The space of logically consistent classical processes without causal order
Classical correlations without predefined causal order arise from processes where parties manipulate random variables, and where the order of these interactions is not predefined. No assumption on the causal order of the parties is made, but the processes are restricted to be logically consistent under any choice of the parties’ operations. It is known that for three parties or more, this set of processes is larger than the set of processes achievable in a predefined ordering of the parties. Here, we model all classical processes without predefined causal order geometrically and find that the set of such processes forms a polytope. Additionally, we model a smaller polytope — the deterministic-extrema polytope — where all extremal points represent deterministic processes. This polytope excludes probabilistic processes that must be — quite unnaturally — fine-tuned, because any variation of the weights in a decomposition into deterministic processes leads to a logical inconsistency.
I Motivation and main result
An assumption often made in physical theories, sometimes implicitly, is the existence of a global time. In particular, quantum theory is formulated with time as an intrinsic parameter. If one relaxes this assumption by requiring local validity of some theory and logical consistency only, then a larger set of correlations can be obtained, called correlations without predefined causal order. The processes that lead to such correlations are called processes without predefined causal order. Two motivations to study such correlations are quantum gravity and quantum non-locality. Quantum gravity motivates this research in the sense that on the one hand, relativity is a deterministic theory equipped with a dynamic spacetime; on the other hand, quantum theory is a probabilistic theory embedded in a fixed spacetime. This suggests that quantum gravity is relaxed in both aspects, i.e., it is a probabilistic theory equipped with a dynamic spacetime Hardy (2007). Quantum non-local correlations Einstein et al. (1935); Bell (1964); Brunner et al. (2014) motivate this study since the possibility of a satisfactory causal explanation Reichenbach (1956) for such correlations is questionable Bell (1964); Suarez and Scarani (1997); Stefanov et al. (2002); Scarani and Gisin (2002); Coretti et al. (2011); Bancal et al. (2012); Barnea et al. (2013); Scarani et al. (2014); Wood and Spekkens (2015). Dropping the notion of a global time or of an a priori spacetime — as has been suggested from different fields of research Leibniz and Clarke (2000); von Weizsäcker (1985); Page and Wootters (1983); Wootters (1984); Bombelli et al. (1987); D’Ariano (2011); Erker (2014); Vedral (2014); Giovannetti et al. (2015); Ranković et al. (2015) — dissolves this paradox. This can be achieved by defining causal relations based on free randomness (see Figure 1) as opposed to defining free randomness based on causal relations Colbeck and Renner (2011); Ghirardi and Romano (2013). Such an approach gives a dynamic character to causality; causal connections are not predefined but are derived from the observed correlations.
Relaxations of quantum theory where the assumption of a global time is dropped have recently been studied widely Hardy (2005, 2007, 2009); Chiribella et al. (2009); Chiribella (2012); Colnaghi et al. (2012); Oreshkov et al. (2012); Baumeler and Wolf (2014); Chiribella et al. (2013); Costa (2013); Baumeler et al. (2014); Brukner (2015); Ibnouhsein and Grinbaum (2015); Ibnouhsein (2014); Morimae (2014); Oreshkov and Cerf (2014); Araújo et al. (2015); Oreshkov and Giarmatzi (2015); Branciard et al. (2016); Feix et al. (2015); Oreshkov and Cerf (2015) (see Ref. Brukner (2014a) for a review). Our work follows the spirit of an operational quantum framework for such correlations developed by Oreshkov, Costa, and Brukner Oreshkov et al. (2012). Some correlations appearing in their quantum framework — for two parties or more — cannot be simulated by assuming a predefined causal order of the parties. Such correlations are termed non-causal. Analogously to non-locality, non-causal correlations could be witnessed by violating so-called causal inequalities Oreshkov et al. (2012); Baumeler and Wolf (2014); Baumeler et al. (2014); Branciard et al. (2016). All causal inequalities in the two-party scenario and for binary inputs and outputs are presented in Ref. Branciard et al. (2016). In a previous work Baumeler et al. (2014), we showed that in the classical limit of the quantum framework, i.e., if it is restricted to probability theory, classical non-causal correlations can arise as well. This result holds for three parties or more. In the present work we follow this path and give a representation of all classical — as opposed to quantum — processes without predefined causal order as polytopes. Such a representation helps in optimizing winning strategies for causal games Oreshkov et al. (2012); Branciard et al. (2016) — the optimization problem can be stated as a linear program —, and for finding new causal games.
First, we present the framework of classical correlations without predefined causal order. Then, we describe the polytope of processes that lead to such correlations implicitly and explicitly for scenarios with up to three parties and binary inputs and outputs. In the general case, we give an implicit description of the polytope. In addition, we construct the smaller polytope of classical processes without predefined causal order where all extremal points describe deterministic processes. We call this polytope the deterministic-extrema polytope. The processes from this polytope can be thought of as being “more physical” in the sense that its extremal points are not proper mixtures of logically inconsistent processes Brukner (2014b), i.e., this set contains processes that can be written as a convex combination of deterministic ones from within the polytope only. Our motivation for this is that some proper mixtures need to be fine-tuned Wood and Spekkens (2015), i.e., tiny variations of the mixtures renders the processes logically inconsistent. The fine-tuned proper mixtures are the probabilistic extremal points of the larger polytope. A qualitative representation of these polytopes is given in Figure 2.
Ii Modelling classical correlations without predefined causal order
ii.1 Causality, predefined causal order, and a framework of classical correlations without predefined causal order
We describe an operational framework without global assumptions (other than logical consistency). Causal relations are defined as in the interventionists’ approach to causality Price (1991); Woodward (2003): Outputs can be correlated to inputs and inputs are manipulated freely (see Figure 1). Defining causality based on free randomness is the converse approach to the one used in recent literature Colbeck and Renner (2011); Ghirardi and Romano (2013); there, free randomness is defined based on causal relations.
Definition 1 (Causality Baumeler et al. (2014)).
For two correlated random variables and , where is an output and is an input, i.e., is chosen freely, we say that is in the causal future of , or equivalently, that is in the causal past of , denoted by or . The negations of these relations are denoted by and .
Consider parties , where party has access to an input random variable and generates an output random variable . This allows us to causally order parties: If is correlated to , then is in the causal past of (). To simplify the presentation, we write and likewise for , , and .
Definition 2 (Two-party predefined causal order).
A two-party predefined causal order is a causal ordering of party with input , output , and party with input , output , such that the distribution can be written as a convex combination of one-way signaling distributions
for some .
A definition for multi-party predefined causal order is given in Ref. Oreshkov and Giarmatzi (2015). Such a definition turns out to be more subtle since a party in the causal past of some other parties can in principle influence everything in her causal future; in particular, can influence the causal order of the parties . We just state a Lemma that follows from such a definition and that is sufficient to prove our claims.
Lemma 1 (Necesarry condition for predefined causal order).
A necessary condition for a predefined causal order is that the probability distribution can be written as a convex combination
with and , such that in every distribution at least one party is not in the causal future of any other party, i.e.,
where stands for the causal relation that is deduced from the distribution .
In the framework without predefined causal order, each party receives a random variable from the environment on which can act. After the interaction with , party outputs a random variable to the environment. Both random variables and are output random variables. The only input random variable a party has is . The operation of is a stochastic process mapping to (see Figure 3). A stochastic process is a probability distribution over the range conditioned on the domain; in this case, the stochastic process of party (which in the following will also be called the local operation of party ) is .
All parties are allowed to apply any possible operation described by probability theory. Furthermore, they are isolated from each other, which means that they can interact only through the environment. Because we do not make global assumptions (beyond logical consistency), the most general picture is that the random variables that are sent from the environment to the parties are the result of a map on the random variables fed back by all parties to the same environment (see Figure 4).
Such a composition of parties with the environment combines states and communication channels in one framework.
A party has access to the four random variables , , , and , where is chosen freely. If we consider all parties together, we should get a probability distribution . Furthermore, we ask the environment to be a multi-linear functional of all local operations. The motivation for this is that linear combinations of local operations should carry through to the probabilities . This brings us to a definition of logical consistency.
Definition 3 (Logical consistency).
An environment is called logically consistent if and only if it is a multi-linear positive map on any choice of local operations of all parties such that the composition of with the local operations results in a probability distribution .
Theorem 1 (Logical consistent environment as stochastic process).
The environment is a stochastic process that maps to .
The environment is a multi-linear positive map on the probabilities (we omit the arguments for the sake of presentation)
that party outputs to the environment and generates conditioned on the setting and on . Therefore, we write
Since is a multi-linear positive map and since it depends on and only, the above probability can be written as
where is a number. This number must be non-negative, as otherwise the above expression (6) is not a probability. By fixing and by summing over , we get
Let us fix the local operations of all parties to be
From the total-probability condition we obtain
By repeating this calculation for different choices of local operations where the parties deterministically output a value, we get
Therefore, is a stochastic process . ∎
The following Corollary follows from Theorem 1.
A logical consistent environment fulfills the property that under any choice of the local operations of all parties, the expression form a conditional probability distribution .
Note that not every conditional distribution is logically consistent. Some stochastic processes lead to grandfather-paradox-type Barjavel (1944) inconsistencies. Consider the following two extreme examples of such inconsistencies. We describe the examples in the single-party scenario as depicted in Figure 5 and where , , , and are binary random variables.
Let the environment as well as the party forward the random variable, i.e., the operation of the environment is
and the operation of the party is
Since the environment and the party forward the random variable, we have . However, it is unclear what value the probability should take. This is also known as the causal-loop paradox.
We alter the local operation of party to negate the binary random variable
Now, we are faced with the grandfather paradox: if party receives from the environment, then she sends the value to the environment. But in that case, she should receive and not .
ii.2 Mathematical model of states, operations, evolution, and composition
Let be the sample space of a random variable with the probability measure .
Definition 4 (States, operations, evolution, and composition).
We represent a state corresponding to a random variable as the probability vector
A stochastic process from to a random variable with sample space describes an operation and is modeled by the stochastic matrix
The result of evolving the random variable through the operation is given by the matrix multiplication
Finally, vectors and matrices are composed in parallel using the Kronecker product .
For example, by this definition, the output of a stochastic process taking two inputs and producing one output is expressed by
ii.3 Set of logically consistent processes without predefined causal order
We derive the conditions on the environment (stochastic process) such that it is logically consistent. For simplicity, we start with the single-party scenario as depicted in Figure 5; the party is denoted by and the environment by .
We can further simplify our picture by fixing the value of to and by summing over :
The stochastic process of the environment is . For now, let us assume that performs a deterministic operation . This assumption is dropped later. The operation applied by can be written as a function
where is a deterministic input value. By embedding into the process of , we get
This can be interpreted as a probability measure of party receiving the value from the environment :
For to represent a probability measure, the values of for every deterministic value must be non-negative and have to sum up to 1:
We express both conditions in the matrix picture. Non-negativity is achieved whenever all entries of the matrix are non-negative. The total-probability conditions are formulated in the following way. The value that is fed into the environment is
The matrix fixed to providing the state to the party is
Therefore, the probability of party observing is
and the law of total probability requires
This condition remains the same if we relax the input to a stochastic input and the operation of to a stochastic process . The reason for this is that any stochastic input can be written as a convex combination of deterministic inputs, and any stochastic process can be written as a convex combination of deterministic operations. Therefore, the logical-consistency requirement asks the environment to be restricted to those processes where, under any choice of the local operation of party , the law of total probability
and the non-negativity condition
hold. Because a stochastic process can be written as a convex mixture of deterministic operations, it is sufficient to ask for
Theorem 2 (Total probability).
The law that the sum of the probabilities over the exclusive states the parties receive is 1 is satisfied if and only if
where represents a deterministic operation of party .
Theorem 3 (Non-negative probabilities).
The law that the probability of the parties observing a state is non-negative is satisfied if and only if
ii.4 Equivalence to the quantum correlations framework in the classical limit
The ingredients of the framework by Oreshkov, Costa, and Brukner Oreshkov et al. (2012) are process matrices and local operations — described by matrices as well. All the matrices are completely-positive trace-preserving quantum maps in the Choi-Jamiołkowski Jamiołkowski (1972); Choi (1975) picture. In the classical limit, the matrices become diagonal in the computational basis Oreshkov et al. (2012); Baumeler et al. (2014). In the single-party scenario, the process matrix is a map from the Hilbert space to the Hilbert space . The party’s local operation then again is a map from the Hilbert space to the Hilbert space . The conditions a process matrix in a single-party scenario has to fulfill Oreshkov et al. (2012) are
where is the set of all completely-positive trace-preserving maps from the space to the space . Intuitively, the condition given by Equation (37) “short-circuits” both maps and enforces the probabilities of the outcomes to sum up to .
Theorem 4 (Equivalence).
The process matrix in the quantum framework corresponds to the stochastic process of the environment in our framework, and the local operations correspond to the stochastic process of the parties. We show a bijection between process matrices and stochastic processes of the environment, and between local operations and stochastic processes of the parties.
A stochastic matrix , representing the environment in our framework, can be translated into the quantum framework by
where and are computational-basis states of the same dimension as , and where the subscripts denote the respective Hilbert spaces. This completely-positive trace-preserving map (expressed in the Choi-Jamiołkowski picture) acts in the same way as the stochastic matrix : The state is mapped to . The function takes the matrix and cancels all off-diagonal terms, i.e.,
We can rewrite as
Analogously, the stochastic matrix of the party can be translated into the quantum framework and becomes
The reverse direction of the bijection follows from the description above.
Now, we show that the conditions (37) and (38) in a single-party scenario on a process matrix coïncide with the conditions (31) and (32) in our framework. The non-negativity condition (38) forces the probabilities of the outputs of to be non-negative; the same holds for the condition (32) in our framework. That the condition (37) coïncides with the condition (31) is shown below. Forcing and to be diagonal in the computational basis gives
Substituting with and with yields
which proves the claim. The multi-party case follows through linearity. ∎
Iii Polytope of classical processes without predefined causal order
Convex polytopes can be represented in two different ways: The -representation is a list of half-spaces where the intersection is the polytope, and the -representation is a list of the extremal points of the polytope. Algorithms like the double-description method Motzkin et al. (1953); Fukuda and Prodon (2005) enumerate all extremal points of the polytope given the -representation. We used cdd+ Fukuda (2003) for vertex enumeration. The inverse problem is solved by its dual: a convex-hull algorithm.
iii.2 Single party, binary input, and binary output
We start with the polytope for one party (see Figure 5) with a binary input and a binary output. In this case, a process is described by a square matrix of dimension . The most general process of the environment is
consisting of variables. The deterministic operations party can apply are
where , produce a constant , , respectively, and where the matrix is the identity and the negation. The equalities
This is shown as follows:
with the non-negativity conditions
This solution set is a one-dimensional polytope with the extremal points and . All solutions describe a state. This implies that all correlations that can be obtained in this framework with a single party and binary input and output, can also be obtained in a framework without feedback, i.e., these correlations can be obtained causally (see Figure 7).
iii.3 Two parties, binary inputs, and binary outputs
In the two-party case with a binary input and a binary output for each party, the process of the environment is described by a square matrix of dimension . The conditions are
With a similar argument as above, one can show that the operation does not need to be considered for either party. The matrix consists of unknowns, out of which are eliminated by the total-probability conditions given by Equation (63). Thus, we are left with unknowns, forming a -dimensional polytope with inequalities.
The resulting -representation of the polytope consists of extremal points, all of which represent deterministic processes:
In the following, we use and . The first four processes represent the four constants as inputs to the parties and . The next four processes represent a constant input to party (processes and produce the constant , and the other two processes produce the constant ) and a channel from party to party ; the processes and describe the identity channel, and and describe the bit-flip channel. The last four processes are analogous, with a channel from to and where party receives a constant.
All these processes act deterministically on bits for two parties where at least one party receives a constant (see Figure 8). Therefore, every such channel can be simulated in a causal fashion. This result generalized to higher dimensions was already shown by taking the classical limit of the framework for quantum correlations without predefined causal order Oreshkov et al. (2012).
iii.4 Three parties, binary inputs, and binary outputs
The process of the environment in a three party setup with binary inputs and outputs is described by a square matrix of dimension . The matrix consists of variables, out of which can be eliminated with the total-probability conditions
resulting in a -dimensional polytope with linear constraints (non-negative probabilities):
Solving this polytope yields extremal points. Only extremal points out of these are deterministic, i.e., consist of 0-1 values; the remaining extremal points are so-called proper mixtures of logically inconsistent processes. Such proper mixtures are not convex combinations of deterministic extremal points inside the polytope, but are convex combinations of deterministic points where some lie outside of the polytope — any process from outside of the polytope leads to logical inconsistencies. Interestingly, this smaller polytope (hence, also the polytope described by the Equations (71) and (72)) consists of processes that cannot be simulated using a predefined causal order, i.e., processes where no party receives a constant, implying that every party causally succeeds some other party. The deterministic extremal points are discussed in Section IV along with the general polytope restricted to the deterministic extremal points.
iii.5 General case
We describe the polytope for logically consistent classical processes without predefined causal order in the general case. Let be the number of parties and let be the dimension of the states entering and leaving every laboratory. This leaves us with a stochastic matrix describing the environment. Every party can perform an operation that is a convex mixture of all deterministic operations. The set of all deterministic operations is denoted by . For every party, under any choice of deterministic operation , the trace of the environment multiplied with the local operations is constrained to give (see Theorem 2). However — as in the binary-input/output case above —, some of these constraints are redundant.
Theorem 5 (Sufficient set for total-probability conditions).
The total-probability conditions to this family of operations
where is output for input and otherwise, imply the total-probability conditions for all remaining deterministic operations of the same dimension, i.e.,
We restrict ourselves to the single-party scenario — the multi-party case follows through linearity. Let be the -dimensional vector with a -entry at position and ’s everywhere else. We can write a -dimensional matrix as
A general deterministic matrix of the same dimension, where is mapped to , is expressed as
On the one hand, using the antecedent above, we have
with . On the other hand, we can rewrite as