if one “blindly” writes down a CSW classical bound for some Bell-KS
expression defined on $O(\Gamma_{18})$, then such a bound is equivalently a
bound for the same Bell-KS expression defined on $\Gamma_{27}$ (where
normalization is restored). Further, the E1 bound on $\Gamma_{18}$ is a ${\rm
CE}^{1}$ bound on $\Gamma_{27}$. The GPT bound happens to agree with the CE1
bound for a particular Bell-KS expression (sum of all probabilities) but
differs for some other Bell-KS expressions defined on this hypergraph.
Consider, for example, the following three expressions (see Fig. 8):

Figure 8: The hypergraph $\Gamma_{27}$ and its subhypergraphs, i.e.,
$\Gamma_{18}$ and $\Gamma_{3}$, appearing in the three Bell-KS expressions of
Eq. (D). The probabilistic model $p$ considered in Eq. (D) is a probabilistic
model on $\Gamma_{27}$, and not on the subhypergraphs. We have illustrated the
subhypergraphs separately only for clarity regarding the subsets of vertices
to which the Bell-KS expressions refer: the probabilities assigned to these
vertices are obtained from probabilistic models on $\Gamma_{27}$. Figure 9:
Going from the orthogonality graph, $G$, of $\Gamma_{18}$ to the hypergraph
$\Gamma_{G}$ (on the right) to which our noise-robust noncontextuality
inequality pertains.

$\displaystyle{\rm Expr}_{1}$ $\displaystyle\equiv$ $\displaystyle\sum_{v\in
V(\Gamma_{18})}p(v),$ $\displaystyle{\rm Expr}_{2}$ $\displaystyle\equiv$
$\displaystyle\sum_{v\in V(\Gamma_{3})}p(v),$ $\displaystyle{\rm Expr}_{3}$
$\displaystyle\equiv$ $\displaystyle\sum_{v\in V(\Gamma_{18})}p(v)+\sum_{v\in
V(\Gamma_{3})}p(v).$ (143)

We have:

$\displaystyle{\rm
Expr}_{1}\overset{\rm\mathcal{C}(\Gamma_{27})}{\leq}8\overset{\mathcal{CE}^{1}(\Gamma_{27})}{<}9\overset{\mathcal{G}(\Gamma_{27})}{=}9,$
$\displaystyle{\rm
Expr}_{2}\overset{\rm\mathcal{C}(\Gamma_{27})}{\leq}1\overset{\mathcal{CE}^{1}(\Gamma_{27})}{=}1\overset{\mathcal{G}(\Gamma_{27})}{<}\frac{3}{2},$
$\displaystyle{\rm
Expr}_{3}\overset{\rm\mathcal{C}(\Gamma_{27})}{\leq}9\overset{\mathcal{CE}^{1}(\Gamma_{27})}{<}10\overset{\mathcal{G}(\Gamma_{27})}{<}10.5.$
(144)

Thus, ${\rm Expr}_{3}$ is a Bell-KS expression that discriminates between
probabilistic models at all three levels of the hierarchy. Indeed, the upper
bound on ${\rm Expr}_{3}$ for $\mathcal{CE}^{1}(\Gamma_{27})$ models can be
saturated by projective quantum realizations of the hypergraph, in particular
the standard realization with 18 rays, with the zero operator for the no-
detection events [51]. The fact that there exists such a Bell-KS expression as
${\rm Expr}_{3}$ means that the ${\rm CE}^{1}$ upper bounds from the CSW
approach can be violated by a general probabilistic model, i.e., the upper
bounds for ${\rm CE}^{1}$ models and general probabilistic models don’t agree,
and we cannot take the graph-theoretic upper bounds of CSW for granted in our
noise-robust noncontextuality inequalities. Indeed, the general probabilistic
upper bound for any Bell-KS expression defined on a contextuality scenario is
a hypergraph invariant — in the sense that it is a property that is shared by
all hypergraphs isomorphic to each other — that may or may not be expressible
as a graph invariant à la CSW.

What, then, do the bounds given by graph invariants of CSW for
$O(\Gamma_{18})$ mean in our generalization of the CSW framework? Following
our approach, outlined in Sec. III.B, we can go from $G=O(\Gamma_{18})$ to the
hypergraph $\Gamma_{G}=\Gamma_{O(\Gamma_{18})}$ (see Fig. 9) for which we have
(by construction) $\mathcal{C}(\Gamma_{O(\Gamma_{18})})\neq\varnothing$ (so
that the underlying hypergraph is no longer KS-uncolourable) and
$\mathcal{CE}^{1}(\Gamma_{O(\Gamma_{18})})=\mathcal{G}(\Gamma_{O(\Gamma_{18})})$
(so that, for any Bell-KS expression, the upper bound given by the fractional
packing number $\alpha^{*}(G,w)$ in the CSW framework agrees with the general
probabilistic upper bound). Since this construction proceeds by converting all
maximal cliques in $\Gamma_{18}$ to hyperedges in $\Gamma_{O(\Gamma_{18})}$
and adding a new vertex to each such hyperedge, it achieves both purposes:
firstly, adding a (no-detection) vertex to every maximal clique that is a
hyperedge in $\Gamma_{18}$ ensures the KS-colourability of
$\Gamma_{O(\Gamma_{18})}$, i.e.,
$\mathcal{C}(\Gamma_{O(\Gamma_{18})})\neq\varnothing$, and secondly, adding a
vertex to every maximal clique that is not a hyperedge in $\Gamma_{18}$
ensures that
$\mathcal{CE}^{1}(\Gamma_{O(\Gamma_{18})})=\mathcal{G}(\Gamma_{O(\Gamma_{18})})$.
Once these two properties are satisfied, the graph invariants of CSW [22]
become applicable to any Bell-KS expression defined for any set of vertices in
the subhypergraph $\Gamma_{18}$ of $\Gamma_{O(\Gamma_{18})}$.

Our noise-robust noncontextuality inequality then applies to the KS-colourable
hypergraph $\Gamma_{O(\Gamma_{18})}$, where the graph invariants of CSW make
sense, rather than the KS-uncolourable hypergraph $\Gamma_{18}$. On the other
hand, an appropriate noise-robust noncontextuality inequality for the KS-
uncolourable hypergraph $\Gamma_{18}$ is, then, the one reported in Ref.
[12].545454The approach for KS-uncolourable hypergraphs will be further
developed in hypergraph-theoretic terms in forthcoming work [34].

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