# Classical causal models cannot faithfully explain Bell nonlocality or
Kochen-Specker contextuality in arbitrary scenarios

J. C. Pearl<EMAIL_ADDRESS>Centre for Quantum Dynamics,
Griffith University, Gold Coast, QLD 4222, Australia  E. G. Cavalcanti
<EMAIL_ADDRESS>Centre for Quantum Dynamics, Griffith University,
Gold Coast, QLD 4222, Australia

###### Abstract

In a recent work, it was shown by one of us (EGC) that Bell-Kochen-Specker
inequality violations in phenomena satisfying the no-disturbance condition (a
generalisation of the no-signalling condition) cannot in general be explained
with a faithful classical causal model—that is, a classical causal model that
satisfies the assumption of _no fine-tuning_. The proof of that claim however
was restricted to Bell scenarios involving 2 parties or Kochen-Specker-
contextuality scenarios involving 2 measurements per context. Here we show
that the result holds in the general case of arbitrary numbers of parties or
measurements per context; the connection between fine-tuning and Bell-KS
inequality violations is generic and not an artefact of the simplest
scenarios. This result unifies, in full generality, Bell nonlocality and
Kochen-Specker contextuality as violations of a fundamental principle of
classical causality.

## Introduction.—

Bell nonlocality Bell (1964) and Kochen-Specker (KS) contextuality Kochen and
Specker (1967) are classically forbidden correlations characteristic of
quantum phenomena. Bell nonlocality can be understood as the impossibility to
explain certain quantum correlations between space-like separated systems
within a classical theory of causality, assuming relativistic causal structure
Wiseman and Cavalcanti (2017). KS-contextuality, on the other hand, can be
understood, within the framework of ontological models Spekkens (2005), as the
incompatibility between the predictions of quantum theory with the joint
assumption of measurement noncontextuality—the assumption that the outcome
statistics of a phenomenon should not depend on the measurement context—and
outcome determinism.

The fundamentally quantum nature of contextual and nonlocal correlations lies
at the heart of many quantum protocols. Bell nonlocality is a key resource for
quantum communication, with applications such as reducing communication
complexity Brukner  _et al._ (2004) and secure communication Ekert (1992).
Since classical simulation of Bell correlations is possible (between time-like
separated system) via the addition of communication channels between the
parties in the Bell test Brassard  _et al._ (1999); Toner and Bacon (2003),
quantum over classical advantages provided by Bell nonlocality can be
understood as quantum protocols having access to correlations that can only be
simulated classically with the aid of extra resources. KS-contextuality, on
the other hand, has been identified as a key resource fuelling quantum over
classical advantages in quantum computation Anders and Browne (2009);
Raussendorf (2013); Howard  _et al._ (2014); Mansfield and Kashefi (2018).

A modern approach is to encode correlations for a set of observed variables in
the framework of causal models, where a causal structure is represented as a
directed acyclic graph (DAG) Pearl (2000); Wood and Spekkens (2015). Recently,
a framework was introduced to unify KS-contextuality and Bell nonlocality as
violations of a fundamental principle of causal models: the principle of _no-
fine-tuning_ , or _faithfulness_ Cavalcanti (2018). In the framework of causal
models, fine-tuning occurs when specific choices of parameters of the model
(such as distributions over latent variables) hide from operational
accessibility some causal connections available in the model. In Wood and
Spekkens (2015) it was shown that representing certain Bell-inequality
violations by classical causal models requires fine-tuning, and this result
was extended to the case of KS contextuality in Cavalcanti (2018). Considering
a classical causal model to be (essentially) a classical simulation of a
quantum phenomenon, this provides a novel approach to understanding the
quantum over classical advantage provided by Bell-KS correlations: fine-tuning
can be considered an _unavoidable resource waste_ in any classical simulation,
relative to the quantum realisation of the same correlations. This causal
perspective also reinforces the program of revising the assumptions underlying
the classical causal models framework, such as Reichenbach’s principle of
common cause Reichenbach (1956); Cavalcanti and Lal (2014)—towards a general
framework of quantum causal models Cavalcanti and Lal (2014); Henson  _et al._
(2014); Pienaar and Brukner (2015); Chaves  _et al._ (2015); Costa and
Shrapnel (2016); Allen  _et al._ (2017); Barrett  _et al._ (2019).

The proofs that classical causal models for Bell-KS correlations require fine-
tuning, however, are so far restricted to bipartite Bell scenarios Wood and
Spekkens (2015) or KS scenarios with two measurements per context Cavalcanti
(2018). As quantum protocols can make use of large numbers of parties or
measurements per context, a general proof is needed for this approach to have
practical merit. Here we generalise the framework of Cavalcanti (2018) to
arbitrary numbers of parties or measurements per context, demonstrating in
full generality the need for fine-tuning in classical causal models for Bell-
KS inequality violations.

## Causal models.—

Causal models have been developed as a tool for connecting causal inferences
and probabilistic observations, with a wide range of applications, from
statistics to epidemiology, economics and computer science Pearl (2000). In
this framework, a causal structure is represented by a graph $\mathcal{G}$
containing a set of observable variables of interest, as well as additional
latent, or hidden, variables. Variables are represented as nodes, with causal
links represented by directed edges (arrows). For a pair of variables
$\\{A,B\\}$, $A$ is considered to be the direct cause of B should the graph
$\mathcal{G}$ contain a directed edge from $A$ to $B$. Topologically ordered
directed graphs (i.e. those that exclude the possibility of paradoxical causal
loops) are known as directed acyclic graphs (DAGs).

Standard terminology will be used to refer to relationships between variables.
If there is is a directed path from $A$ to $B$, then $A$ is said to be an
ancestor of $B$, and $B$ is a descendent of $A$. If $A$ has a directed edge to
$B$ (i.e. $A$ is a direct cause of $B$), then $A$ is said to be the parent of
$B$. The set of all parents (all direct causes) for $B$ is denoted by
$\mathrm{Pa}(B)$; the set of all non-descendents of $B$ is denoted by
$\mathrm{Nd}(B)$. The _Causal Markov Condition_ is the assumption that a
variable $X$ is conditionally independent of its non-descendents, given its
parents. This conditional independence (c.i.) is denoted as
$(X\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\mathrm{Nd}(X)\mid\mathrm{Pa}(X))$, meaning
that $P(X\mid\mathrm{Nd}(X),\mathrm{Pa}(X))=P(X\mid\mathrm{Pa}(X))$. For a DAG
$\mathcal{G}$ containing variables $\\{X_{1},\dots,X_{n}\\}$, the Causal
Markov Condition implies that any probability distribution _compatible_ with
$\mathcal{G}$ factorises as

$P(X_{1},\dots,X_{n})=\prod_{j}P(X_{j}\mid\mathrm{Pa}(X_{j})).$ (1)

A procedure called _d-separation_ (directional separation) can be used to
obtain c.i. relations from a graph Pearl (2000). A path $p$ connecting a set
of nodes $\bm{X}$ with a set of nodes $\bm{Y}$ is blocked (d-separated) by a
set of nodes $\bm{Z}$ if and only if

  1. 1.

$p$ contains a chain $A\rightarrow B\rightarrow C$ or a fork $A\leftarrow
B\rightarrow C$ such that the middle node $B$ is in $\bm{Z}$, or

  2. 2.

$p$ contains an inverted fork (collider) $A\rightarrow B\leftarrow C$ such
that the middle node $B$ is not in $\bm{Z}$ and such that no descendant of $B$
is in $\bm{Z}$.

A set $\bm{Z}$ is said to d-separate $\bm{X}$ from $\bm{Y}$ (denoted
$(\bm{X}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{Y}\mid\bm{Z})_{d}$ if and only if $\bm{Z}$
blocks every path from a node in $\bm{X}$ to a node in $\bm{Y}$.

The d-separation condition is a _sound_ and _complete_ criteria for
conditional independence. Sound: if the d-separation condition
$(\bm{X}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{Y}\mid\bm{Z})_{d}$ is satisfied by a graph
$\mathcal{G}$, then all probability distributions compatible with
$\mathcal{G}$ satisfy the c.i. relation $(\bm{X}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{Y}\mid\bm{Z})$; complete: if all
probability distributions compatible with $\mathcal{G}$ satisfy
$(\bm{X}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{Y}\mid\bm{Z})$, then $\mathcal{G}$
satisfies the d-separation condition $(\bm{X}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{Y}\mid\bm{Z})_{d}$. Note that d-separation
refers to a relation between $\bm{X}$, $\bm{Y}$ and $\bm{Z}$ relative to a
graph, and can also be applied to a subgraph $\mathcal{S}$ of a graph
$\mathcal{G}$. A d-separation condition obeyed by a subgraph $\mathcal{S}$ is
not necessarily obeyed by $\mathcal{G}$ however.

Conditional independence relations satisfy certain properties called _graphoid
axioms_ Pearl (2000):
Symmetry,

$(\bm{X}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{Y}\mid\bm{Z})\bm{\Leftrightarrow}(\bm{Y}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{X}\mid\bm{Z})\,,$ (2)

Decomposition,

$(\bm{X}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{YW}\mid\bm{Z})\bm{\Rightarrow}(\bm{X}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{Y}\mid\bm{Z})\,,$ (3)

Weak union,

$(\bm{X}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{YW}\mid\bm{Z})\bm{\Rightarrow}(\bm{X}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{Y}\mid\bm{ZW})\,,$ (4)

Contraction,

$(\bm{X}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{Y}\mid\bm{Z})\textbf{ \&
}(\bm{X}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{W}\mid\bm{ZY})\\\
\bm{\Rightarrow}(\bm{X}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{YW}\mid\bm{Z})\,,$ (5)

Intersection,

$(\bm{X}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{W}\mid\bm{ZY})\textbf{ \&
}(\bm{X}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{Y}\mid\bm{ZW})\\\
\bm{\Rightarrow}(\bm{X}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{YW}\mid\bm{Z}).$ (6)

## Causal framework for contextuality & nonlocality.—

The framework used here generalises that of Cavalcanti (2018), where
traditional ontological models for Bell-nonlocality and contextuality were
translated into the language of causal models. Some of the terminology follows
that of Abramsky and Brandenburger (2011).

A _measurement scenario_ , or _contextuality scenario_ , is specified by a set
of measurements $\mathcal{M}=\\{m_{1},\dots,m_{k}\\}$, a set $\mathcal{O}$ of
possible outcomes for each measurement, and a _compatibility structure_
$\mathcal{C}$, defined to contain all subsets of jointly measurable members of
$\mathcal{M}$: a subset $c\subseteq\mathcal{M}$ is said to be _jointly
measurable_ , _compatible_ , or to represent a _measurement context_ iff
$c\in\mathcal{C}$. A special class of contextuality scenarios are $n$-partite
_Bell-nonlocality scenarios_ , where $\mathcal{M}$ can be decomposed into $n$
subsets $\\{\mathcal{M}_{1},\dots,\mathcal{M}_{n}\\}$ such that each context
$c\in\mathcal{C}$ contains at most one element from each subset.

Here we consider a general class of measurement scenarios, with no restriction
on the number of measurements per context. For simplicity, however, and
without loss of generality, we augment all contexts, where needed, with
trivial measurements (that always give the same outcome), so that all contexts
contain exactly the same number of measurements
$n=\mathrm{max}_{c\in\mathcal{C}}|c|$. Similarly, there is no loss in
generality by assigning the same outcome set $\mathcal{O}$ to every
measurement, as $\mathcal{O}$ can be made large enough to include all possible
outcomes of all $m_{i}\in\mathcal{M}$.

Given a measurement scenario, in each _test_ , that is, in each run of the
experiment, a set of $n$ compatible measurements—i.e. a measurement context—is
chosen to be performed, via a set of random variables
$\bm{X}=\\{X_{1},X_{2},...,X_{n}\\}\in\mathcal{C}$. The respective outcomes
are recorded by the set of random variables
$\bm{A}=\\{A_{1},A_{2},...,A_{n}\\}$. Measurement-outcome pairs are
represented as ordered pairs $\\{A_{i},X_{i}\\}$ for all
$i\in\mathcal{I}=\\{1,2,...,n\\}$. For convenience, we denote an index subset
by $\gamma\subseteq\mathcal{I}$ such that $\bm{A}_{\gamma}\subseteq\bm{A}$ and
$\bm{X}_{\gamma}\subseteq\bm{X}$. We then introduce the shortcut notations
$\bm{A}_{\setminus\gamma}=\bm{A}\setminus\bm{A}_{\gamma}$ and
$\bm{X}_{\setminus\gamma}=\bm{X}\setminus\bm{X}_{\gamma}$.

A _phenomenon_ is specified by a probability distribution
$\mathcal{P}(\bm{AX})$ for all allowed values of the observable variables.
Note that the formalism so far is independent of any causal structure. We now
define a (classical) causal model for a phenomenon.

######  Definition 1 (Classical causal model).

A _classical causal model_ $\Gamma$ for a phenomenon $\mathcal{P}$ consists of
a (possibly empty) set of latent variables $\bm{\Lambda}$, a DAG $\mathcal{G}$
with nodes $\\{\bm{A,X},\bm{\Lambda}\\}$, and a probability distribution
$P(\bm{AX}\bm{\Lambda})$ compatible with $\mathcal{G}$, such that
$\mathcal{P}(\bm{AX})=\sum_{\bm{\Lambda}}P(\bm{AX}\bm{\Lambda})$.

If the probability distribution for any compatible subset of measurement
outcomes is independent of the context in which they are performed, the
phenomenon is said to satisfy the condition of _no-disturbance_.

######  Definition 2 (No-disturbance).

A phenomenon is said to satisfy _no-disturbance_ iff
$\mathcal{P}(\bm{A}_{\gamma}|\bm{X})=\mathcal{P}(\bm{A}_{\gamma}|\bm{X}_{\gamma})$
for all values of the variables $\\{A_{i},X_{i}\\}$ for which those
conditionals are defined, for all $\gamma\subseteq\mathcal{I}$ and for all
$i\in\gamma$.

In the language of causal models, the no-disturbance condition is denoted by
$(\bm{A}_{\gamma}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{X}_{\setminus\gamma}\mid\bm{X}_{\gamma})$.
The decomposition axiom (3) can then be used to derive less general no-
disturbance conditions for subsets of $\bm{X}_{\setminus\gamma}$. In Bell
scenarios, when each measurement in $\bm{X}$ is space-like separated from all
others, the no-disturbance condition is called the _no-signalling_ condition.

It is important to note that no-disturbance and no-signalling are defined as
properties of phenomena, that is, they are defined operationally. The
following definitions instead deal with properties of causal models for a
phenomenon—that is, they can be understood as _ontological_ properties. Bell-
locality and KS-noncontextuality can be translated to the language of causal
models through the condition of factorisability as:

######  Definition 3 (Factorisability).

A causal model for a phenomenon $\mathcal{P}$ is said to satisfy
_factorisability_ iff $\forall\,\bm{A},\bm{X},$
$\mathcal{P}(\bm{A}|\bm{X})=\sum_{\bm{\Lambda}}\prod_{i}P(\bm{\Lambda})P(A_{i}|X_{i}\bm{\Lambda})$.

By the Fine-Abramsky-Brandenburger theorem Fine (1982); Abramsky and
Brandenburger (2011), the assumption of KS-noncontextuality is equivalent to
the existence of a factorisable model for a phenomenon satisfying no-
disturbance. Bell-locality is the special case of KS-noncontextuality in a
Bell scenario.

######  Definition 4 (KS-noncontextuality).

A causal model for a contextuality scenario is said to satisfy _KS-
noncontextuality_ iff it is factorisable.

The set of KS-noncontextual phenomena for each scenario is bounded by the _KS
inequalities_ Klyachko  _et al._ (2008); Cabello (2008), which can be derived
as the facets of a convex polytope Abramsky and Hardy (2012); Pitowsky (1989)
induced by the factorisability condition. These inequalities reduce to _Bell
inequalities_ Peres (1999) in Bell scenarios.

The equivalence between factorisability and Bell-locality for Bell scenarios
is justified by Bell’s notion of local causality (for a review, see Wiseman
and Cavalcanti (2017)). The equivalence for contextuality scenarios, where
measurements are not spacelike separated, rests on more controversial grounds,
requiring an assumption of outcome determinism that is arguably unjustified
when formulated within the language of ontological models Spekkens (2005,
2014). Here we show that, fortunately, this controversy can be avoided within
the framework of causal models, as the condition of factorisability is implied
by the principle of _no fine-tuning_ , or _faithfulness_ , a fundamental
principle of causal models, without the need to invoke outcome determinism.

######  Definition 5 (Faithfulness (no fine-tuning)).

A causal model $\Gamma$ is said to satisfy _no fine-tuning_ or be _faithful_
relative to a phenomenon $\mathcal{P}$ iff every conditional independence
$(C\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}D|E)$ in $\mathcal{P}$ corresponds to a
d-separation $(C\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}D|E)_{d}$ in the causal graph $\mathcal{G}$ of
$\Gamma$.

Consider a phenomenon that is known to satisfy the c.i. relation
$(A\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}B\mid C)$ (corresponding, for example, to a
no-signalling condition). If the causal structure does not satisfy the
d-separation $(A\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}B\mid C)_{d}$, then the observed conditional
independence can only arise due to specially _fine-tuned_ values of the causal
parameters. These fine-tuned parameters act to “hide” causal connections (for
example, faster-than-light causation), creating the illusion of a c.i.
relation at the operational level. A faithful causal model is then best
understood to be a causal model with no hidden causal connections.

The motivation for no fine-tuning is analogous to that for Leibniz’s principle
of the identity of indiscernibles Spekkens (2019), which states that a theory
should avoid to postulate distinctions at the ontological level that are not
reflected in operational distinctions. It can also be understood as the
methodological principle underlying Einstein’s principles of relativity and of
equivalence Spekkens (2019). In Cavalcanti (2018) (following Wood and Spekkens
(2015)), it was shown that no-fine-tuning leads to KS-noncontextuality for any
phenomenon satisfying no-disturbance. The proof however was restricted to
contextuality scenarios with two measurements per context (and bipartite Bell
scenarios as a special case). Here, we show that this result holds in general
for an arbitrary numbers of parties or measurements per context:

######  Theorem 1.

Every faithful causal model for a phenomenon satisfying no-disturbance in an
arbitrary contextuality scenario is factorisable.

The proof proceeds by deriving a set of d-separation conditions that must be
obeyed by any graph which satisfies no-disturbance and no-fine-tuning, even
considering the possibility of extra latent variables. These d-separation
conditions then imply new c.i. relations, beyond those defined by no-
disturbance alone, in any joint distribution compatible with any faithful
causal graph satisfying no-disturbance. These c.i. relations will then be
shown to imply factorisability in the joint distribution for any number of
parties or measurements per context.

Theorem 1 leads to the following immediate corollaries:

######  Corollary 1.

No fine-tuning and no-disturbance (no-signalling) imply KS noncontextuality
(Bell-locality) in arbitrary scenarios.

######  Corollary 2.

Every classical causal model that reproduces the violation of a Bell-KS
inequality in a no-disturbance phenomenon in an arbitrary Bell-KS scenario
requires fine-tuning.

To aid in the proof of Theorem 1, we introduce the graphical notations in Fig.
1 to represent sets of causal connections. A diagram using these shortcut
notations represents the set of all DAGs compatible with all shortcut
notations. A dashed line represents a connection of the type indicated or no
connection.

Figure 1: Shortcut graphical notations for causal connections between X and Y.

The proof will make use of the following Lemma:

######  Lemma 1.

Let a _chained graph_ be a graph of the form below (Fig. 2), where A,B,C,D
represent sets of vertices, and the connections between two such sets
represent possible connections between any pairs of elements in each set. Note
that all paths between elements of non-adjacent sets go through the
intermediate set, e.g. all paths from A to C go through at least one element
of B.

Figure 2: A chained graph $\mathcal{V}_{c}$.

If $\mathcal{V}_{c}$ satisfies $(\bm{A}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{C}\mid\bm{B})_{d}$, then it also satisfies
$(\bm{A}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{CD}\mid\bm{B})_{d}$ and
$(\bm{A}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{D}\mid\bm{BC})_{d}$.

## Proof of Lemma 1.

To satisfy $(\bm{A}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{C}\mid\bm{B})_{d}$, $\bm{B}$ must block
all paths between $\bm{A}$ and $\bm{C}$. If $\bm{B}$ blocks all paths between
$\bm{A}$ and $\bm{C}$, it follows that $\bm{B}$ blocks all paths between
$\bm{A}$ and $\bm{D}$ (as all paths must pass through at least one element of
$\bm{B}$, and through an element of $\bm{C}$ before reaching $\bm{D}$).
Likewise $\bm{B}$ blocks all paths between $\bm{A}$ and $\bm{C}\bm{D}$. We can
then write $(\bm{A}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{D}\mid\bm{B})_{d}$ and
$(\bm{A}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{CD}\mid\bm{B})_{d}$. From the second of
these and the weak union axiom (4) we can then derive
$(\bm{A}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{D}\mid\bm{BC})_{d}$. $\square$

## Proof of Theorem 1.—

The no-disturbance condition, when combined with the assumption of no fine-
tuning, leads to the d-separation conditions,

$(\bm{A}_{\gamma}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{X}_{\setminus\gamma}\mid\bm{X}_{\gamma})_{d}.$
(7)

## Step 1a.—

The class of DAGs we need to consider are those that include latent variables
as common causes for observable variables or direct causal connections between
them. There is no point considering latent variables as intermediaries or as
common effects between variables, since adding those has no effect on the
allowed probability distributions over the observable variables.

From (7), we can exclude any causal link between $\\{A_{i},\bm{X}_{\setminus
i}\\}$ for all $i\in\mathcal{I}=\\{1,2,...,n\\}$. This leaves us with the
possibility of any causal link between $\\{A_{i},X_{i}\\}$,
$\\{A_{i},\bm{A}_{\setminus i}\\}$ and $\\{X_{i},\bm{X}_{\setminus i}\\}$ as
shown in Fig. 3. Without loss of generality, we introduce $\bm{\Lambda}$ and
$\bm{\Omega}$ as the sets of latent variables such that
$(\bm{\Lambda}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{X})$ and
$(\bm{\Omega}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{A})$.

Figure 3: Remaining class of DAGs after Step 1a.

## Step 1b.—

For every DAG represented by Fig. 3, and without loss of generality, all
members of $\bm{A}$ and $\bm{X}$ can be grouped into subsets depending on the
existence of certain causal connections, as shown in Fig 4.

Figure 4: A shortcut representation of the class of DAGs in Fig. 3. Each node
now represents a set of variables. Dashed circles represent the possibility of
an empty set. $\bm{B}\subseteq\bm{A}$ is the subset containing all members of
$\bm{A}$ with no causal connection to $\bm{X}$ while $\bm{C}\subseteq\bm{A}$
is the subset containing all members of $\bm{A}$ with some causal connection
to $\bm{X}$. Since nodes now represent sets, connections represent all
possible connections of the type indicated between members of each node.

## Step 2a.—

From (7), we can derive $(\bm{B}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{Z}\mid\bm{Y})_{d}$. Note that any path
between $\bm{B}$ and $\bm{Z}$ must pass through at least one element of
$\bm{C}$. Therefore, for any such path, $\bm{C}$ acts as a middle node that is
not in $\bm{Y}$. For $\bm{B}$ to be d-separated from $\bm{Z}$ given $\bm{Y}$,
$\bm{C}$ must act as a collider in any path between $\bm{B}$ and $\bm{Z}$.
Since every member of $\bm{C}$ has a connection to one and only one member of
$\bm{Z}$, any member of $\bm{C}$ with a direct causal connection to $\bm{B}$
would be a non-collider middle node between $\bm{B}$ and $\bm{Z}$. Thus,
direct connections from $\bm{C}$ to $\bm{B}$ would violate
$(\bm{B}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{Z}\mid\bm{Y})_{d}$, and are excluded, as
shown in Fig. 5.

Figure 5: Elimination of direct links from $\bm{C}$ to $\bm{B}$.

## Step 2b.—

From Fig. 5 we see that $\bm{Y}$ cannot act as a middle node in paths between
$\bm{B}$ and $\bm{Z}$. Thus $(\bm{B}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{Z}\mid\bm{Y})_{d}$ implies
$(\bm{B}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{Z})_{d}$. Therefore $\bm{B}$ is
d-separated from $\bm{Z}$ given any variable that is not a collider in a path
between them. As $\bm{\Lambda}$ satisfies this condition, we find that

$(\bm{B}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{Z}\mid\bm{\Lambda})_{d}.$ (8)

Using (7) again, we can write the d-separation condition
$(\bm{C}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{Y}\mid\bm{Z})_{d}$. From the symmetry
axiom (2), this can be rewritten as $(\bm{Y}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{C}\mid\bm{Z})_{d}$. From Lemma 1, it
follows that all graphs in Fig. 5 compatible with
$(\bm{Y}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{C}\mid\bm{Z})_{d}$ must also satisfy
$(\bm{Y}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{CB}\bm{\Lambda}\mid\bm{Z})_{d}$. From the
weak union axiom (4), this can be rewritten as
$(\bm{Y}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{CB}\mid\bm{Z}\bm{\Lambda})_{d}$.
Reapplying the symmetry axiom (2) and rewriting $\bm{BC}=\bm{A}$, we arrive at
the condition

$(\bm{A}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{Y}\mid\bm{Z}\bm{\Lambda})_{d}.$ (9)

## Step 3a.—

Now we consider the causal connections between two arbitrary variables
$\\{C_{i},C_{j}\\}\in\bm{C}$. From (7) and the decomposition axiom (3), we can
write the condition $(C_{j}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Z_{i}\mid Z_{j})_{d}$. Consider the path
$(Z_{i}-C_{i}-C_{j})$ for arbitrary $i,j$. For this path to be blocked by
$Z_{j}$, the middle node $C_{i}$ must be a collider. Thus, we can eliminate
direct connections $C_{i}\rightarrow C_{j}$ between any two members of
$\bm{C}$, as shown in Fig. 6.

## Step 3b.—

We now consider what d-separation conditions can be found between members of
$\bm{C}$, as this will be required for the final step. To aid in this process,
we represent the remaining set of DAGs as shown in Fig. 6. All paths between
$C_{i}$ and $\bm{C}_{\setminus i}$ can be divided in two classes: (i) those
paths that go through $Z_{i}$ (the bottom half of Fig. 6) and (ii) those that
go through $\bm{B}\bm{\Lambda}$ (the top half).

Figure 6: Representation of the set of graphs with $\bm{C}$ separated into
$C_{i}$ and $\bm{C}_{\setminus i}$ and $\bm{Z}$ separated into $Z_{i}$ and
$\bm{Z}_{\setminus i}$.

Consider the paths in (i). From (7) and the decomposition axiom (3), we can
write $(C_{i}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{Z}_{\setminus i}\mid Z_{i})_{d}$. That is,
$Z_{i}$ blocks all paths between $C_{i}$ and $\bm{Z}_{\setminus i}$. Thus
$Z_{i}$ blocks all paths between $C_{i}$ and $\bm{C}_{\setminus i}$ that go
through $Z_{i}$. Therefore $\bm{Z}$ blocks all paths in (i) between $C_{i}$
and $\bm{C}_{\setminus i}$.

The paths in (ii) are blocked by conditioning on $\bm{B}\bm{\Lambda}$, as all
such paths are chains or forks with $\bm{B}$ and/or $\bm{\Lambda}$ as a middle
node. Thus, we find that all DAGs in Fig. 6 that satisfy condition (7) also
satisfy $(C_{i}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{C}_{\setminus i}\mid\bm{ZB\Lambda})_{d}$.

Now note that no observable variable outside $\bm{B}$ can have a direct causal
link to $\bm{B}$. Without loss of generality, we can thus let $\bm{\Lambda}$
determine $\bm{B}$. Any information about $\bm{B}$ is then known given
$\bm{\Lambda}$, and we arrive at the condition

$(C_{i}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{C}_{\setminus
i}\mid\bm{Z}\bm{\Lambda})_{d}.$ (10)

Any path through $\bm{\Lambda}$ is blocked by $\bm{\Lambda}$ because it is a
fork. From $(C_{i}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{Z}_{\setminus i}\mid Z_{i})_{d}$ we can
then write

$(C_{i}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{Z}_{\setminus i}\mid
Z_{i}\bm{\Lambda})_{d},$ (11)

as conditioning on $\bm{\Lambda}$ cannot make $C_{i}$ and $\bm{Z}_{\setminus
i}$ dependent.

## Step 4.—

The d-separation conditions derived in (8), (9), (10) and (11) imply the
corresponding c.i. conditions

$\displaystyle(\bm{B}$ $\displaystyle\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{Z}\mid\bm{\Lambda})\,,$ (12)
$\displaystyle(\bm{A}$ $\displaystyle\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{Y}\mid\bm{Z\bm{\Lambda}})\,,$ (13)
$\displaystyle(C_{i}$ $\displaystyle\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{C}_{\setminus
i}\mid\bm{Z\bm{\Lambda}})\,,$ (14) $\displaystyle(C_{i}$
$\displaystyle\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{Z}_{\setminus i}\mid
Z_{i}\bm{\Lambda})\,.$ (15)

From the definition of conditional probability, we can write the observable
joint distribution as

$\mathcal{P}(\bm{A}\mid\bm{X})=\sum_{\bm{\Lambda}\,\bm{\Omega}}P(\bm{A\bm{\Omega}}\mid\bm{X}\bm{\Lambda})P(\bm{\Lambda}\mid\bm{X})\,.$
(16)

Summing over $\bm{\Omega}$, writing $\bm{X}=\bm{YZ}$ and since by definition
$(\bm{\Lambda}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\bm{X})$, we can write

$\mathcal{P}(\bm{A}\mid\bm{X})=\sum_{\bm{\Lambda}}P(\bm{A}\mid\bm{YZ}\bm{\Lambda})P(\bm{\Lambda})\,.$
(17)

Substituting $\bm{A}=\bm{BC}$, and using (12) and (13),

$\mathcal{P}(\bm{A}\mid\bm{X})=\sum_{\bm{\Lambda}}P(\bm{C}\mid\bm{BZ}\bm{\Lambda})P(\bm{B}\mid\bm{\Lambda})P(\bm{\Lambda})\,.$
(18)

Since $\bm{\Lambda}$ determines $\bm{B}$,
$P(\bm{C}\mid\bm{BZ}\bm{\Lambda})=P(\bm{C}\mid\bm{Z}\bm{\Lambda})$. Using the
definition of conditional probability,

$P(\bm{C}\mid\bm{Z}\bm{\Lambda})=\prod_{j}P(C_{j}\mid\bm{C}\setminus\\{C_{1},C_{2},\dots,C_{j}\\}\bm{Z}\bm{\Lambda}).$
(19)

From (14), all $C_{j}\in\bm{C}$ are independent given $\bm{Z}\bm{\Lambda}$.
From (15), all $C_{j}\in\bm{C}$ are independent of $\bm{Z}_{\setminus j}$
given $Z_{j}\bm{\Lambda}$. Thus

$P(\bm{C}\mid\bm{Z}\bm{\Lambda})=\prod_{j}P(C_{j}\mid Z_{j}\bm{\Lambda})\,.$
(20)

Applying this procedure to $P(\bm{B}\mid\bm{\Lambda})$, and since by
definition $\bm{\Lambda}$ determines $\bm{B}$, we can similarly write

$P(\bm{B}\mid\bm{\Lambda})=\prod_{k}P(B_{k}\mid\bm{\Lambda})\,.$ (21)

We can finally write the observable joint distribution as:

$\displaystyle\mathcal{P}(\bm{A}\mid\bm{X})=\sum_{\bm{\Lambda}}\prod_{j,k}P(C_{j}\mid
Z_{j}\bm{\Lambda})P(B_{k}\mid\bm{\Lambda})P(\bm{\Lambda})\,,$ (22)

which is a factorisable model. $\square$

## Conclusion.—

In summary, we have shown that Kochen-Specker contextuality and Bell-
nonlocality, in fully general scenarios with arbitrary numbers of measurements
per context or parties, and arbitrary numbers of outcomes per measurement, can
both be understood as phenomena for which it is impossible to construct a
faithful classical causal model. This means that these key quantum phenomena
can be understood in a unified way as violations of the classical framework of
causality.

This result has several important consequences. Firstly, from a foundational
perspective, it generalises the results of Wood and Spekkens (2015);
Cavalcanti (2018), confirming that this relationship between fine-tuning and
Bell-KS inequality violations is fully general, and not an artefact of the
simplest scenarios. This adds extra motivation for the program of quantum
causal models Cavalcanti and Lal (2014); Henson  _et al._ (2014); Pienaar and
Brukner (2015); Chaves  _et al._ (2015); Costa and Shrapnel (2016); Allen  _et
al._ (2017); Barrett  _et al._ (2019), in which the classical framework of
causality is extended into a framework where quantum correlations can be
potentially explained _without_ fine-tuning, thus removing the objectionable
property that, according to the present result, is required of classical
causal models for all Bell-KS correlations.

Secondly, as alluded to in the introduction, and in Cavalcanti (2018), this
result gives a general motivation for the idea of quantifying quantum
advantage via fine tuning, as it shows that this is a property of all
classical simulations of phenomena displaying Bell-KS contextuality—key
resources for quantum communication and computation protocols.

Another avenue for further research is to extend the principle of no fine-
tuning to accommodate phenomena that do not satisfy no-disturbance, as is the
case in non-ideal experiments. In Cavalcanti (2018), a _generalised principle
of no fine-tuning_ was proposed, whereby a causal model should not allow
causal connections stronger than needed to explain the observed deviations
from the no-disturbance condition. A recent work Jones (2019) has implemented
a version of this principle, and shown that models that satisfy this property
(dubbed “M-noncontextuality”) are equivalent to models that satisfy the
property of CbD-noncontextuality defined in the “Contextuality by Default”
approach Dzhafarov and Kujala (2017). However, the class of causal models
considered in Jones (2019) is not as general as the ones considered here: it
only considers a minimal relaxation from the default causal structure,
allowing for causal influences from the contexts to the measurement outcomes.
This excludes by fiat a large class of candidate causal models. It would be
interesting to know whether this relationship between generalised no-fine-
tuning and CbD-contextuality holds in general, and whether this can lead to
robust experimental tests.

###### Acknowledgements.

ECG acknowledges Shane Mansfield for useful feedback on a draft of this paper.
This work was supported by the Australian Research Council Future Fellowship
FT180100317, and grant number FQXi-RFP-1807 from the Foundational Questions
Institute and Fetzer Franklin Fund, a donor advised fund of Silicon Valley
Community Foundation.

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# Resonance excitations in 7Be(d,p)8Be∗ to address the cosmological lithium
problem

Sk M. Ali1<EMAIL_ADDRESS>D. Gupta1<EMAIL_ADDRESS>K. Kundalia1
Swapan K. Saha1 Former faculty  O. Tengblad2 J.D. Ovejas2 A. Perea2 I. Martel3
J. Cederkall4 J. Park4 Present Address: Center for Exotic Nuclear Studies,
Institute for Basic Science, 34126 Daejeon, South Korea  S. Szwec5,6
1Department of Physics, Bose Institute, 93/1 APC Road, Kolkata 700009, India
2Instituto de Estructura de la Materia $-$ CSIC, Serrano 113 bis, ES-28006
Madrid, Spain  3University of Huelva, Av. Fuerzas Armadas s/n. Campus “El
Carmen”, 21007, Huelva, Spain  4Department of Physics, Lund University, Box
118, SE-221 00 Lund, Sweden  5Accelerator Laboratory, Department of Physics,
University of Jyv$\ddot{a}$skyl$\ddot{a}$, FI-40014
Jyv$\ddot{a}$skyl$\ddot{a}$, Finland  6Helsinki Institute of Physics,
University of Helsinki, FIN-00014 Helsinki, Finland

###### Abstract

The anomaly in the lithium abundance is a well-known unresolved problem in
nuclear astrophysics. Recent revisit to the problem tried the avenue of
resonance enhancement to account for the primordial 7Li abundance in standard
Big-Bang Nucleosynthesis (BBN). Prior measurements of the 7Be(d,p)8Be*
reaction could not account for the individual contributions of the different
excited states involved, particularly at higher energies close to the Q-value
of the reaction. We carried out an experiment at HIE-ISOLDE, CERN to study
this reaction at Ecm = 7.8 MeV, populating excitations up to 22 MeV in 8Be for
the first time. The angular distributions of the several excited states have
been measured and the contributions of the higher excited states in the total
cross section at the relevant Big Bang energies were obtained by extrapolation
to the Gamow window using the TALYS code. The results show that by including
the contribution of the 16.63 MeV state, the maximum value of the total
S-factor inside the Gamow window comes out to be 167 MeV b as compared to
earlier estimate of 100 MeV b. However, this still does not account for the
lithium discrepancy.

Resonance, Folding, Isospectral potential

###### pacs:

25.40.Hs, 26.35.+c, 25.40.Ny, 25.60.-t, 29.30.Ep

The cosmological lithium problem CO04 ; BO10 ; FI11 ; CY16  is a decades-old
and yet unresolved problem in nuclear astrophysics. The problem delineates a
pronounced anomaly in abundance of 7Li between observation and prediction of
the BBN theory. Using a baryon-to-photon ratio $\eta=(6.104\pm 0.058)\times
10^{-10}$ AG18 , the BBN theory predicts a 7Li abundance about 3 times higher
than observation. There is however, good agreement for 2H and 4He. Several
avenues were searched unsuccessfully to solve the anomaly from observation
errors or new physics beyond the standard model FI11 . The primordial 7Li
mostly originated from the decay of 7Be (T1/2 = 53.22d TI02 ). Consequently
the production and destruction channels of 7Be are linked to the lithium
problem.

Previous experiments AN05 ; RI19  on 7Be destruction with deuterium tried to
search for a solution based on incomplete nuclear physics input to BBN
calculations. The reaction 7Be(d,p)8Be(2$\alpha$) has a high Q value of 16.67
MeV. The earliest cross-section measurements of this reaction by Kavanagh KA60
were limited to excitations of 11 MeV. Later, Parker PA72  estimated the
contributions from higher excited states of 8Be, not observed in KA60 . He
multiplied the differential cross sections by 4$\pi$, and further by a factor
of $\sim$ 3 to account for higher excited states. The d + 7Be rate used in BBN
calculations CA88  was based on PA72 . Later work by Angulo AN05  found even a
smaller rate of d + 7Be. In addition to the ground state (gs) and first
excited states of 8Be, the 11.4 MeV state in 8Be was observed and cross
sections were measured up to excitations of 13.8 MeV AN05 . It was concluded
AN05  that higher energy states not observed in KA60  contribute $\sim$ 1/3 of
the total S-factor instead of 2/3 estimated in PA72 .

Now, the destruction reaction might proceed through intermediate states; in
8Be by 7Be(d,p)8Be∗; in 5Li by 7Be(d,$\alpha$)5Li∗ or in a democratic three-
particle decay of the 9B compound system RI19 . Rijal stated in RI19  that the
(d,$\alpha$) yield dominated over (d,p) yield. They claim to have found a new
resonance inside the Gamow window ($T=0.5-1$ GK, $E_{cm}=0.11-0.56$ MeV), that
reduces the predicted abundance of 7Li but not sufficiently to solve it.
However, Gai GA19  pointed out that Rijal’s new d + 7Be rate is nearly
identical to PA72  in the BBN region. Also the rates are uncertain by a factor
of 10 due to uncertainty of the resonance energy around 16.8 MeV GA19 . In
this context, no data extracting contributions of the excited states around 16
MeV exist in the 7Be + d channel.

The above mentioned 16.8 MeV state of 9B , may also decay by proton emission
to a highly excited state in 8Be at 16.626 MeV, subsequently breaking into two
$\alpha$-particles KI11 . To determine fully the contribution of 7Be(d,p)8Be
reaction to the 7Li abundance, it needs to be measured for 8Be excitations
around 16 MeV. In order to measure that with good statistics, the experiment
might be done at higher energies. The data are then used to normalize the
excitation function calculated with TALYS KO19 , to ascertain contributions of
the higher excited states at the Gamow window.

In this Letter, we discuss our findings from the reaction 7Be(d,p)8Be∗,
measuring states of 8Be∗ up to 22 MeV. The experiment was carried out at CERN
HIE-ISOLDE HIE , using a 5 MeV/u 7Be beam with energy spread of $\sim$ 168
keV. A uranium carbide target was irradiated offline ST19  and the activated
target was heated up during the experiment. Using the RILIS RILIS  laser ion-
source, the 7Be was extracted and post-accelerated. When the residual gas in
the REX-EBIS charge breeder is ionized REX , the stable beam contaminants
produced are mainly 14N4+ and 21Ne6+, having the same A/q as 7Be2+. A
stripping foil and a dipole before the experimental station was used to clean
the beam to 7Be4+. Any 7Li accelerated as 7Li2+ was fully stripped in the foil
and removed. The intensity of 7Be beam was $\sim$ 5 $\times$ 105 pps impinging
on a 15 $\mu$m CD2 target. We also used a 15 $\mu$m CH2 and a 1 mg/cm2 208Pb
target for background measurements and normalization respectively, by taking
short consecutive runs with 208Pb and CD2 targets.

Figure 1:  The detector setup for the 7Be + d experiment at 5 MeV/u.

The setup (Fig. 1) in the scattering chamber SEC , consisted of 5 double-sided
Micron 16$\times$16 silicon strip detectors (DSSD) of thickness 60 $\mu$m
(W1). These were backed by unsegmented 1500 $\mu$m silicon pad detectors
(MSX25). The $\Delta E-E$ telescopes were set up in a pentagon geometry
covering 40∘ $-$ 80∘ in lab. Each W1 was symmetrically placed with respect to
the target center at a distance of 59 mm. The angles 8∘ $-$ 25∘ were covered
by a 1000 $\mu$m annular detector (S3), placed at a distance of 74.1 mm from
the target. The angles 127∘ $-$ 165∘ were covered by two 32$\times$32 DSSDs of
thickness 60 $\mu$m and 140 $\mu$m (BB7), placed right and left of the beam
direction respectively. The distance of the left (right) BB7 are 102 (110) mm
from the target. These were backed by unsegmented 1500 $\mu$m silicon pad
detectors (MSX40). The total solid angle coverage of the detectors is $\sim$
32$\%$ of 4$\pi$.

Figure 2:  The $\Delta E-E$ spectrum of protons, deuterons, 3He and $\alpha$,
detected at W1 + MSX25 telescopes, from 5 MeV/u 7Be on CD2 target.

The detectors were calibrated with a 148Gd$-^{239}$Pu$-^{241}$Am$-^{244}$Cm
mixed $\alpha$-source. For S3, due to higher dynamic range, we also used the
elastic peaks from Rutherford scattering of ${5}$ MeV/u 7Be and ${5.15}$ MeV/u
12C beams on 208Pb. The light charged particles emitted from 7Be + d reaction,
were identified by $\Delta E-E$ telescopes from the energy loss spectra (Fig.
2). Elastic 7Be scattered from Carbon of CD2 was observed in S3.

Data from both W1 and BB7 were used to study excitations in 8Be. At BB7, we
detected the gs and 3.03, 11.35, 16.63 MeV excited states of 8Be. After
selecting the protons from $\Delta E-E$ spectra, the gs and 3.03 MeV could be
identified from the lab energy ($E$) vs scattering angle ($\theta$) plots
(Fig.3). The protons corresponding to 11.35 and 16.63 MeV states were
completely stopped in $\Delta E$ of BB7 and were extracted from the excitation
energy spectrum (Fig. 3 inset) after background subtraction. Since Fig. 3
shows only protons selected from $\Delta E-E$ spectrum, we do not observe
kinematic bands corresponding to these two states at
$130^{\circ}-170^{\circ}$. The higher excited states above 16 MeV were
obtained from W1 using $E$ vs $\theta$ plot (Fig. 3) of protons detected in
coincidence with $\alpha$-particles. To obtain data for forward scattered
protons, the energy-energy correlation of two coincident $\alpha$-particles at
W1 with one hit at S3 were considered. The $\alpha$-p coincidence efficiency
at W1 drops sharply for excitation energy below 16 MeV. Hence the kinematic
signatures corresponding to states below 16 MeV are absent at
$40^{\circ}-80^{\circ}$.

Table 1: Excitations (Ex) of 8Be listed in Tilley et al TI04 , and excitations (E${}_{x}^{fit}$) and widths ($\Gamma^{fit}$) obtained from fitting the excitation energy spectrum in the inset of Fig. 3. Ex (MeV $\pm$ keV) |  $\Gamma$ (keV) | E${}_{x}^{fit}$ (MeV)111The average uncertainty in the peak centroids is around $10\%$. |  $\Gamma^{fit}$ (MeV)
---|---|---|---
0.0 |  $5.57\pm 0.25$ eV | 0.02 | $0.71\pm 0.04$
$3.03\pm 10$ | $1513\pm 15$ | 3.51 | $2.00\pm 0.50$
$11.35\pm 150$ | 3500 | 11.31 | $3.77\pm 0.27$
$16.626\pm 3$ | $108.1\pm 0.5$ | 16.50 | $1.18\pm 0.51$
$16.922\pm 3$ | $74.0\pm 0.4$ | 16.99 | $1.17\pm 0.51$
$17.640\pm 1.0$ | $10.7\pm 0.5$ | 17.50 | $0.71\pm 0.05$
$18.150\pm 4$ | $138\pm 6$ | 18.19 | $0.71\pm 0.05$
18.91 | 122 | 18.80 | $0.67\pm 0.01$
$19.07\pm 30$ | $270\pm 20$ | 19.00 | $0.90\pm 0.25$
$19.235\pm 10$ | $227\pm 16$ | 19.28 | $0.71\pm 0.03$
19.40 | 645 | 19.58 | $0.95\pm 0.27$
$19.86\pm 50$ | $700\pm 100$ | 19.79 | $0.95\pm 0.25$
20.10 | $880\pm 20$ | 20.00 | $1.10\pm 0.22$
20.20 | $720\pm 20$ | 20.29 | $1.41\pm 0.69$
20.90 | $1600\pm 200$ | 20.84 | $1.75\pm 0.15$
21.50 | 1000 | 21.44 | $1.60\pm 0.60$
$22.05\pm 100$ | $270\pm 70$ | 22.10 | $0.94\pm 0.28$

The energy resolution $\sim$ 660 keV due to beam, target straggling and
detectors, limits the separation of narrowly spaced high lying resonances at
16.63 and 16.92 MeV, and resonances around $17-22$ MeV. Earlier works WI65 ;
BE17  suggest that the 16.63 MeV state is populated considerably more than
16.92 MeV. Hence, we refer to this doublet as 16.63 MeV. The inset of Fig. 3
shows excitation energy spectrum of 8Be from protons detected at W1 (blue) and
BB7 (red). The corresponding fits are shown by dark-blue and brown dashed
lines. The counts from BB7 are multiplied by two for clarity. The $16-22$ MeV
excited states are populated for the first time from 7Be(d,p)8Be∗. We did not
find any evidence of (d,$\alpha$) channel. The peaks are fitted simultaneously
by Gaussian functions, with input of excitation energies and their widths TI04
. The arrows show the excitation energies used in the fitting. The peak
centroids and widths from the resultant fits are summarized in Table 1. The
fitted widths represent the quadratic sum of 660 keV experimental resolution
and total decay width of the excited states.

Figure 3:  $E$ vs $\theta$ for protons at W1 and BB7. The kinematic lines for
different excited states of 8Be are shown for 7Be(d,p)8Be∗ at 5 MeV/u. Inset
shows the excitation energy spectrum of 8Be.

The angular distribution for 7Be + d elastic scattering from the present work
is shown in Fig. 4(a). The scattering is quasi elastic as the 0.43 MeV state
of 7Be could not be separated. However, the inelastic contribution is expected
to be small DA06 . The 7Be(d,p)8Be* angular distributions for 16.63 MeV
($\times 10^{-1}$ for clarity) and gs, 3.03, 11.35 MeV ($\times 10$ for
clarity) excited states are shown in Fig. 4(a)(b). At BB7, corresponding to
excited states up to 16.63 MeV, the proton counts are obtained from background
subtracted excitation energy histogram with an angular interval of two
degrees. For excited states $>11.35$ MeV detected at W1, we considered the
coincidence efficiency to correct for proton counts detected in coincidence
with $\alpha$-particles using NPTool MA16 . The errors in cross sections
mainly arise from statistical uncertainties, systematic uncertainties in
target thickness ($\sim 10\%$) and beam intensity ($\sim 10\%$). The optical
model potential (OMP) fit for (d,d) and finite range DWBA calculations for
(d,p) are carried out by FRESCO TH88 , shown by dashed and dotted lines in
Fig. 4. The DWBA calculations require the OMP for the entrance channel d +
7Be, exit channel p + 8Be, and the core-core p + 7Be interactions. Woods-Saxon
interaction potentials are used PE76 .

Table 2: Optical model parameters used in the present work. $V$ and $W$ are the real and imaginary depths in MeV, $r$ and $a$ are the radius and diffuseness in fm. $R_{x}=r_{x}A^{1/3}$ fm ($x=V,S,SO,C$). Channel | $V$ | $r_{V}$ | $a_{V}$ | $W_{S}$ | $r_{S}$ | $a_{S}$ | $V_{SO}$ | $r_{SO}$ | $a_{SO}$ | $r_{C}$
---|---|---|---|---|---|---|---|---|---|---
d + 7Be | 80.98 | 1.35 | 0.83 | 36.91 | 2.21 | 0.10 | 2.08 | 0.49 | 0.42 | 1.30
p + 7Be | 92.07 | 0.87 | 0.89 | 1.23 | 0.10 | 0.10 | 16.82 | 1.34 | 0.12 | 1.14
p + 8Be | 82.60 | 1.10 | 0.41 | 1.97 | 1.10 | 1.30 | 5.54 | 1.14 | 0.57 | 1.14
Figure 4:  Angular distributions for 7Be(d,d) and 7Be(d,p) to 16.63 MeV (a);
to 0.0, 3.03 and 11.35 MeV states (b). The corresponding FRESCO TH88
calculations are shown by dashed and dotted lines respectively. Figure 5: (a)
Excitation function for 7Be(d,p)8Be∗. The solid triangle, diamond, square and
circles correspond to total cross-sections due to gs, gs + 3.03, gs + 3.03 +
11.35 and gs + 3.03 + 11.35 + 16.63 MeV states respectively. The data in
green, brown, blue and magenta are the measurements of KA60 ; AN05 ; RI19  and
the present work. The violet (gs), cyan (gs + 3.03), yellow (gs + 3.03 +
11.35) and red (gs + 3.03 + 11.35 + 16.63) MeV bands are TALYS calculations
normalized to the present data at 7.8 MeV (green vertical line). The bands do
not include systematic uncertainty due to extrapolation. (b) The S-factor
representation of the excitation function. The red dotted line is the estimate
by Parker PA72 .

We started with OMP for d + 7Be PA17 , p + 7Be WA69  and used SFRESCO TH88  to
arrive at the final OMP (Table 2) by minimizing $\chi^{2}$. Since p + 8Be
elastic data are not available, we used OMP of p + 8Li PA17  (Table 2) for the
exit channel. The n + 7Be binding potential SC71  has a Wood-Saxon shape with
$r=1.36$ fm and $a=0.55$ fm. Its depth is adjusted to reproduce effective
neutron separation energy for each state of 8Be, while a spin-orbit potential
with same geometry and fixed well depth of 9 MeV is included. A Gaussian
interaction $V_{np}(r)=-V_{0}\exp(-r^{2}/r^{2}_{0})$ is used for the n$-$p
system with $V_{0}=72.15$ MeV and $r_{0}=1.484$ fm AU87 . The DWBA
calculations are in good agreement with the data for gs, 3.03 and 11.35 MeV
states in a limited angular range while for the 16.63 MeV state, only forward
angles up to $\sim 40^{\circ}$ could be reproduced. Consideration of compound
nuclear contributions, coupled-channel calculations incorporating collective
excitations of 7Be, inclusion of deuteron breakup and a d + 7Be adiabatic
potential might result in a better fit at large angles. The experimental
spectroscopic factors $C^{2}S$, are extracted by normalizing the calculated
DWBA cross-sections to measured cross-sections. The average values of $C^{2}S$
for the excited states of 8Be are compared with the OXBASH Br04  shell model
calculations Ku74  in Table 3. Assuming full isotropy for the gs, 3.03 and
11.35 MeV states, we obtain the total cross section $\sigma$ for these states.
For 16.63 MeV, $\sigma$ is obtained by connecting the differential cross
sections and integrating (Table 3).

Table 3: The spectroscopic factors $C^{2}S$ from the present work as compared to the theoretical predictions of Ku74 . Ex (MeV) | T | $nl_{j}$ | $C^{2}S$ | KumarKu74  |  $\sigma$ (mb)
---|---|---|---|---|---
0.0 | 0 | $1p_{3/2}$ | $1.19\pm 0.22$ | 1.51 | $14.2\pm 2.6$
3.03 | 0 | $1p_{3/2}$ | $1.11\pm 0.20$ | 0.91 | $65.9\pm 12.1$
11.35 | 0 | $1f_{7/2}$ | $3.20\pm 0.59$ | $-$ | $82.3\pm 15.1$
16.63 | $0+1$ | $1p_{3/2}$ | $0.35\pm 0.07$ | 0.34 | $87.5\pm 9.2$

To obtain the excitation functions of 7Be(d,p)8Be∗, TALYS-1.95 KO19
calculations are carried out and normalized to present experimental data. In
Fig. 5(a), the normalized excitation functions from TALYS for gs, gs + 3.03,
gs + 3.03 + 11.35 and gs + 3.03 + 11.35 + 16.63 MeV (colored bands to include
error bars of the present data) have been compared to the data of KA60 ; AN05
; RI19 . The calculations agree very well with KA60 ; AN05  outside the Gamow
window, for Ecm $>$ 0.7 MeV. The data RI19  for gs and 3.03 MeV are well below
corresponding TALYS calculations (cyan). However, the data within and near
Gamow window have relatively large errors. Inside Gamow window, TALYS
calculations for higher excited states (cyan, yellow, red) overestimate AN05 ;
RI19  except at 0.22 MeV, within error bars. The systematic uncertainties due
to extrapolation of the total S-factor from Ecm = 7.8 MeV to BBN energies is
$\sim$ 48$\%$. This arises due to various phenomenological and microscopic
models for level densities ($\sim$ 46$\%$), choice of global deuteron ($\sim$
10$\%$) and proton ($\sim$ 10$\%$) OMP in TALYS. In Fig. 5(b), the
corresponding astrophysical S-factor is shown. The S-factor band with
contributions from gs + 3.03 + 11.35 MeV (yellow) converges to an average
value of $\sim 95.6$ MeV b in the Gamow window which is close to Parker’s
estimate of 100 MeV b. The S-factor band (red) with the contributions of all
the states including 16.63 MeV gives a maximum of 167 MeV b inside the Gamow
window, close to 162.0 MeV b for the claimed new resonance at 0.36 MeV RI19 .
If we assume a constant S-factor of 167 MeV b, then the ratio of reaction rate
from the present work with CA88  at the relevant BBN energies is less than 2
whereas for solving the Li problem this ratio needs to be around 100 CO04 .
Therefore, the BBN calculations of RI19  seem to overestimate the importance
of the 7Be + d reaction. Even the maximum S-factor inferred from this
measurement would reduce the primordial Li abundance by less than 1$\%$ FI20 ;
DA20  with respect to previous expectation CA88  and thereby fail to alleviate
the discrepancy between theory and observation.

In summary, the present experiment reports the first measurement of all
resonances in the 7Be(d,p)8Be∗ channel up to 22 MeV. In particular, the 16.63
MeV state is analyzed in the context of the cosmological lithium problem. The
measurement was carried out at a much higher center-of-mass energy of 7.8 MeV
compared to the Gamow window. This facilitated populating the previously
unseen higher excitations of 8Be with good statistics and their contributions
to the total cross section are studied. The existing data within Gamow window
has large error bars in energy as well as cross sections. The TALYS
calculations normalized to the present data give an estimate of the
contributions of resonance excitations in the (d,p) channel. It is apparent
that inclusion of the 16.63 MeV state may lead to a maximum S-factor of 167
MeV b, higher than earlier used value of 100 MeV b PA72 . But it does not
fully account for the Lithium anomaly and our present understanding of nuclear
physics may not solve this problem.

###### Acknowledgements.

The authors thank the ISOLDE engineers in charge, RILIS team and Target Group
at CERN for their support. D. Gupta acknowledges research funding from the
European Union’s Horizon 2020 research and innovation programme under grant
agreement no. 654002 (ENSAR2) and ISRO, Government of India under grant no.
ISRO/RES/2/378/15$-$16\. O. Tengblad would like to acknowledge the support by
the Spanish Funding Agency (AEI/FEDER, EU) under the project
PID2019-104390GB-I00. I. Martel would like to acknowledge the support by the
Ministry of Science, Innovation and Universities of Spain (Grant No.
PGC2018-095640-B-I00). J. Cederkall acknowledges grants from the Swedish
Research Council (VR) under contract numbers VR-2017-00637 and VR-2017-03986
as well as grants from the Royal Physiographical Society. J. Park would like
to acknowledge the support by Institute for Basic Science (IBS-R031-D1). S.
Szwec acknowledges support by the Academy of Finland (Grant No. 307685).

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