# A mathematical framework for operational fine tunings

Lorenzo Catani and Matthew Leifer  Schmid College of Science and Technology,
Chapman University, One University Drive, Orange, CA, 92866, USA
<EMAIL_ADDRESS>
###### Abstract

In the framework of ontological models, the features of quantum mechanics that
emerge as inherently nonclassical always involve properties that are fine
tuned, _i.e._ properties that hold at the operational level but break at the
ontological level (they only hold for fine tuned values of the ontic
parameters). Famous examples of such features are contextuality and
nonlocality. We here develop a precise theory-independent mathematical
framework for characterizing _operational_ fine tunings. These are distinct
from _causal_ fine tunings – already introduced by Wood and Spekkens in
[NJP,$\mathbb{17}$ 033002(2015)] – as they do not involve any assumption on
the underlying causal structure. We show how all the already known examples of
operational fine tunings fit into our framework, we discuss possibly new fine
tunings and we use the framework to shed new light on the relation between
nonlocality and generalized contextuality, where the former can involve a
causal fine tuning too, unlike the latter. The framework is also formulated in
the language of category theory and functors.

##  1 Introduction

Although quantum mechanics is about a century old, there is still not a
universal consensus on what is the nature of the reality it describes.
Multiple interpretations are still on the table, each giving a radically
different account of the world [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. In order
to obtain an uncontroversial understanding of how nature behaves according to
quantum theory, it is useful to first understand which of its features are
inherently nonclassical. A rigorous way to achieve this is to formalize
precise notions of classicality and formulate theorems that prove their
inconsistency with the theory. This approach is successfully implemented in
the framework of ontological models [12], where assuming natural notions of
classicality (that in fact hold in classical Hamiltonian mechanics [13]), like
local causality [14] and noncontextuality [15, 16], leads to contradictions
with the statistics of quantum theory. As a consequence, nonlocality and
contextuality emerge as truly nonclassical features. A related one, which has
recently been discovered and that we here consider, is the breaking of time
symmetry [17]. No-go theorems of these kind do not exist for other phenomena
usually listed as characteristic and surprising novelties of quantum theory,
like superposition, entanglement, interference and no cloning. It is indeed
possible to show that they can be reproduced, still allowing for a natural
interpretation, in classical-like theories in the phase space such as
Spekkens’ toy model [18, 19].

In this work we focus on the aspect that is common among all such inherently
nonclassical features of quantum theory and we develop a mathematical
framework for it. More precisely, the mystery characterizing these features
can be distilled in the fact that they involve fine tuned properties – or, for
simplicity, _fine tunings_. These are defined as properties that hold at the
operational level, but cannot hold at the ontological level (they only hold by
fine tuning of the ontic parameters). Let us consider a paradigmatic example.
Bell’s theorem can be explained by appealing to the failure of parameter
independence, one of the components in the assumption of Bell’s local
causality [14]. Accepting this explanation would mean that no signaling holds
at the level of the operational statistics, but it does not hold at the
ontological level, due to the presence of superluminal influence. In this case
no signaling would emerge as a fine tuned property, because it would only
arise from a fine tuning of the ontic parameters. The intuition that the
conflict between the fundamental nonlocality of quantum mechanics and the
operational validity of Einstein’s relativity is resolved at the price of
introducing what we here call a fine tuning was originally highlighted by
Valentini [20]. Let us consider a second example. The generalized notion of
noncontextuality [16] states that experimental procedures that are, in
principle, operationally indistinguishable must be represented by the same
probability distribution at the ontological level. Accepting that quantum
mechanics is contextual would mean to accept that the operational equivalences
are a result of a fine tuning of the ontic parameters, that makes two distinct
ontological probability distributions look the same at the operational level.

Accepting fine tuned properties is problematic in a scientific theory, because
they characterize nature with a conspiratorial connotation, meaning that there
are certain features that are not available for us to use and experience, even
if they are present in nature. This denies the core idea of science and its
empiricist roots.

In 2015 Wood and Spekkens [21] introduced the notion of fine tunings in the
framework of classical causal models [22] and showed that there are no
explanations of Bell’s theorem free of fine tunings. The requirement of no
causal fine tuning reads as a criterion for the most natural causal
explanation of the observed conditional independencies between the variables.
In contrast, in this work we refer to fine tunings that are purely
operational, without any assumption on the underlying causal structure. In
brief, the condition of no operational fine tuning requires the operational
equivalences between the statistics of experiments to be preserved at the
ontological level. Here the ontological level is defined by the notion of
ontic extension [17], which removes the causal assumptions from the standard
ontological model framework (often called the hidden variable model framework)
[12]. Common examples of operational fine tunings are the ones described
above, that arise from the violation of parameter independence and generalized
noncontextuality. Underlying the distinction, so far ignored, between causal
and operational fine tunings allows us to further dissect the assumptions
present in the no go theorems and to obtain insights regarding the relations
between nonclassical features, as we will show in the following.

The requirement of no operational fine tuning can be seen as a requirement of
structure preservation between the operational and the ontological level. The
branch of mathematics that deals with structures preservation is the one of
category theory and functors [23], that we here adopt. More precisely, we
define the operational and ontological categories. The former refers to all
the possible experimental statistics associated to experiments, while the
latter refers to the corresponding ontological representations. An operational
theory, like quantum mechanics, puts restrictions on the possible experimental
statistics and it is associated to a subcategory of the operational category.
A property, represented by an equation between different states in such
subcategory, is no fine tuned with respect to a functor – that is associated
to the ontological representation of the operational theory – if the functor
maps it to the analogous equation in the ontological category.

In summary, we provide a rigorous mathematical framework, developed also in
the language of category theory and functors, that characterizes operational
fine tunings. In addition to accounting for all the known operational fine
tunings – associated to violations of generalized noncontextuality, parameter
independence and time symmetry – the framework describes more general ones,
thus setting the ground for formulations of further no-go theorems. In light
of our framework and the distinction we draw between operational and causal
fine tunings, we analyze the notion of Bell’s local causality, that can be
conceived in two ways. On the one hand, in the classical causal model
framework, it can be seen as a requirement of no causal fine tuning [21]; on
the other hand, it can be decomposed into the assumptions of parameter
independence – justified by a requirement of no operational fine tuning – and
outcome independence – an assumption of purely causal nature [24]. Through
this consideration we deepen the understanding of the relation between
nonlocality and generalized contextuality, where the former is not just an
example of the latter – as usually stated if considering the Kochen-Specker
notion [15] – because it can be obtained, unlike contextuality, by involving a
purely causal fine tuning.

The article is structured as follows. We define the ontological framework in
section 2. We report the already known examples of operational fine tunings in
section 3. We construct our generalization of operational fine tunings in
section 4, and we provide its categorical formulation in section 5. We discuss
the future avenues and the applications of the framework, _e.g._ possible new
fine tunings and the analysis of the relation between nonlocality and
contextuality, in section 6.

##  2 The ontological framework

In this section we develop the operational description of arbitrary
experiments and we relate it to the underlying ontology associated with them.
Let us start by considering arbitrary experimental scenarios, in general
composed by a set of variables, $r_{1},\dots,r_{m},$ that an experimenter
controls and a set of variables that she observes, $u_{1},\dots,u_{n}$. We
denote lists of variables with bold letters, $\mathbb{r}\equiv
r_{1},\dots,r_{m}$ and $\mathbb{u}\equiv u_{1},\dots,u_{n}.$ Controlled
variables, for example, can refer to the choice of setting of a certain
apparatus, like turning the knob for selecting the frequency of a laser or the
orientation of a Stern-Gerlach magnet, while observable variables, that are
not under the control of the experimenter, are about detectors clicking and
showing the outcomes of the experiment. These operational variables are
associated with localized regions of space time as in figure 1 and we assume
that the resulting pattern on space time can be repeated multiple times. In
summary, we associate an experiment with a fixed set of operational variables
and a pattern in space time that can be repeated in different regions.

The role of an operational theory $\mathcal{T}$, like classical or quantum
theory, is to provide a rule to predict the probabilities of the observed
variables given the controlled ones, $p_{E}(\mathbb{u}|\mathbb{r}),$ for every
experiment $E$. Unless strictly needed, we will omit the subscript $E$ that
labels the experiment in order to soften the notation. A natural way of
explaining why the operational theory works is to assume that there are actual
systems with physical properties that are the subjects of the experiment. We
call these physical properties of the system, existing even if no observer is
present and no experiment is performed, as the ontic states – denoted with
$\lambda$ – and the space where they live (a measurable set) as the ontic
space – denoted with $\Lambda.$

Figure 1: Generic experimental scenario. The operational variables can be
divided into the controlled ones (in red) and the observed ones (in black).

We further impose the following assumptions.

  1. 1.

_(Single world) Realism_. The operational and ontic variables take a definite
value on each run of the experiment.

  2. 2.

_Independence_. All the runs of the experiment are independent and identically
distributed. This implies the existence of a joint conditional probability
distribution associated to the operational and ontic variables,
$p(\mathbb{u},\boldsymbol{\lambda}|\mathbb{r}).$ This must be consistent with
the operational theory, and so reproduce the operational statistics,
$p(\mathbb{u}|\mathbb{r})=\int_{\Lambda}p(\mathbb{u},\boldsymbol{\lambda}|\mathbb{r})d\boldsymbol{\lambda}.$

  3. 3.

_Free choice_. The experimenter can choose $p(\mathbb{r})$ however she likes.

We say that an operational theory $\mathcal{T}$ admits an _ontic extension_
$\mathcal{T}_{\Lambda}$ if there exists an ontic space $\Lambda$ with the
above properties associated to every experiment in the theory. This notion of
ontic extension does not coincide with the standard ontological model
framework that is usually considered in the literature [12]. The latter
imposes extra assumptions, that we here analyze. Let us consider simple kinds
of experiments involving controlled variables that correspond to some choices
$r_{1}$ of preparation $P$ and $r_{2}$ of measurement M, and the observable
variable $u$ which is the outcome of the measurement $M,$ as in figure 2.111We
could consider also transformations in these scenarios, but this would involve
a larger number of variables to be specified and more convoluted definitions
for stating the assumptions of interest, without adding any crucial insight.
An _ontological model_ of an operational theory $\mathcal{T}$ is an ontic
extension $\mathcal{T}_{\Lambda}$ for the theory with the following extra
assumptions.

Figure 2: Prepare and measure scenario. The controlled variables $r_{1}$ and
$r_{2}$ denote the choices of preparation and measurement on the system,
respectively. The outcome $u$ is the observed variable.

  1. 1.

_$\lambda$ -mediation_. The ontic states of the system mediate any correlation
between the preparation and the measurement. More precisely,

$p(u|\lambda,r_{1},r_{2})=p(u|\lambda,r_{2}),$ (1)

where $\lambda$ is the ontic variable that mediates between the preparation
and measurement (see figure 3).

  2. 2.

_Measurement independence_ , _i.e._

$p(\lambda|r_{1},r_{2})=p(\lambda|r_{1}).$ (2)

This assumption is often motivated as an assumption of no-retrocausality[17].

Figure 3: Causal structure of the prepare and measure scenario in the standard
ontological model framework. The directed acyclic graph above represents the
causal relations between the operational and ontic variables. The extra
assumptions of the standard ontological model framework [12] involve
restrictions on the possible causal structures associated to an experiment.
These assumptions can be also obtained by imposing the so called causal Markov
condition on the graph above [22].

Notice that the above assumptions are of causal nature, as also explicated by
the causal network in figure 3. The reason why, in general, we do not include
them in our notion of ontology is because we want to separate the assumptions
on the underlying causal structure of the experiments from the independent
assumption of no operational fine tuning that we address in this work.

Just to provide an example of ontological models, in classical Hamiltonian
mechanics [13] the ontic space is the phase space, whose points – specified by
a position and momentum - are the ontic states. In quantum theory the same
simple picture cannot be drawn. In particular, in the next section we list
some properties that an ontological model of quantum theory cannot possess.

##  3 Examples of operational fine tunings

We here focus on the known examples of operational fine tunings, that arise
from breaking one of the following requirements.

  * •

_Preparation noncontextuality_.222The notion can be defined analogously for
transformations and measurements too. A preparation noncontextual ontological
model of a physical theory is a model where the ontological conditional
probability distributions representing operationally equivalent preparations
are the same distribution,

$p(u|r_{1},r_{2})=p(u|r^{\prime}_{1},r_{2})\;\;\forall\;\;r_{2},u.$ (3)
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\Downarrow$
$\displaystyle
p(u,\lambda|r_{1},r_{2})=p(u,\lambda|r^{\prime}_{1},r_{2})\;\;\;\forall\;\lambda\in\Lambda,\;\forall\;r_{2},u,$
(4)

where $r_{1},r_{1}^{\prime}$ denote the two choices of preparations, $r_{2}$
the choice of measurement, $\Lambda$ is the ontic space where the ontic states
$\lambda$ live in, and the above implication must hold for every two
operationally equivalent preparations (_i.e._ satisfying equation (3)). The
condition above could be written more explicitly, for example for equation
(3), as
$p(\text{u}=u|\text{r}_{1}=r_{1},\text{r}_{2}=r_{2})=p(\text{u}=u|\text{r}_{1}=r^{\prime}_{1},\text{r}_{2}=r_{2})$
in order to underline that with $u,r_{1},r^{\prime}_{1},r_{2}$ we refer to the
actual values taken by the variables $\text{u},\text{r}_{1},\text{r}_{2}.$
However, we avoid this extended notation because it is too cumbersome. The
idea motivating the notion of preparation noncontextuality is that the best
explanation for two preparation procedures that cannot be operationally
distinguished in principle, by any measurement or transformation, is that they
are the same object. It can be proven that quantum theory is inconsistent with
a preparation noncontextual ontological model [16].

  * •

_Parameter independence_. The no operational fine tuning requirement
associated to parameter independence states that the no signaling condition
must be preserved at the ontological level. More precisely, this is usually
stated in Bell’s scenario [14], where there are two parties, Alice and Bob,
who perform one of two measurements on their local system. We here adopt the
standard notation for Bell’s scenario: Alice’s and Bob’s choices of
measurements are denoted by the variables $x,y\in\\{0,1\\}$ and their outcomes
are $a,b\in\\{0,1\\},$ respectively. These are analogous to the variables
$r_{1},r_{2}$ and $u_{1},u_{2}$, respectively, that we have used so far. The
no signaling condition means that the choice of Alice’s measurement cannot
influence Bob’s outcome and vice versa. This means no signaling as Alice and
Bob may be, in principle, space-like separated. Parameter independence – the
fact that the no signaling condition must hold at the ontological level
(equation (6)) – can be justified by a requirement of no operational fine
tuning:

$\left\\{\begin{array}[]{lr}p(a|x,y=0)=p(a|x,y=1)\\\
p(b|x=0,y)=p(b|x=1,y)\end{array}\right.$ (5)
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\Downarrow$
$\left\\{\begin{array}[]{lr}p(a|\lambda,x,y=0)=p(a|\lambda,x,y=1)\\\
p(b|\lambda,x=0,y)=p(b|\lambda,x=1,y)\end{array}\right.$ (6)

The requirement above can be straightforwardly extended to the multiparty case
too. Notice that the assumption of Bell’s local causality, which is the one
considered in Bell’s theorem, assumes both parameter independence and outcome
independence [24]. The latter can be expressed as follows,

$\left\\{\begin{array}[]{lr}p(a|\lambda,b=0,x,y)=p(a|\lambda,b=1,x,y)\\\
p(b|\lambda,a=0,x,y)=p(b|\lambda,a=1,x,y).\end{array}\right.$ (7)

Bell’s theorem states that the statistical predictions of quantum theory are
inconsistent with the predictions of an ontological model satisfying Bell’s
local causality [14].

  * •

_Time symmetry_. An _operationally_ time symmetric theory is a theory where
the predictive and retrodictive formalisms are mathematically identical, thus
providing the same statistics (operational equivalence). The assumption of
time symmetry requires this property to also hold at the ontological level.
More precisely, let us denote with $r_{1}$ the input to a preparation $P$ and
$u_{1}$ its output, and with $r_{2}$ the input to a measurement $M$ and
$u_{2}$ its output. An operational theory $\mathcal{T}$ has _operational_ time
symmetry if for every experiment $E$ involving a choice of preparation
$\text{r}_{1}=r_{1}$ with associated outcome $\text{u}_{1}=u_{1}$ and a choice
of measurement $\text{r}_{2}=r_{2}$ with associated outcome
$\text{u}_{2}=u_{2}$, there exists another experiment $E^{\prime},$ where the
choice of preparation is $\text{r}_{1}=r_{2}$ with associated outcome
$\text{u}_{1}=u_{2}$ and the choice of measurement is $\text{r}_{2}=r_{1}$
with associated outcome $\text{u}_{2}=u_{1},$ that provides the same
statistics of $E,$ as depicted in figure 4. Quantum theory is an example of an
operational time symmetric theory. Time symmetry requires that operational
time symmetry implies ontological time symmetry,

$p_{E}(u_{1},u_{2}|r_{1},r_{2})=p_{E^{\prime}}(u_{2},u_{1}|r_{2},r_{1}).$ (8)
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\Downarrow$
$p_{E}(u_{1},u_{2},\lambda|r_{1},r_{2})=p_{E^{\prime}}(u_{2},u_{1},k(\lambda)|r_{2},r_{1}),$
(9)

where $k$ is a one-to-one map $k:\Lambda\rightarrow\Lambda^{\prime},$ with
$\Lambda,\Lambda^{\prime}$ two ontic spaces that provide the ontic extensions
of the two experiments $E$ and $E^{\prime},$ respectively. It is proven in
[17] that no time symmetric ontological model is consistent with quantum
theory.

Figure 4: Operational time symmetry. An operational theory is operational time
symmetric if for every experiment $E$ there exists another experiment
$E^{\prime}$, where the inputs and outputs of the preparations and
measurements are inverted, that provides the same statistics of $E$.

##  4 Generalizing operational fine tunings

The three requirements of no operational fine tunings listed in the previous
section can be seen as instances of a general requirement to preserve a given
operational equivalence between the statistics of two experiments, subjected
to certain classical processing, at the ontological level.

With classical processing we refer to processes that consist of a pre-
processing and a post-processing. The pre-processing denotes the process
acting on the input variables $\mathbb{s}$ of the experimental apparatus and
leading to the controlled variables $\mathbb{r}$. The post-processing denotes
the process acting on the output variables $\mathbb{t}$ of the experiment and
yielding to the observable variables $\mathbb{u}.$ Mathematically, we describe
a classical processing $f_{0}$ as a map from the conditional probability
distribution $p(\mathbb{t}|\mathbb{s})$ to $p(\mathbb{u}|\mathbb{r})$ acting
as
$p(\mathbb{u}|\mathbb{r})=\sum_{\mathbb{s},\mathbb{t}}p_{f_{0}}(\mathbb{u}|\mathbb{t})p(\mathbb{t}|\mathbb{s})p_{f_{0}}(\mathbb{s}|\mathbb{r}).$
The processing can consist of taking convex mixtures of the conditional
probabilities associated to different sets of values of the variables
$\mathbb{s},\mathbb{t}$, discarding variables or it can act as a simple
function. More precisely, let us describe the classical processing involved in
the examples of the previous section. In the case of noncontextuality, both
the pre and post processing are trivial, they just act as identity functions,
and the different choices of preparations are specified by the labels of the
experiments $E$ and $E^{\prime}.$ In the case of parameter independence the
pre processing is a function that sets one of the input variables to a precise
value, while the post processing discards one of the two output variables. In
the case of time symmetry, one experiment is subjected to the trivial
processing while the other has both the pre and post processing corresponding
to symmetry group transformations that permute the preparation and the
measurement procedures, respectively.

Despite it is not the case for the examples that we consider, in the most
general scenarios it may be possible that the pre and post processing are
correlated, _i.e._ they depend on each other. We represent these correlations
by introducing an extra variable $\mathbb{c}$ that connects the pre and post
processing. The action of the general classical processing
$f_{0}:\\{p(\mathbb{t}|\mathbb{s})\\}\rightarrow\\{p(\mathbb{u}|\mathbb{r})\\}$
reads as

$p(\mathbb{u}|\mathbb{r})=\sum_{\mathbb{s},\mathbb{t},\mathbb{c}}p_{f_{0}}(\mathbb{u}|\mathbb{c},\mathbb{t})p(\mathbb{t}|\mathbb{s})p_{f_{0}}(\mathbb{c},\mathbb{s}|\mathbb{r}).$

Let us state now the general no operational fine tuning condition.

######  Definition 1.

_No fine tuning condition_. A property of the operational theory
$\mathcal{T}$, defined as an equation involving classical processing
$f_{0},f_{1}:\\{p(\mathbb{t}|\mathbb{s})\\}\rightarrow\\{p(\mathbb{u}|\mathbb{r})\\},$
on the statistics of two experiments labelled with $E,E^{\prime},$
respectively, is no fine tuned with respect to a given ontic extension
$\mathcal{T}_{\Lambda}$ for the operational theory, if there exist classical
processing
$h_{0},h_{1}:\\{p(\mathbb{t},\boldsymbol{\lambda}|\mathbb{s})\\}\rightarrow\\{p(\mathbb{u},\boldsymbol{\omega}|\mathbb{r})\\},$
such that the consistency conditions (i), (ii), (iii) listed below are
satisfied and such that the operational equation is preserved in
$\mathcal{T}_{\Lambda}$,

$\displaystyle\sum_{\mathbb{s},\mathbb{t},\mathbb{c}}p_{f_{0}}(\mathbb{u}|\mathbb{c},\mathbb{t})p_{E}(\mathbb{t}|\mathbb{s})p_{f_{0}}(\mathbb{c},\mathbb{s}|\mathbb{r})=\sum_{\mathbb{s},\mathbb{t},\mathbb{c}}p_{f_{1}}(\mathbb{u}|\mathbb{c},\mathbb{t})p_{E^{\prime}}(\mathbb{t}|\mathbb{s})p_{f_{1}}(\mathbb{c},\mathbb{s}|\mathbb{r})$
(10)
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\Downarrow$
$\displaystyle\sum_{\boldsymbol{\lambda},\mathbb{s},\mathbb{t},\mathbb{c}}p_{h_{0}}(\mathbb{u},\boldsymbol{\omega}|\boldsymbol{\lambda},\mathbb{c},\mathbb{t})p_{E}(\boldsymbol{\lambda},\mathbb{t}|\mathbb{s})p_{h_{0}}(\mathbb{c},\mathbb{s}|\mathbb{r})=\sum_{\boldsymbol{\lambda},\mathbb{s},\mathbb{t},\mathbb{c}}p_{h_{1}}(\mathbb{u},\boldsymbol{\omega}|\boldsymbol{\lambda},\mathbb{c},\mathbb{t})p_{E^{\prime}}(\boldsymbol{\lambda},\mathbb{t}|\mathbb{s})p_{h_{1}}(\mathbb{c},\mathbb{s}|\mathbb{r})$
(11)

where $\mathbb{r},\mathbb{s},\mathbb{t},\mathbb{u}$ are operational variables
belonging to the sets $R,S,T,U$ and $\boldsymbol{\lambda},\boldsymbol{\omega}$
are ontological variables belonging to the ontic spaces $\Lambda,\Omega.$ The
variable $\mathbb{c}$ serves to represent the possible dependence between the
pre and post processing.

The following consistency conditions must hold between the processing $f_{i}$
and $h_{i},$ where $i\in{0,1},$ and between the ontic variables
$\boldsymbol{\lambda}$ and $\boldsymbol{\omega}.$

  1. 1.

_Ontic extension condition_. The ontological processing $h_{i}$ must be an
ontic extension of the operational one, $f_{i},$

$\sum_{\boldsymbol{\omega}}p_{h_{i}}(\mathbb{u},\boldsymbol{\omega}|\boldsymbol{\lambda},\mathbb{r})=p_{f_{i}}(\mathbb{u}|\mathbb{r}).$
(12)

This condition guarantees that the ontological equivalence necessarily implies
the operational one.

  2. 2.

_Structure preserving conditions_. The ontological processing must preserve
the structure of the operational processing, meaning that

$f_{i}=\mathbb{I}\Rightarrow h_{i}=\mathbb{I}_{\Lambda}$ $f_{i}=f_{a}\circ
f_{b}\Rightarrow h_{i}=h_{a}\circ_{\Lambda}h_{b}$ (13) $f_{i}=\sigma
f_{a}+(1-\sigma)f_{b}\Rightarrow h_{i}=\sigma h_{a}+(1-\sigma)h_{b},$

where $\mathbb{I}_{\Lambda}$ is the identity processing on the operational
plus ontological variables, $\circ$ and $\circ_{\Lambda}$ are composition
operations for the operational processing denoted with $f_{a},f_{b}$ and
corresponding ontological processing denoted with $h_{a},h_{b},$ respectively,
and $\sigma\in[0,1].$ The above conditions guarantee that the identity
processing, the action of composing processing and the convex structure are
preserved between the operational and ontological level.

  3. 3.

_Sufficient statistics condition_. The ontic variables $\boldsymbol{\omega}$
must be a sufficient statistics for $\boldsymbol{\lambda}.$ This means that

$p_{h_{i}}(\mathbb{u}|\boldsymbol{\omega},\mathbb{r})=p_{h_{i}}(\mathbb{u}|\boldsymbol{\lambda},\mathbb{r}).$
(14)

Without imposing this condition we could imagine an ontological processing
that simply erases all the information about the correlations between the
ontological variable $\boldsymbol{\lambda}$ and the operational variables,
thus trivially satisfying the no fine tuning condition, independent of the
operational condition. In other words, with this condition we want to prevent
$p_{h_{i}}(\mathbb{u}|\boldsymbol{\omega},\mathbb{r})$ from being the uniform
distribution, _i.e._ to trash out the information of $\boldsymbol{\lambda}$
and set a new trivial ontic variable, that would still satisfy the condition
(i). This condition guarantees that after the ontological processing $h_{i}$
all the information about $\lambda$ which is relevant for the fine tuning is
still present.

A property of an operational theory $\mathcal{T}$ is said to be fine tuned
with respect to a given ontic extension $\mathcal{T}_{\Lambda}$ of the
operational theory if it does not satisfy the no fine tuning condition 1.
Figure 5 schematically represents the definition above.

Figure 5: General operational no fine tuning condition. The figure above
represents in a schematic way the condition of no operational fine tuning in
the generalized sense. It demands that an operational equivalence, possibly
involving pre and post processing (represented in the figure as _combs_) on
two experiments $E,E^{\prime},$ implies a consistent (in the sense specified
in conditions (12), (13) and (14)) ontological equivalence. Notice that the
classical processing act on the conditional probabilities, despite, for
simplicity, the figure illustrates them as acting on variables. The above
scheme should be read as in equations (10) and (11).

Notice that trivial ontic extensions, where the ontic and operational
variables are uncorrelated, _e.g._
$p(\mathbb{u},\boldsymbol{\lambda}|\mathbb{r})=p(\mathbb{u}|\mathbb{r})p(\boldsymbol{\lambda}),$
always make the properties of operational theories no fine tuned. Arguably,
these should not be considered as valid ontological explanation of the
operational theory. However, we do not explicitly impose extra conditions to
prevent them. In the no go theorems considered in this article trivial ontic
extensions are ruled out by the extra causal assumptions included in the
definition of the standard ontological model framework. In the simple case of
prepare and measure scenarios these can be ruled out by the assumption of
$\lambda-$mediation, while in more general scenarios the question is more
complicated and we leave its study for future research. For the purpose of the
current work – which consists of defining operational fine tunings and pin
down the subtle assumptions underlying the known no-go theorems – this
generalization is not needed.

Let us now show how the equations (10) and (11) reduce to the known ones for
the examples considered in the previous section. Figure 6 schematically
represents them in light of the new framework.

(a)  (b)

(c)

Figure 6: Three examples of no operational fine tuning requirements in the
generalized framework. Figure LABEL:a represents the requirement of
noncontextuality (notice that the different values of $r_{1}$ are taken into
account by the different labels, $E$ and $E^{\prime}$) , figure LABEL:b of
parameter independence and figure LABEL:c of time symmetry.

  * •

_Preparation noncontextuality_. In this case there are no pre and post
processing, $f_{0}=f_{1}\equiv\mathbb{I},$ which means that
$p_{\mathbb{I}}(u|c,t)=\delta_{u,t},$ and
$p_{\mathbb{I}}(c,s|r)=\delta_{s,r}\frac{1}{N_{c}},$ where $N_{c}$ is a
constant such that $\sum_{c}N_{c}^{-1}=1.$ Equation (10) reduces to the
equation (3), where the variable $r$ specifies which preparation is
considered. The different choices of preparations are specified by the labels
of the experiments $E$ and $E^{\prime}.$ By imposing the no fine tuning
condition 1 we obtain

$\sum_{\lambda}p_{h_{0}}(\omega|\lambda)p_{E}(u,\lambda|r)=\sum_{\lambda}p_{h_{1}}(\omega|\lambda)p_{E^{\prime}}(u,\lambda|r).$

Given the structure preserving condition (ii), the post processing including
ontological variables must be $h_{0}=h_{1}=\mathbb{I}_{\Lambda}.$ This means
that $p_{h_{i}}(\omega|\lambda)=\delta_{\omega,\lambda}\;\;\forall
i\in\\{0,1\\}$ and that $\omega\equiv\lambda.$ Therefore equation (11) reduces
to equation (4) and the requirement of noncontextuality is obtained.

  * •

_Parameter independence_. Let us focus only on the first of the two equations
in (5). The argument runs analogously for the second equation. We can rewrite
the left hand side of equation (10) in this case as,

$\sum_{\alpha,\beta,\gamma,c}p_{Y_{0}}(a,b|\alpha,\beta,c)p_{E}(\alpha,\beta|x,\gamma)p_{Y_{0}}(c,\gamma|y).$

We have here identified $r_{1}=x,r_{2}=y,u_{1}=a,u_{2}=b$ and we have denoted
$f_{0}$ with $Y_{0}$. The right hand side of equation (10) reads the same
apart from $f_{1}\equiv Y_{1}$ instead of $f_{0}.$ The pre-processing that
sets $y$ to 0 reads as
$p_{Y_{0}}(c,\gamma|y)=\delta_{\gamma=0,y}\frac{1}{N_{c}},$ and the post-
processing that corresponds to marginalize out $b$ corresponds to setting

$\displaystyle p_{Y_{0}}(a,b|\alpha,\beta,c)$
$\displaystyle=p_{Y_{0}}(b|\alpha,\beta,c)p_{Y_{0}}(a|\alpha,\beta,c)$
$\displaystyle=p_{Y_{0}}(b|\beta)p_{Y_{0}}(a|\alpha)$
$\displaystyle=\frac{1}{N_{b}}p_{Y_{0}}(a|\alpha),$

where $N_{b}$ is a constant such that $\sum_{b}N_{\beta}^{-1}=1.$ Therefore,
by the classical law of total probability,
$\sum_{\alpha}p_{Y0}(a|\alpha)p_{E}(\alpha|x,y=0)=\sum_{\alpha}p_{Y1}(a|\alpha)p_{E}(\alpha|x,y=1),$
we obtain the operational equation (5). The no fine tuning condition 1 leads
to

$\displaystyle\sum_{\lambda,\alpha}p_{h_{0}}(\omega,a|\lambda,\alpha)p(\alpha,\lambda|x,y=0)=\sum_{\lambda,\alpha}p_{h_{0}}(\omega,a|\lambda,\alpha)p(\alpha,\lambda|x,y=1).$

The classical law of total probability then yields to

$p(a,\omega|x,y=0)=p(a,\omega|x,y=1).$ (15)

Notice that if we sum over $a$ on both sides we obtain that
$p(\omega|x,y=0)=p(\omega|x,y=1).$ By the analogous procedure on the second
equation in (5) we obtain $p(\omega|x=0,y)=p(\omega|x=1,y).$ Together these
two equations means that $p(\omega|x,y)=p(\omega).$ This condition is useful
for our purposes because we can divide both sides of equation (15) by it and
obtain

$p(a|\omega,x,y=0)=p(a|\omega,x,y=1).$

The condition of parameter independence in equation (6) then follows from the
condition (iii) of sufficient statistics.

  * •

_Time symmetry_. In this case $f_{0}=\mathbb{I}$ and so for obtaining the left
hand side of equation (8) we can run the same argument of preparation
noncontextuality. For the right hand side we need to permute the two inputs
and the two outputs. We denote $f_{1}$ with $\Pi,$ which represents the
permutation,

$\sum_{s,t,c}p_{\Pi}(u_{1},u_{2}|c,t_{1},t_{2})p_{E^{\prime}}(t_{1},t_{2}|s_{1},s_{2})p_{\Pi}(c,s_{1},s_{2}|r_{1},r_{2}),$

where
$p_{\Pi}(u_{1},u_{2}|c,t_{1},t_{2})=\delta_{u_{1},t_{2}}\delta_{u_{2},t_{1}}$
and
$p_{\Pi}(c,s_{1},s_{2}|r_{1},r_{2})=\delta_{s_{1},r_{2}}\delta_{s_{2},r_{1}}\frac{1}{N_{c}},$
thus yielding equation (8). By imposing the no fine tuning condition 1 we
obtain

$\displaystyle\sum_{s,t,c,\lambda}p_{h_{0}}(u_{1},u_{2},\omega|\lambda,c,t_{1},t_{2})p_{E}(t_{1},t_{2},\lambda|s_{1},s_{2})p_{h_{0}}(c,s_{1},s_{2}|r_{1},r_{2})=$
$\displaystyle\sum_{s,t,c,\lambda}p_{h_{1}}(u_{1},u_{2},\omega|\lambda,c,t_{1},t_{2})p_{E^{\prime}}(t_{1},t_{2},\lambda|s_{1},s_{2})p_{h_{1}}(c,s_{1},s_{2}|r_{1},r_{2}).$

The left hand side can be treated in the same way as we treated the left hand
side of the preparation noncontextuality case. For the right hand side we need
to use the structure preserving condition (ii) for the composition of
processing. More precisely, let us notice that, by definition of permutation,
$f_{1}$ is an involution, _i.e._ $f_{1}\circ f_{1}=\Pi\circ\Pi=\mathbb{I}.$
Therefore, from conditions (ii) and (i),
$h_{1}\circ_{\Lambda}h_{1}=\mathbb{I}_{\Lambda},$ which means that $h_{1}$ is
an involution too. This guarantees that the ontological post-processing reads
as

$\displaystyle
p_{h_{1}}(u_{1},u_{2},\omega|\lambda,c,t_{1},t_{2})=p_{k}(\omega|\lambda)p_{\Pi}(u_{1},u_{2}|t_{1},t_{2})=\delta_{\omega,k(\lambda)}\delta_{u_{1},t_{2}}\delta_{u_{2},t_{1}}$

where $k$ is a one to one function, $k:\Lambda\rightarrow\Omega,$ and that the
pre-processing reads as
$p_{h_{1}}(c,s_{1},s_{2}|r_{1},r_{2})=p_{\Pi}(c,s_{1},s_{2}|r_{1},r_{2}).$ The
condition of time symmetry in equation (9) is thus recovered.

##  5 The categorical framework

The previous construction for generalizing operational fine tunings finds a
natural formulation in the framework of category theory [23]. Let us briefly
refresh the basics of category theory. A category
$\mathcal{C}=(\mathcal{O},\mathcal{M})$ is defined as a set $\mathcal{O}$ of
objects and a set $\mathcal{M}$ of morphisms acting on them. Given two objects
$a,b\in\mathcal{C},$ a morphism between them is denoted with $a\rightarrow b.$
By definition, in a category there is a notion of composition of morphisms – a
binary operation $\circ$ – that satisfies the properties of associativity,
_i.e._ if $f:a\rightarrow b,\;g:b\rightarrow c$ and $h:c\rightarrow d,$ then
$h\circ(g\circ f)=(h\circ g)\circ f,$ and the existence of the identity
morphism for every object $x$, _i.e._ $\mathbb{I}_{x}:x\rightarrow x$ such
that for every morphism $f:a\rightarrow b$ we have $\mathbb{I}_{b}\circ
f=f=f\circ\mathbb{I}_{a}.$ A map between categories that preserves their
structure is defined as a functor. More precisely, a functor $\mathcal{G}$
between two categories $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ associates an
object $x\in\mathcal{C}_{1}$ to an object $\mathcal{G}(x)\in\mathcal{C}_{2}$
and associates to each morphism $f:x\rightarrow y\in\mathcal{C}_{1}$ a
morphism $\mathcal{G}(f):\mathcal{G}(x)\rightarrow\mathcal{G}(y)$ that
preserves the identity morphism,
$\mathcal{G}(\mathbb{I}_{x})=\mathbb{I}_{\mathcal{G}(x)}\;\;\forall\;x\in\mathcal{C}_{1},$
and the composition rule, $\mathcal{G}(g\circ
f)=\mathcal{G}(g)\circ\mathcal{G}(f)\;\;\forall\;f:x\rightarrow
y,\;g:y\rightarrow z\in\mathcal{C}_{1}.$ We further assume that the functor
$\mathcal{G}$ is convexity preserving, meaning that it maps $\sigma
f+(1-\sigma)g\in\mathcal{C}_{1},$ where $f,g\in\mathcal{C}_{1}$ and
$\sigma\in[0,1],$ to
$\sigma\mathcal{G}(f)+(1-\sigma)\mathcal{G}(g)\in\mathcal{C}_{2},$ where
$\mathcal{G}(f),\mathcal{G}(g)\in\mathcal{C}_{2}.$ This assumption requires
that the morphisms in the categories $\mathcal{C}_{1},\mathcal{C}_{2}$ form a
convex set. Notice the similarity between the properties of a functor and the
structure preserving conditions (ii) of the previous section.

Let us now define the operational and ontological categories. The former
refers to all the possible experimental statistics associated to experiments
(without any operational constraint), while the latter refers to the
corresponding ontological representations.

######  Definition 2.

_Operational Category_. The operational category
$\mathcal{O}p=(\mathcal{P},\mathcal{F})$ is a category where the objects are
sets of conditional probability distributions of observable variables given
controlled variables, $\\{p(\mathbb{u}|\mathbb{r})\\}\in\mathcal{P}.$ Each set
includes conditional probability distributions with the same number of
observable variables and controlled variables, and those variables have the
same cardinality. The morphisms in the category are classical processing
between those sets of probabilities,

$f_{i}:\\{p(\mathbb{t}|\mathbb{s})\\}\rightarrow\\{p(\mathbb{u}|\mathbb{r})\\}.$

We denote the set of such morphisms as $\mathcal{F}.$

The motivation for defining an object in the operational category as a set of
conditional probability distributions with the same number of observable
variables $\mathbb{u}$ and controlled variables $\mathbb{r}$, with the same
cardinality, is that, operationally, conditional probability distributions
$p(\mathbb{u}|\mathbb{r})$ within the same set provide equal amount of
information about the nature of the system subjected to the experiment. As a
clarifying example, let us imagine, in quantum theory, an experiment $E_{1}$
involving two preparations of a qubit with a two-outcome measurement and an
experiment $E_{2}$ on a qutrit, again involving two preparations and a two-
outcome measurement. According to definition 2, the probability distributions
$p_{E_{1}}(u|r)$ and $p_{E_{2}}(u|r)$ belong to the same object in the
category. This may seem counterintuitive, because these probabilities refer to
experiments involving different kinds of systems. However, the crucial point
is that, from an operational perspective, these two probabilities do not allow
to infer such distinction. In particular, we would need extra variables and/or
greater cardinality to realize that $E_{2}$ involves a qutrit.

Notice, from definition 2, that a precise conditional probability distribution
$p(\mathbb{u}|\mathbb{r})$ associated to a particular experimental setup can
be defined in the categorical language as a morphism from the trivial object
of the category – which is the conditional probability distribution with one
element $u$ and one element $r,$ $p(u|r),$ both with cardinality one – to the
desired conditional probability distribution $p(\mathbb{u}|\mathbb{r}).$ These
morphisms are called _states_. An operational theory $\mathcal{T}$ is
associated to a subcategory $\mathcal{O}p_{\mathcal{T}}$ of the operational
category $\mathcal{O}p,$ where only a set of objects
$\mathcal{P}_{\mathcal{T}}\subset\mathcal{P}$ and morphisms
$\mathcal{F}_{\mathcal{T}}\subset\mathcal{F}$ that map between them are
considered. For example, quantum theory is associated to the subcategory of
the operational category with the constraint that objects are composed by sets
of conditional probabilities consistent with the statistics predicted by
quantum theory, where the controlled and observable variables refer to events
in experimental scenarios that are possible in quantum theory – _e.g._ turning
a knob to select the orientation of a Stern-Gerlach magnet and detect where,
on the screen, the particle outputs.

######  Definition 3.

_Ontological Category_. The ontological category
$\mathcal{O}n=(\mathcal{P}_{\Lambda},\mathcal{H})$ is a category where the
objects are sets of conditional probability distributions of observable and
ontic variables given controlled variables,
$\\{p(\mathbb{u},\boldsymbol{\lambda}|\mathbb{r})\\}\in\mathcal{P}_{\Lambda}.$
Each set includes conditional probability distributions with the same number
of observable variables, ontic variables and controlled variables, and those
variables have the same cardinality. The morphisms in the category are
classical processing between those probabilities,

$\displaystyle
h_{i}:\\{p(\mathbb{t},\boldsymbol{\lambda}|\mathbb{s})\\}\rightarrow\\{p(\mathbb{u},\boldsymbol{\omega}|\mathbb{r})\\}.$

These morphisms must satisfy the sufficient statistics condition as in
condition (iii), meaning that

$\displaystyle
p(\mathbb{u}|\boldsymbol{\omega},\mathbb{r})=p(\mathbb{u}|\boldsymbol{\lambda},\mathbb{r})$

for every
$p(\mathbb{u}|\boldsymbol{\omega},\mathbb{r})\in\\{p(\mathbb{u}|\boldsymbol{\omega},\mathbb{r})\\}$
and for every object $\\{p(\mathbb{u}|\boldsymbol{\omega},\mathbb{r})\\}.$ We
denote the set of such morphisms as $\mathcal{H}.$

Given the definitions above, the notion of ontic extension
$\mathcal{T}_{\Lambda}$ of an operational theory $\mathcal{T}$ is associated
to the existence of a functor from the operational subcategory
$\mathcal{O}p_{\mathcal{T}}$ to the ontological category $\mathcal{O}n$.
Therefore we can define the no fine tuning condition as follows.

######  Definition 4.

_No fine tuning condition in category theory_. A property of the operational
theory $\mathcal{T}$ – defined as an equation between different states in the
operational subcategory $\mathcal{O}p_{\mathcal{T}}$ – is no fine tuned with
respect to a functor
$\mathcal{G}:\mathcal{O}p_{\mathcal{T}}\rightarrow\mathcal{O}n$ if this maps
to the same equation in the ontological category $\mathcal{O}n.$

Let us now review again the known examples of operational fine tuning in light
of the categorical framework. We adopt a compact notation that consists of
denoting states in the operational category with a capital letter, _e.g._
$E\equiv p_{E}(\mathbb{u}|\mathbb{r}),$ and with a subscript on the left/right
hand side of the capital letter the extra variables that are included in the
conditional probabilities, _e.g._ ${}_{\boldsymbol{\omega}}E\equiv
p_{E}(\mathbb{u},\boldsymbol{\omega}|\mathbb{r})$ and
$E_{\boldsymbol{\omega}}\equiv
p_{E}(\mathbb{u}|\boldsymbol{\omega},\mathbb{r}).$ Applications of morphisms
to the objects of the category can also be written with this notation, for
example a permutation of the inputs and the outputs in the case of time
symmetry is denoted as $\Pi(E)$ and reads as
$\Pi(E)=p_{E}(u_{2},u_{1}|r_{2},r_{1}),$ for
$E=p_{E}(u_{1},u_{2}|r_{1},r_{2}).$ This notation, which is just a concise way
to write what we already developed throughout the manuscript, does not specify
the number of variables involved and the actual action of the morphisms, that
must be declared separately.

  * •

_Preparation noncontextuality_.

$\displaystyle
E=E^{\prime}\xRightarrow{\mathcal{G}_{NC}}{}_{\omega}E={}_{\omega}E^{\prime}.$
(16)

Notice that the functor $\mathcal{G}_{NC}$ acts as the identity functor,
$\mathcal{G}_{NC}(E)={}_{\omega}E,$ and
$\mathcal{G}_{NC}(E^{\prime})={}_{\omega}E^{\prime},$ and that it acts on
every state of $\mathcal{O}p.$ Therefore we do not need to add
“$\forall\;E,E^{\prime}\in\mathcal{F}\;\;s.t.\;E=E^{\prime}$” to recover the
noncontextuality requirement. This is already entailed by the categorical
framework. Lastly, the standard noncontextuality condition of equation (4)
that involves $\lambda$ can be obtained from equation (16) by the fact that a
functor preserves the identity morphism (compare this with the first condition
(ii) in the previous section), thus yielding
${}_{\omega}E={}_{\omega}Q\Rightarrow{}_{\lambda}E={}_{\lambda}E^{\prime}.$

  * •

_Parameter independence_.

$\displaystyle
Y_{0}(E)=Y_{1}(E)\xRightarrow{\mathcal{G}_{PI}}{}_{\omega}Y_{0}(E)={}_{\omega}Y_{1}(E),$
(17)

where $Y_{0},Y_{1}$ are the two stochastic processes already defined in the
previous section. Equation (17) can be reduced to equation (6) by noticing
that $p(\omega)=p(\omega|x,y)$ as we showed in the previous section and by the
sufficient statistics property of the ontological category $\mathcal{O}n$,

${}_{\omega}Y_{0}(E)={}_{\omega}Y_{1}(E)\Rightarrow
Y_{\omega,0}(E)=Y_{\omega,1}(E)\Rightarrow Y_{\lambda,0}(E)=Y_{\lambda,1}(E).$

  * •

_Time symmetry_.

$\displaystyle
E=\Pi(E^{\prime})\xRightarrow{\mathcal{G}_{TS}}{}_{\omega}E={}_{\omega}\Pi(E^{\prime}),$
(18)

where, by the fact that a functor preserves the composition of morphisms, we
obtain the standard time symmetry equation (9),
${}_{\omega}E={}_{\omega}\Pi(E^{\prime})\Rightarrow{}_{\lambda}E={}_{\lambda}\Pi(E^{\prime}).$

The reason why the categorical framework is natural for defining operational
fine tunings is that the functors, by definition, preserve the structure
between the operational and ontological categories and allow to easily include
the consistency conditions we listed in the previous section.

##  6 Discussion and conclusion

In the framework of ontological models the mystery of quantum theory is
distilled in the notion of fine tunings. We have here provided a general
mathematical framework for characterizing such mystery. More precisely, we
have rigorously defined operational fine tunings. In the language of category
theory, a property of an operational theory – defined as an equation between
different states in the corresponding operational category – is fine tuned
with respect to a functor – associated to an ontic extension of the
operational theory – if the functor does not preserve the equation to the
ontological category. Our framework both accommodates the already known
operational fine tunings – arising from the violation of generalized
noncontextuality, parameter independence and time symmetry – and provides the
mathematical ground for more general ones, in the sense of accounting for fine
tunings involving any generic classical processing on the experiments under
consideration. We leave the study of novel operational fine tunings for future
work. However, we already argue that interesting classes of possible new ones
are group symmetry fine tunings, with a particular focus on the Lorentz
symmetry group. When considering Bell’s scenario, Lorentz invariance has
already been shown to break at the ontological level for particular
ontological models, like the ones of Bohmian mechanics [25] and collapse
theories [26]. It would be interesting to study whether this must be the case
for any ontological model.

An application and insight entailed by our framework regards the relation
between nonlocality and contextuality. In the Kochen Specker formulation,
contextuality is usually regarded, at least formally, as a generalization of
nonlocality [27]. For example, in the standard Bell scenario, the idea is that
the context is selected by Alice’s and Bob’s choices of local measurements and
the violation of a Bell inequality is seen as a proof of contextuality.
However, when considering the generalized notion of contextuality [16], our
framework shows a neat difference between the locality and the
noncontextuality assumptions, because the former cannot be expressed just as a
requirement of no operational fine tuning. As already discussed, Bell’s local
causality can be decomposed into the assumptions of parameter independence
(equation (6)) and outcome independence (equation (7)). Notice that these
assumptions, together, are equivalent to the Markov condition that embodies
the assumption of Bell’s local causality in the causal model framework [21].
In this article we showed that parameter independence can be justified by a
requirement of no operational fine tuning. In this sense it represents the
“noncontextuality part” of the assumption of locality. However, the additional
assumption of outcome independence, which is crucially needed for obtaining
Bell’s local causality, has a purely causal nature. It cannot be justified by
a requirement of no operational fine tuning. Therefore nonlocality – the
breaking of Bell’s local causality – can result from breaking either the
operational noncontextuality-like assumption of parameter independence or the
purely causal assumption of outcome independence. On the contrary,
contextuality involves a purely operational fine tuning, as shown by our
framework. Among other consequences, this consideration provides an
alternative interpretation to the claim made in [28] about contextuality
involving a causal fine tuning.

In this work we have related functors in category theory with operational
theories and ontological models, which was already proposed by Gheorghiu and
Heunen in [29]. However, our approach differs from theirs both in the goal and
in the mathematical formulation. Their goal was to address, in the categorical
framework, the $\psi-$ontic versus $\psi-$epistemic issue on the reality of
the quantum state [30], and, in order to do so, they defined operational
categories in a much more structured way (also involving a notion of topology)
than what we do here.

The present work originates a proper research program, where the next step
consists of formulating a resource theory for operational fine tunings. This
would also allow us to witness and quantify the presence of fine tunings in
information processing tasks and quantum computational protocols. As many
results are showing [31, 32, 33, 34, 35, 36, 37, 38, 39], the quantum
phenomena proven to be responsible for the quantum computational advantages
are so far dependent on the model and scenario considered, and this because,
by construction, these phenomena arise only in certain setups. In this
respect, the benefit of adopting the notion of fine tunings, is that it
captures the aspect that is common and inherently nonclassical about all such
physical phenomena. Therefore, it may be possible that a certain amount of
fine tuning, independent on which actual phenomenon is manifested in the setup
considered, is necessary (or even sufficient!) for quantum computational
advantages. For these reasons, we believe that the notion of fine tunings is
more promising than the ones so far explored in order to understand what
powers quantum computers and technologies. Finally, the very foundational
motivation for studying and characterizing fine tunings is to ultimately
develop a new ontological framework for quantum theory free of fine tunings,
or, alternatively, explain them as emergent from physical phenomena. 333An
example of emergent fine tuning has been proposed in [17], where the fine
tuning associated to retrocausality can be thought as originating from
thermodynamics reasons similar to the ones proposed for explaining the arrow
of time. Another famous example is present in Valentini’s version of Bohmian
mechanics [40].

##  7 Acknowledgments

The authors thank David Schmid and John Selby for helpful discussions. This
research was supported by the Fetzer Franklin Fund of the John E. Fetzer
Memorial Trust.

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