# Analysing causal structures in generalised probabilistic theories

Mirjam Weilenmann<EMAIL_ADDRESS>Institute for Quantum Optics
and Quantum Information (IQOQI) Vienna, Austrian Academy of Sciences,
Boltzmanngasse 3, 1090 Vienna, AT  Department of Mathematics, University of
York, Heslington, York, YO10 5DD, UK  Roger Colbeck<EMAIL_ADDRESS>Department of Mathematics, University of York, Heslington, York, YO10 5DD, UK

###### Abstract

Causal structures give us a way to understand the origin of observed
correlations. These were developed for classical scenarios, but quantum
mechanical experiments necessitate their generalisation. Here we study causal
structures in a broad range of theories, which include both quantum and
classical theory as special cases. We propose a method for analysing
differences between such theories based on the so-called measurement entropy.
We apply this method to several causal structures, deriving new relations that
separate classical, quantum and more general theories within these causal
structures. The constraints we derive for the most general theories are in a
sense minimal requirements of any causal explanation in these scenarios. In
addition, we make several technical contributions that give insight for the
entropic analysis of quantum causal structures. In particular, we prove that
for any causal structure and for any generalised probabilistic theory, the set
of achievable entropy vectors form a convex cone.

Given a set of observed variables, some of which may be correlated, a causal
structure gives a more detailed picture of how the correlations come about.
Depending on the situation, this causal structure may posit the existence of
hidden common causes and the nature of these depends on the physical theory.
For instance, the experimental violation of a Bell inequality [1] can be
explained either by adapting the causal structure within the realm of
classical physics (at the expense of resorting to fine-tuning [2]) or by
allowing hidden systems to be non-classical.

Causal structures also provide a suitable basis for analysing the features of
different theories by allowing us to phrase communication and cryptographic
protocols in terms of the dependencies among the involved systems. They help
us predict the success of players engaged in a protocol when restricted
according to different theories, for example, in random access coding and the
related principle of Information Causality [3, 4].

The differences between the observable correlations that can be achieved with
classical and quantum resources within a given causal structure have been
extensively analysed, starting with the derivation of several classical
constraints and their quantum violations [5, 6], and progressing to a
systematic analysis [7, 8, 9, 10, 11]. Less work has been dedicated to
understanding the limitations of quantum systems [12, 9, 13] and of the
behaviour of theories beyond. For the latter, there have been analyses of the
implications of the no-signalling principle on causal structures [14, 15].
More generally, understanding the differences of generalised probabilistic
theories (GPTs) with respect to different tasks may inform the search for
principles that single out quantum mechanics.

In this work, we introduce a technique for deriving constraints on the
observable correlations that are achievable in different causal structures
according to different GPTs, with the aim of moving towards a systematic
analysis of the differences between such theories. Our approach is based on
_measurement entropy_ [16, 17] and inspired by entropic approaches to
analysing causal structures involving classical and quantum resources [18, 19,
20, 8, 9, 21, 22]. That such a generalisation is possible, was not at all
clear, since work regarding the definition of entropy in GPTs showed that
there is no entropy measure that retains the relevant properties of the von
Neumann entropy [16, 17]. In particular, the additivity of entropy under the
composition of different systems is not retained by the proposed measures,
which in previous entropic approaches for analysing causal structures was
crucial for encoding causal constraints.

One of the key points that allows us to overcome these issues is to explicitly
include the conditional entropy in the analysis. Nevertheless, since our final
results are stated in terms of the Shannon entropy they can be directly
compared to those obtained with previous entropic techniques.

We apply our method to various causal structures, generating a series of
entropic constraints that exclude certain causal explanations of observed
correlations when restricted by arbitrary GPTs. This allows us to compare
different causal structures with respect to the correlations they allow in
different theories (in particular, we compare classical, quantum and arbitrary
GPTs). In some cases we find the same sets of entropic description regardless
of the theory (here known quantum inequalities also apply to GPTs), while in
others we can show an entropic separation. For instance, we apply our
technique to Information Causality [3], a candidate principle for singling out
quantum theory, showing that our method improves upon that of [15], yielding
the stronger inequalities of [23]. Although the maximally non-local GPT, _box-
world_ does not satisfy the notion of Information Causality, we identify
minimal notions of causation that are satisfied.

In addition to providing a method for analysing causal structures with GPT
resources, we make technical contributions by showing that any set of
achievable entropy vectors for the observed variables in a causal structure
involving quantum or other generalised probabilistic resources is a convex
cone. Previously this had only been shown for the entropy vectors of classical
resources [24, 20, 8, 22]. This insight allows for easy comparison of the
entropic sets within different theories, and in some cases enables us to prove
that a given characterisation is complete by showing that all extremal points
are achievable. We also give some insights into the entropic analysis of
quantum causal structures.

##  1 Preliminaries

For every system $A$ in a GPT, there is an associated state space
$\mathcal{S}_{A}$, a compact convex subset of a real vector space $V$ and an
associated space of effects, $\mathcal{F}_{A}$. An _effect_
$e\in\mathcal{F}_{A}$ is a linear map $\mathcal{S}_{A}\to[0,1]$ (thus, $e$ is
a vector in the dual space to $V$). There is a special effect,
$u_{A}\in\mathcal{F}_{A}$, called the unit effect, with the property that
$u_{A}(s)=1$ for all $s\in\mathcal{S}_{A}$. A _measurement_ $M$ is a
collection of effects whose sum is the unit effect, i.e., we can write
$M=\\{e^{x}\in\mathcal{F}_{A}:\sum_{x}e^{x}=u_{A}\\}$. We use
$\mathcal{E}_{A}$ to represent the set of allowed measurements on $A$. The
interpretation of $e^{x}(s)$ is the probability of outcome $x$ when $M$ is
performed on a system in state $s$.

Consider two measurements on $A$: $M=\\{e^{x}\\}_{x\in\mathcal{R}_{M}}$ and
$N=\\{f^{y}\\}_{y\in\mathcal{R}_{N}}$. If there exists a map
$F:\mathcal{R}_{M}\to\mathcal{R}_{N}$ such that

$\sum_{x\in\mathcal{R}_{M}:F(x)=y}e^{x}=f^{y}\ \forall y\in\mathcal{R}_{N}$
(1)

we say that $M$ is a _refinement_ of $N$ (equivalently, $N$ is a _coarse-
graining_ of $M$).111If $F$ is bijective, (1) is a _relabelling_. The set of
measurements $\mathcal{E}_{A}$ is assumed to be closed under relabelling and
coarse-graining. A refinement is _trivial_ if for all $x\in\mathcal{R}_{M}$,
$e^{x}=c(x)\,f^{F(x)}$ for some $c(x)\in\mathbbm{R}_{>0}$. The subset of
_fine-grained measurements_ , $\mathcal{E}^{*}_{A}$, are those for which there
are no non-trivial refinements. Throughout this article we restrict to GPTs
where there is at least one finite-outcome fine-grained measurement (in
classical and quantum theory this is a restriction to finite-dimensional
systems).

Transformations of systems are represented by linear maps between state
spaces, $T:\mathcal{S}_{A}\to\mathcal{S}_{B}$ and the set of such
transformations is denoted $\mathcal{T}_{A\to B}$. The set $\mathcal{T}_{A\to
A}$, contains the identity transformation, $I_{A}$, and is closed under
composition. Furthermore, a transformation followed by a measurement is a
valid measurement.

Two systems $A$ and $B$ can be thought of as parts of a single joint system
$AB$. We do not specify precisely what the joint state space is, but a minimal
requirement is that if $s_{A}\in\mathcal{S}_{A}$ and $s_{B}\in\mathcal{S}_{B}$
then $s_{A}\otimes s_{B}\in\mathcal{S}_{AB}$. States that can be written as
$s_{A}\otimes s_{B}$ are called _product_ and convex combinations thereof are
_separable_. Analogously, measurements
$M=\\{e^{x}\\}_{x\in\mathcal{R}_{M}}\in\mathcal{E}_{A}$ and
$N=\\{f^{y}\\}_{y\in\mathcal{R}_{N}}\in\mathcal{E}_{B}$ can be composed into
$L=M\otimes N\in\mathcal{E}_{AB}$ with outcome set
$\mathcal{R}_{L}=\mathcal{R}_{M}\times\mathcal{R}_{N}$ and effects
$g^{x,y}=e^{x}\otimes f^{y}$. Product effects act on product states according
to ${(e^{x}\otimes f^{y})(s_{A}\otimes s_{B})}=e^{x}(s_{A})f^{y}(s_{B})$.

This implies that we have non-signalling theories: Suppose
$\\{e_{a}^{x}\\}_{x}\in\mathcal{E}_{A}$ and
$\\{e_{b}^{y}\\}_{y}\in\mathcal{E}_{B}$ are measurements for
$a=1,\ldots,n_{a}$ and $b=1,\ldots,n_{b}$, then, for example,

$\displaystyle p(y|a,b)=\sum_{x}(e_{a}^{x}\otimes
f_{b}^{y})(s_{AB})=(u_{A}\otimes f_{b}^{y})(s_{AB})\,,$

which is independent of $a$.

We also assume that there are well-defined reduced states: $\forall
s_{AB}\in\mathcal{S}_{AB}\ \exists s_{A}\in\mathcal{S}_{A}\ \mathrm{s.t.}\
\forall e\in\mathcal{F}_{A},e(s_{A})=(e\otimes u_{B})s_{AB}$. The post-
measurement state on $A$ after a measurement on $B$ with outcome $x$ is

$s_{A|x}=\frac{(I_{A}\otimes e^{x})(s_{AB})}{e^{x}(s_{B})}.$ (2)

If the system $A$ is classical, then $\mathcal{S}_{A}$ is a simplex and (up to
relabelling) there is only one fine-grained measurement that is not a trivial
refinement of another fine-grained measurement. We call this a _standard
classical measurement_. Note that classical systems can be represented in any
GPT and composing them maintains separability.

_Box world_ [25] is the GPT in which the joint state space of several systems
is in one-to-one correspondence with the set of no-signalling distributions
amongst those systems, i.e., its state space in this sense the largest
possible within the framework.

##  2 Measurement entropy and its properties

The approach to analysing causal structures that we use in this work is based
on measurement entropy. In this section we introduce this and outline some of
its properties.

Measurement entropy was first introduced in [23, 17]; we follow the exposition
of [23] here. The _measurement entropy_ , $H_{+}$, is the minimal Shannon
entropy of the outcome distribution after a fine-grained measurement, i.e.,
for $s_{A}\in\mathcal{S}_{A}$,

$H_{+}\\!\left(A\right)=\inf_{\\{e^{x}\\}\in\mathcal{E}_{A}^{*}}-\sum_{x}e^{x}(s_{A})\log_{2}e^{x}(s_{A}).$
(3)

Several ways to define the conditional measurement entropy have been proposed
[23, 17], of which we use the following [23]. For any state
$s_{AB}\in\mathcal{S}_{AB}$ with reduced state $s_{B}\in\mathcal{S}_{B}$, the
_conditional measurement entropy_ is

$H_{+}\\!\left(A|B\right)=\inf_{\\{f^{y}\\}\in\mathcal{E}_{B}}\sum_{y}f^{y}(s_{B})H_{+}\\!\left(A_{|y}\right)\,,$
(4)

where $H_{+}\\!\left(A_{|y}\right)$ is the entropy of the state on $A$ after a
measurement on $B$ with outcome $y$, $s_{A|y}$. For classical systems these
entropies coincide with the Shannon entropy, $H$.

The measurement entropy satisfies a list of properties that are useful to this
work. Some of these have previously been derived in [23, 17], others are new
to this work. In the remainder of this section, $\mathcal{S}_{A}$,
$\mathcal{S}_{AB}$ etc. refer to state spaces within an arbitrary GPT. For the
proofs of the first two properties we refer to [23].

######  Property 1 (Positivity [23]).

$H_{+}\\!\left(A\right)\geq 0$ for all ${s_{A}\in\mathcal{S}_{A}}$ and
$H_{+}\\!\left(A|B\right)\geq 0$ for all $s_{AB}\in\mathcal{S}_{AB}$.

######  Property 2 (Reduction to Shannon entropy [23, 17]).

Let $A$ and $B$ be classical systems and $s_{AB}\in\mathcal{S}_{AB}$, then
$H_{+}\\!\left(A\right)=H(A)$ and $H_{+}\\!\left(A|B\right)=H(A|B)$.

######  Property 3 (Data processing).

Let $s_{AB}\in\mathcal{S}_{AB}$, $T\in\mathcal{T}_{B\to C}$ and
$s_{AC}=(I_{A}\otimes T)(s_{AB})$. Then ${H_{+}\\!\left(A|B\right)\leq
H_{+}\\!\left(A|C\right)}$.

###### Proof.

If $\\{f^{j}\\}$ form a measurement on $C$, then $\\{g^{j}\\}$ form a
measurement on $B$, where $g^{j}:s_{B}\mapsto f^{j}(T(s_{B}))$. It follows
that

$\displaystyle H_{+}\\!\left(A|C\right)$
$\displaystyle=\inf_{\\{f^{j}\\}\in\mathcal{E}_{C}}\sum_{j}f^{j}(s_{C})H_{+}\\!\left(A_{|j}\right)$
$\displaystyle=\inf_{\\{f^{j}\\}\in\mathcal{E}_{C}}\sum_{j}g^{j}(s_{B})H_{+}\\!\left(A_{|j}\right)$
$\displaystyle\geq\inf_{\\{g^{j}\\}\in\mathcal{E}_{B}}\sum_{j}g^{j}(s_{B})H_{+}\\!\left(A_{|j}\right)$
$\displaystyle=H_{+}\\!\left(A|B\right)\qed$

######  Property 4 (Independence).

If two systems $A$ and $B$ are independent, i.e., $s_{AB}=s_{A}\otimes s_{B}$,
then $H_{+}\\!\left(A|B\right)=H_{+}\\!\left(A\right)$.

###### Proof.

If $A$ and $B$ are independent, after any measurement
$\\{e^{j}\\}\in\mathcal{E}_{B}$ on $B$ the post-measurement state on $A$ is
$s_{A}$, independent of the outcome of the measurement. Therefore
$H_{+}\\!\left(A|B\right)=H_{+}\\!\left(A\right)$. ∎

######  Property 5 (Classical subsystem inequalities).

For a joint state $s_{ABC}\in\mathcal{S}_{ABC}$ with classical subsystems $A$
and $B$, $H_{+}\\!\left(AB|C\right)\geq H_{+}\\!\left(A|C\right)$.

###### Proof.

For any measurement $\\{f^{j}\\}\in\mathcal{E}_{C}$ we have

$\displaystyle\sum_{j}f^{j}(s_{C})H(AB)_{s_{AB}^{j}}$
$\displaystyle\geq\sum_{j}f^{j}(s_{C})H(A)_{s_{A}^{j}}$ $\displaystyle\geq
H_{+}\\!\left(A|C\right)\,.$

Applying this to a sequence of measurements that converge to
$H_{+}\\!\left(AB|C\right)$ establishes the claimed result. ∎

######  Property 6 (Subadditivity [23, 17]).

In GPTs for which $M\in\mathcal{E}^{*}_{A}$ and $N\in\mathcal{E}^{*}_{B}$
imply $M\otimes N\in\mathcal{E}^{*}_{AB}$ (which holds for locally tomographic
theories, including box-world)
${H_{+}\\!\left(A\right)+H_{+}\\!\left(B\right)\geq H_{+}\\!\left(AB\right)}$.

We refer to [23] or Appendix A for a proof of Property 6. Note further that
Property 6 is the only one that does not hold in arbitrary GPTs.

######  Property 7 (Lemma C2 of [23]).

For $s_{ABC}\in\mathcal{S}_{ABC}$ where $A$ and $B$ are classical systems,
$H_{+}\\!\left(A|BC\right)\geq
H_{+}\\!\left(AB|C\right)-H_{+}\\!\left(B\right)$.

We refer to [23] or Appendix A for a proof of Property 7.

######  Property 8.

For $s_{ABC}\in\mathcal{S}_{ABC}$, where $B$ is a classical subsystem,
${H_{+}\\!\left(A|BC\right)\leq
H_{+}\\!\left(AB|C\right)-H_{+}\\!\left(B|C\right)}$. If $C$ is also classical
then this holds with equality.

We prove this property in Appendix A. Note that Properties 7 and 8 are both
relaxations of the chain rule, $H(A|BC)=H(AB|C)-H(B|C)$, that holds for
Shannon and von Neumann entropy.

##  3 Entropy vector method for causal structures in GPTs

A _causal structure_ is a set of nodes arranged in a directed acyclic graph,
some of which are labelled observed. Each observed node has a corresponding
random variable, while the other, unobserved nodes correspond to resources
from a GPT. For a causal structure $C$ we use $C^{\mathrm{C}}$,
$C^{\mathrm{Q}}$, $C^{\mathrm{B}}$ or $C^{\mathrm{G}}$ depending on whether
the resources are classical, quantum, box-world systems or from some
unspecified GPT respectively. For each unobserved node we associate a
subsystem with each of its outgoing edges. An example for this is displayed in
Figure 1.

A direct arrow from a node $A$ in a causal structure to a node $Z$ means that
$A$ is a _parent_ of $Z$; a directed path from $A$ to $Z$ means that $A$ is an
_ancestor_ of $Z$. For an unobserved node $A$, all subsystems associated with
its outgoing edges are considered parents/ancestors of each its
children/descendants. Given a causal structure, a _coexisting set_ of systems
[9, 22] is one for which a joint state can be defined. In general, no
coexisting set includes all nodes, since there is no joint state of a system
and the output obtained from a measurement on it (unless the system is
classical).

Our method to generate new inequalities for causal structures with GPT
resources begins by considering an _entropy vector_ whose components are the
entropies and conditional entropies of all coexisting sets. Conditional
entropies composed entirely of classical subsystems are excluded because they
are linear combinations of other entropies (e.g., $H(X|Y)=H(XY)-H(Y)$).

We then impose a system of linear (in)equalities that are necessary for a
vector to be realisable as an entropy vector in a causal structure. These
inequalities are constructed using the properties of the measurement entropy
explained earlier and strong subadditivity in the cases where the measurement
entropy reduces to the Shannon entropy. In the case of locally-tomographic
GPTs, such as box-world, there is one additional property (Property 6) that
does not hold in all GPTs. Further constraints come from the causal structure:
two sets of nodes are independent if they do not share any ancestors in the
causal structure. In general, there may be further independencies among the
observed variables (see Theorem 22(i) of [14] and Appendix D). This system of
inequalities constrains a polyhedral cone, which can be projected to a
_marginal cone_ that contains no components involving unobserved systems. The
projection is performed with a Fourier-Motzkin elimination algorithm [26]. An
example that illustrates this procedure in detail is provided at the beginning
of Section 5.1.

When dealing with causal structures for which computing all entropy
inequalities for the marginal scenarios of interest is computationally
impractical or even not possible with the computational resources at hand, due
to the scaling of Fourier-Motzkin elimination [27], we can still derive valid
entropy inequalities by marginalising subsets of all valid inequalities.

Furthermore, given a particular observed distribution that we suspect to be
incompatible with a causal structure (either for classical theory, quantum
theory or boxworld), it is not necessary to go through the marginalisation
procedure discussed here. Instead we can look for a certificate of
incompatibility using a linear program. This program can be set up by
computing the entropy vector for the distribution in question and then adding
this as a list of equalities (one for each of its components) to the list of
valid entropy inequalities for the causal structure. If the resulting system
of linear (in)equalities is infeasible, then the distribution in question is
certified as incompatible with the causal structure within the theory under
consideration.

For some causal structures, the entropic constraints derived using Property 6
are also valid for GPTs that are not locally tomographic, which is the content
of the following proposition.

######  Proposition 1.

Let $C$ be a causal structure in which there are no nodes with two or more
unobserved parents. For any GPT $G$, any correlations achievable with a finite
number of finite-outcome measurements in $C^{\mathrm{G}}$ are achievable in
$C^{\mathrm{B}}$.

This proposition follows from the insight presented in the proof of Lemma 2 in
[28] that any joint measurement on a classical and a GPT system can be written
as a measurement on the classical system followed by an outcome-dependent
measurement on the other (see also Lemma 4 in Appendix A).

###### Proof.

Since each node has at most one unobserved parent, by Lemma 4 at each node we
can assume a standard classical measurement on the classical subsystems
followed by a measurement on the GPT subsystem depending on the result.
Consider then
$s_{A_{1}A_{2}\ldots}\in\mathcal{S}^{\mathrm{G}}_{A_{1}A_{2}\ldots}$ and let
$\\{e^{x}_{a}\\}_{x}\in\mathcal{E}_{A_{1}}$ for $a\in\\{1,2,\ldots,m_{a}\\}$,
$\\{f^{y}_{b}\\}_{y}\in\mathcal{E}_{A_{2}}$ for $b\in\\{1,2,\ldots,m_{b}\\}$
etc. Since the outcome distribution $p(x,y,\ldots|a,b,\ldots)$ is no-
signalling and since (by definition) all no-signalling distributions can be
realised by states in box-world, there exists
$s^{\prime}_{A_{1}A_{2}\ldots}\in\mathcal{S}^{\mathrm{B}}_{A_{1}A_{2}\ldots}$
and box-world measurements on $A_{1}$, $A_{2}$, $\ldots$ that give rise to the
same correlations. ∎

Note that the same argument does not hold if there are multiple unobserved
parents at a single node. This is because some joint measurements cannot be
expressed as a measurement on one system followed by a measurement on the
other conditioned on the first (cf. Lemma 4), for example a measurement in the
Bell basis in quantum mechanics.

The method presented in this section recovers previous entropic approaches for
describing classical [20, 8] and quantum [9] causal structures as special
cases. In the classical case, the measurement entropy and its conditional
version coincide with the Shannon entropy and all variables in a causal
structure (observed and unobserved ones) coexist. In this case, our method is
equivalent to that of [20, 8].222In this case Property 5 imposes that the
entropy is strongly subadditive. For the quantum case, the recovery of the
method proposed in [9] from ours is less obvious and is explained in Section
4.

When considering different causal structures, convexity of the sets of
achievable entropy vectors of the observed variables is useful for their
comparison: for instance, it allows us to prove that the achievable entropies
in one case are contained in those of another by considering only the extreme
points. The following theorem (proven in Appendix B) establishes convexity in
general and is therefore an important structural insight.

######  Theorem 1.

For any causal structure $C^{\mathrm{G}}$ the closure of the set of achievable
entropy vectors of the observed variables is a convex cone.

###  3.1 Causal structures and post-selection

For a causal structure involving a parentless observed node $X$ that takes
values $1,2,\ldots,n$, we can also analyse an adapted causal structure where
each descendant of $X$ is split into $n$-variables and $X$ is dropped, e.g., a
descendant $Y$ is split into $Y_{|X=1},\ldots,Y_{|X=n}$. The resulting causal
structure is said to be _post-selected_ on $X$ (see Figure 5 for an example,
and [9, 22] for further details of this procedure). For some causal
structures, $C$, post-selection is necessary for deriving entropy inequalities
that distinguish between $C^{\mathrm{C}}$, $C^{\mathrm{Q}}$ and
$C^{\mathrm{B}}$ [18, 9, 29]. When post-selecting on parentless nodes,
convexity of the set of achievable entropy vectors of the observed variables
also holds by the following corollary of Theorem 1, which is also proven in
Appendix B.

######  Corollary 1.

For any causal structure $C^{\mathrm{G}}$ in which one or more nodes have been
split into alternatives by post-selecting on parentless observed nodes, the
closure of the set of achievable entropy vectors for the coexisting observed
variables is a convex cone.

##  4 Quantum causal structures

For quantum causal structures there is an entropy vector method in which the
components are the unconditional von Neumann entropies, $H$, of all coexisting
sets [9]. Conditional entropies are not explicitly included, but, because
$H(A|B)=H(AB)-H(B)$, relations involving conditional entropy can still be
encoded. It is natural to ask whether the technique introduced earlier in this
paper could yield different results to the existing entropic approach to
quantum causal structures. In this section, we consider this question.

Following the approach outlined earlier in the paper we consider including all
entropy inequalities from [9] (see Appendix D for a full description of the
method employed in [9]). These are automatically part of our approach, since
the unconditional measurement entropy—for which we include all inequalities
valid in the theory at hand—coincides with the von Neumann entropy in the
quantum case. In addition, we also take into account inequalities for
conditional measurement entropies, which are always positive and differ from
the conditional von Neumann entropy. Thus, our approach could lead to more
restrictive inequalities than the previous quantum one. With the following
proposition we show that the previous method for quantum causal structures [9]
can be refined in a way that makes the additional variables corresponding to
conditional measurement entropies superfluous.

######  Proposition 2.

Consider a causal structure $C^{\mathrm{Q}}$ and suppose that, in addition to
any causal constraints, we use positivity of unconditional entropies, strong
subadditivity and additionally impose positivity of conditional entropies for
all combinations of variables that occur in a co-existing set. If we then
eliminate all variables corresponding to unobserved systems, the resulting
entropic inequalities for the observed variables are all valid.

As a result, all valid inequalities for conditional measurement entropy can be
imposed for the conditional von Neumann entropy instead, and can hence be
encoded as linear constraints on the (unconditional) von Neumann entropy. In
other words, although the conditional von Neumann entropy can be negative for
some quantum states, by constraining it to be positive and eliminating
unobserved variables, we obtain valid entropy inequalities for the observed
variables in $C^{\mathrm{Q}}$. This was not used in previous entropic analyses
of quantum causal structures [9].

Previous quantum methods [9, 22] instead analysed quantum causal structures by
considering the von Neumann entropy of coexisting sets and imposing positivity
of the entropy, strong-subadditivity as well as the weak monotonicity
constraints that for any state $\rho_{XYZ}$, $H(X|Y)+H(X|Z)\geq 0$ [30]. Weak
monotonicity constraints are not needed in the statement of Proposition 2
because they are implied by the positivity of conditional entropies. See
Appendix C for the full proof of Proposition 2 and for a complete account of
previous quantum methods.

This also gives an important insight into the entropy vector method: the
difference between the inequalities that result from using the entropy vector
method in the classical [20, 8] and quantum [9] cases is entirely due to the
fact that in the quantum case not all variables coexist and does not arise
from the different properties of the Shannon and von Neumann entropy (see also
Appendix C for further discussion).

For most causal structures of interest we can prove that our refined entropy
vector method does not allow us to find any tighter entropy inequalities than
that of [9] (see Lemma 8 in Appendix D). Nonetheless, the possibility of using
positivity of conditional entropy instead of weak monotonicity simplifies the
quantum method even in these cases.

##  5 Applications: Entropic characterisation of causal structures with GPT
resources

In this section we illustrate the techniques introduced above through a series
of examples that are chosen to show the different types of result that can
occur. We summarize our findings here, before presenting these cases in
detail.

First, in Section 5.1, we consider the case without post-selection and give
three types of example:

  * •

A causal structure in which the actual entropic cones for classical, quantum
and GPTs are provably the same is in given in Section 5.1.1.

  * •

A causal structure in which our methods lead to the same outer approximations
to the entropy cones in the classical, quantum and GPTs cases is given in
Section 5.1.2.

  * •

Two causal structures for which our methods yield the same outer
approximations of the entropy cones in the classical and quantum cases, but a
different outer approximation is obtained for GPTs is given in Sections 5.1.3
and 5.1.4. In these cases we are not aware of any GPT correlations that
violate the classical/quantum inequalities.

Note that we already know of a case (the triangle causal structure) where we
have different outer approximations of the entropy cones in the classical and
quantum cases [10]. We are currently unaware of a case without post-selection
in which the outer approximations of the entropy cones between two theories
are different and in which we know of correlations in one theory that violate
those in another. In other words, it remains possible that the gaps we find in
the outer approximations are a symptom of the method used, rather than
features of the actual entropy cones.

Then, in Section 5.2 we move to the use of post-selection and show that:

  * •

For the bilocality causal structure there are different entropy cones for
classical, quantum and locally tomographic GPTs (Section 5.2.1). We also
identify distributions that certify that the true entropy cones are indeed
different for all three cases.

  * •

We can use our methods to obtain entropic inequalities for the Information
Causality scenario in arbitrary GPTs (Section 5.2.2).

###  5.1 Analysis of causal structures without post-selection

In this section we illustrate that our method for GPTs can recover the
classically valid entropy inequalities for some causal structures (the first
two examples), and that for other causal structures we recover different
inequalities (the last two examples).

####  5.1.1 Instrumental Scenario

In the classical literature, one well-studied causal structure is the
_instrumental causal structure_ [31] of Figure 1. The quantum version of this
causal structure has also been studied [32, 13].

$Y$$Z$$A_{Y}$$A_{Z}$$X$$A$

Figure 1: The instrumental causal structure. The nodes labelled $X$, $Y$ and
$Z$ correspond to observations, modelled as random variables. The node $A$
labels a resource-system with subsystems $A_{Y}$ and $A_{Z}$ associated to its
outgoing edges.

In this case, the coexisting sets are all subsets of
$\left\\{A_{Y},A_{Z},X\right\\}$, $\left\\{A_{Y},X,Z\right\\}$ and
$\left\\{X,Y,Z\right\\}$. The second set implies that the entropy vector
includes components corresponding to $H_{+}\\!\left(A_{Y}\right)$,
$H_{+}\\!\left(X\right)$, $H_{+}\\!\left(Z\right)$,
$H_{+}\\!\left(A_{Y}X\right)$, $H_{+}\\!\left(A_{Y}Z\right)$,
$H_{+}\\!\left(XZ\right)$, $H_{+}\\!\left(A_{Y}XZ\right)$,
$H_{+}\\!\left(A_{Y}|X\right)$, $H_{+}\\!\left(A_{Y}|Z\right)$,
$H_{+}\\!\left(A_{Y}|XZ\right)$, $H_{+}\\!\left(A_{Y}X|Z\right)$,
$H_{+}\\!\left(A_{Y}Z|X\right)$, $H_{+}\\!\left(X|A_{Y}\right)$,
$H_{+}\\!\left(X|A_{Y}Z\right)$, $H_{+}\\!\left(XZ|A_{Y}\right)$,
$H_{+}\\!\left(Z|A_{Y}\right)$, $H_{+}\\!\left(Z|A_{Y}X\right)$, and similarly
for the other two, leading to a vector with 35 components.

For these we impose all entropy inequalities of Properties 1 to 8 as well as
the independencies of the subsystems of $A$ and $X$, such as
$H_{+}\\!\left(A_{Y}A_{Z}|X\right)=H_{+}\\!\left(A_{Y}A_{Z}\right)$ or
$H_{+}\\!\left(X|A_{Y}A_{Z}\right)=H_{+}\\!\left(X\right)$.

Eliminating all other variables in order to obtain inequalities that only
involve the components $(H(X)$, $H(Y)$, $H(Z)$, $H(XY)$, $H(XZ)$, $H(YZ)$,
$H(XYZ))$, we obtain the _Shannon inequalities_ (positivity of entropy and
conditional entropy, and strong subadditivity) for three variables and

$I(X:YZ)\leq H(Z)\,,$ (5)

which form a polyhedral cone $\Gamma$. Valid entropy vectors for distributions
compatible with the instrumental scenario for a system $A$ of some locally
tomographic GPT are necessarily within $\Gamma$. For this causal structure, it
is known that being in $\Gamma$ is necessary and sufficient for being in the
closure of the set of valid entropy vectors when $A$ is classical or quantum
[33, 14, 10]. Since classical systems are a special case of systems in a GPT,
it follows that membership of $\Gamma$ is also sufficient for locally
tomographic GPTs. We have hence found all valid entropy inequalities in this
scenario for such theories.

According to Proposition 1, (5) holds in any (not necessarily locally
tomographic) GPT333It is already clear that it holds in any GPT satisfying the
premise of Property 6. Note that Proposition 1 implies that the set of
correlations in $C^{\mathrm{B}}$ is at least as large as those in
$C^{\mathrm{G}}$, so any valid restriction that holds for correlations in
$C^{\mathrm{B}}$ holds also for correlations in $C^{\mathrm{G}}$.. Thus, in
the instrumental causal structure, $\Gamma$ completely characterises the set
of achievable entropy vectors independently of whether system $A$ is
classical, quantum or any GPT system.

####  5.1.2 Causal structure of Figure 2

Applied to the causal structure of Figure 2, our method leads to the following
entropy inequalities for the observed variables when $A$ and $B$ are taken to
be box-world systems:

$\displaystyle I(C:E|D)$ $\displaystyle=0$ (6) $\displaystyle I(C:DEF)$
$\displaystyle\leq H(D)$ (7) $\displaystyle I(C:EF)$ $\displaystyle\leq H(E)$
(8)

and the Shannon inequalities.

$F$$E$$D$$C$$A$$B$

Figure 2: Causal Structure where our outer approximations for classical,
quantum and box-world resources coincide.

In [14] the same inequalities were derived for classical $A$ and $B$. It
follows from Proposition 2 (see below), that these constraints also hold in
the quantum case. Thus, for all three theories we obtain the same outer
approximation of the respective entropy cones444This may change if we were to
take non-Shannon inequalities into account prior to Fourier-Motzkin
elimination.. Violation of any of these inequalities excludes this causal
structure as a possible explanation of the observed correlations, irrespective
of the nature of the pre-shared resources.

In this example, these outer approximations are not tight: there are further
valid entropy inequalities for the classical systems $C$, $D$, $E$ and $F$—so-
called non-Shannon inequalities—that lead to tighter approximations, e.g. the
following inequality (from [24]):

$\displaystyle{I(D:E|F)}+{I(D:E|C)}+{I(F:C)}-{I(D:E)}$
$\displaystyle+{I(D:F|E)}+{I(E:F|D)}+{I(D:E|F)}\geq 0.$ (9)

####  5.1.3 Causal structure of Figure 3(a)

With resources $A$ and $B$ that are allowed in box-world we obtain the Shannon
inequalities and

$\displaystyle I(C:EF)$ $\displaystyle\leq H(E),$ (10) $\displaystyle
I(C:DEF)$ $\displaystyle\leq H(D).$ (11)

Classical and quantum resources $A$ and $B$ lead to slightly tighter
inequalities, namely the Shannon inequalities and

$\displaystyle I(C:EF)$ $\displaystyle\leq H(E)$ (12) $\displaystyle
I(C:DEF)+I(D:F|E)$ $\displaystyle\leq H(D).$ (13)

The question of whether or not there exist box-world distributions that
violate one of these inequalities remains open555In general, the methods used
here lead to outer approximations to the entropy cone of a causal structure.
In some cases we can show these to be tight, see e.g. [29, 34], but not
always. It could thus be the case that while there is a gap between the outer
approximations of these cones, there is still no gap between the true cones..

####  5.1.4 Causal structure of Figure 3(b)

With resources $A$ and $B$ that are allowed according to the theory of box-
world we obtain the Shannon inequalities and

$\displaystyle I(C:D)$ $\displaystyle=0,$ (14) $\displaystyle I(F|CE)$
$\displaystyle\leq H(CF|DE),$ (15) $\displaystyle I(D:CEF)$ $\displaystyle\leq
H(E).$ (16)

Classical and quantum resources lead to slightly tighter inequalities, namely
the Shannon inequalities, Equation (14), Inequality (15) and

$\displaystyle I(D:CEF)$ $\displaystyle\leq H(E|C).$ (17)

Note that the classical case was treated in [14].

In general, such comparisons are interesting for analysing the nature of
causation in different theories. In a sense the box-world inequalities can be
thought of as minimal requirements for a theory with a reasonable notion of
causation. Developing a systematic understanding of this may hint at ways to
find a physical principle that singles out quantum correlations in general
scenarios.

$(a)$$F$$E$$D$$C$$A$$B$$(b)$$F$$E$$D$$C$$A$$B$

Figure 3: Causal structures where there are different entropic bounds
depending on the nature of the unobserved $A$ and $B$.

###  5.2 Analysis of causal structures with post-selection

In this section we apply our method to post-selected causal structure and show
how this allows us to distinguish the correlations obtained in different GPTs.
We give two examples for this.666These constraints can be useful even in the
classical case, where there is already the inflation technique for deriving
incompatibility constraints [11, 35], if the cardinality of the variables is
high, for instance.

####  5.2.1 Bilocality

The bilocal causal structure, first analysed in [36, 37], characterises the
situation we encounter in scenarios where we rely on entanglement swapping
[38], see Figure 4(a). Technologically, this scenario is encountered, for
instance, in quantum repeaters [39] or event-ready detection schemes [40].

The entropy vector method provides us with a convenient means to compare the
observable correlations when the sources $L_{1}$ and $L_{2}$ come from
different theories. Applying the entropy vector method to the bilocal causal
structure with sources from a locally tomographic GPT, apart from the Shannon
inequalities we find only

$H(X_{0}Z_{0})=H(X_{0})+H(Z_{0})$ (18)

up to symmetry (exchanging $X_{0}$ and $X_{1}$ as well as $Z_{0}$ and
$Z_{1}$).

Applying the entropy vector method to the bilocal causal structure in the
quantum case we find the Shannon inequalities as well as

$\displaystyle H(X_{0}Z_{0})$ $\displaystyle=H(X_{0})+H(Z_{0})$ (19)
$\displaystyle I(X_{0}Y_{0}:Z_{0})$ $\displaystyle\leq H(Y_{0}|X_{1})$ (20)
$\displaystyle I(X_{1}:Z_{1}|Y_{0})$ $\displaystyle\leq
H(Y_{0}|X_{0})+H(Y_{0}|Z_{0})-H(Y_{0}),$ (21)

up to symmetry. [There are $4$ instances of (19) (obtained by exchanging
$X_{0}$ and $X_{1}$ as well as $Z_{0}$ and $Z_{1}$), $16$ of (20) (obtained by
exchanging $X_{0}$ and $X_{1}$, $Y_{0}$ and $Y_{1}$, or $Z_{0}$ and $Z_{1}$
and by exchanging the roles of $X$ and $Z$) and $8$ of (21) (obtained by
exchanging $X_{0}$ and $X_{1}$, $Y_{0}$ and $Y_{1}$, or $Z_{0}$ and $Z_{1}$).]

The equality constraints are found in both the quantum and GPT case, while the
quantum description is tighter, and in this case the gap is provably
significant. An example of a box-world distribution that violates (21) is
obtained by taking the systems $L_{1}$ and $L_{2}$ to be PR-boxes,
$A\in\left\\{0,1\right\\}$ is the uniform input and $X\in\left\\{0,1\right\\}$
the output on the left, and analogously for $C$ and $Z$ on the right.
$B\in\left\\{0,1\right\\}$ is with probability $\frac{1}{2}$ input into the
first and with probability $\frac{1}{2}$ input into the second box and the
respective output serves as an input for the other, where $Y$ is equal to the
outputs of the two boxes. This distribution also violates the classical
inequalities below and is, to the best of our knowledge, the first
manifestation of such a violation with a GPT correlation that is proven to be
achievable in the bilocal causal structure. [Note that when the violation of
the classical inequalities of [20] was discussed, a tripartite box was
considered directly (and without proof that it can be generated from a GPT in
the bilocal causal structure)777Note also that we have a slight difference
from [20]: they did not consider $B$..]

For classical sources $L_{1}$ and $L_{2}$, we obtain a convex cone constrained
by Shannon inequalities and $53$ other independent classes of linear
inequalities. We list one representative of each of these $53$ classes in
Appendix E. Note that the (in)equalities obtained in the quantum case (cf.
(19)–(21)) are present. A quantum distribution that leads to an entropy vector
outside this cone and which is thus not achievable with classical resources in
this causal structure is, for instance, the following: Suppose $L_{1}$ and
$L_{2}$ share a singlet state each and at the nodes the output is made
according to projective measurements $\Pi_{\theta}$ in the basis
$\left\\{\cos\theta|0\rangle+\sin\theta|1\rangle,\sin\theta|0\rangle-\cos\theta|1\rangle\right\\}$.
The angles are chosen as follows: for $X_{0}$, $\theta=x$, for $X_{1}$,
$\theta=3x$, for $Z_{0}$, $\theta=0$, for $Z_{1}$, $\theta=2x$ and for $Y_{0}$
we consider $\theta=0$ on the state shared by $L_{1}$ and $\theta=(2y_{0}+1)x$
on the other, for $Y_{1}$ we consider $\theta=2x$ on the state shared by
$L_{1}$ and $\theta=(2y_{0}+1)x$ on the other system, where the output is made
up of the outputs of both measurements in both cases and where $x=0.1$ and
$y_{0}$ denotes the outcome of the first measurement. This distribution
violates several of our inequalities, for instance one inequality from class
31 from Appendix E). Note also that for the entropic inequalities derived in
[20] for the bilocal causal structure with classical sources, no quantum
violations are known, so, as far as we know, the characterisation reported
here is the first entropic one that is provably able to resolve this gap.

$(a)$$(b)$$X$$A$$Y$$B$$Z$$C$$L_{1}$$L_{2}$$X_{1}$$X_{0}$$Y_{0}$$Y_{1}$$Z_{0}$$Z_{1}$$L_{1}$$L_{2}$

Figure 4: Bilocal causal structure. (a) The nodes labelled $A$, $B$, $C$, $X$,
$Y$ and $Z$ correspond to observations, modelled as random variables. The
nodes $A$ and $B$ label a resource-system which in this case we take to be
quantum. (b) Post-selected version of the bilocal causal structure, where
$X_{0}$ stands for $X_{|A=0}$ and similarly for $X_{1}$ as well as the
corresponding $Y$ and $Z$.

####  5.2.2 Information Causality

Information causality [3] is a candidate principle for singling out quantum
theory. Roughly speaking the principle is that that sending $n$ bit of
classical information from one party to another cannot give the recipient
access to more than $n$ bits of previously unknown information regardless of
any pre-shared resources the parties may have. The associated causal structure
is shown in Figure 5, and information causality is obeyed if

$\displaystyle I(X_{1}:Y_{|R=1})+I(X_{2}:Y_{|R=2})\leq H(Z)\,.$ (22)

This relation is known to hold in both classical and quantum theory, while it
is violated in box-world.888In fact, tighter entropy inequalities have since
been shown to hold in the quantum (and classical) case [9].

Using the technique of the present paper, we find that the following relations
hold for underlying box-world systems

$\displaystyle I(X_{1}X_{2}:Y_{|R=1}Z)$ $\displaystyle\leq H(Z)$ (23)
$\displaystyle I(X_{1}X_{2}:Y_{|R=2}Z)$ $\displaystyle\leq H(Z)\,.$ (24)

These are valid in any GPT (see Proposition 1) for the entropy vectors
${\bf{H}}$ containing the entropies of all $23$ subsets of the coexisting sets
$\\{X_{1},X_{2},Y_{|R=1},Z\\}$ and $\\{X_{1},X_{2},Y_{|R=2},Z\\}$. Thus,
although information causality does not hold in general, some minimal notion
of causation remains (beyond no-signalling).

We remark that the information causality scenario in boxworld was also
considered in [15], but in a slightly different way. There the relations
${I(X_{1}:Y_{|R=1})}\leq H(Z)$ and ${I(X_{2}:Y_{|R=2})}\leq H(Z)$ were
postulated. We are able to recover these with our approach in the following
way. If, instead of considering all joint entropies of coexisting variables,
only the restricted entropy vectors with components ${\bf{H}_{R}}=(H(X_{1})$,
$H(X_{2})$, $H(Y_{|R=1})$, $H(Y_{|R=2})$, $H(Z)$, $H(X_{1}Y_{|R=1})$,
$H(X_{2}Y_{|R=2}))$ are considered, the inequalities ${I(X_{1}:Y_{|R=1})}\leq
H(Z)$ and ${I(X_{2}:Y_{|R=2})}\leq H(Z)$ emerge. Since all extremal vertices
of the entropy cone of vectors ${\bf{H}_{R}}$ are achievable (as was shown in
[15]) and, according to Corollary 1, the (closure of the) set of achievable
entropy vectors ${\bf{H}_{R}}$ is convex, this is the true entropy cone for
this restricted marginal scenario. However, we don’t see a clear motivation
for excluding the additional observed entropies.

The two inequalities (23) and (24) were already derived in [23] for box-world,
but here they emerge systematically from our method (and our results imply
that they hold for any GPT). They are the only inequalities (other than
Shannon inequalities) that follow from our method as it was introduced above.
However, further inequalities hold for this scenario, for instance the non-
Shannon inequality [24]

$\displaystyle I(X_{2}:Y_{|R=1}|Z)+I(X_{2}:Y_{|R=1}|X_{1})+I(Z:X_{1})$
$\displaystyle-I(X_{2}\\!:\\!Y_{|R=1})\\!+\\!I(X_{2}\\!:\\!Z|Y_{|R=1})\\!+\\!I(Y_{|R=1}\\!:\\!Z|X_{2})\\!$
$\displaystyle+I(X_{2}:Y_{|R=1}|Z)\geq 0\,.$

Because a complete set of non-Shannon inequalities is not known, we do not
have a complete characterisation of the entropy cone of vectors ${\bf{H}}$ for
this scenario (which may require infinitely many linear inequalities).

$(a)$$Y$$R$$Z$$X_{1}$$X_{2}$$A$$(b)$$Y_{|R=1}$$Y_{|R=2}$$Z$$X_{1}$$X_{2}$$A$

Figure 5: Causal structure of the Information Causality scenario. (a) Alice
holds two pieces of information $X_{1}$ and $X_{2}$ and is allowed to send a
message $Z$ to Bob. Bob then has to make a guess of either $X_{1}$ or $X_{2}$,
depending on the request of a referee, $R=1$ or $R=2$. $A$ is a pre-shared
resource the parties may use. (b) We divide $Y$ into two variables, $Y_{|R=1}$
and $Y_{|R=2}$, depending on the question $R$. While for classical $A$ Bob can
always compute the value of both $Y_{|R=1}$ and $Y_{|R=2}$, more generally
these have to be understood as alternatives, of which only one is generated.

##  6 Limitations and directions to overcome them

As is the case for previous entropic methods [20, 8, 9], there are causal
structures for which this method does not imply any entropic constraints for
the observed variables (except Shannon inequalities), an example being the
triangle causal structure [19, 7, 14, 9, 11, 10, 41].999However, the triangle
causal structure with inputs can be treated with our method. Furthermore, all
known strategies that certify incompatibility of entropy vectors relying on
GPT resources with classical and quantum scenarios rely on post-selection (cf.
Figure 5). If post-selection is necessary for this, for some causal structures
(such as the triangle causal structure) current entropic techniques cannot
certify this distinction. This is not a severe limitation, since most
experimentally interesting causal structures involve measurement settings,
which we can post-select on.

Considering entropy vectors rather than the corresponding joint probability
distributions gives a computational advantage and provides constraints that
are valid independently of the dimension of the involved resources. However,
this advantage comes with restricted precision (see for instance [29]). In
particular, there are distributions between observed variables that are
realisable with box-world resources but not with classical or quantum systems
but which the method cannot certify as such. How to overcome this remains an
open question, although the ideas in [42] may form a useful starting point
(notwithstanding the limitations of the ideas of [42] identified in [43]).

While we found that our method strictly improves on previous entropic methods
[15], another promising research avenue is to generalise the inflation
technique to GPTs. This research has been started in [11], where it was
pointed out that certain inflations are valid for any GPT and hence some
general constraints can be derived with it. The question of how these
constraints relate to the ones found with our method is left for future work.
One key difference is that the method presented in [11] does not distinguish
what GPT the latent systems are described by (e.g. whether they are quantum
and box-world systems). Considering the latent variables explicitly allows us
to make this distinction and to certify that different sets of correlations
are produced within different GPTs (e.g. quantum theory and box-world).

## Acknowledgements

We thank Peter J. Brown for comments on an earlier version of this manuscript.
This work was supported by an EPSRC First grant (grant number EP/P016588/1),
the EPSRC’s Quantum Communications Hub (grant number M013472/1) and by the
Austrian Science fund (FWF) stand-alone project P 30947. The majority of this
work was carried out while MW was based at the University of York.

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##  Appendix A Further details regarding the inequalities for the measurement
entropy in GPTs

Before getting to the additional properties, we need a few lemmas. The first
is the concavity of $H_{+}$, proven in [23, 17], which follows from the
concavity of the Shannon entropy.

######  Lemma 1.

For $s_{1}$, $s_{2}\in\mathcal{S}_{A}$ and $0\leq p\leq 1$,
$H_{+}\\!\left(A\right)_{ps_{1}+(1-p)s_{2}}\geq
pH_{+}\\!\left(A\right)_{s_{1}}+(1-p)H_{+}\\!\left(A\right)_{s_{2}}$.

The next lemma says that the infimum in the definition of conditional
measurement entropy can be restricted to fine-grained measurements.

######  Lemma 2.

Let $s_{AB}\in\mathcal{S}_{AB}$, then
$H_{+}\\!\left(A|B\right)=\inf_{\\{f^{y}\\}\in\mathcal{E}^{*}_{B}}\sum_{y}f^{y}(s_{B})H_{+}\\!\left(A_{|y}\right)$.

###### Proof.

Let $\\{f^{y}\\}_{y=1}^{n}\in\mathcal{E}_{B}$ and consider the coarse-graining
that combines the last two effects in $\\{f^{y}\\}_{y=1}^{n}$ to give a new
measurement $\\{g^{y}\\}_{y=1}^{n-1}$ where $g^{y}=f^{y}$ for $y=1,\ldots,n-2$
and $g^{n-1}=f=f^{n-1}+f^{n}$. We have

$\displaystyle\sum_{y=1}^{n-1}$ $\displaystyle
g^{y}(s_{B})H_{+}\\!\left(A_{|y}\right)=\sum_{y=1}^{n-2}f^{y}(s_{B})H_{+}\\!\left(A_{|y}\right)+(f^{n-1}(s_{B})+f^{n}(s_{B}))H_{+}\\!\left(\alpha
s_{A|n-1}+(1-\alpha)s_{A|n}\right)\,,$

where $\alpha=f^{n-1}(s_{B})/(f^{n-1}(s_{B})+f^{n}(s_{B}))$. Using concavity,
we have

$\displaystyle\sum_{y=1}^{n-1}g^{y}(s_{B})H_{+}\\!\left(A\right)_{s_{A|y}}$
$\displaystyle\geq\sum_{y=1}^{n-2}f^{y}(s_{B})H_{+}\\!\left(A\right)_{s_{A|y}}+f^{n-1}(s_{B})H_{+}\\!\left(s_{A|n-1}\right)+f^{n}(s_{B})H_{+}\\!\left(s_{A|n}\right)$
$\displaystyle=\sum_{y=1}^{n}f^{y}(s_{B})H_{+}\\!\left(A\right)_{s_{A|y}},$

i.e., for this coarse-graining the measurement on Bob cannot decrease the
expected measurement entropy on Alice conditioned on the result. Since all
coarse-grainings can be formed by a sequence of such combinations, it follows
that the infimum on Bob’s measurements can be restricted to fine-grained
measurements. ∎

It is also worth noting the following.

######  Lemma 3.

Let $\\{f^{y}\\}\in\mathcal{E}_{B}$ be a trivial refinement of
$\\{e^{x}\\}\in\mathcal{E}_{B}$. Then
$\sum_{y}f^{y}(s_{B})H_{+}\\!\left(A_{|y}\right)=\sum_{x}e^{x}(s_{B})H_{+}\\!\left(A_{|x}\right)$.

###### Proof.

Consider the case in which one of the effects in $\\{e^{x}\\}_{x=1}^{n}$ is
split into two to form $\\{f^{y}\\}_{y=1}^{n+1}$ with $f^{y}=e^{y}$ for
$y=1,\ldots,n-1$ and $f^{n}+f^{n+1}=e^{n}$. We have
$H_{+}\\!\left(A_{|n+1}\right)=H_{+}\\!\left(A_{|n}\right)$ and hence in this
case the claim follows from $f^{n+1}+f^{n}=e^{n}$. Since any trivial
refinement can be formed by combining such splittings, the result generalizes
to all trivial refinements. ∎

The following lemma is in essence a restatement of part of the proof of Lemma
2 from [28].

######  Lemma 4.

Let $A$ be classical and $B$ be a system from an arbitrary GPT. For any
measurement $M\in\mathcal{E}_{AB}$ there exists an $n$-outcome measurement
$N_{A}\in\mathcal{E}_{A}$ and measurements $N^{x}_{B}\in\mathcal{E}_{B}$ for
$x=1,2,\ldots,n$ such that $M$ is equivalent to performing $N_{A}$, then
performing $N^{x}_{B}$ (where $x$ is the result of $N_{A}$).

###### Proof.

Let $M=\\{e^{r}\\}$ and let $\\{f^{x}\\}$ be a standard classical measurement
on $A$. We can define a new set of effects on $B$ that act as
$e^{r}_{x}:s_{B}\mapsto e^{r}(s_{A}^{x}\otimes s_{B})$, where
$\\{s_{A}^{x}\\}$ are a set of states on $A$ for which
$f^{x}(s_{A}^{y})=\delta_{x,y}$. For each $x$, the set $\\{e^{r}_{x}\\}_{r}$
form a measurement on $B$:

$\sum_{r}e^{r}_{x}(s_{B})=\sum_{r}e^{r}(s_{A}^{x}\otimes
s_{B})=1\quad\text{for all }s_{B}\in\mathcal{S}_{B}\,.$

If we take $N_{A}=\\{f^{x}\\}$ and $N_{B}^{x}=\\{e_{x}^{r}\\}_{r}$, then this
is equivalent to measuring $M$:

$\displaystyle p(r)$
$\displaystyle=\sum_{x}p(x)p(r|x)=\sum_{x}p(x)e_{x}^{r}(s_{B|x})=\sum_{x}f^{x}(s_{A})e^{r}(s_{A}^{x}\otimes
s_{B|x})=\sum_{x}e^{r}(s_{A}^{x}\otimes(f^{x}\otimes I_{B})(s_{AB}))$
$\displaystyle=\sum_{x}e^{r}((s_{A}^{x}f^{x}\otimes
I_{B})(s_{AB}))=e^{r}(s_{AB})\,,$

where we have used that $\sum_{x}s_{A}^{x}f^{x}$ is the identity
transformation on the classical system $A$. ∎

We will in particular rely on the following corollary of this lemma.

######  Corollary 2.

Let $A$ be classical and $B$ be a system from an arbitrary GPT. For any fine-
grained measurement $M\in\mathcal{E}^{*}_{AB}$ there exists an $n$-outcome
fine-grained measurement $N_{A}\in\mathcal{E}^{*}_{A}$, and fine-grained
measurements $N^{x}_{B}\in\mathcal{E}^{*}_{B}$ for $x=1,2,\ldots,n$ such that
$M$ is equivalent to performing $N_{A}$, then performing $N^{x}_{B}$ (where
$x$ is the result of $N_{A}$).

###### Proof.

We have already shown that $N_{A}$ can be taken to be fine-grained. Suppose
$N_{B}^{1}$ is not fine grained, and consider $\hat{N}_{B}^{1}$ in which
$e_{1}^{t}$ is split into other effects $e_{1}^{t_{1}}$ and $e_{1}^{t_{2}}$
satisfying $e_{1}^{t_{1}}+e_{1}^{t_{2}}=e_{1}^{t}$ in a non-trivial way, i.e.,
with $e_{1}^{t_{1}}$ and $e_{1}^{t_{2}}$ not proportional to one another or to
any other effect $e_{1}^{j}$ with $j\neq t$. The measurement that involves
measuring $N_{A}$ and then $\hat{N}_{B}^{1}$ if $x=1$ and otherwise
$N_{B}^{x}$ depending on the result is a non-trivial fine-graining of $M$.
Thus, if $M$ is fine-grained, so are $N^{x}_{B}$ for all $x$. ∎

###### Proof of Property 6.

For any measurements $\\{e^{x}\\}\in\mathcal{E}^{*}_{A}$ and
$\\{f^{y}\\}\in\mathcal{E}^{*}_{B}$ and any state $s_{AB}\in\mathcal{S}_{AB}$
we have

$\displaystyle-\sum_{x}e^{x}(s_{A})\log(e^{x}(s_{A}))-\sum_{y}f^{y}(s_{B})\log(f^{y}(s_{B}))$
$\displaystyle\geq-\sum_{xy}(e^{x}\otimes f^{y})(s_{AB})\log((e^{x}\otimes
f^{y})(s_{AB}))$ $\displaystyle\geq H_{+}\\!\left(AB\right)\,,$

where the first inequality uses subadditivity of the Shannon entropy and the
last line follows because $\\{e^{x}\otimes f^{y}\\}_{x,y}$ is a measurement on
$AB$ and is hence at least as large as the infimum over joint measurements in
$H_{+}\\!\left(AB\right)$. Then the fact that $H_{+}\\!\left(A\right)$ and
$H_{+}\\!\left(B\right)$ are infima over fine-grained measurements means that
$\\{e^{x}\\}$ and $\\{f^{y}\\}$ can be chosen such that
$-\sum_{x}e^{x}(s_{A})\log(e^{x}(s_{A}))-\sum_{y}f^{y}(s_{B})\log(f^{y}(s_{B}))$
is arbitrarily close to $H_{+}\\!\left(A\right)+H_{+}\\!\left(B\right)$
implies result. ∎

###### Proof of Property 7.

Using Corollary 2, the measurement on $BC$ in $H_{+}\\!\left(A|BC\right)$ can
be decomposed into a standard measurement on $B$, yielding $y$, followed by a
fine grained measurement on $C$ depending on the value of $y$ obtained, i.e.,

$H_{+}\\!\left(A|BC\right)=\sum_{y}e^{y}(s_{B})\inf_{\\{h^{z}_{y}\\}_{z}\in\mathcal{E}_{C}^{*}}\sum_{z}h^{z}_{y}(s_{C|y})H(A_{|yz})\,.$

For some fixed set $\\{h^{z}_{y}\\}_{z}\in\mathcal{E}_{C}^{*}$, the right hand
side (without the $\inf$) takes the form

$\displaystyle\sum_{yz}p(y,z)H(A_{|yz})$
$\displaystyle=H(A|YZ)=H(AY|Z)-H(Y|Z)\geq H(AB|Z)-H(Y)$
$\displaystyle=H(AB|Z)-H_{+}\\!\left(B\right)\geq
H_{+}\\!\left(AB|C\right)-H_{+}\\!\left(B\right)\,,$

where we have used that there is a standard measurement achieving the infimum
for classical systems in $H_{+}\\!\left(B\right)$ and the subadditivity of the
Shannon entropy. The result follows because we can choose
$\\{h^{z}_{y}\\}_{z}\in\mathcal{E}_{C}^{*}$ such that the left hand side is
arbitrarily close to $H_{+}\\!\left(A|BC\right)$. ∎

###### Proof of Property 8.

We start from the definition of $H_{+}\\!\left(AB|C\right)$, and use Corollary
2 to give

$\displaystyle H_{+}\\!\left(AB|C\right)\\!$
$\displaystyle=\\!\\!\\!\inf_{\\{f^{z}\\}\in\mathcal{E}_{C},\\{e^{x}_{z}\\}_{x}\in\mathcal{E}_{AB}^{*}}\\!\\!\\!\\!\\!-\sum_{z}f^{z}(s_{C})\sum_{x}e_{z}^{x}(s_{AB|z})\log(e_{z}^{x}(s_{AB|z}))$
$\displaystyle=\\!\\!\\!\inf_{\\{f^{z}\\}\in\mathcal{E}_{C},\\{g^{x}_{yz}\\}_{x}\in\mathcal{E}_{A}^{*}}\\!\\!\\!\\!\\!-\sum_{z}f^{z}(s_{C})\sum_{xy}h^{y}(s_{B|z})g_{yz}^{x}(s_{A|yz})\log(h^{y}(s_{B|z})g_{yz}^{x}(s_{A|yz}))$
$\displaystyle=\\!\\!\\!\inf_{\\{f^{z}\\}\in\mathcal{E}_{C},\\{g^{x}_{yz}\\}_{x}\in\mathcal{E}_{A}^{*}}\\!\\!\\!\\!\\!-\\!\sum_{z}f^{z}(s_{C})\\!\\!\left[\\!\sum_{y}\\!h^{y}(s_{B|z})\log(h^{y}(s_{B|z}))\\!+\\!\\!\sum_{xy}\\!h^{y}(s_{B|z})g_{yz}^{x}(s_{A|yz})\log(g_{yz}^{x}(s_{A|yz}))\\!\right]$
$\displaystyle\geq\inf_{\\{f^{z}\\}\in\mathcal{E}_{C}}-\sum_{z}f^{z}(s_{C})\sum_{y}h^{y}(s_{B|z})\log(h^{y}(s_{B|z}))\quad+$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\inf_{\\{f^{z}\\}\in\mathcal{E}_{C},\\{g^{x}_{yz}\\}_{x}\in\mathcal{E}_{A}^{*}}-\sum_{z}f^{z}(s_{C})\sum_{xy}h^{y}(s_{B|z})g_{yz}^{x}(s_{A|yz})\log(g_{yz}^{x}(s_{A|yz}))$
$\displaystyle\geq H_{+}\\!\left(B|C\right)+H_{+}\\!\left(A|BC\right)\,.$

In the last inequality we use that a measurement on $C$ followed by a fine-
grained measurement on $A$ is a joint measurement on $s_{AC}$, so the infimum
over all joint measurements cannot be larger than this term.

If $C$ is classical then we can drop $\inf_{\\{f^{z}\\}\in\mathcal{E}_{C}}$
and take $C$ to always be measured with a standard classical measurement. This
gives equality in both inequalities in the above proof. ∎

##  Appendix B Proof of Theorem 1 and Corollary 1

The proof relies on the following Lemmas.

######  Lemma 5.

For any causal structure $C^{\mathrm{G}}$ where all observed variables share
one observed ancestor the topological closure of the set of achievable entropy
vectors of the observed variables is convex.

###### Proof.

We first prove convexity, and then show that the set form a cone.

Let $C^{\mathrm{G}}$ have $n$ observed variables and $m$ unobserved ones. Let
$\bf{H_{1}}$ and $\bf{H_{2}}$ be two achievable entropy vectors for the $n$
observed variables in $C^{\mathrm{G}}$. In the following, we show that for any
$0\leq p\leq 1$, there is a sequence of entropy vectors $\bf{H^{\prime}_{k}}$
within $C^{\mathrm{G}}$, such that
$\lim_{k\rightarrow\infty}{\bf{H^{\prime}_{k}}}=p{\bf{H_{1}}}+(1-p){\bf{H_{2}}}$.

For $i=1,2$, suppose that ${\bf H_{i}}$ is generated by using states
$\\{Y_{j}^{i}\\}_{j=1}^{m}$ for the $m$ unobserved nodes and that the observed
random variables are $\\{X_{j}^{i}\\}_{j=1}^{n}$. The strategy for achieving
the convex combination is as follows. The common observed variable is taken to
be $X_{1}^{\prime}(k)=(A,Z)$ where

$(A,Z)=\begin{cases}(0,0)&\text{ with probability }\frac{k-1}{k}\\\
(1,(X_{1}^{1})^{k})&\text{ with probability }\frac{p}{k}\\\
(2,(X_{1}^{2})^{k})&\text{ with probability }\frac{1-p}{k}\end{cases}$

and where $X^{k}$ denotes $k$ i.i.d. copies of a random variable $X$. Each of
the unobserved nodes is prepared in state

$Y^{\prime}_{j}=(Y_{j}^{1})^{\otimes k}\otimes(Y_{j}^{2})^{\otimes k}\,.$

Each of the other observed nodes then behaves as follows. The children of
$X_{1}$ have access to $A$. If $A=0$ they output $X_{j}^{\prime}(k)=(0,0)$. If
$A=1$ they perform the operation that would have led to ${\bf H_{1}}$ $k$
times independently, acting on the first $k$ subsystems of any GPT resources
they have access to. They then output $X_{j}^{\prime}(k)=(1,(X_{j}^{1})^{k})$.
If $A=2$, the procedure is the same except that the operation that would have
led to ${\bf H_{2}}$ is repeated $k$ times by acting on the second $k$
subsystems of any GPT resources and the output is
$X_{j}^{\prime}(k)=(2,(X_{j}^{2})^{k})$. Note that the first part of the
argument is equal to $A$, so, in this way, the value of $A$ is transferred to
all descendants. An analogous strategy is then used for subsequent
generations.

For any subset $S$ of the observed random variables
$\\{X_{j}^{\prime}(k)\\}_{j=1}^{n}$ we have

$H_{k}^{\prime}(S)=H_{k}^{\prime}(A)+pH^{1}(S)+(1-p)H^{2}(S)\,,$

where $H_{k}^{\prime}$ refers to the entropy in the new strategy and $H^{1}$
and $H^{2}$ refer to the entropies in the original strategies (i.e., according
to ${\bf H_{1}}$ or ${\bf H_{2}}$). Noting that

$H_{k}^{\prime}(A)=-\frac{k-1}{k}\log\frac{k-1}{k}-\frac{p}{k}\log\frac{p}{k}-\frac{1-p}{k}\log\frac{1-p}{k}$

tends to $0$ as $k$ tends to $\infty$, we have
$\lim_{k\rightarrow\infty}{\bf{H^{\prime}_{k}}}=p{\bf{H_{1}}}+(1-p){\bf{H_{2}}}$.

If $\bf{H_{1}}$ and $\bf{H_{2}}$ are themselves only achievable as limits of
entropy vectors the above argument can be followed for each vector in the
corresponding sequences tending to $\bf{H_{1}}$ and $\bf{H_{2}}$ respectively
and thus also holds for $\bf{H_{1}}$ and $\bf{H_{2}}$. This shows that the
closure of the set of entropy vectors is convex. ∎

The next lemma extends this beyond the case where there is a common observed
ancestor.

######  Lemma 6.

For any causal structure $C^{\mathrm{G}}$ the topological closure of the set
of achievable entropy vectors of the observed variables is convex.

###### Proof.

If all observed variables in $C^{\mathrm{G}}$ have a common observed ancestor,
the statement follows by Lemma 5. Otherwise, there are $1<l\leq n$ observed
nodes without any observed ancestors, which we label $X_{1},\ldots,X_{l}$ (all
other observed nodes ($X_{l+1},\ldots,X_{n}$) are descendants of at least one
of these nodes). We construct a larger causal structure $C^{\prime}$ by
introducing an observed parent node $A_{i}$ for each $X_{i}$ with
$i=1,\ldots,l$, where $A_{i}$ has no direct link to any variable except for
$X_{i}$. Note that a distribution over the observed variables
$X_{1},\ldots,X_{n}$ is compatible with $C^{\mathrm{G}}$ if and only if it is
the marginal of a distribution over $X_{1},\ldots,X_{n},A_{1},\ldots,A_{l}$
that is compatible with $C^{\prime}$.

Now let $C^{\prime\prime}$ be another causal structure that is constructed
from $C^{\prime}$ by adding a directed link from $A_{1}$ to all other $A_{i}$
with $2\leq i\leq l$. A distribution over
$X_{1},\ldots,X_{n},A_{1},\ldots,A_{l}$ is compatible with $C^{\prime}$ if and
only if it is compatible with $C^{\prime\prime}$ and, at the same time, obeys
$I(A_{1}:A_{i})=0$ for all $2\leq i\leq l$.

The “if” condition follows because any distribution in $C^{\prime}$ obeys
$I(A_{1}:A_{i})=0$ for all $2\leq i\leq l$ and it can be realised in
$C^{\prime\prime}$ without using the additional causal links $A_{1}\rightarrow
A_{i}$.

For the “only if”, we use that $I(A_{1}:A_{i})=0$ holds if and only if
$p(a_{i}a_{1})=p(a_{i})p(a_{1})$101010This can for instance be seen by writing
the mutual information in terms of the relative entropy and using Klein’s
inequality [44]., so that any distribution in $C^{\prime\prime}$ obeying
$I(A_{1}:A_{i})=0$ for all $2\leq i\leq l$ can be written as

$p(x_{1},\ldots,x_{n},a_{1},\ldots,a_{l})=p(x_{l+1},\ldots,x_{n}|x_{1},\ldots,x_{l})p(x_{1}|a_{1})\ldots
p(x_{l}|a_{l})p(a_{1})\ldots p(a_{l})\,,$ (25)

with the right hand side compatible with $C^{\prime}$.

Hence, a distribution over $X_{1},\ldots,X_{n}$ is compatible with
$C^{\mathrm{G}}$ if and only if it is the marginal of a distribution over
$X_{1},\ldots X_{n},A_{1},\ldots,A_{l}$ that is compatible with
$C^{\prime\prime}$ and obeys $I(A_{1}:A_{i})=0$ for all $2\leq i\leq l$.

The closure of the set of entropy vectors of the observed variables that are
compatible with $C^{\prime\prime}$ (without any additional constraints) is
convex by Lemma 5. The closure of the set of achievable entropy vectors in
$C^{\mathrm{G}}$ is the closure of the set of achievable entropy vectors in of
$C^{\prime\prime}$ restricted by the linear equalities $I(A_{1}:A_{i})=0$ for
all $2\leq i\leq l$ and projected to the marginals involving only
$X_{1},\ldots,X_{n}$. Because these operations preserve convexity, the closure
of the set of achievable entropy vectors of $C^{\mathrm{G}}$ is convex. ∎

The main theorem of this section now follows as a corollary.

###### Proof of Theorem 1.

Convexity of the set of achievable entropy vectors follows by Lemma 6. That it
is a cone follows because if ${\bf H}$ is an achievable entropy vector, then
$k{\bf H}$ for $k\in\mathbb{N}$ is achievable by taking $k$ independent copies
of all systems in the strategy achieving ${\bf H}$. Furthermore, in any causal
structure $C^{\mathrm{G}}$, ${\bf H}={\bf 0}$ is achievable by taking all
observed variables to be $0$ with probability 1. Hence, by taking convex
combinations, if ${\bf H}$ is achievable, so is $\lambda{\bf H}$ for any
$\lambda\in\mathbb{R}_{\geq 0}$. ∎

Corollary 1 then follows in a similar way.

###### Proof of Corollary 1.

Consider first postselecting on one of the parentless variables in
$C^{\mathrm{G}}$, and suppose that this variable has $k$ possible values. Let
$X_{1},\ldots,X_{n}$ be the set of all the observed descendants of the
variable that has been postselected on and $Y_{1},\ldots,Y_{m}$ be the set of
all other observed nodes. In other words, for a fixed distribution,
$P_{Y_{1},\ldots,Y_{m}}$, of all other observed nodes we consider $k$
different ways to form $X_{1},\ldots,X_{n}$ to give $P_{X_{1}^{1}\cdots
X_{n-1}^{1}X_{n}^{1}Y_{1}^{1},\cdots,Y_{m}^{1}},\ldots,P_{X_{1}^{k}\cdots
X_{n-1}^{k}X_{n}^{k}Y_{1}^{k},\cdots,Y_{m}^{k}}$ respectively. For these
distributions we define an entropy vector with $k(2^{n+m}-1)$ components by
concatenating the entropy vectors of each of them. Since the marginal
distributions obey $P_{Y_{1}^{1}\cdots Y_{m}^{1}}=\cdots=P_{Y_{1}^{k}\cdots
Y_{m}^{k}}$, $(k-1)(2^{m}-1)$ components can be removed from the vector. If
${\bf{H_{1}}}$ and ${\bf{H_{2}}}$ are two such achievable entropy vectors,
then for any $0\leq p\leq 1$, $p{\bf{H_{1}}}+(1-p){\bf{H_{2}}}$ is also such
an achievable entropy vector. This follows by applying the technique used to
prove Lemma 6 separately to the causal structure including only one of the $k$
alternatives (i.e., the causal structure formed from the post-selected causal
structure by removing all nodes associated with other alternatives). This
strategy leads to the same distribution on $Y_{1},\ldots,Y_{m}$ for each
alternative and hence the overall entropy vector has the same entropies for
all subsets of $\\{Y_{1},\ldots,Y_{m}\\}$. We can then postselect on further
parentless variables in a similar way. ∎

##  Appendix C Proof of Proposition 2

We use $X_{1},\ldots,X_{n}$ for the observed variables and
$Y_{1},\ldots,Y_{m}$ for the unobserved nodes in $C$. For each unobserved node
$Y_{i}$ we use $Y_{i}^{j}$ with $1\leq j\leq k_{i}$ for the subsystems
associated with the $k_{i}$ outgoing edges, sometimes using
$Y=\\{Y_{i}^{j}\\}_{i=1,j=1}^{m,\ \ k_{i}}$ and $X=X_{1},\ldots,X_{n}$ as a
shorthand. For any unobserved node $Y_{i}$ with $1\leq i\leq m$ and for any
$1\leq j\leq k_{i}$, we show how to modify $Y_{i}^{j}$ to
${\tilde{Y}_{i}^{j}}$ such that, if ${\tilde{Y}_{i}^{j}}$ is shared along the
$j^{\textrm{th}}$ outgoing edge instead of ${{Y}_{i}^{j}}$, the same
distributions among the observed variables are obtained. This construction of
${\tilde{Y}_{i}^{j}}$ will make all conditional entropies of unobserved
systems positive.

If $H({Y}_{i}^{j}|Y\cup X\setminus{Y}_{i}^{j})\geq 0$, we set
$\tilde{Y}_{i}^{j}={Y}_{i}^{j}$. Otherwise, we let
$\tilde{Y}_{i}^{j}=Y_{i}^{j}\otimes\alpha_{i}^{j}$, where
$\alpha_{i}^{j}=\sum_{a}p_{i}^{j}(a)|a\rangle\\!\langle a|$ is a system that
is uncorrelated with any other system and obeys
$H(\alpha_{i}^{j})=H(Y_{i}^{j})$. Then,
$H(\tilde{Y}_{i}^{j})=H({Y}_{i}^{j})+H(\alpha_{i}^{j})$ and
$\tilde{Y}_{i}^{j}$ can be used to produce the same observed distributions as
${Y}_{i}^{j}$, since $\alpha_{i}^{j}$ may be ignored when processing the
unobserved systems $\tilde{Y}_{i}^{j}$ to obtain observed variables.
Furthermore, due to the independence of the $\alpha_{i}^{j}$ and by weak
monotonicity, for any $\tilde{Y}_{i}^{j}$ and any set of variables, $S$,
coexisting with $\tilde{Y}_{i}^{j}$,

$\displaystyle
H(\tilde{Y}_{i}^{j}|S)=H({Y}_{i}^{j}|S)+H(\alpha_{i}^{j})\geq-H(Y_{i}^{j})+H(\alpha_{i}^{j})=0,$
(26)

where the last equality follows by construction. (Note that for an observed
variable $X$ and a set $S$ coexisting with $X$, the analogous relation
$H(X|S)\geq 0$ already holds.)

We now show that for any two coexisting sets $S,T\subset U$ with $S\cap
T=\emptyset$ and where $U$ is a maximal coexisting set, the conditional
entropy $H(S|T)$ is positive. First of all, by strong subadditivity,

$H(S|T)\geq H(S|U\setminus S).$ (27)

Positivity of $H(S|U\setminus S)$ can be shown inductively in the cardinality
of $S$. For cardinality $1$ this is implied by (27) and (26). Assuming that
this holds for any set with cardinality $q$, the following shows that it also
holds for any set with cardinality $q+1$. Let there be a set of variables
$S\subseteq U$ of a maximal coexisting set $U$ with cardinality $q+1$ and a
one element subset $S_{1}\subseteq S$, Writing $S=S_{1}\bar{S}_{1}$, then

$\displaystyle H(S|U\setminus S)$ $\displaystyle=H(S_{1}|U\setminus
S)+H(\bar{S}_{1}|(U\setminus S)\cup S_{1})$ $\displaystyle=H(S_{1}|U\setminus
S)+H(\bar{S}_{1}|U\setminus\bar{S}_{1}),$

which is at least 0 by the inductive hypothesis. It then follows from (27)
that $H(S|T)\geq 0$ for all $T\subseteq(U\setminus S)$.

##  Appendix D Remarks on the quantum entropy vector method

This appendix gives additional information regarding the role of Proposition 2
for the quantum entropy vector method introduced in [9] (see also [22] for a
review). For completeness, we first briefly introduce the details of this
method.

The _quantum entropy vector method_ is based on the von Neumann entropy. For
any joint state of coexisting systems associated with some of the nodes (and
edges) of a causal structure a joint entropy can be defined, where the notion
of coexisting sets is the one discussed for the measurement entropy in the
main text. However, the quantum method does not take the conditional entropies
as separate variables (these would be redundant because $H(X|Y)=H(XY)-H(Y)$).
For all variables within a coexisting set, the following inequalities
hold,111111In both inequalities $X$, $Y$, $Z$ are all coexisting and may each
be made up of subsystems associated with several nodes.

  * •

Strong subadditivity: For any state $\rho_{XYZ}$, $H(XYZ)+H(Z)\leq
H(XZ)+H(YZ)$.

  * •

Weak monotonicity: For any state $\rho_{XYZ}$, $H(X|Y)+H(X|Z)\geq 0$.

Note that whenever there is no entanglement between two subsystems $X$ and $Y$
of a state $\rho_{XY}$, the stronger monotonicity statement $H(X|Y)\geq 0$
holds. Since it is always possible to purify an unobserved quantum state
$\rho_{A_{1}\cdots A_{n}}$, we can impose the following.

  * •

Purification for unobserved systems: For an unobserved system in state
$\rho_{A_{1}\cdots A_{n}}$ we can take $H(A_{1}\cdots A_{n})=0$ and for any
subsystem $S\subset\left\\{A_{1},\ldots,A_{n}\right\\}$ we can take
$H(S)={H(\left\\{A_{1},\ldots,A_{n}\right\\}\setminus S)}$.121212This can be
seen by considering the Schmidt decomposition of the purified state [22].

Among the variables of different coexisting sets data processing inequalities
hold.

  * •

Data Processing: Let $\rho_{XY}$ be the joint state of two sets of coexisting
nodes $X$ and $Y$ and let $\mathcal{E}$ be a completely positive trace
preserving map taking $Y$ to $Z$ such that
$(\mathcal{I}\otimes\mathcal{E})(\rho_{XY})=\rho_{XZ}$, then $H(X|Y)\leq
H(X|Z)$.131313For a discussion on which data processing inequalities are
relevant for computing constraints on entropy vectors we refer to [22].

In addition, the causal structure will in general imply independence
constraints among observed as well as among unobserved systems. These are
based on the notion of _d-separation_ : for three pairwise disjoint sets of
variables $X$, $Y$ and $Z$, $X$ and $Y$ are d-separated by $Z$, if $Z$ blocks
any path from any node in $X$ to any node in $Y$. A path is blocked by $Z$, if
it contains one of the following: $i\rightarrow z\rightarrow j$ or
$i\leftarrow z\rightarrow j$ for nodes $i$, $j$ and a node $z\in Z$ in that
path, or if it contains $i\rightarrow k\leftarrow j$, where $k\notin Z$. Note
that it is possible that $Z=\emptyset$.

  * •

Independences (following Theorem 22 (i) from [14]): For three pairwise
disjoint sets of observed variables, $X$, $Y$ and $Z$, if $X$ and $Y$ are
d-separated by $Z$, then $H(X|Z)=H(X|YZ)$. (Note that $Z=\emptyset$ is
allowed.)

We show with the following Lemma that weak monotonicity constraints are not
relevant in this approach when considering causal structures where none of the
unobserved nodes have any parents. These are the scenarios that are usually
considered in the literature.

######  Lemma 7.

For any causal structure $C^{\mathrm{Q}}$ in which the unobserved quantum
nodes do not have any parents, all weak monotonicity inequalities are implied
by the other inequalities, i.e., for any two coexisting sets $S_{1},S_{2}$
with $S_{1}\cap S_{2}\neq\emptyset$,

$\displaystyle H(S_{1}\cap S_{2}|S_{1}\setminus S_{2})+H(S_{1}\cap
S_{2}|S_{2}\setminus S_{1})\geq 0$ (28)

is redundant.

###### Proof.

Let $A$ denote the collection of subsystems of all unobserved nodes and $S$ be
the maximal coexisting set that includes all unobserved systems $A$. Then for
the coexisting sets $S_{1},S_{2}$ with $S_{1}\cap S_{2}\neq\emptyset$ we
divide into three cases.

Case 1: $S_{1},S_{2}\subseteq S$. Let $R_{1}=S_{1}\cap A$ and $R_{2}=S_{2}\cap
A$. We use the purification of unobserved systems to rewrite,141414As the
unobserved nodes do not have any parents the systems associated with a
particular node are independent of those associated with another node as well
as independent of other variables coexisting with them.

$\displaystyle H$ $\displaystyle(S_{1}\cap S_{2}|S_{1}\setminus
S_{2})\\!+\\!H(S_{1}\cap S_{2}|S_{2}\setminus S_{1})$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!{\small=}\\!\\!H(R_{1})\\!-\\!H(R_{1}\\!\setminus\\!R_{2})\\!+\\!H(S_{1}\\!\setminus\\!R_{1})\\!-\\!H((S_{1}\\!\setminus\\!R_{1})\\!\\!\setminus\\!S_{2})\\!+\\!H(R_{2})\\!-\\!H(R_{2}\\!\setminus\\!R_{1})\\!+\\!H(S_{2}\\!\setminus\\!R_{2})\\!-\\!H((S_{2}\\!\setminus\\!R_{2})\\!\\!\setminus\\!S_{1})$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!{\small=}\\!\\!H(R_{1})\\!\\!-\\!\\!H(R_{1}\\!\setminus\\!R_{2})\\!\\!+\\!\\!H(A\\!\setminus\\!R_{2})\\!\\!-\\!\\!H(A\\!\setminus\\!\\!(R_{2}\\!\setminus\\!R_{1}))\\!\\!+\\!\\!H((S_{1}\\!\cap\\!S_{2})\\!\\!\setminus\\!A|(S_{1}\\!\setminus\\!S_{2})\\!\\!\setminus\\!A)\\!\\!+\\!\\!H((S_{1}\\!\cap\\!S_{2})\\!\setminus\\!\\!A|(S_{2}\\!\setminus\\!S_{1})\\!\\!\setminus\\!A),$
(29)

where to obtain the first equality we used that
$\rho_{S_{1}}=\rho_{R_{1}}\otimes\rho_{S_{1}\setminus R_{1}}$ and
$\rho_{S_{2}}=\rho_{R_{2}}\otimes\rho_{S_{2}\setminus R_{2}}$ and
$R_{1}\setminus S_{2}=R_{1}\setminus R_{2}$ and $R_{2}\setminus
S_{1}=R_{2}\setminus R_{1}$; in the second line we used the purity of
$\rho_{A}$. The last two terms in (29) are positive because these are
classical conditional entropies (none of the sets contain elements of $A$).
The sum of the remaining four terms is then positive by strong subadditivity.

Case 2: $S_{1}\cap S_{2}\subseteq S$ and either $S_{1}\not\subseteq S$ or
$S_{2}\not\subseteq S$, or both. In this case we use data-processing to give

$\displaystyle H(S_{1}\cap S_{2}|S_{1}\setminus S_{2})+H(S_{1}\cap
S_{2}|S_{2}\setminus S_{1})\geq H(S_{1}\cap S_{2}|T_{1})+H(S_{1}\cap
S_{2}|T_{2}),$ (30)

where $T_{1},T_{2}\subseteq S$ are the sets of variables that are processed to
$S_{1}\setminus S_{2}$ and $S_{2}\setminus S_{1}$ respectively. Since
$S_{1}\cap S_{2}\subseteq S$, $S_{1}\cap S_{2}=T_{1}\cap T_{2}$ and so
positivity of the remaining expression follows using Case 1.

Case 3: $S_{1}\cap S_{2}\not\subseteq S$. In this case we can find
$R_{1}\subseteq S$ and $R_{2}$ with $R_{2}\cap S=\emptyset$ such that
$S_{1}\cap S_{2}=R_{1}\cup R_{2}$, and rewrite

$\displaystyle
H(S_{1}\\!\cap\\!S_{2}|S_{1}\\!\setminus\\!S_{2})\\!+\\!H(S_{1}\\!\cap\\!S_{2}|S_{2}\\!\setminus\\!S_{1})\\!=\\!H(R_{1}|S_{1}\\!\setminus\\!S_{2})\\!+\\!H(R_{2}|S_{1}\\!\setminus\\!R_{2})\\!+\\!H(R_{1}|S_{2}\\!\setminus\\!S_{1})\\!+\\!H(R_{2}|S_{2}\\!\setminus\\!R_{2})\,.$
(31)

The second and fourth terms are positive since $R_{2}$ is classical. The first
and third terms correspond to a weak monotonicity inequality like that
considered in case Case 2, and so their sum is also positive. ∎

By Proposition 2, instead of purifying the unobserved systems and dropping
weak monotonicity, we could alternatively replace all weak monotonicity
constraints by monotonicity (doing so prevents us from purifying the
unobserved quantum systems). The question then arises as to the implications
of each for deriving new entropy inequalities for the observed variables. The
following lemma shows that the quantum approach outlined in this section
(which takes the purification of unobserved systems into account) leads to
entropy inequalities that are at least as tight as those obtained by
considering monotonicity instead.

Constraints on the observed variables are usually derived starting from:

  1. (1)

The Shannon constraints for the observed variables.

  2. (2)

All independences among observed and unobserved variables that $C$ implies.

  3. (3)

Data processing inequalities.

  4. (4)

Positivity of all entropies.

  5. (5)

Positivity of the conditional mutual information (strong subadditivity).

  6. (6)

Positivity of the conditional entropy (monotonicity) between subsets that
cannot be entangled.

  7. (7)

Weak monotonicity between subsets that can be entangled.

Proposition 2 implies that we can add

(8) Monotonicity between subsets that can be entangled.

and Lemma 7 implies that in the case where all the quantum nodes are
parentless, instead of (8) we can add

($8^{\prime}$) The unobserved systems originating at a node are in a pure
state, e.g., for a node $A$ with subsystems $A_{1},A_{2}$ they obey
$H(A_{1},A_{2})=0$ and $H(A_{1})=H(A_{2})$.

######  Lemma 8.

Let $C^{\mathrm{Q}}$ be a causal structure in which each unobserved node is
parentless and has at most two children. Consider starting with the
constraints (1)–(7) and performing a Fourier-Motzkin elimination to give a set
of constraints on the entropies of observed variables. Call the resulting cone
$\Gamma$. Consider also $\Gamma_{2}$ formed analogously but with (8) in
addition to (1)–(7) and likewise $\Gamma_{2}^{\prime}$ with ($8^{\prime}$) in
addition to (1)–(7). We have $\Gamma\subseteq\Gamma_{2}$ and
$\Gamma\subseteq\Gamma_{2}^{\prime}$.

In other words, including either (8) or ($8^{\prime}$) does not give a tighter
approximation on the set of achievable entropy vectors of the observed
variables.

###### Proof.

For any unobserved node $A$ in $C$, the subsystems $A_{1}$ and $A_{2}$ only
occur jointly in coexisting sets whose state can be written as
$\rho_{A_{1}A_{2}}\otimes\rho_{R}$ where $R$ contains all other systems in
that coexisting set. This implies that for any $R_{1}\subseteq R$,

$\displaystyle H(A_{1}A_{2}R_{1})$ $\displaystyle=H(A_{1}A_{2})+H(R_{1})$ (32)
$\displaystyle H(A_{1}R_{1})$ $\displaystyle=H(A_{1})+H(R_{1})$ (33)
$\displaystyle H(A_{2}R_{1})$ $\displaystyle=H(A_{2})+H(R_{1}).$ (34)

and hence that $H(A_{1}A_{2}R_{1})$, $H(A_{1}R_{1})$ and $H(A_{2}R_{1})$ can
be eliminated from all valid inequalities. Strong-subadditivity and
monotonicity inequalities that include any of these three are redundant since
they decompose into terms that only involve $A_{1}$ and $A_{2}$ and terms that
do not involve those variables, both of which are separately implied by
another valid inequality.151515Since monotonicity is only included for cq-
states, the reduction does not lead to any decompositions that would require
positivity of $H(A_{1}|A_{2})$ or $H(A_{2}|A_{1})$ to be implied.

The remaining types of inequalities involving both $A_{1}$ and $A_{2}$ are
those with the form (up to exchange of $A_{1}$ and $A_{2}$)

$\displaystyle H(A_{1})+H(A_{2})$ $\displaystyle\geq H(A_{1}A_{2})$ (35)
$\displaystyle H(T_{1}|A_{1}S_{2}R_{2}^{\prime})$ $\displaystyle\geq
H(T_{1}|A_{1}A_{2}R_{2})$ (36) $\displaystyle
H(A_{1}T_{1}|S_{2}R_{2}^{\prime})$ $\displaystyle\geq
H(A_{1}T_{1}|A_{2}R_{2})\,,$ (37)

where $R_{1},R_{2},T_{1}\subseteq R$, $S_{2}$ is obtained by processing
$A_{2}$ and $R_{2}$, the set $T_{1}$ coexists with $S_{2}$, and
$R_{2}^{\prime}$ is the subset of all observed variables in $R_{2}$.

By (32) and (34), the inequalities of types (36) and (37) are equivalent to

$\displaystyle H(T_{1}|A_{1}S_{2}R_{2}^{\prime})$ $\displaystyle\geq
H(T_{1}|R_{2}),$ (38) $\displaystyle H(A_{1}T_{1}|S_{2}R_{2}^{\prime})$
$\displaystyle\geq H(T_{1}|R_{2})+H(A_{1}A_{2})-H(A_{2}).$ (39)

For (8) we have the additional inequalities

$\displaystyle H(A_{1}A_{2})$ $\displaystyle\geq H(A_{1})$ (40) $\displaystyle
H(A_{1}A_{2})$ $\displaystyle\geq H(A_{2})$ (41)

and for ($8^{\prime}$),

$\displaystyle H(A_{1}A_{2})$ $\displaystyle=0$ (42) $\displaystyle H(A_{1})$
$\displaystyle=H(A_{2}).$ (43)

In the following we show that neither (40) and (41) nor (42) and (43) imply
any inequalities for the observed variables other than the ones that follow
without them.

For (8), the only remaining inequalities containing $H(A_{1}A_{2})$ are (35)
and (39), both of which have $H(A_{1}A_{2})$ as a lower bound to other
entropies as well as (40) and (41), where $H(A_{1}A_{2})$ is an upper bound.
After eliminating $H(A_{1}A_{2})$, we hence obtain $H(A_{1})+H(A_{2})\geq
H(A_{1})$ and $H(A_{1})+H(A_{2})\geq H(A_{2})$ as well as
$H(A_{1}T_{1}|S_{2}R_{2}^{\prime})\geq H(T_{1}|R_{2})$ and
$H(A_{1}T_{1}|S_{2}R_{2}^{\prime})\geq H(T_{1}|R_{2})+H(A_{1})-H(A_{2})$,
where the first two immediately follow from positivity of the entropy and the
third is implied by (38) and
$H(A_{1}T_{1}|S_{2}R_{2}^{\prime})=H(T_{1}|A_{1}S_{2}R_{2}^{\prime})+H(A_{1}|S_{2}R_{2}^{\prime})$.

If we were not to impose the inequalities (40) and (41), the variable
elimination would lead to $H(A_{1})+H(A_{2})\geq 0$ and
$H(A_{1}T_{1}|S_{2}R_{2}^{\prime})\geq H(T_{1}|R_{2})-H(A_{2})$, the first of
which is implied by positivity and the second by (38) and monotonicity for cq-
states. We now show that after the elimination of $H(A_{2})$ and $H(A_{1})$
the additional inequalities we obtained from (40) and (41),

$\displaystyle H(A_{2})$
$\displaystyle\geq-(H(A_{1}T_{1}|S_{2}R_{2}^{\prime})-H(T_{1}|R_{2}))+H(A_{1})$
(44) $\displaystyle H(A_{1})$
$\displaystyle\geq-(H(A_{2}T_{1}|S_{1}R_{1}^{\prime})-H(T_{1}|R_{1}))+H(A_{2})$
(45)

become redundant (here we have put back the relation where $A_{1}$ and $A_{2}$
are interchanged). To see this, assume that in addition the constraint
$H(A_{2})\geq H(A_{1})$ holds (which could always be achieved by adding an
independent system in a maximally mixed state to $A_{2}$ and which also
preserves $H(A_{1}|A_{2})\geq 0$ and $H(A_{2}|A_{1})\geq 0$). This implies all
inequalities (44). Since it is then the only inequality where $H(A_{2})$ upper
bounds other entropies, it is in the elimination of $H(A_{2})$ combined with
all inequalities that involve $H(A_{2})$. It furthermore renders (45)
redundant after elimination (since only the inequalities (44) have $H(A_{2})$
as an upper bound). Since there is no inequality left with $H(A_{1})$ as an
upper bound, the subsequent elimination of $H(A_{1})$ leads to the same
inequalities as we obtain without including $H(A_{1}|A_{2})\geq 0$,
$H(A_{2}|A_{1})\geq 0$ and $H(A_{2})\geq H(A_{1})$.

For ($8^{\prime}$), using $H(A_{1}A_{2})=0$ changes (35) and (39) to
$H(A_{1})+H(A_{2})\geq 0$ and $H(A_{1}T_{1}|S_{2}R_{2}^{\prime})\geq
H(T_{1}|R_{2})-H(A_{2})$ respectively, which we have seen to be the
inequalities that also follow upon elimination of $H(A_{1}A_{2})$ if (42) is
not imposed and which we have also seen to be redundant.

Now, in all inequalities where $H(A_{1})$ or $H(A_{2})$ occur, they are lower
bounds (see (38)) or as $H(A_{1})\geq 0$ or $H(A_{2})\geq 0$. Thus, setting
$H(A_{2})=H(A_{1})$ and then eliminating $H(A_{1})$ is equivalent to
eliminating them each separately. ∎

##  Appendix E Inequalities for the bilocal causal structure with classical
variables

In the following we give one representative of each of the $53$ types of
entropy inequalities for the bilocal causal structure. The remaining
inequalities can be generated by using the symmetries under exchange of
$X_{0}$ and $X_{1}$, $Y_{0}$ and $Y_{1}$ as well as $Z_{0}$ and $Z_{1}$ and
the exchange of the pairs of variables $(X_{0},X_{1})$ with $(Z_{0},Z_{1})$.
We list them in terms of the coefficients of the entropies in the inequalities
such that each row understood as a vector ${\bf v}$ imposes an inequality
${\bf v}.{\bf H}\geq 0$ for the entropy vectors ${\bf H}$.

$\begin{array}[]{c|ccccccccccccccc}\text{\\#}&\\!\\!H(X_{0})\\!\\!&\\!\\!H(X_{1})\\!\\!&\\!\\!H(Y_{0})\\!\\!&\\!\\!H(Y_{1})\\!\\!&\\!\\!H(Z_{0})\\!\\!&\\!\\!H(Z_{1})\\!\\!&\\!\\!H(X_{0}Y_{0})\\!\\!&\\!\\!H(X_{0}Y_{1})\\!\\!&\\!\\!H(X_{0}Z_{0})\\!\\!&\\!\\!H(X_{0}Z_{1})\\!\\!&\\!\\!H(X_{1}Y_{0})\\!\\!&\\!\\!H(X_{1}Y_{1})\\!\\!&\\!\\!H(X_{1}Z_{0})\\!\\!&\\!\\!H(X_{1}Z_{1})\\!\\!&\\!\\!H(Y_{0}Z_{0})\\\
\hline\cr\\\ 1&0&-1&0&0&0&-1&0&0&0&0&0&0&0&1&0\\\
2&0&-1&0&0&0&-1&0&0&0&0&0&0&0&0&0\\\ 3&0&0&0&0&0&0&0&0&0&0&0&-1&0&0&0\\\
4&0&-1&0&0&0&-1&0&0&0&0&0&0&0&0&0\\\ 5&0&-1&0&0&0&-1&0&0&0&0&0&0&0&0&0\\\
6&0&-1&0&0&0&-1&0&0&0&0&0&0&0&0&0\\\ 7&0&-1&0&0&0&-1&0&0&0&0&0&0&0&0&0\\\
8&0&-1&0&0&0&-1&0&0&0&0&0&0&0&0&0\\\ 9&0&-1&0&0&0&-1&0&0&0&0&0&0&0&0&0\\\
10&0&-1&0&0&0&-1&0&0&0&0&0&0&0&0&0\\\ 11&0&-1&0&0&0&-1&0&0&0&0&0&-1&0&0&-1\\\
12&0&-1&0&0&0&-1&0&-1&0&0&0&1&0&0&0\\\ 13&0&-1&0&0&0&-1&0&-1&0&0&0&1&0&0&-1\\\
14&0&-1&0&0&0&-1&0&-1&0&0&0&1&0&0&-1\\\ 15&0&-1&0&0&0&-1&0&0&0&0&0&0&0&0&0\\\
16&0&-1&0&0&0&-1&0&0&0&0&0&0&0&0&0\\\ 17&0&-1&0&0&0&-1&0&0&0&0&0&0&0&0&0\\\
18&0&-1&0&0&0&-1&0&-1&0&0&0&1&0&0&-2\\\ 19&0&-1&0&0&0&-1&0&0&0&0&0&-1&0&0&0\\\
20&0&-1&0&0&0&-1&0&0&0&0&0&-1&0&0&0\\\ 21&0&-1&0&0&0&-1&0&0&0&0&0&-1&0&0&0\\\
22&0&-1&0&0&0&-1&0&0&0&0&0&-1&0&0&-1\\\
23&0&-1&0&0&0&-1&0&0&0&0&0&-1&0&0&-1\\\
24&0&-1&0&0&0&-1&0&-1&0&0&0&1&0&0&-1\\\
25&0&-1&0&0&0&-1&0&-1&0&0&0&1&0&0&-1\\\
26&0&-1&0&0&0&-1&0&-1&0&0&0&1&0&0&-1\\\
27&0&-1&0&0&0&-1&0&-1&0&0&0&1&0&0&-1\\\ 28&0&-1&0&0&0&-1&0&-1&0&0&0&0&0&0&0\\\
29&0&-1&-1&0&0&-1&0&0&0&0&0&0&0&0&0\\\ 30&0&-1&-1&0&0&-1&0&0&0&0&0&0&0&0&0\\\
31&0&-1&0&0&0&-1&0&0&0&0&0&-1&0&0&0\\\ 32&0&-1&0&0&0&-1&0&0&0&0&0&-1&0&0&0\\\
33&0&-1&0&0&0&-1&0&-1&0&0&0&1&0&0&0\\\ 34&0&-1&0&0&0&-1&0&-1&0&0&0&1&0&0&-2\\\
35&0&-1&0&0&0&-1&0&-1&0&0&0&1&0&0&-2\\\ 36&0&-1&0&0&0&-1&0&-1&0&0&0&0&0&0&0\\\
37&0&-1&-1&0&0&-1&0&0&0&0&0&0&0&0&0\\\ 38&0&-1&-1&0&0&-1&0&0&0&0&0&0&0&0&0\\\
39&0&0&0&-1&0&0&0&0&0&-1&0&0&-1&0&0\\\ 40&0&0&0&-1&0&0&0&0&0&-1&0&0&-1&0&0\\\
41&0&-1&0&0&0&-1&0&0&0&0&0&-1&0&0&-1\\\
42&0&-1&0&0&0&-1&0&0&0&0&0&-1&0&0&-1\\\
43&0&-1&0&0&0&-1&0&0&0&0&0&-1&0&0&-1\\\
44&0&-1&0&0&0&-1&0&0&0&0&0&-1&0&0&-1\\\
45&0&-1&0&0&0&-1&0&-1&0&0&0&0&0&0&-1\\\
46&0&-1&0&0&0&-1&0&-1&0&0&0&0&0&0&-1\\\
47&0&-1&0&0&0&-1&0&-1&0&0&0&0&0&0&-1\\\
48&0&-1&0&0&0&-1&0&-1&0&0&0&0&0&0&-1\\\
49&0&-1&0&0&0&-1&0&-1&0&0&-1&1&0&0&0\\\
50&0&-1&0&0&0&-1&0&-1&0&0&-1&1&0&0&0\\\
51&0&-1&0&0&0&-1&0&-1&0&0&-1&1&0&0&-1\\\
52&0&-1&0&0&0&-1&0&-1&0&0&-1&1&0&0&-1\\\
53&0&-1&0&0&0&-1&-1&-1&0&0&1&0&0&0&0\\\ \end{array}$

$\begin{array}[]{c|ccccccccccc}\text{\\#}&H(Y_{0}Z_{1})\\!\\!&\\!\\!H(Y_{1}Z_{0})\\!\\!&\\!\\!H(Y_{1}Z_{1})\\!\\!&\\!\\!H(X_{0}Y_{0}Z_{0})\\!\\!&\\!\\!H(X_{0}Y_{0}Z_{1})\\!\\!&\\!\\!H(X_{0}Y_{1}Z_{0})\\!\\!&\\!\\!H(X_{0}Y_{1}Z_{1})\\!\\!&\\!\\!H(X_{1}Y_{0}Z_{0})\\!\\!&\\!\\!H(X_{1}Y_{0}Z_{1})\\!\\!&\\!\\!H(X_{1}Y_{1}Z_{0})\\!\\!&\\!\\!H(X_{1}Y_{1}Z_{1})\\\
\hline\cr\\\ 1&0&0&0&0&0&0&0&0&0&0&0\\\ 2&0&-1&1&0&0&0&0&0&0&1&0\\\
3&0&0&-1&0&0&-1&1&0&0&1&1\\\ 4&0&0&-1&0&0&-1&1&0&0&1&1\\\
5&0&0&-1&0&-1&0&1&0&1&0&1\\\ 6&0&0&-1&-1&0&0&1&1&0&0&1\\\
7&0&-1&0&0&-1&1&0&0&1&0&1\\\ 8&0&-1&0&0&-1&0&1&0&1&1&0\\\
9&0&-1&0&-1&0&1&0&1&0&0&1\\\ 10&0&-1&0&-1&0&0&1&1&0&1&0\\\
11&1&0&0&0&0&0&0&0&0&1&1\\\ 12&0&-1&1&0&0&1&0&0&0&0&0\\\
13&0&0&0&1&0&0&0&0&1&0&0\\\ 14&0&0&0&0&1&0&0&1&0&0&0\\\
15&0&-2&1&0&0&1&-1&0&0&1&1\\\ 16&0&-2&1&0&-1&1&0&0&1&1&0\\\
17&0&-2&1&-1&0&1&0&1&0&1&0\\\ 18&1&0&0&1&0&0&0&1&0&0&0\\\
19&0&0&-1&-1&0&0&1&0&1&1&1\\\ 20&0&-1&1&0&0&1&-1&0&0&1&1\\\
21&0&-1&0&-1&0&1&0&0&1&1&1\\\ 22&1&0&0&1&-1&0&0&0&0&1&1\\\
23&1&0&0&0&0&0&0&1&-1&1&1\\\ 24&0&0&0&1&0&1&0&0&1&-1&0\\\
25&0&0&0&1&0&0&1&0&1&0&-1\\\ 26&0&0&0&0&1&1&0&1&0&-1&0\\\
27&0&0&0&0&1&0&1&1&0&0&-1\\\ 28&0&-1&0&-1&0&1&1&0&1&1&0\\\
29&0&-1&1&1&0&0&-1&0&1&1&0\\\ 30&0&-1&1&0&1&0&-1&1&0&1&0\\\
31&0&0&-1&-2&1&0&1&1&0&1&1\\\ 32&0&-1&0&-2&1&1&0&1&0&1&1\\\
33&0&-2&1&0&0&1&1&0&0&1&-1\\\ 34&1&0&0&1&0&1&0&1&0&-1&0\\\
35&1&0&0&1&0&0&1&1&0&0&-1\\\ 36&0&-1&0&-2&1&1&1&1&0&1&0\\\
37&0&-1&1&1&0&1&-2&0&1&0&1\\\ 38&0&-1&1&0&1&1&-2&1&0&0&1\\\
39&0&0&0&0&1&-1&1&1&-1&1&1\\\ 40&0&0&0&-1&1&1&1&1&-1&1&0\\\
41&1&0&-1&0&-1&0&1&0&1&1&1\\\ 42&1&0&-1&-1&0&0&1&1&0&1&1\\\
43&1&-1&0&0&-1&1&0&0&1&1&1\\\ 44&1&-1&0&-1&0&1&0&1&0&1&1\\\
45&1&0&-1&0&-1&1&1&0&1&0&1\\\ 46&1&0&-1&-1&0&1&1&1&0&0&1\\\
47&1&-1&0&0&-1&1&1&0&1&1&0\\\ 48&1&-1&0&-1&0&1&1&1&0&1&0\\\
49&-1&0&0&0&1&0&1&1&1&-1&0\\\ 50&-1&-1&1&0&1&0&0&1&1&0&0\\\
51&0&0&0&1&0&0&1&1&1&-1&0\\\ 52&0&-1&1&1&0&0&0&1&1&0&0\\\
53&0&0&-1&0&1&1&1&-1&0&0&1\\\ \end{array}$