the square and pentagon in Figure 10(c) and Figure 10(d). The entropic
constraints associated with $\mathcal{C}_{n}$ are the mutual independence of
the ancestors $H(\lambda_{\Omega})=\sum_{i\in\Omega}H(\lambda_{i})$ and the
local Markov conditions

$\displaystyle
0=I(X_{i}:X_{\Omega-\\{i\\}},\lambda_{\Omega-\operatorname{Pa}_{i}}\ooalign{$\phantom{{},{}}$\cr$|$}\lambda_{\operatorname{Pa}_{i}}),$

for all observable variables $X_{i}$, where $\Omega=[n]$ and
$\operatorname{Pa}_{i}=\\{i,\,(i\kern-2.0pt\mod n)+1\\}$.

In Fritz and Chaves (2013), the authors employ a Fourier-Motzkin elimination
using the PORTA open source software package in an attempt to compute the
projection of $\mathcal{C}_{3}$ to the marginal scenario
$\mathcal{M}=2^{X_{\Omega}}$ that includes all observable terms. However,
their computations did not complete – which shows that even such a simple
model can prove difficult for FME if not using proper protection redundancy
elimination. This particular problem is solved in Chaves  _et al._ (2014a),
showing that there are only three non-trivial classes of inequalities:

$\displaystyle 0$ $\displaystyle\leq
4H_{12}+4H_{13}+4H_{23}-2H_{123}-5H_{1}-5H_{2}-5H_{3},$ $\displaystyle 0$
$\displaystyle\leq
3H_{12}+2H_{13}+2H_{23}-\phantom{2}H_{123}-3H_{1}-3H_{2}-3H_{3},$
$\displaystyle 0$ $\displaystyle\leq H_{12}+H_{23}-H_{1}-H_{2}-H_{3}.$

We now extend these results by computing the full projection of the square
$\mathcal{C}_{4}$ and pentagon $\mathcal{C}_{5}$. The results are obtained
using an FM elimination with full LP-based redundancy removal. The
implementation can be found in Gläßle (2016b). All classes of facets can be
found online Gläßle (2016a) and are additionally listed in Section XVI. In
total, the marginal cone of $\mathcal{C}_{4}$ has 12 distinct classes of
facets, the $\mathcal{C}_{5}$ marginal cone has 39 classes of facets.

Starting from a system with no redundancies, the computations take
$7.4\text{\,}\mathrm{s}\text{/}$ and $340\text{\,}\mathrm{s}\text{/}$ for
$\mathcal{C}_{4}$ and $\mathcal{C}_{5}$, respectively (on a usual desktop PC).
This underlines how LP-based redundancy removal can be a major advantage in
performing FM eliminations.

###  XVI List of inequalities

In the following we list the facets that were found using the techniques
discussed in the main text. Note that we list expressions $I_{k}$ which should
be read as inequalities $I_{k}\geq 0$. Furthermore, if multiple inequalities
are equivalent by a symmetry of the cone (i.e. relabeling of variables), we
list only one representative. The data can also be found online in the
repository of the pystif software package Gläßle (2016a).

####  XVI.1 Machine readable data

We include quick response (QR) codes to allow easy data acquisition. Due to
the capacity limitations of QR codes the data is gzipped. The raw data files
can be unpacked with a wide range of popular compression software such as 7z
on Windows or the prevalent gunzip tool on UNIX-like systems:

    gunzip -c raw.dat > friendly.txt

The unzipped data is also directly attached to the PDF and can be accessed by
clicking on the QR code images. The uncompressed data is a self-documenting
text file that can e.g. be directly used as input file for any of the pystif
utilities. However, the full pystif stack is not required – the text format is
suitable for machine reading and easy to parse. For example, in python you can
use

⬇

>>> numpy.loadtxt(’friendly.txt’)

to obtain a numpy array whose row vectors $\boldsymbol{I}$ correspond to
linear inequalities $\boldsymbol{I}\cdot\boldsymbol{h}\geq 0$ where
$\boldsymbol{h}$ is an entropy vector. The column order is generally based on
number of arguments and party affiliation but is also documented within the
file.

####  XVI.2 8D cone

Full characterization of the marginal cone with all one-body terms for the
tripartite Bell scenario.

$I_{0}=H(A_{1}\,B_{1}\,C_{2})$
$I_{1}=H(A_{1}\,B_{2}\,C_{1})+H(A_{2}\,B_{1}\,C_{2})-H(A_{1}\,B_{2}\,C_{2})$
$I_{2}=H(A_{1}\,B_{1}\,C_{2})+H(A_{2}\,B_{2}\,C_{2})-H(A_{2}\,B_{1}\,C_{2})$
$I_{3}=H(A_{1}\,B_{2}\,C_{2})+H(A_{2}\,B_{1}\,C_{2})+H(A_{2}\,B_{2}\,C_{1})-H(A_{1}\,B_{1}\,C_{1})-H(A_{2}\,B_{2}\,C_{2})$
$I_{4}=H(A_{1}\,B_{1}\,C_{2})+H(A_{1}\,B_{2}\,C_{1})+H(A_{2}\,B_{2}\,C_{2})-2H(A_{1}\,B_{2}\,C_{2})$
$I_{5}=H(A_{1}\,B_{1}\,C_{2})+H(A_{1}\,B_{2}\,C_{1})+H(A_{2}\,B_{2}\,C_{2})-H(A_{1}\,B_{2}\,C_{2})-H(A_{2}\,B_{2}\,C_{1})$
$I_{6}=2H(A_{1}\,B_{1}\,C_{1})+2H(A_{2}\,B_{1}\,C_{2})+2H(A_{2}\,B_{2}\,C_{1})-H(A_{1}\,B_{1}\,C_{2})-H(A_{1}\,B_{2}\,C_{1})-H(A_{2}\,B_{1}\,C_{1})-H(A_{2}\,B_{2}\,C_{2})$

Figure 11: 8D cone

####  XVI.3 12D cone

Full characterization of the marginal cone with all two-body terms for the
tripartite Bell scenario.

$I_{0}=H(B_{1}\,C_{2})+H(B_{2}\,C_{1})-H(B_{2}\,C_{2})$
$I_{1}=H(A_{1}\,B_{2})+H(B_{1}\,C_{2})-H(A_{1}\,C_{2})$
$I_{2}=H(A_{1}\,B_{2})+H(B_{1}\,C_{2})-H(B_{2}\,C_{2})$
$I_{3}=H(A_{1}\,B_{1})+H(B_{1}\,C_{2})-H(A_{1}\,C_{2})$
$I_{4}=H(A_{1}\,B_{1})+H(A_{2}\,B_{2})+H(B_{2}\,C_{1})-H(A_{1}\,B_{2})-H(A_{2}\,C_{1})$
$I_{5}=H(A_{1}\,B_{1})+H(A_{2}\,C_{2})+H(B_{1}\,C_{2})-H(A_{2}\,B_{1})-H(A_{1}\,C_{2})$
$I_{6}=H(A_{1}\,B_{1})+H(A_{2}\,C_{2})+H(B_{2}\,C_{2})-H(A_{2}\,B_{2})-H(A_{1}\,C_{2})$
$I_{7}=H(A_{1}\,B_{1})+H(A_{2}\,B_{2})+H(A_{2}\,C_{2})+H(B_{1}\,C_{2})-H(A_{2}\,B_{1})-H(A_{1}\,C_{2})-H(B_{2}\,C_{2})$
$I_{8}=H(A_{1}\,B_{1})+H(A_{2}\,C_{1})+H(B_{1}\,C_{2})+H(B_{2}\,C_{1})-H(A_{2}\,B_{2})-H(A_{1}\,C_{2})-H(B_{1}\,C_{1})$
$I_{9}=H(A_{1}\,B_{1})+H(A_{2}\,B_{1})+H(A_{2}\,B_{2})+H(A_{1}\,C_{1})+H(A_{1}\,C_{2})+H(B_{2}\,C_{1})+H(B_{2}\,C_{2})-3H(A_{1}\,B_{2})-H(A_{2}\,C_{1})-H(B_{1}\,C_{2})$
$I_{10}=2H(A_{1}\,B_{1})+2H(B_{1}\,C_{2})+H(A_{1}\,C_{1})+H(A_{2}\,C_{1})+H(A_{2}\,C_{2})-3H(A_{1}\,C_{2})-H(A_{2}\,B_{1})-H(B_{1}\,C_{1})$
$I_{11}=3H(A_{1}\,B_{1})+2H(B_{1}\,C_{2})+H(A_{1}\,C_{1})+H(A_{2}\,C_{1})+H(A_{2}\,C_{2})+H(B_{2}\,C_{1})+H(B_{2}\,C_{2})-4H(A_{1}\,C_{2})-2H(B_{1}\,C_{1})-H(A_{2}\,B_{2})$
$I_{12}=2H(A_{2}\,B_{2})+2H(A_{1}\,C_{1})+H(A_{1}\,B_{2})+H(A_{2}\,B_{1})+H(A_{1}\,C_{2})+H(A_{2}\,C_{1})+H(B_{1}\,C_{1})+H(B_{1}\,C_{2})+H(B_{2}\,C_{2})-4H(B_{2}\,C_{1})-2H(A_{1}\,B_{1})-2H(A_{2}\,C_{2})$
$I_{13}=3H(B_{2}\,C_{2})+2H(A_{2}\,B_{2})+2H(A_{1}\,C_{2})+H(A_{1}\,B_{1})+H(A_{2}\,B_{1})+H(A_{1}\,C_{1})+H(A_{2}\,C_{1})-4H(A_{2}\,C_{2})-3H(A_{1}\,B_{2})-H(B_{1}\,C_{1})$

Figure 12: 12D cone

####  XVI.4 14D cone

Full characterization of the marginal cone with all one- and three-body terms
for the tripartite Bell scenario.

$I_{0}=H(B_{1})$ $I_{1}=H(A_{1})+H(B_{1})+H(C_{2})-H(A_{1}\,B_{1}\,C_{2})$
$I_{2}=H(A_{2}\,B_{2}\,C_{2})-H(B_{2})$
$I_{3}=H(A_{2}\,B_{1}\,C_{2})+H(A_{1})-H(A_{1}\,B_{1}\,C_{2})$
$I_{4}=H(A_{1}\,B_{2}\,C_{2})+H(A_{2}\,B_{2}\,C_{1})-H(A_{1}\,B_{2}\,C_{1})-H(B_{2})$
$I_{5}=H(A_{1}\,B_{2}\,C_{2})+H(A_{2}\,B_{1}\,C_{2})+H(A_{2}\,B_{2}\,C_{2})-H(A_{1}\,B_{1}\,C_{2})-H(A_{2})-H(B_{2})$
$I_{6}=H(A_{1}\,B_{2}\,C_{1})+H(A_{2}\,B_{1}\,C_{1})+H(A_{2}\,B_{2}\,C_{2})-H(A_{1}\,B_{1}\,C_{2})-H(B_{2})-H(C_{1})$
$I_{7}=H(A_{1}\,B_{1}\,C_{1})+H(A_{2}\,B_{2}\,C_{1})+H(A_{2}\,B_{2}\,C_{2})-H(A_{1}\,B_{1}\,C_{2})-H(B_{2})-H(C_{1})$
$I_{8}=H(A_{1}\,B_{1}\,C_{1})+H(A_{2}\,B_{2}\,C_{1})+H(A_{2}\,B_{2}\,C_{2})-H(A_{2}\,B_{1}\,C_{2})-H(B_{2})-H(C_{1})$
$I_{9}=2H(A_{2}\,B_{2}\,C_{2})+H(A_{1}\,B_{2}\,C_{1})+H(A_{2}\,B_{1}\,C_{1})-H(A_{1}\,B_{1}\,C_{1})-H(A_{2}\,B_{2}\,C_{1})-H(A_{2})-H(B_{2})$
$I_{10}=2H(A_{2}\,B_{2}\,C_{2})+H(A_{1}\,B_{2}\,C_{1})+H(A_{2}\,B_{1}\,C_{1})-H(A_{1}\,B_{1}\,C_{2})-H(A_{2}\,B_{2}\,C_{1})-H(A_{2})-H(B_{2})$
$I_{11}=H(A_{1}\,B_{1}\,C_{1})+H(A_{1}\,B_{2}\,C_{2})+H(A_{2}\,B_{1}\,C_{2})+H(A_{2}\,B_{2}\,C_{2})+H(C_{2})-H(A_{1}\,B_{1}\,C_{2})-H(A_{1}\,B_{2}\,C_{1})-H(A_{2})-H(B_{2})$
$I_{12}=H(A_{1}\,B_{1}\,C_{1})+H(A_{1}\,B_{2}\,C_{2})+H(A_{2}\,B_{1}\,C_{1})+H(A_{2}\,B_{1}\,C_{2})+H(B_{1})-H(A_{1}\,B_{1}\,C_{2})-H(A_{2}\,B_{2}\,C_{1})-H(A_{2})-H(C_{1})$
$I_{13}=2H(A_{1}\,B_{1}\,C_{1})+H(A_{1}\,B_{2}\,C_{2})+H(A_{2}\,B_{1}\,C_{1})+H(A_{2}\,B_{1}\,C_{2})-H(A_{1}\,B_{1}\,C_{2})-H(A_{2}\,B_{2}\,C_{1})-H(A_{1})-H(A_{2})-H(C_{1})$
$I_{14}=2H(A_{1}\,B_{1}\,C_{1})+H(A_{1}\,B_{2}\,C_{2})+H(A_{2}\,B_{1}\,C_{1})+H(A_{2}\,B_{1}\,C_{2})-H(A_{1}\,B_{1}\,C_{2})-H(A_{2}\,B_{2}\,C_{2})-H(A_{1})-H(A_{2})-H(C_{1})$
$I_{15}=2H(A_{2}\,B_{1}\,C_{1})+H(A_{1}\,B_{2}\,C_{1})+H(A_{1}\,B_{2}\,C_{2})+H(A_{2}\,B_{2}\,C_{2})-2H(A_{1}\,B_{1}\,C_{2})-H(A_{2})-H(B_{2})-H(C_{1})$
$I_{16}=H(A_{1}\,B_{1}\,C_{1})+H(A_{1}\,B_{2}\,C_{2})+H(A_{2}\,B_{1}\,C_{1})+H(A_{2}\,B_{1}\,C_{2})+H(A_{2}\,B_{2}\,C_{2})-H(A_{1}\,B_{1}\,C_{2})-H(A_{1}\,B_{2}\,C_{1})-H(A_{2})-H(B_{2})-H(C_{1})$
$I_{17}=2H(A_{1}\,B_{1}\,C_{1})+2H(A_{1}\,B_{2}\,C_{2})+2H(A_{2}\,B_{2}\,C_{1})-2H(A_{2}\,B_{1}\,C_{2})-H(A_{1}\,B_{2}\,C_{1})-H(A_{1})-H(B_{2})-H(C_{1})$
$I_{18}=H(A_{1}\,B_{1}\,C_{1})+H(A_{1}\,B_{2}\,C_{1})+H(A_{1}\,B_{2}\,C_{2})+H(A_{2}\,B_{1}\,C_{1})+H(A_{2}\,B_{1}\,C_{2})+H(A_{2}\,B_{2}\,C_{2})-3H(A_{1}\,B_{1}\,C_{2})-H(A_{2})-H(B_{2})-H(C_{1})$
$I_{19}=H(A_{1}\,B_{1}\,C_{1})+H(A_{1}\,B_{2}\,C_{1})+H(A_{1}\,B_{2}\,C_{2})+H(A_{2}\,B_{1}\,C_{1})+H(A_{2}\,B_{1}\,C_{2})+H(A_{2}\,B_{2}\,C_{2})-2H(A_{2}\,B_{2}\,C_{1})-H(A_{1}\,B_{1}\,C_{2})-H(A_{2})-H(B_{2})-H(C_{1})$
$I_{20}=2H(A_{1}\,B_{1}\,C_{1})+2H(A_{1}\,B_{2}\,C_{2})+2H(A_{2}\,B_{1}\,C_{2})+H(A_{2}\,B_{1}\,C_{1})-2H(A_{2}\,B_{2}\,C_{1})-H(A_{1}\,B_{1}\,C_{2})-H(A_{1})-H(A_{2})-H(C_{1})-H(C_{2})$
$I_{21}=2H(A_{1}\,B_{2}\,C_{2})+2H(A_{2}\,B_{2}\,C_{1})+H(A_{1}\,B_{1}\,C_{1})+H(A_{2}\,B_{1}\,C_{1})+H(A_{2}\,B_{1}\,C_{2})-3H(A_{1}\,B_{1}\,C_{2})-2H(B_{2})-H(A_{2})-H(C_{1})$

Figure 13: 14D cone

####  XVI.5 18D cone

Listing of discovered facets of the marginal cone of the tripartite Bell
scenario where the marginal scenario contains all one- and two-body terms.
These facets do not provide a full characterization.

$I_{0}=H(B_{1})+H(C_{1})-H(B_{1}\,C_{1})$ $I_{1}=H(A_{1}\,B_{2})-H(B_{2})$
$I_{2}=H(A_{1}\,B_{2})+H(A_{1}\,C_{1})-H(B_{2}\,C_{1})-H(A_{1})$
$I_{3}=H(A_{1}\,C_{1})+H(A_{2}\,C_{1})+H(A_{2}\,C_{2})-H(A_{1}\,C_{2})-H(A_{2})-H(C_{1})$
$I_{4}=H(A_{1}\,B_{2})+H(A_{1}\,C_{2})+H(A_{2}\,C_{1})+H(A_{2}\,C_{2})+H(B_{2}\,C_{1})-H(A_{1}\,C_{1})-H(B_{2}\,C_{2})-H(A_{1})-H(A_{2})-H(C_{1})$
$I_{5}=H(A_{1}\,B_{2})+H(A_{1}\,C_{1})+H(A_{2}\,C_{1})+H(A_{2}\,C_{2})+H(B_{2}\,C_{2})-H(A_{2}\,B_{2})-H(A_{1}\,C_{2})-H(A_{2})-H(B_{2})-H(C_{1})$
$I_{6}=H(A_{1}\,B_{2})+H(A_{2}\,B_{2})+H(A_{1}\,C_{1})+H(A_{2}\,C_{1})+H(A_{2}\,C_{2})-H(A_{1}\,C_{2})-H(B_{2}\,C_{1})-H(A_{2})-H(B_{2})-H(C_{1})$
$I_{7}=H(A_{1}\,C_{1})+H(A_{1}\,C_{2})+H(A_{2}\,C_{1})+H(B_{1}\,C_{1})+H(B_{1}\,C_{2})-H(A_{1}\,B_{1})-H(A_{2}\,C_{2})-2H(C_{1})-H(C_{2})$
$I_{8}=H(A_{1}\,B_{1})+H(A_{1}\,B_{2})+H(A_{2}\,B_{1})+H(A_{2}\,B_{2})+H(A_{1}\,C_{2})+H(B_{1}\,C_{1})-H(A_{2}\,C_{2})-H(B_{2}\,C_{1})-2H(A_{1})-H(A_{2})-H(B_{1})$
$I_{9}=H(A_{1}\,B_{2})+H(A_{2}\,B_{2})+H(A_{1}\,C_{2})+H(A_{2}\,C_{1})+H(B_{1}\,C_{1})+H(B_{1}\,C_{2})-H(A_{1}\,C_{1})-H(A_{2}\,C_{2})-H(A_{1})-H(B_{1})-H(B_{2})-H(C_{1})$
$I_{10}=H(A_{1}\,B_{1})+H(A_{1}\,B_{2})+H(A_{2}\,B_{2})+H(A_{1}\,C_{2})+H(A_{2}\,C_{1})+H(A_{2}\,C_{2})+H(B_{1}\,C_{1})-H(A_{2}\,B_{1})-H(A_{1}\,C_{1})-H(B_{2}\,C_{2})-2H(A_{1})-H(A_{2})-H(C_{1})$
$I_{11}=H(A_{1}\,B_{2})+H(A_{2}\,B_{1})+H(A_{2}\,B_{2})+H(A_{1}\,C_{2})+H(B_{1}\,C_{1})+H(B_{1}\,C_{2})+H(B_{2}\,C_{1})-H(A_{1}\,B_{1})-H(A_{2}\,C_{1})-H(B_{2}\,C_{2})-H(A_{1})-H(A_{2})-H(B_{1})-H(C_{1})$
$I_{12}=H(A_{1}\,B_{1})+H(A_{1}\,B_{2})+H(A_{2}\,B_{2})+H(A_{2}\,C_{1})+H(A_{2}\,C_{2})+H(B_{1}\,C_{1})+H(B_{1}\,C_{2})-H(A_{2}\,B_{1})-H(A_{1}\,C_{1})-H(B_{2}\,C_{2})-H(A_{1})-H(A_{2})-H(B_{1})-H(C_{1})$
$I_{13}=H(A_{1}\,B_{2})+H(A_{2}\,B_{2})+H(A_{1}\,C_{2})+H(A_{2}\,C_{1})+H(A_{2}\,C_{2})+H(B_{1}\,C_{1})+H(B_{1}\,C_{2})-H(A_{2}\,B_{1})-H(A_{1}\,C_{1})-H(B_{2}\,C_{2})-H(A_{1})-H(A_{2})-H(C_{1})-H(C_{2})$
$I_{14}=H(A_{1}\,B_{2})+H(A_{2}\,B_{1})+H(A_{2}\,B_{2})+H(A_{1}\,C_{1})+H(B_{1}\,C_{1})+H(B_{1}\,C_{2})+H(B_{2}\,C_{2})-H(A_{1}\,C_{2})-H(A_{2}\,C_{2})-H(B_{2}\,C_{1})-2H(B_{1})-H(A_{1})-H(B_{2})$
$I_{15}=H(A_{1}\,B_{2})+H(A_{2}\,B_{1})+H(A_{2}\,B_{2})+H(A_{1}\,C_{1})+H(A_{1}\,C_{2})+H(B_{1}\,C_{1})+H(B_{1}\,C_{2})-H(A_{1}\,B_{1})-H(A_{2}\,C_{1})-H(A_{2}\,C_{2})-H(A_{1})-H(B_{1})-H(B_{2})-H(C_{1})$
$I_{16}=H(A_{1}\,B_{2})+H(A_{2}\,B_{1})+H(A_{2}\,B_{2})+H(B_{1}\,C_{1})+H(B_{1}\,C_{2})+H(B_{2}\,C_{1})+H(B_{2}\,C_{2})-H(A_{1}\,B_{1})-H(A_{2}\,C_{1})-H(A_{2}\,C_{2})-2H(B_{2})-H(B_{1})-H(C_{1})$
$I_{17}=2H(B_{1}\,C_{2})+H(A_{1}\,B_{2})+H(A_{2}\,B_{1})+H(A_{2}\,B_{2})+H(A_{1}\,C_{2})+H(B_{1}\,C_{1})+H(B_{2}\,C_{1})-H(A_{1}\,B_{1})-H(A_{2}\,C_{1})-H(A_{2}\,C_{2})-2H(B_{1})-H(A_{1})-H(B_{2})-H(C_{1})$
$I_{18}=2H(A_{1}\,B_{2})+2H(B_{1}\,C_{1})+H(A_{1}\,B_{1})+H(A_{2}\,B_{2})+H(A_{1}\,C_{2})+H(A_{2}\,C_{1})+H(A_{2}\,C_{2})+H(B_{1}\,C_{2})+H(B_{2}\,C_{1})-2H(A_{1}\,C_{1})-2H(B_{2}\,C_{2})-H(A_{2}\,B_{1})-2H(A_{1})-2H(C_{1})-H(A_{2})-H(B_{1})$
$I_{19}=2H(A_{2}\,B_{2})+2H(A_{2}\,C_{2})+2H(B_{1}\,C_{1})+H(A_{1}\,B_{1})+H(A_{1}\,B_{2})+H(A_{1}\,C_{2})+H(A_{2}\,C_{1})+H(B_{1}\,C_{2})+H(B_{2}\,C_{1})-2H(A_{2}\,B_{1})-2H(A_{1}\,C_{1})-H(B_{2}\,C_{2})-2H(B_{2})-2H(C_{1})-2H(C_{2})-H(A_{1})$
$I_{20}=2H(A_{2}\,C_{1})+2H(A_{2}\,C_{2})+2H(B_{1}\,C_{1})+H(A_{1}\,B_{1})+H(A_{1}\,B_{2})+H(A_{2}\,B_{2})+H(A_{1}\,C_{2})+H(B_{1}\,C_{2})+H(B_{2}\,C_{1})-2H(A_{2}\,B_{1})-2H(A_{1}\,C_{1})-H(B_{2}\,C_{2})-3H(C_{1})-2H(C_{2})-H(A_{1})-H(B_{2})$
$I_{21}=3H(B_{1}\,C_{1})+2H(A_{1}\,B_{1})+2H(A_{1}\,B_{2})+2H(A_{2}\,B_{2})+2H(A_{2}\,C_{1})+2H(A_{2}\,C_{2})+H(A_{1}\,C_{2})+H(B_{1}\,C_{2})+H(B_{2}\,C_{1})-3H(A_{2}\,B_{1})-3H(A_{1}\,C_{1})-H(B_{2}\,C_{2})-3H(C_{1})-2H(A_{1})-2H(B_{2})-2H(C_{2})$

Figure 14: 18D cone

####  XVI.6 20D cone

Listing of discovered facets of the marginal cone of the tripartite Bell
scenario where the marginal scenario contains all two- and three-body terms.
These facets do not provide a full characterization.

$I_{0}=H(A_{2}\,B_{2})+H(A_{2}\,C_{2})+H(B_{2}\,C_{2})-2H(A_{2}\,B_{2}\,C_{2})$
$I_{1}=H(A_{2}\,B_{2})+H(A_{2}\,C_{1})+H(B_{2}\,C_{2})-H(A_{2}\,B_{2}\,C_{1})-H(A_{2}\,B_{2}\,C_{2})$
$I_{2}=H(A_{2}\,B_{1})+H(A_{1}\,C_{1})+H(B_{1}\,C_{2})-H(A_{1}\,B_{1}\,C_{1})-H(A_{2}\,B_{1}\,C_{2})$
$I_{3}=H(A_{1}\,B_{1})+H(A_{2}\,B_{2})+H(A_{1}\,C_{2})+H(B_{2}\,C_{1})-H(A_{1}\,B_{1}\,C_{2})-H(A_{2}\,B_{2}\,C_{1})-H(A_{1}\,B_{2})$
$I_{4}=H(A_{1}\,B_{1})+H(A_{1}\,C_{1})+H(B_{1}\,C_{2})+H(B_{2}\,C_{1})-H(A_{1}\,B_{1}\,C_{2})-H(A_{1}\,B_{2}\,C_{1})-H(B_{1}\,C_{1})$
$I_{5}=H(A_{2}\,B_{1}\,C_{1})-H(A_{2}\,C_{1})$
$I_{6}=H(A_{1}\,B_{1}\,C_{2})+2H(A_{2}\,B_{1})+2H(A_{2}\,C_{2})+H(B_{1}\,C_{1})+H(B_{2}\,C_{1})+H(B_{2}\,C_{2})-2H(A_{2}\,B_{1}\,C_{2})-H(A_{1}\,B_{2}\,C_{1})-H(A_{2}\,B_{2})-H(A_{2}\,C_{1})-H(B_{1}\,C_{2})$
$I_{7}=H(A_{2}\,B_{1}\,C_{2})+2H(A_{1}\,B_{1})+2H(A_{1}\,C_{2})+H(B_{1}\,C_{1})+H(B_{2}\,C_{1})+H(B_{2}\,C_{2})-2H(A_{1}\,B_{1}\,C_{2})-H(A_{2}\,B_{1}\,C_{1})-H(A_{1}\,B_{2})-H(A_{1}\,C_{1})-H(B_{1}\,C_{2})$
$I_{8}=H(A_{1}\,B_{1}\,C_{1})+H(A_{1}\,B_{1}\,C_{2})+H(A_{2}\,B_{1}\,C_{2})-H(A_{2}\,B_{1}\,C_{1})-H(A_{1}\,B_{1})-H(B_{1}\,C_{2})$
$I_{9}=H(A_{1}\,B_{2}\,C_{1})+H(A_{1}\,B_{2}\,C_{2})+H(A_{2}\,B_{2}\,C_{2})+2H(A_{2}\,B_{2})+H(A_{2}\,C_{1})+H(B_{1}\,C_{1})+H(B_{1}\,C_{2})+H(B_{2}\,C_{1})-3H(A_{2}\,B_{2}\,C_{1})-H(A_{1}\,B_{1}\,C_{1})-H(A_{1}\,B_{2})-H(A_{2}\,B_{1})-H(B_{2}\,C_{2})$
$I_{10}=H(A_{1}\,B_{1}\,C_{2})+H(A_{1}\,B_{2}\,C_{2})+H(A_{2}\,B_{1}\,C_{2})+2H(A_{2}\,B_{2})+2H(A_{2}\,C_{2})+H(B_{1}\,C_{1})+H(B_{2}\,C_{1})-3H(A_{2}\,B_{2}\,C_{2})-H(A_{1}\,B_{2}\,C_{1})-H(A_{2}\,B_{1})-H(A_{1}\,C_{2})-H(A_{2}\,C_{1})$
$I_{11}=H(A_{1}\,B_{2}\,C_{1})+H(A_{1}\,B_{2}\,C_{2})+H(A_{2}\,B_{1}\,C_{1})+H(A_{2}\,B_{2}\,C_{1})-H(A_{2}\,B_{1}\,C_{2})-H(A_{1}\,B_{2})-H(A_{2}\,C_{1})-H(B_{2}\,C_{1})$
$I_{12}=H(A_{1}\,B_{2}\,C_{1})+H(A_{1}\,B_{2}\,C_{2})+H(A_{2}\,B_{1}\,C_{1})+H(A_{2}\,B_{2}\,C_{2})-H(A_{1}\,B_{1}\,C_{2})-H(A_{1}\,B_{2})-H(A_{2}\,C_{1})-H(B_{2}\,C_{2})$
$I_{13}=H(A_{1}\,B_{1}\,C_{2})+H(A_{1}\,B_{2}\,C_{1})+H(A_{2}\,B_{1}\,C_{2})+H(A_{2}\,B_{2}\,C_{2})-H(A_{2}\,B_{2}\,C_{1})-H(A_{1}\,B_{2})-H(A_{2}\,C_{2})-H(B_{1}\,C_{2})$
$I_{14}=H(A_{1}\,B_{2}\,C_{1})+H(A_{2}\,B_{1}\,C_{1})+H(A_{2}\,B_{2}\,C_{1})+H(A_{2}\,B_{2}\,C_{2})-H(A_{1}\,B_{1}\,C_{2})-H(A_{2}\,B_{2})-H(A_{2}\,C_{1})-H(B_{2}\,C_{1})$
$I_{15}=2H(A_{1}\,B_{2}\,C_{1})+H(A_{1}\,B_{1}\,C_{2})+H(A_{1}\,B_{2}\,C_{2})+H(A_{2}\,B_{1}\,C_{1})+H(A_{2}\,B_{2}\,C_{1})-H(A_{1}\,B_{1}\,C_{1})-H(A_{2}\,B_{2}\,C_{2})-H(A_{1}\,B_{2})-H(A_{1}\,C_{2})-H(A_{2}\,C_{1})-H(B_{2}\,C_{1})$
$I_{16}=2H(A_{2}\,B_{2}\,C_{2})+H(A_{1}\,B_{1}\,C_{2})+H(A_{1}\,B_{2}\,C_{1})+H(A_{2}\,B_{1}\,C_{2})+H(A_{2}\,B_{2}\,C_{1})-H(A_{1}\,B_{1}\,C_{1})-H(A_{1}\,B_{2}\,C_{2})-H(A_{2}\,B_{2})-H(A_{2}\,C_{2})-H(B_{1}\,C_{2})-H(B_{2}\,C_{1})$

Figure 15: 20D cone

####  XVI.7 26D cone

Listing of discovered facets of the marginal cone of the tripartite Bell
scenario where the marginal scenario contains all one-, two-, and three-body
terms. These facets do not provide a full characterization.

$I_{0}=H(B_{1})+H(C_{1})-H(B_{1}\,C_{1})$
$I_{1}=H(A_{1}\,B_{1})+H(A_{1}\,B_{2})+H(A_{2}\,B_{1})-H(A_{2}\,B_{2})-H(A_{1})-H(B_{1})$
$I_{2}=H(A_{1}\,C_{2})+H(B_{2}\,C_{2})-H(A_{1}\,B_{2}\,C_{2})-H(C_{2})$
$I_{3}=H(A_{1}\,B_{1})+H(A_{2}\,B_{1})+H(A_{1}\,C_{1})+H(A_{1}\,C_{2})+H(A_{2}\,C_{1})-H(A_{2}\,B_{1}\,C_{1})-H(A_{2}\,C_{2})-H(A_{1})-H(B_{1})-H(C_{1})$
$I_{4}=H(A_{1}\,B_{2})+H(A_{2}\,B_{1})+H(A_{2}\,B_{2})+H(A_{1}\,C_{2})+H(A_{2}\,C_{2})-H(A_{2}\,B_{2}\,C_{2})-H(A_{1}\,B_{1})-H(A_{2})-H(B_{2})-H(C_{2})$
$I_{5}=H(A_{1}\,B_{2}\,C_{2})-H(B_{2}\,C_{2})$
$I_{6}=H(A_{2}\,B_{1}\,C_{1})+H(A_{1}\,B_{1})+H(A_{1}\,B_{2})+H(A_{2}\,B_{2})-H(A_{2}\,B_{2}\,C_{1})-H(A_{2}\,B_{1})-H(A_{1})-H(B_{2})$
$I_{7}=H(A_{1}\,B_{2}\,C_{2})+H(A_{1}\,C_{1})+H(A_{2}\,C_{1})+H(A_{2}\,C_{2})-H(A_{2}\,B_{2}\,C_{1})-H(A_{1}\,C_{2})-H(A_{2})-H(C_{1})$
$I_{8}=H(A_{1}\,B_{2}\,C_{2})+H(A_{1}\,B_{1})+H(A_{2}\,B_{1})+H(A_{1}\,C_{1})+H(A_{2}\,C_{1})-H(A_{1}\,B_{1}\,C_{1})-H(A_{2}\,B_{2}\,C_{2})-H(A_{1})-H(B_{1})-H(C_{1})$
$I_{9}=H(A_{2}\,B_{1}\,C_{1})+H(A_{1}\,B_{1})+H(A_{1}\,B_{2})+H(B_{1}\,C_{2})+H(B_{2}\,C_{2})-H(A_{1}\,B_{2}\,C_{2})-H(A_{2}\,B_{2}\,C_{1})-H(A_{1})-H(B_{1})-H(C_{2})$
$I_{10}=H(A_{2}\,B_{1}\,C_{1})+H(A_{1}\,B_{1})+H(A_{1}\,B_{2})+H(A_{2}\,B_{2})+H(B_{1}\,C_{2})+H(B_{2}\,C_{2})-H(A_{1}\,B_{2}\,C_{2})-H(A_{2}\,B_{2}\,C_{1})-H(A_{2}\,B_{1})-H(A_{1})-H(B_{2})-H(C_{2})$
$I_{11}=H(A_{2}\,B_{1}\,C_{2})+H(A_{2}\,B_{1})+H(A_{2}\,B_{2})+H(B_{1}\,C_{1})+H(B_{2}\,C_{1})+H(B_{2}\,C_{2})-H(A_{2}\,B_{1}\,C_{1})-H(A_{2}\,B_{2}\,C_{2})-H(B_{1}\,C_{2})-H(A_{2})-H(B_{2})-H(C_{1})$
$I_{12}=H(A_{1}\,B_{1}\,C_{1})+H(A_{1}\,B_{2})+H(A_{2}\,B_{1})+H(A_{2}\,B_{2})+H(B_{1}\,C_{2})+H(B_{2}\,C_{2})-H(A_{1}\,B_{2}\,C_{1})-H(A_{2}\,B_{1}\,C_{2})-H(A_{1}\,B_{1})-H(A_{2})-H(B_{2})-H(C_{2})$
$I_{13}=H(A_{1}\,B_{1}\,C_{2})+H(A_{1}\,B_{2})+H(A_{2}\,B_{1})+H(A_{2}\,B_{2})+H(A_{1}\,C_{2})+H(A_{2}\,C_{2})-H(A_{2}\,B_{1}\,C_{2})-H(A_{2}\,B_{2}\,C_{2})-H(A_{1}\,B_{1})-H(A_{2})-H(B_{2})-H(C_{2})$
$I_{14}=H(A_{1}\,B_{1}\,C_{2})+H(A_{2}\,B_{1}\,C_{2})+H(A_{1}\,B_{2})+H(A_{2}\,B_{2})+H(A_{1}\,C_{1})-H(A_{2}\,B_{2}\,C_{2})-H(A_{2}\,C_{1})-H(B_{1}\,C_{2})-H(A_{1})-H(B_{2})$
$I_{15}=H(A_{1}\,B_{2}\,C_{1})+H(A_{2}\,B_{2}\,C_{2})+H(A_{1}\,B_{1})+H(A_{2}\,B_{1})+H(A_{1}\,C_{1})+H(A_{1}\,C_{2})+H(A_{2}\,C_{1})-H(A_{2}\,B_{1}\,C_{1})-H(A_{2}\,B_{2}\,C_{1})-H(A_{1}\,B_{2})-H(A_{2}\,C_{2})-H(A_{1})-H(B_{1})-H(C_{1})$
$I_{16}=H(A_{1}\,B_{1}\,C_{2})+H(A_{1}\,B_{2}\,C_{2})+H(A_{1}\,B_{1})+H(A_{2}\,B_{1})+H(A_{2}\,B_{2})+H(A_{1}\,C_{1})+H(A_{2}\,C_{1})-H(A_{2}\,B_{1}\,C_{1})-H(A_{2}\,B_{1}\,C_{2})-H(A_{1}\,B_{2})-H(A_{1}\,C_{2})-H(A_{2})-H(B_{1})-H(C_{1})$
$I_{17}=H(A_{1}\,B_{1}\,C_{1})+H(A_{1}\,B_{1}\,C_{2})+H(A_{2}\,B_{1}\,C_{1})-H(A_{2}\,B_{1}\,C_{2})-H(A_{1}\,B_{1})-H(B_{1}\,C_{1})$
$I_{18}=H(A_{1}\,B_{1}\,C_{1})+H(A_{1}\,B_{2}\,C_{1})+H(A_{1}\,B_{2}\,C_{2})+H(A_{2}\,C_{1})+H(A_{2}\,C_{2})-H(A_{1}\,B_{1}\,C_{2})-H(A_{2}\,B_{2}\,C_{1})-H(A_{1}\,B_{2})-H(A_{2})-H(C_{1})$
$I_{19}=H(A_{1}\,B_{1}\,C_{1})+H(A_{1}\,B_{2}\,C_{1})+H(A_{1}\,B_{2}\,C_{2})+H(A_{2}\,C_{1})+H(A_{2}\,C_{2})-H(A_{1}\,B_{1}\,C_{2})-H(A_{2}\,B_{2}\,C_{2})-H(A_{1}\,B_{2})-H(A_{2})-H(C_{1})$
$I_{20}=H(A_{1}\,B_{1}\,C_{1})+H(A_{1}\,B_{2}\,C_{1})+H(A_{1}\,B_{2}\,C_{2})+H(A_{1}\,B_{1})+H(A_{2}\,B_{1})+H(A_{2}\,B_{2})-H(A_{1}\,B_{1}\,C_{2})-H(A_{2}\,B_{1}\,C_{1})-H(A_{1}\,B_{2})-H(A_{1}\,C_{1})-H(A_{2})-H(B_{1})$
$I_{21}=H(A_{1}\,B_{1}\,C_{1})+H(A_{1}\,B_{2}\,C_{1})+H(A_{1}\,B_{2}\,C_{2})+H(A_{1}\,B_{1})+H(A_{2}\,B_{1})+H(A_{2}\,B_{2})-H(A_{1}\,B_{1}\,C_{2})-H(A_{2}\,B_{2}\,C_{1})-H(A_{1}\,B_{2})-H(A_{1}\,C_{1})-H(A_{2})-H(B_{1})$
$I_{22}=H(A_{1}\,B_{1}\,C_{1})+H(A_{1}\,B_{2}\,C_{1})+H(A_{2}\,B_{1}\,C_{2})+H(A_{2}\,B_{1})+H(A_{2}\,B_{2})+H(B_{2}\,C_{2})-H(A_{2}\,B_{1}\,C_{1})-H(A_{2}\,B_{2}\,C_{2})-H(A_{1}\,C_{1})-H(B_{1}\,C_{2})-H(A_{2})-H(B_{2})$
$I_{23}=H(A_{1}\,B_{1}\,C_{2})+H(A_{1}\,B_{2}\,C_{2})+H(A_{2}\,B_{1}\,C_{1})+H(A_{2}\,B_{1})+H(A_{2}\,B_{2})+H(B_{2}\,C_{1})-H(A_{2}\,B_{2}\,C_{1})-H(A_{2}\,B_{2}\,C_{2})-H(A_{1}\,C_{2})-H(B_{1}\,C_{1})-H(A_{2})-H(B_{2})$
$I_{24}=H(A_{1}\,B_{1}\,C_{2})+H(A_{1}\,B_{2}\,C_{2})+H(A_{2}\,B_{1}\,C_{1})+H(A_{1}\,B_{2})+H(A_{2}\,B_{2})+H(A_{1}\,C_{1})+H(A_{2}\,C_{1})+H(A_{2}\,C_{2})-2H(A_{2}\,B_{2}\,C_{1})-2H(A_{1}\,C_{2})-H(A_{2}\,B_{1})-H(A_{2})-H(B_{2})-H(C_{1})$
$I_{25}=H(A_{1}\,B_{1}\,C_{2})+H(A_{1}\,B_{2}\,C_{1})+H(A_{2}\,B_{1}\,C_{1})+2H(A_{1}\,B_{2})+2H(A_{2}\,B_{2})+2H(A_{1}\,C_{2})+2H(A_{2}\,C_{2})+2H(B_{2}\,C_{2})+H(A_{2}\,B_{1})+H(A_{2}\,C_{1})+H(B_{1}\,C_{2})+H(B_{2}\,C_{1})-4H(A_{2}\,B_{2}\,C_{2})-H(A_{1}\,B_{1}\,C_{1})-H(A_{1}\,B_{2}\,C_{2})-H(A_{1}\,B_{1})-H(A_{1}\,C_{1})-3H(A_{2})-3H(B_{2})-3H(C_{2})$
$I_{26}=H(A_{1}\,B_{1}\,C_{1})+H(A_{1}\,B_{1}\,C_{2})+H(A_{1}\,B_{2}\,C_{1})+H(A_{2}\,B_{1}\,C_{1})-H(A_{2}\,B_{2}\,C_{2})-H(A_{1}\,B_{1})-H(A_{1}\,C_{1})-H(B_{1}\,C_{1})$
$I_{27}=H(A_{1}\,B_{1}\,C_{1})+H(A_{1}\,B_{2}\,C_{1})+H(A_{2}\,B_{1}\,C_{1})+H(A_{2}\,B_{2}\,C_{2})-H(A_{1}\,B_{1}\,C_{2})-H(A_{2}\,B_{2})-H(A_{1}\,C_{1})-H(B_{1}\,C_{1})$
$I_{28}=H(A_{1}\,B_{1}\,C_{1})+H(A_{1}\,B_{2}\,C_{1})+H(A_{2}\,B_{1}\,C_{1})+H(A_{2}\,B_{2}\,C_{2})-H(A_{1}\,B_{2}\,C_{2})-H(A_{2}\,B_{2})-H(A_{1}\,C_{1})-H(B_{1}\,C_{1})$
$I_{29}=H(A_{1}\,B_{1}\,C_{1})+H(A_{1}\,B_{2}\,C_{1})+H(A_{2}\,B_{2}\,C_{1})+H(A_{2}\,B_{2}\,C_{2})-H(A_{2}\,B_{1}\,C_{2})-H(A_{2}\,B_{2})-H(A_{1}\,C_{1})-H(B_{2}\,C_{1})$
$I_{30}=H(A_{1}\,B_{1}\,C_{1})+H(A_{1}\,B_{2}\,C_{1})+H(A_{1}\,B_{2}\,C_{2})+H(A_{2}\,B_{1}\,C_{2})+H(A_{1}\,B_{1})+H(A_{2}\,B_{1})+H(A_{2}\,B_{2})-H(A_{2}\,B_{1}\,C_{1})-H(A_{2}\,B_{2}\,C_{2})-H(A_{1}\,B_{2})-H(A_{1}\,C_{1})-H(B_{1}\,C_{2})-H(A_{2})-H(B_{1})$
$I_{31}=H(A_{1}\,B_{1}\,C_{2})+H(A_{1}\,B_{2}\,C_{1})+H(A_{1}\,B_{2}\,C_{2})+H(A_{2}\,B_{1}\,C_{1})+H(A_{1}\,B_{1})+H(A_{2}\,B_{1})+H(A_{2}\,B_{2})-H(A_{2}\,B_{2}\,C_{1})-H(A_{2}\,B_{2}\,C_{2})-H(A_{1}\,B_{2})-H(A_{1}\,C_{2})-H(B_{1}\,C_{1})-H(A_{2})-H(B_{1})$
$I_{32}=H(A_{1}\,B_{1}\,C_{1})+H(A_{1}\,B_{1}\,C_{2})+H(A_{1}\,B_{2}\,C_{2})+H(A_{2}\,B_{1}\,C_{1})+H(A_{1}\,B_{2})+H(A_{2}\,B_{1})+H(A_{2}\,B_{2})-H(A_{2}\,B_{2}\,C_{1})-H(A_{2}\,B_{2}\,C_{2})-H(A_{1}\,B_{1})-H(A_{1}\,C_{2})-H(B_{1}\,C_{1})-H(A_{2})-H(B_{2})$
$I_{33}=2H(A_{1}\,B_{1}\,C_{1})+2H(A_{1}\,B_{2}\,C_{1})+H(A_{1}\,B_{2}\,C_{2})+H(A_{1}\,B_{1})+H(A_{2}\,B_{1})+H(A_{2}\,B_{2})+H(A_{1}\,C_{2})+H(A_{2}\,C_{2})-2H(A_{2}\,B_{1}\,C_{1})-H(A_{1}\,B_{1}\,C_{2})-2H(A_{1}\,B_{2})-2H(A_{1}\,C_{1})-H(A_{2})-H(B_{1})-H(C_{2})$
$I_{34}=2H(A_{1}\,B_{1}\,C_{1})+H(A_{1}\,B_{1}\,C_{2})+H(A_{1}\,B_{2}\,C_{2})+H(A_{2}\,B_{1}\,C_{1})+H(A_{2}\,B_{2}\,C_{1})-H(A_{1}\,B_{2}\,C_{1})-H(A_{2}\,B_{1}\,C_{2})-H(A_{1}\,B_{1})-H(A_{1}\,C_{2})-H(A_{2}\,C_{1})-H(B_{1}\,C_{1})$
$I_{35}=2H(A_{1}\,B_{2}\,C_{2})+H(A_{1}\,B_{1}\,C_{1})+H(A_{1}\,B_{2}\,C_{1})+H(A_{2}\,B_{1}\,C_{2})+H(A_{2}\,B_{2}\,C_{2})-H(A_{1}\,B_{1}\,C_{2})-H(A_{2}\,B_{1}\,C_{1})-H(A_{1}\,B_{2})-H(A_{1}\,C_{1})-H(A_{2}\,C_{2})-H(B_{2}\,C_{2})$
$I_{36}=2H(A_{1}\,B_{1}\,C_{1})+2H(A_{2}\,B_{1}\,C_{2})+2H(A_{2}\,B_{2}\,C_{1})+H(A_{1}\,B_{1}\,C_{2})+H(A_{1}\,B_{2}\,C_{1})+H(A_{2}\,B_{2}\,C_{2})-2H(A_{1}\,B_{2}\,C_{2})-H(A_{2}\,B_{1}\,C_{1})-H(A_{1}\,B_{1})-H(A_{2}\,B_{2})-H(A_{1}\,C_{1})-H(A_{2}\,C_{2})-H(B_{1}\,C_{2})-H(B_{2}\,C_{1})$

Figure 16: 26D cone

####  XVI.8 The square $\mathcal{C}$4

Full characterization of the marginal cone of the pairwise hidden ancestor
model $\mathcal{C}_{4}$ with 4 hidden ancestors and 4 observable variables.

$I_{0}=H_{24}-H_{2}-H_{4}$
$I_{1}=H_{123}+H_{124}+H_{234}-H_{12}-H_{23}-H_{14}-H_{34}$
$I_{2}=H_{123}+H_{124}+H_{134}+2H_{13}-H_{12}-H_{23}-H_{14}-H_{34}-2H_{1}-2H_{3}$
$I_{3}=2H_{124}+2H_{134}+2H_{234}+H_{13}-2H_{12}-2H_{23}-2H_{14}-2H_{34}-H_{1}-H_{3}$
$I_{4}=3H_{124}+3H_{234}+2H_{123}+2H_{134}-H_{1234}-3H_{12}-3H_{23}-3H_{14}-3H_{34}$
$I_{5}=3H_{123}+3H_{234}+2H_{124}+2H_{134}-H_{1234}-3H_{12}-3H_{23}-3H_{14}-3H_{34}$
$I_{6}=3H_{123}+3H_{134}+2H_{124}+2H_{234}+H_{13}-H_{1234}-3H_{12}-3H_{23}-3H_{14}-3H_{34}-H_{1}-H_{3}$
$I_{7}=3H_{123}+3H_{124}+2H_{134}+2H_{234}+H_{13}-H_{1234}-3H_{12}-3H_{23}-3H_{14}-3H_{34}-H_{1}-H_{3}$
$I_{8}=3H_{124}+3H_{134}+2H_{123}+2H_{234}+2H_{13}-H_{1234}-3H_{12}-3H_{23}-3H_{14}-3H_{34}-2H_{1}-2H_{3}$
$I_{9}=5H_{234}+4H_{123}+4H_{124}+4H_{134}-2H_{1234}-5H_{12}-5H_{23}-5H_{14}-5H_{34}$
$I_{10}=5H_{124}+4H_{123}+4H_{134}+4H_{234}+H_{13}-2H_{1234}-5H_{12}-5H_{23}-5H_{14}-5H_{34}-H_{1}-H_{3}$
$I_{11}=5H_{123}+4H_{124}+4H_{134}+4H_{234}+H_{13}-2H_{1234}-5H_{12}-5H_{23}-5H_{14}-5H_{34}-H_{1}-H_{3}$
$I_{12}=6H_{123}+6H_{124}+6H_{134}+6H_{234}+H_{13}-3H_{1234}-7H_{12}-7H_{23}-7H_{14}-7H_{34}-H_{1}-H_{3}$

Figure 17: The square $\mathcal{C}_{4}$

####  XVI.9 The pentagon $\mathcal{C}$5

Full characterization of the marginal cone of the pairwise hidden ancestor
model $\mathcal{C}_{5}$ with 5 hidden ancestors and 5 observable variables.

$I_{0}=H_{35}-H_{3}-H_{5}$ $I_{1}=H_{134}-H_{34}-H_{1}$
$I_{2}=H_{235}+H_{3}-H_{23}-H_{35}$
$I_{3}=H_{124}+H_{245}-H_{12}-H_{24}-H_{45}$
$I_{4}=H_{1234}+H_{1245}+H_{1345}+H_{2345}+H_{124}+H_{245}+H_{35}-H_{123}-H_{234}-H_{125}-H_{145}-H_{345}-H_{12}-H_{24}-H_{45}-H_{3}-H_{5}$
$I_{5}=H_{1234}+H_{1235}+H_{1345}+H_{2345}+3H_{124}+2H_{35}+3H_{2}-H_{123}-H_{234}-H_{125}-H_{145}-H_{345}-3H_{12}-3H_{24}-2H_{3}-2H_{5}$
$I_{6}=H_{1234}+H_{1235}+H_{1245}+H_{1345}+4H_{124}+H_{134}+H_{135}+H_{235}+4H_{2}+H_{3}+H_{5}-H_{123}-H_{234}-H_{125}-H_{145}-H_{345}-4H_{12}-4H_{24}-2H_{35}-H_{23}-H_{34}-H_{15}-H_{1}$
$I_{7}=H_{1234}+H_{1235}+H_{1245}+H_{2345}+7H_{124}+H_{35}+7H_{2}-H_{123}-H_{234}-H_{125}-H_{145}-H_{345}-7H_{12}-7H_{24}-H_{3}-H_{5}$
$I_{8}=2H_{1235}+2H_{1245}+2H_{1345}+2H_{2345}+5H_{124}+2H_{245}+H_{135}+3H_{2}+H_{5}-2H_{123}-2H_{234}-2H_{125}-2H_{145}-2H_{345}-5H_{12}-5H_{24}-2H_{45}-H_{15}-H_{35}$
$I_{9}=3H_{1245}+3H_{1345}+3H_{2345}+2H_{1234}+2H_{1235}+3H_{124}+H_{245}+2H_{2}-H_{12345}-3H_{123}-3H_{234}-3H_{125}-3H_{145}-3H_{345}-3H_{12}-3H_{24}-H_{45}$
$I_{10}=3H_{1235}+3H_{1345}+3H_{2345}+2H_{1234}+2H_{1245}+5H_{124}+H_{35}+5H_{2}-H_{12345}-3H_{123}-3H_{234}-3H_{125}-3H_{145}-3H_{345}-5H_{12}-5H_{24}-H_{3}-H_{5}$
$I_{11}=3H_{1234}+3H_{1345}+3H_{2345}+2H_{1235}+2H_{1245}+4H_{124}+H_{245}+2H_{35}+3H_{2}-H_{12345}-3H_{123}-3H_{234}-3H_{125}-3H_{145}-3H_{345}-4H_{12}-4H_{24}-H_{45}-2H_{3}-2H_{5}$
$I_{12}=3H_{1234}+3H_{1235}+3H_{1345}+2H_{1245}+2H_{2345}+5H_{124}+H_{235}+H_{35}+5H_{2}-H_{12345}-3H_{123}-3H_{234}-3H_{125}-3H_{145}-3H_{345}-5H_{12}-5H_{24}-H_{23}-2H_{5}-H_{3}$
$I_{13}=3H_{1234}+3H_{1245}+3H_{1345}+2H_{1235}+2H_{2345}+5H_{124}+H_{134}+H_{235}+H_{245}+4H_{2}-H_{12345}-3H_{123}-3H_{234}-3H_{125}-3H_{145}-3H_{345}-5H_{12}-5H_{24}-H_{23}-H_{34}-H_{45}-H_{1}-H_{5}$
$I_{14}=3H_{1235}+3H_{1245}+3H_{1345}+2H_{1234}+2H_{2345}+6H_{124}+H_{134}+H_{235}+6H_{2}+H_{3}-H_{12345}-3H_{123}-3H_{234}-3H_{125}-3H_{145}-3H_{345}-6H_{12}-6H_{24}-H_{23}-H_{34}-H_{35}-H_{1}$
$I_{15}=3H_{1235}+3H_{1245}+3H_{2345}+2H_{1234}+2H_{1345}+8H_{124}+8H_{2}-H_{12345}-3H_{123}-3H_{234}-3H_{125}-3H_{145}-3H_{345}-8H_{12}-8H_{24}$
$I_{16}=3H_{1234}+3H_{1245}+3H_{2345}+2H_{1235}+2H_{1345}+7H_{124}+H_{245}+H_{35}+6H_{2}-H_{12345}-3H_{123}-3H_{234}-3H_{125}-3H_{145}-3H_{345}-7H_{12}-7H_{24}-H_{45}-H_{3}-H_{5}$
$I_{17}=3H_{1234}+3H_{1235}+3H_{1245}+2H_{1345}+2H_{2345}+8H_{124}+H_{235}+8H_{2}+H_{3}-H_{12345}-3H_{123}-3H_{234}-3H_{125}-3H_{145}-3H_{345}-8H_{12}-8H_{24}-H_{23}-H_{35}$
$I_{18}=3H_{1234}+3H_{1235}+3H_{2345}+2H_{1245}+2H_{1345}+9H_{124}+H_{35}+9H_{2}-H_{12345}-3H_{123}-3H_{234}-3H_{125}-3H_{145}-3H_{345}-9H_{12}-9H_{24}-H_{3}-H_{5}$
$I_{19}=5H_{1345}+5H_{2345}+4H_{1234}+4H_{1235}+4H_{1245}+6H_{124}+H_{245}+H_{35}+5H_{2}-2H_{12345}-5H_{123}-5H_{234}-5H_{125}-5H_{145}-5H_{345}-6H_{12}-6H_{24}-H_{45}-H_{3}-H_{5}$
$I_{20}=5H_{1235}+5H_{1345}+4H_{1234}+4H_{1245}+4H_{2345}+7H_{124}+H_{235}+7H_{2}-2H_{12345}-5H_{123}-5H_{234}-5H_{125}-5H_{145}-5H_{345}-7H_{12}-7H_{24}-H_{23}-H_{5}$
$I_{21}=5H_{1245}+5H_{1345}+4H_{1234}+4H_{1235}+4H_{2345}+7H_{124}+H_{134}+H_{235}+H_{245}+6H_{2}+H_{3}-2H_{12345}-5H_{123}-5H_{234}-5H_{125}-5H_{145}-5H_{345}-7H_{12}-7H_{24}-H_{23}-H_{34}-H_{35}-H_{45}-H_{1}$
$I_{22}=5H_{1245}+5H_{2345}+4H_{1234}+4H_{1235}+4H_{1345}+9H_{124}+H_{245}+8H_{2}-2H_{12345}-5H_{123}-5H_{234}-5H_{125}-5H_{145}-5H_{345}-9H_{12}-9H_{24}-H_{45}$
$I_{23}=5H_{1234}+5H_{1245}+4H_{1235}+4H_{1345}+4H_{2345}+9H_{124}+H_{235}+H_{245}+8H_{2}-2H_{12345}-5H_{123}-5H_{234}-5H_{125}-5H_{145}-5H_{345}-9H_{12}-9H_{24}-H_{23}-H_{45}-H_{5}$
$I_{24}=5H_{1235}+5H_{1245}+4H_{1234}+4H_{1345}+4H_{2345}+10H_{124}+H_{235}+10H_{2}+H_{3}-2H_{12345}-5H_{123}-5H_{234}-5H_{125}-5H_{145}-5H_{345}-10H_{12}-10H_{24}-H_{23}-H_{35}$
$I_{25}=5H_{1235}+5H_{2345}+4H_{1234}+4H_{1245}+4H_{1345}+11H_{124}+11H_{2}-2H_{12345}-5H_{123}-5H_{234}-5H_{125}-5H_{145}-5H_{345}-11H_{12}-11H_{24}$
$I_{26}=5H_{1234}+5H_{1235}+4H_{1245}+4H_{1345}+4H_{2345}+11H_{124}+H_{235}+11H_{2}-2H_{12345}-5H_{123}-5H_{234}-5H_{125}-5H_{145}-5H_{345}-11H_{12}-11H_{24}-H_{23}-H_{5}$
$I_{27}=7H_{1245}+6H_{1234}+6H_{1235}+6H_{1345}+6H_{2345}+11H_{124}+H_{235}+H_{245}+10H_{2}+H_{3}-3H_{12345}-7H_{123}-7H_{234}-7H_{125}-7H_{145}-7H_{345}-11H_{12}-11H_{24}-H_{23}-H_{35}-H_{45}$
$I_{28}=7H_{1235}+6H_{1234}+6H_{1245}+6H_{1345}+6H_{2345}+13H_{124}+H_{235}+13H_{2}+H_{3}-3H_{12345}-7H_{123}-7H_{234}-7H_{125}-7H_{145}-7H_{345}-13H_{12}-13H_{24}-H_{23}-H_{35}$
$I_{29}=10H_{1234}+10H_{1345}+8H_{1235}+8H_{1245}+8H_{2345}+15H_{124}+2H_{134}+2H_{235}+2H_{245}+H_{35}+13H_{2}-4H_{12345}-10H_{123}-10H_{234}-10H_{125}-10H_{145}-10H_{345}-15H_{12}-15H_{24}-2H_{23}-2H_{34}-2H_{45}-3H_{5}-2H_{1}-H_{3}$
$I_{30}=10H_{1234}+10H_{2345}+8H_{1235}+8H_{1245}+8H_{1345}+19H_{124}+2H_{245}+3H_{35}+17H_{2}-4H_{12345}-10H_{123}-10H_{234}-10H_{125}-10H_{145}-10H_{345}-19H_{12}-19H_{24}-2H_{45}-3H_{3}-3H_{5}$
$I_{31}=14H_{1345}+12H_{1234}+12H_{1235}+12H_{1245}+12H_{2345}+19H_{124}+2H_{134}+2H_{235}+2H_{245}+17H_{2}+H_{3}-6H_{12345}-14H_{123}-14H_{234}-14H_{125}-14H_{145}-14H_{345}-19H_{12}-19H_{24}-2H_{23}-2H_{34}-2H_{45}-H_{35}-2H_{1}-H_{5}$
$I_{32}=14H_{2345}+12H_{1234}+12H_{1235}+12H_{1245}+12H_{1345}+23H_{124}+2H_{245}+H_{35}+21H_{2}-6H_{12345}-14H_{123}-14H_{234}-14H_{125}-14H_{145}-14H_{345}-23H_{12}-23H_{24}-2H_{45}-H_{3}-H_{5}$
$I_{33}=14H_{1234}+12H_{1235}+12H_{1245}+12H_{1345}+12H_{2345}+23H_{124}+2H_{235}+2H_{245}+H_{35}+21H_{2}-6H_{12345}-14H_{123}-14H_{234}-14H_{125}-14H_{145}-14H_{345}-23H_{12}-23H_{24}-2H_{23}-2H_{45}-3H_{5}-H_{3}$
$I_{34}=16H_{1234}+16H_{1235}+16H_{1245}+16H_{1345}+16H_{2345}+27H_{124}+2H_{235}+2H_{245}+25H_{2}+H_{3}-8H_{12345}-18H_{123}-18H_{234}-18H_{125}-18H_{145}-18H_{345}-27H_{12}-27H_{24}-2H_{23}-2H_{45}-H_{35}-H_{5}$
$I_{35}=H_{12345}+H_{124}+H_{235}+H_{245}-H_{1235}-H_{12}-H_{23}-H_{24}-H_{45}-H_{5}$
$I_{36}=H_{12345}+3H_{124}+2H_{35}+3H_{2}-H_{1245}-3H_{12}-3H_{24}-2H_{3}-2H_{5}$
$I_{37}=H_{12345}+2H_{124}+H_{135}+H_{245}+H_{2}+H_{5}-H_{1234}-2H_{12}-2H_{24}-H_{15}-H_{35}-H_{45}$
$I_{38}=H_{12345}+5H_{124}+5H_{2}-H_{1345}-5H_{12}-5H_{24}$
$I_{39}=H_{12345}+3H_{124}+H_{134}+H_{135}+H_{235}+3H_{2}+H_{3}+H_{5}-H_{2345}-3H_{12}-3H_{24}-2H_{35}-H_{23}-H_{34}-H_{15}-H_{1}$

Figure 18: The pentagon $\mathcal{C}_{5}$

###  XVII Details about the numerical search for the violation of entropic
inequalities

We search for quantum states that violate a particular Bell inequality
$I_{k}\geq 0$ via numeric optimization that minimizes the value of $I_{k}$ by
varying the state and measurement operators. In order to do this, we have to
choose a suitable parametrization for these objects.

Let the Hilbert space of the composite system be
$\mathcal{H}=\mathbbm{C}_{n}^{\otimes 3}$. First note that the subsystem
dimensions give the maximum possible number of outcomes of any measurement on
the respective subsystem, in this case $n$. Considering only non-degenerate
measurements, the results can be labelled $\\{1,\ldots,n\\}$. A single-party
observable corresponds to a hermitian operator and can thus be seen as a
unitary rotation $UDU^{\dagger}$ of a diagonal matrix $D$. Specifying a
measurement hence amounts to specifying a unitary matrix $U$ which can be done
with $n^{2}$ real parameters of which $n$ correspond to phase transformations
on the components of the state vector. When optimizing over both the state and
measurements, only the relative phase matters and so it not necessary to
specify the phases in both parameter sets. A convenient parametrization that
allows to effortlessly leave out the parameters corresponding to phase
transformations is given in Jarlskog (2005).

Pure states $\psi\in\mathcal{H}=\mathbbm{C}_{n}^{\otimes 3}$ can be
represented as unit vectors with $n^{3}$ complex components. A straightforward
approach to sample a random state is to either normalize a vector created from
$2n^{3}$ unconstrained (normally distributed) real parameters; or to directly
create a unit vector from $2n^{3}{-}1$ angles using a spherical coordinate
system. While the latter uses one less parameter, we employ the former option
as this involves less complexity and treats every parameter equally.

We were unable to enforce the PPT constraint exactly using _pure_ states. To
avoid this issue, we removed the PPT constraint violations by replacing the
state vector with a density matrix and mixing with “white noise” in a second
optimization pass:

$\displaystyle\psi\rightarrow\rho=\lambda\ket{\psi}\bra{\psi}+(1-\lambda)\left(\tfrac{1}{n^{3}}\mathbbm{1}\right),$

where $\lambda\in[0,1]$ is the mixing parameter. This adds one additional
optimization parameter to the set.

In the special case of a three-qubit system $\mathbbm{C}_{2}^{\otimes 3}$ more
physically recognized parametrizations are available: Since we are only
interested in states that show non-local behaviour we only need to consider
entangled states. This can be achieved with a parametrization in terms of
entanglement parameters such as described in Acín  _et al._ (2000). The
subsystems are essentially spin one-half systems with spin measurements
$\boldsymbol{\sigma}\cdot\boldsymbol{v}$ along an arbitrary direction
$\boldsymbol{v}$ where $\boldsymbol{\sigma}$ is the vector of Pauli matrices.
A measurement can thus be specified in terms of two angles.

###  XVIII A structural constraint on facets

As can be seen in Appendix XVI, the listed inequalities fall into groups that
share a distinct structural similarity. For instance, the $I_{26}$, $I_{27}$,
$I_{28}$, and $I_{29}$ inequalities of the 26D local cone have the same number
of positive and negative $N$-body terms. Similar observations hold true for
groups of facets in all of the computed projections. This suggests to
investigate the underlying structure.

We start noting that any valid Shannon-type inequality $I$ can be written as a
non-negative linear combination of elemental forms:

$\displaystyle
I=\sum_{i}c_{i}\,H(X_{i}\ooalign{$\phantom{{},{}}$\cr$|$}X_{\Omega-\\{i\\}})+\sum_{i<j}\sum_{\omega\subset\Omega}c_{ij\omega}\,I(X_{i}:X_{j}\ooalign{$\phantom{{},{}}$\cr$|$}X_{\omega}).$
(38)

Such a combination can be retrieved as the dual solution of the LP that proves
the inequality, and hence we call such a combination a _proof_ for the
inequality. For example, the facet $I_{17}$ of the 26D local cone,

$\displaystyle I_{17}^{\mathrm{26D}}=$ $\displaystyle
H(A_{1}\,B_{1}\,C_{1})+H(A_{1}\,B_{1}\,C_{2})+H(A_{2}\,B_{1}\,C_{1})$
$\displaystyle-H(A_{2}\,B_{1}\,C_{2})-H(A_{1}\,B_{1})-H(B_{1}\,C_{1}),$

can be proven by the following combination of elemental forms:

$\displaystyle I$ ${}_{17}^{\mathrm{26D}}=$
$\displaystyle+H(A_{1}\ooalign{$\phantom{{},{}}$\cr$|$}A_{2},B_{1},B_{2},C_{1},C_{2})$
$\displaystyle{}+H(B_{1}\ooalign{$\phantom{{},{}}$\cr$|$}A_{1},A_{2},B_{2},C_{1},C_{2})$
$\displaystyle+I(A_{1}:A_{2}\ooalign{$\phantom{{},{}}$\cr$|$}B_{1},B_{2},C_{1},C_{2})$
$\displaystyle{}+I(B_{1}:C_{2}\ooalign{$\phantom{{},{}}$\cr$|$}A_{1},A_{2},B_{2},C_{1})$
$\displaystyle+I(A_{1}:B_{1}\ooalign{$\phantom{{},{}}$\cr$|$}A_{2},B_{2},C_{1})$
$\displaystyle{}+I(A_{1}:C_{2}\ooalign{$\phantom{{},{}}$\cr$|$}B_{1},B_{2},C_{1})$
$\displaystyle+I(B_{1}:B_{2}\ooalign{$\phantom{{},{}}$\cr$|$}A_{1},C_{1})$
$\displaystyle{}+I(A_{2}:B_{2}\ooalign{$\phantom{{},{}}$\cr$|$}B_{1},C_{1}).$

This expression is highly structured, and not by chance. In fact, every
entropic Bell inequality must have a similar looking proof: There must be the
same number $k$ of elemental forms of every kind until reaching the output
space. This leads to the following result about the structure of facets of the
$(2n^{2})$-dimensional marginal cone that contains all one- and two-body terms
in the multipartite Bell scenario with two measurements per party.

####  XVIII.1 Definition of the constraint

Consider linear information inequalities on $N$ random variables of the form

$\displaystyle 0$ $\displaystyle\leq
I=\sum_{i}a_{i}H(X_{i})+\sum_{i<j}b_{ij}H(X_{i},X_{j}).$ (39)

Then, for $N\geq 2$ every minimal Shannon type inequality of this form falls
into one of two possible categories up to a positive scale factor. The first
one being the mutual informations among different random variables, i.e.:

$\displaystyle I=I(X_{i}:X_{j})=H(X_{i})+H(X_{j})-H(X_{i},X_{j}).$

In order to describe the second category, first rewrite the coefficients in
(39) as the difference of two non-negative numbers $a_{i}=a_{i}^{+}-a_{i}^{-}$
and $b_{ij}=b_{ij}^{+}-b_{ij}^{-}$ where only one of them is allowed to be
non-zero. Then all remaining facets can be written with integer coefficients
fulfilling

$\displaystyle a_{i}^{+}$ $\displaystyle=0,$
$\displaystyle\sum_{i<j}b_{ij}^{+}$ $\displaystyle=m+k,$
$\displaystyle\sum_{i}a_{i}^{-}$ $\displaystyle=k,$
$\displaystyle\sum_{i<j}b_{ij}^{-}$ $\displaystyle=m,$

where $0\leq m\leq k$ are integers.

For example, in

$I_{9}^{\mathrm{18D}}=H(A_{1}\,B_{2})+H(A_{2}\,B_{2})+H(A_{1}\,C_{2})+H(A_{2}\,C_{1})+H(B_{1}\,C_{1})+H(B_{1}\,C_{2})-H(A_{1}\,C_{1})-H(A_{2}\,C_{2})-H(A_{1})-H(B_{1})-H(B_{2})-H(C_{1})$

we have:

$\displaystyle a_{i}^{+}$ $\displaystyle=0,$
$\displaystyle\sum_{i<j}b_{ij}^{+}$ $\displaystyle=6,$
$\displaystyle\sum_{i}a_{i}^{-}$ $\displaystyle=4,$
$\displaystyle\sum_{i<j}b_{ij}^{-}$ $\displaystyle=2,$

consistent with this statement.

####  XVIII.2 Intuition

Recall that the elemental inequalities are simply the non-negativities of the
so-called elemental forms

$\displaystyle 0\leq
H(X_{i}\ooalign{$\phantom{{},{}}$\cr$|$}X_{\Omega-\\{i\\}})$
$\displaystyle=H(X_{\Omega})-H(X_{\Omega-\\{i\\}})$ and $\displaystyle 0\leq
I(X_{i}:X_{j}\ooalign{$\phantom{{},{}}$\cr$|$}X_{\omega})$
$\displaystyle=-H(X_{\omega\cup\\{i,j\\}})-H(X_{\omega})$
$\displaystyle\quad+H(X_{\omega\cup\\{i\\}})+H(X_{\omega\cup\\{j\\}}),$

where $\Omega=[N]$ is the set of all indices and
$\omega\subset\Omega-\\{i,j\\}$. The unconditional mutual informations
($\omega=\emptyset$) between two measurements of different parties lie
directly in the output space and therefore make up one category of its facets.

The true conditional mutual informations ($\omega\neq\emptyset$) have a
negative term containing $\lvert\omega\rvert+2\geq 3$ random variables.
Clearly, this term is not part of the output space, and if such a term appears
in the _proof_ of a facet, it has to be counteracted by another elemental
form. For $\lvert\omega\rvert+2<N$ the only way to compensate the negative
high-order term is to add another conditional mutual information (CMI) with
$\lvert\omega^{\prime}\rvert=\lvert\omega\rvert+1$. For
$\lvert\omega\rvert+2=N$ this requires a conditional entropy since there are
no CMIs with $\lvert\omega^{\prime}\rvert>\lvert\omega\rvert=N-2$. Hence,
writing a facet as a sum of elemental forms leads to a chain of equally many
CMIs for each $\lvert\omega\rvert$ and equally many conditional entropies.

####  XVIII.3 Proof

For $N=2$ the claim holds trivially since the output cone coincides with the
Shannon cone $\Gamma_{2}$ whose only facets are the elemental inequalities
$I(X_{1}:X_{2})\geq 0$ and $H(X_{1},X_{2})-H(X_{i})\geq 0$. We consider $N\geq
3$ in the following.

Let $I\geq 0$ be a valid Shannon-type inequality of the form (39), then

$\displaystyle
I=\sum_{i}c_{i}\,H(X_{i}\ooalign{$\phantom{{},{}}$\cr$|$}X_{\Omega-\\{i\\}})+\sum_{i<j}\sum_{\omega\subset\Omega}c_{ij\omega}\,I(X_{i}:X_{j}\ooalign{$\phantom{{},{}}$\cr$|$}X_{\omega}),$
(44)

where all the coefficients are non-negative and $c_{ij\omega}=0$ if
$i\in\omega$ or $j\in\omega$. Define

$\displaystyle\Delta_{p}$
$\displaystyle=\sum_{i<j}\sum_{\lvert\omega\rvert=p}c_{ij\omega}\,I(X_{i}:X_{j}\ooalign{$\phantom{{},{}}$\cr$|$}X_{\omega})$
(47) $\displaystyle I_{p}$ $\displaystyle=I-\sum_{i<p}\Delta_{i}.$ (48)

The first bit is to show that for $p\geq 2$ the following invariant holds:

$\displaystyle I_{p}$
$\displaystyle=\sum_{\lvert\alpha\rvert=p+1}c_{\alpha}^{+}H(X_{\alpha})-\sum_{\lvert\beta\rvert=p}c_{\beta}^{-}H(X_{\beta})$
(49)

for some non-negative coefficients with $\sum c_{\alpha}^{+}=\sum
c_{\beta}^{-}=k$.

Note that $I_{N-1}$ contains only the conditional entropy terms and hence (49)
holds trivially for $p=N-1\geq 2$. If $N=3$, we are finished. Otherwise,
assume that (49) is the case for $p\geq 3$ and consider $I_{p-1}$. Clearly,
the entropy terms in $I_{p}$ have at least $3$ arguments and must therefore be
cancelled in the final $I$. It is a straightforward observation that this must
be achieved by the terms in $\Delta_{p-1}$ and that the term counts must
match:

$\displaystyle\sum_{i<j}\sum_{\lvert\omega\rvert=p-1}c_{ij\omega}=\sum_{\lvert\alpha\rvert=p+1}c^{+}_{\alpha}=k.$
(50)

Conversely, this means that $k$ of the negative $(p{+}1)$-body terms and $k$
of the positive $p$-body terms in $\Delta_{p+1}$ are sucked up by $I_{p}$ –
leaving only $k$ positive $p$-body terms and $k$ negative $(p{-}1)$-body terms
in $I_{p-1}$. This proves the claim (49) for $p^{\prime}=p-1\geq 2$.

The situation changes slightly when eliminating terms from $I_{2}$ which
consists only of two- and three-body terms. Just as before, the three-body
terms must be compensated by terms in $\Delta_{1}$ but the two-body entropies
are in the output space and can therefore be left untouched. Assuming that $m$
of the two-body terms are nevertheless cancelled, $I_{1}$ is of the form

$\displaystyle I_{1}=\sum_{i<j}b_{ij}^{+}$ $\displaystyle H(X_{i},X_{j})$
$\displaystyle{}-{}\sum_{i<j}b_{ij}^{-}$ $\displaystyle H(X_{i},X_{j})$
$\displaystyle{}-{}\sum_{i}a_{i}^{+}$ $\displaystyle H(X_{i})$ with non-
negative coefficients fulfilling $\displaystyle\sum_{i<j}b_{ij}^{+}$
$\displaystyle=2k-m,$ $\displaystyle\sum_{i<j}b_{ij}^{-}$ $\displaystyle=k-m,$
$\displaystyle\sum_{i}a_{i}^{-}$ $\displaystyle=k.$

Finally, $\Delta_{0}$ consists of unconditional mutual informations. This
means that both $I_{1}$ and $\Delta_{0}$ are valid Shannon type inequalities
in the output space and therefore an inequality $0\leq I=I_{1}+\Delta_{0}$ can
only be minimal if $\Delta_{0}=0$ or $I_{1}=0$. Furthermore, the mutual
informations (having only positive one-body terms) can not be obtained as non-
negative combinations of facets of the other category and must therefore be
facets on their own.

This concludes the proof for the output space that includes _all_ one- and
two-body entropy terms. Furthermore, this result extends naturally to the
local cone of _simultaneously accessible_ quantities in a multipartite Bell
scenario. These inequalities make up a subset of the general case by grouping
the $X_{i}$ into parties $P_{j}=(X_{2j},X_{2j+1})$ and forcing that no two-
body terms of a single party may be present. In order to achieve this, two
issues must be considered:

First, $\Delta_{1}$ must cancel all negative two-body terms in $I_{2}$ if they
contain variables of the same party. This puts an additional constraint on
$\Delta_{1}$ but doesn’t affect the argument otherwise.

Second, we must take care for inaccessible two-body terms added by
$\Delta_{1}$ because they would need to be cancelled by a non-zero
$\Delta_{0}$ term. Without loss of generality this means that the expansion
(44) for $I$ contains a subexpression of the form:

$\displaystyle I$
$\displaystyle(X_{2}:X_{3}\ooalign{$\phantom{{},{}}$\cr$|$}X_{1})+I(X_{1}:X_{2})$
$\displaystyle=-H(X_{1},X_{2},X_{3})+H(X_{1},X_{3})+H(X_{2}).$

However, observe that the resulting inequality can not be minimal since an
alternative expansion that eliminates the same three-body terms but doesn’t
require a non-zero $\Delta_{0}$ is

$\displaystyle I$
$\displaystyle(X_{1}:X_{2}\ooalign{$\phantom{{},{}}$\cr$|$}X_{3})$
$\displaystyle=-H(X_{1},X_{2},X_{3})+H(X_{1},X_{3})+H(X_{2},X_{3})-H(X_{3})$
$\displaystyle\leq-H(X_{1},X_{2},X_{3})+H(X_{1},X_{3})+H(X_{2})$
$\displaystyle=\phantom{-{}}I(X_{2}:X_{3}\ooalign{$\phantom{{},{}}$\cr$|$}X_{1})+I(X_{1}:X_{2}).$

The inequality sign in the second line is a consequence of the independence
bound $H(X_{2},X_{3})\leq H(X_{2})+H(X_{3})$.

####  XVIII.4 Bounding $k$

Looking at the obtained inequalities, we can see that that for all discovered
facets, expansions in the sense of (38) were available with relatively low
coefficients $c\lesssim 3$. For most quantities even $c=1$. That is, when
eliminating $v$ variables from a system of linear inequalities, every minimal
inequality in the resulting system can be obtained as a linear combination of
at most $v+1$ of the original facets. This observation suggests upper bounds
on the maximum value of $k$.

####  XVIII.5 Enumerating constraints

Based on the insights, one can implement an algorithm that enumerates all
possible linear combinations leading to constraints in the desired marginal
space. This is achieved in a top-down manner where at each step we only need
to consider those combinations of CMIs that eliminate the according terms of
the current step. This enumeration can be done exhaustively only for low $k$
(number of elemental forms at each layer).

To show the practical relevance of these observations, we use this method to
ensure that the listed descriptions of the 18D and 26D cones contain every
facet with $k\leq 4$. For the 26D cone this allowed to find 14 of the 37
facets discovered in total. Note that this works more efficiently for the 26D
cone than for the 18D, since this requires one less recursion.