The fact that logical consistency does not imply a fixed acyclic causal structure is shown via studying correlations among “parties.”

#### Definition 1 (Party, causal, and non-causal correlations).

A *party* ${\mathit{S}}_{\mathit{j}}=\mathrm{(}{\mathit{A}}_{\mathit{j}}\mathrm{,}{\mathit{X}}_{\mathit{j}}\mathrm{,}{\mathit{L}}_{\mathit{j}}\mathrm{)}$ is a tuple consisting of two random variables ${\mathit{A}}_{\mathit{j}}\mathrm{,}{\mathit{X}}_{\mathit{j}}$ and a local operation ${\mathit{L}}_{\mathit{j}}$. The random variable ${\mathit{A}}_{\mathit{j}}$ is the *setting* and the random variable ${\mathit{X}}_{\mathit{j}}$ the *outcome*.

For *k* parties, the correlations ${\mathit{P}}_{{\mathit{X}}_{1}\mathrm{,}{\mathit{X}}_{2}\mathrm{,}\dots \mathrm{,}{\mathit{X}}_{\mathit{k}}\mid {\mathit{A}}_{1}\mathrm{,}{\mathit{A}}_{2}\mathrm{,}\dots \mathrm{,}{\mathit{A}}_{\mathit{k}}}$ are called *causal* if and only if they can be simulated by arranging the parties on a fixed acyclic causal structure; otherwise, they are called *non-causal*.

*Causal* two-party correlations ${\mathit{P}}_{{\mathit{X}}_{1}\mathrm{,}{\mathit{X}}_{2}\mid {\mathit{A}}_{1}\mathrm{,}{\mathit{A}}_{2}}$ can hence be decomposed as
$$\begin{array}{r}\mathit{p}{\mathit{P}}_{{\mathit{X}}_{1}\mid {\mathit{A}}_{1}}{\mathit{P}}_{{\mathit{X}}_{2}\mid {\mathit{A}}_{1}\mathrm{,}{\mathit{A}}_{2}\mathrm{,}{\mathit{X}}_{1}}+\mathrm{(}1-\mathit{p}\mathrm{)}{\mathit{P}}_{{\mathit{X}}_{2}\mid {\mathit{A}}_{2}}{\mathit{P}}_{{\mathit{X}}_{1}\mid {\mathit{A}}_{1}\mathrm{,}{\mathit{A}}_{2}\mathrm{,}{\mathit{X}}_{2}}\phantom{\rule{0.1667em}{0ex}}\mathrm{,}\end{array}$$
where *p* is some probability: With probability *p* party ${\mathit{S}}_{1}$ acts *before* ${\mathit{S}}_{2}$. Note that convex mixtures of fixed acyclic causal structures remain fixed and acyclic: The probability *p* might arise due to some “ignorance.”

Now, we describe how the correlations are obtained based on the local operations ${\mathit{L}}_{\mathit{j}}$. Following Oreshkov, Costa, and Brukner [24], we do not assume an underlying fixed acyclic causal structure of the parties. Instead, we assume that the parties are *isolated* (A1), every party acts *once* (A2), the correlations are *linear* in the choice of local operations (A3), and *logical consistency* (A4). Condition (A1) means that the local operation ${\mathit{L}}_{\mathit{j}}$ of any party ${\mathit{S}}_{\mathit{j}}$ can be described independently of the other parties’ local operations. Then again, (A2) says that every local operation is applied once; this could be generalized, but as it stands, makes calculations easier and is sufficient for our claims. The requirement (A3) of linearity is natural: It translates to the requirement that convex combinations of local operations transform to convex combinations of correlations. Last but not least, (A4) is the core assumption in order to have a consistent theory.

Since we are interested in the case where the underlying theory is classical probability theory (as opposed to quantum theory), the most general form of the local operation ${\mathit{L}}_{\mathit{j}}$ is ${\mathit{P}}_{{\mathit{O}}_{\mathit{j}}\mathrm{,}{\mathit{X}}_{\mathit{j}}\mid {\mathit{I}}_{\mathit{j}}\mathrm{,}{\mathit{A}}_{\mathit{j}}}$ for some random variables ${\mathit{O}}_{\mathit{j}}\mathrm{,}{\mathit{I}}_{\mathit{j}}$: a stochastic channel that depends on the *setting* ${\mathit{A}}_{\mathit{j}}$ and produces an *outcome* ${\mathit{X}}_{\mathit{j}}$. Hence, up to the *logical consistency* assumption that we formulate shortly, the most general correlations ${\mathit{P}}_{{\mathit{X}}_{1}\mathrm{,}{\mathit{X}}_{2}\mathrm{,}\dots \mathrm{,}{\mathit{X}}_{\mathit{k}}\mid {\mathit{A}}_{1}\mathrm{,}{\mathit{A}}_{2}\mathrm{,}\dots \mathrm{,}{\mathit{A}}_{\mathit{k}}}$ – without assuming an underlying fixed acyclic causal structure – are given by
$$\begin{array}{r}\mathit{f}\mathrm{(}{\mathit{L}}_{1}\mathrm{,}{\mathit{L}}_{2}\mathrm{,}\dots \mathrm{,}{\mathit{L}}_{\mathit{k}}\mathrm{)}\phantom{\rule{0.1667em}{0ex}}\mathrm{,}\end{array}$$
for some multi-linear function *f*. Note that *f* can be interpreted as a “supermap:” It maps operations to an operation ${\mathit{P}}_{{\mathit{X}}_{1}\mathrm{,}{\mathit{X}}_{2}\mathrm{,}\dots \mathrm{,}{\mathit{X}}_{\mathit{k}}\mid {\mathit{A}}_{1}\mathrm{,}{\mathit{A}}_{2}\mathrm{,}\dots \mathrm{,}{\mathit{A}}_{\mathit{k}}}$. *Logical consistency*, finally, asks that for any choice of local operations, the distribution ${\mathit{P}}_{{\mathit{X}}_{1}\mathrm{,}{\mathit{X}}_{2}\mathrm{,}\dots \mathrm{,}{\mathit{X}}_{\mathit{k}}\mid {\mathit{A}}_{1}\mathrm{,}{\mathit{A}}_{2}\mathrm{,}\dots \mathrm{,}{\mathit{A}}_{\mathit{k}}}$ is well defined. This means that the parties are *unrestricted* in what local operation they want to perform.

#### Definition 2 (Logical consistency).

A supermap *f* is called *logically consistent* if and only if $\forall {\mathit{L}}_{1}\mathrm{,}{\mathit{L}}_{2}\mathrm{,}\dots \mathrm{,}{\mathit{L}}_{\mathit{k}}:\mathit{f}\mathrm{(}{\mathit{L}}_{1}\mathrm{,}{\mathit{L}}_{2}\mathrm{,}\dots \mathrm{,}{\mathit{L}}_{\mathit{k}}\mathrm{)}$ is a probability distribution of the form ${\mathit{P}}_{{\mathit{X}}_{1}\mathrm{,}{\mathit{X}}_{2}\mathrm{,}\dots \mathrm{,}{\mathit{X}}_{\mathit{k}}\mid {\mathit{A}}_{1}\mathrm{,}{\mathit{A}}_{2}\mathrm{,}\dots \mathrm{,}{\mathit{A}}_{\mathit{k}}}$.

Given a supermap ${\mathit{f}}^{\prime}$ that is *not* logically consistent, a set of local operations of the parties exists such that the result is not a probability distribution (the “probabilities” are not normalized or negative).

Note that *causal correlations* satisfy all the assumptions (A1–A4). The question we are interested in now is: Do the four assumptions (A1–A4) imply causal correlations? If the underlying theory is quantum theory, then Oreshkov, Costa, and Brukner [24] answered this question negatively. In the same work the authors showed that if the underlying theory is classical, then for *two parties* the question is answered *positively*. This suggested that fixed acyclic causal structures might be the result from quantum-to-classical transitions (the quantum world may be causally exotic, yet the classical is not – supporting our everyday view on nature). In the dissertation we show that the latter is not true; classical theories allow for non-causal correlations if more than two parties are involved. Before we illustrate this result with an example, we present two theorems that help to understand logically consistent supermaps *f*.

#### Theorem 1 (Logically consistent supermaps (informal) [4]).

*Logically consistent supermaps represent (stochastic) channels E from the* ${\{{\mathit{O}}_{\mathit{j}}\}}_{\mathit{J}}$ *variables to the* ${\{{\mathit{I}}_{\mathit{j}}\}}_{\mathit{J}}$ *variables (see Figure* *2**).*

Following this theorem, the resulting distribution is calculated according to probability theory by taking the product over all local operations and the channel *E*, and by marginalizing over the ${\{{\mathit{O}}_{\mathit{j}}\mathrm{,}{\mathit{I}}_{\mathit{j}}\}}_{\mathit{J}}$ variables. Note that some channels *E* do *not* lead to a logically consistent supermap, *e. g.,* if we would plug in the identity channel, then a set of local operations exists such that the result is *not* a valid probability distribution as required.

Figure 2 A box ${\mathit{L}}_{\mathit{j}}$ represents the local operation ${\mathit{L}}_{\mathit{j}}$ of party ${\mathit{S}}_{\mathit{j}}$ which can be understood as a channel. Likewise, logically consistent supermaps can be understood as stochastic channels. The average number of deterministic fixed points of $\mathit{E}\circ \mathrm{(}{\mathit{L}}_{1}\mathrm{,}{\mathit{L}}_{2}\mathrm{,}{\mathit{L}}_{3}\mathrm{,}\dots \phantom{\rule{0.1667em}{0ex}}\mathrm{)}$ is 1.

#### Theorem 2 (Fixed-point characterization (informal) [5]).

*A supermap f is logically consistent if and only if the average number of fixed points of* ${\mathit{P}}_{{\mathit{I}}_{1}\mathrm{,}{\mathit{I}}_{2}\mathrm{,}\dots \mid {\mathit{O}}_{1}\mathrm{,}{\mathit{O}}_{2}\mathrm{,}\dots}$ *concatenated with any choice of deterministic local operations is 1 (see Figure* *2**).*

A corollary of the latter theorem is that a *deterministic* supermap is logically consistent if and only if for any choice of deterministic local operations, a *unique* fixed point exists. This result is tantamount to avoiding the grandfather as well as the information antinomy discussed in the introduction.

**Example.** Let ${\mathit{S}}_{1}\mathrm{,}{\mathit{S}}_{2}\mathrm{,}{\mathit{S}}_{3}$ be parties with local operations ${\mathit{L}}_{\mathit{j}}={\mathit{P}}_{{\mathit{O}}_{\mathit{j}}\mathrm{,}{\mathit{X}}_{\mathit{j}}\mid {\mathit{I}}_{\mathit{j}}\mathrm{,}{\mathit{A}}_{\mathit{j}}}$, where all random variables are binary and where the *settings* $\{{\mathit{A}}_{1}\mathrm{,}{\mathit{A}}_{2}\mathrm{,}{\mathit{A}}_{3}\}$ are uniformly distributed. To decide whether correlations are *non-causal*, we make use of *causal inequalities* [24], [13]. If for a given distribution ${\mathit{P}}_{{\mathit{X}}_{1}\mathrm{,}{\mathit{X}}_{2}\mathrm{,}{\mathit{X}}_{3}\mid {\mathit{A}}_{1}\mathrm{,}{\mathit{A}}_{2}\mathrm{,}{\mathit{A}}_{3}}$ the inequality
$$\begin{array}{rl}Pr\mathrm{(}& \mathrm{(}{\mathit{X}}_{1}=\neg {\mathit{A}}_{2}\wedge {\mathit{A}}_{3}\mathrm{)}\phantom{\rule{0.1667em}{0ex}}\wedge \phantom{\rule{0.1667em}{0ex}}\mathrm{(}{\mathit{X}}_{2}=\neg {\mathit{A}}_{3}\wedge {\mathit{A}}_{1}\mathrm{)}\phantom{\rule{0.1667em}{0ex}}\wedge \phantom{\rule{0.1667em}{0ex}}\\ & \mathrm{(}{\mathit{X}}_{3}=\neg {\mathit{A}}_{1}\wedge {\mathit{A}}_{2}\mathrm{)}\mathrm{)}\le 3\mathrm{/}4\end{array}$$
is violated, then this distribution must be *non-causal*. The reason for this is that for *causal* correlations (the parties are positioned on a fixed acyclic causal structure), *at least one party* ${\mathit{S}}_{\mathit{j}}$ has no other parties in her past and therefore no access to the settings $\{{\mathit{A}}_{1}\mathrm{,}{\mathit{A}}_{2}\mathrm{,}{\mathit{A}}_{3}\}\setminus \{{\mathit{A}}_{\mathit{j}}\}$ of the other parties; hence, ${\mathit{X}}_{\mathit{j}}$ takes the correct value with probability at most 3/4.

Yet, in the framework presented, the supermap constructed from
$$\begin{array}{r}\mathit{E}\mathrm{(}{\mathit{i}}_{1}\mathrm{,}{\mathit{i}}_{2}\mathrm{,}{\mathit{i}}_{3}\mathrm{,}{\mathit{o}}_{1}\mathrm{,}{\mathit{o}}_{2}\mathrm{,}{\mathit{o}}_{3}\mathrm{)}=\left\{\begin{array}{ll}1\phantom{\rule{1em}{0ex}}& \mathrm{(}{\mathit{i}}_{1}=\neg {\mathit{o}}_{2}\wedge {\mathit{o}}_{3}\mathrm{)}\wedge \\ \phantom{\rule{1em}{0ex}}& \mathrm{(}{\mathit{i}}_{2}=\neg {\mathit{o}}_{3}\wedge {\mathit{o}}_{1}\mathrm{)}\wedge \\ \phantom{\rule{1em}{0ex}}& \mathrm{(}{\mathit{i}}_{3}=\neg {\mathit{o}}_{1}\wedge {\mathit{o}}_{2}\mathrm{)}\\ 0\phantom{\rule{1em}{0ex}}& \text{otherwise,}\end{array}\right.\end{array}$$
along with the local operations
$$\begin{array}{r}{\mathit{L}}_{\mathit{j}}\mathrm{(}{\mathit{o}}_{\mathit{j}}\mathrm{,}{\mathit{x}}_{\mathit{j}}\mathrm{,}{\mathit{i}}_{\mathit{j}}\mathrm{,}{\mathit{a}}_{\mathit{j}}\mathrm{)}=\left\{\begin{array}{ll}1\phantom{\rule{1em}{0ex}}& {\mathit{o}}_{\mathit{j}}={\mathit{a}}_{\mathit{j}}\wedge {\mathit{i}}_{\mathit{j}}={\mathit{x}}_{\mathit{j}}\\ 0\phantom{\rule{1em}{0ex}}& \text{otherwise,}\end{array}\right.\end{array}$$
allow to violate the above inequality maximally: It is *impossible* to arrange the parties on a fixed acyclic causal structure in order to obtain the same correlations. By consulting the second theorem above, the reader can convince herself/himself that *E* indeed leads to a *logically consistent* supermap. The underlying causal structure is *cyclic* (see Figure 3).

Figure 3 The value of the variable ${\mathit{O}}_{\mathit{j}}$ is determined by the local operation ${\mathit{L}}_{\mathit{j}}$ and ${\mathit{I}}_{\mathit{j}}$. The value of ${\mathit{I}}_{\mathit{j}}$, then again, is a function of $\{{\mathit{O}}_{1}\mathrm{,}{\mathit{O}}_{2}\mathrm{,}{\mathit{O}}_{3}\}\setminus \{{\mathit{O}}_{\mathit{j}}\}$. For simplicity this figure is blind of the ${\mathit{X}}_{\mathit{j}}\mathrm{,}{\mathit{A}}_{\mathit{j}}$ variables.

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