The space of logically consistent classical processes without causal order

The space of logically consistent classical processes without causal order

Abstract

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.

Figure 1: If the random variable  is an input (here, visualized by a knob), the random variable  is an output, and  is correlated to , then  can signal to  which implies that  is in the causal future of  ().

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.

causal deterministicextremal point

proper-mixture extremal point(fine-tuned)

non-causal deterministic extremal point

Figure 2: A qualitative representation of processes without predefined causal order studied in this work is given. The dashed region describes all processes that are achievable in a predefined causal order — it also forms a polytope Oreshkov and Giarmatzi (2015); Branciard et al. (2016). The polytope with the dashed-dotted lines is the polytope of processes without predefined causal order. The region in-between marked with the solid lines is the polytope of processes without predefined causal order restricted to deterministic extremal points.

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

(1)

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

(2)

with  and , such that in every distribution  at least one party is not in the causal future of any other party, i.e.,

(3)

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 .

Figure 3: A single party  describes a stochastic process . The variables , and  model the input and the output. The variable  is obtained from the environment ; the party  feeds the variable  into the same environment.

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).

Figure 4: The box  describes the environment. Because no predefined causal order is assumed between the parties, the random variable obtained by the parties is the result of  applied to the outgoing random variable of all parties. This picture combines states and channels, i.e., signaling and no-signaling correlations. For example, assume that  is in the causal past all other parties. In that case, the random variable  is constant, whereas the random variable  could depend on . For three parties or more, this framework gives rise to a new quality:  can describe a map where no  is a constant, yet where no contradiction arises. Such correlations are called non-causal. Similarly to the parties, the box  is a stochastic process .

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 .

The linearity and positivity conditions from Definition 3 imply Theorem 1, which states that the environment must be a stochastic process (conditional probability distribution).

Theorem 1 (Logical consistent environment as stochastic process).

The environment  is a stochastic process  that maps  to .

Proof.

The environment is a multi-linear positive map  on the probabilities (we omit the arguments for the sake of presentation)

(4)

that party  outputs  to the environment and generates  conditioned on the setting  and on . Therefore, we write

(5)

Since  is a multi-linear positive map and since it depends on  and  only, the above probability can be written as

(6)

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

(7)
(8)
(9)

where

(10)

Let us fix the local operations  of all parties to be

(11)

From the total-probability condition we obtain

(12)

By repeating this calculation for different choices of local operations where the parties deterministically output a value, we get

(13)

Therefore,  is a stochastic process . ∎

The following Corollary follows from Theorem 1.

Corollary 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.

Example 1.

Let the environment as well as the party  forward the random variable, i.e., the operation of the environment is

(14)

and the operation of the party  is

(15)

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.

Example 2.

We alter the local operation of party  to negate the binary random variable

(16)

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

(17)

A stochastic process  from  to a random variable  with sample space  describes an operation and is modeled by the stochastic matrix

(18)

The result  of evolving the random variable  through the operation  is given by the matrix multiplication

(19)

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

(20)

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 .

Figure 5: Party  is described by  and the environment  is .

We can further simplify our picture by fixing the value of  to  and by summing over :

(21)

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

(22)

where  is a deterministic input value. By embedding  into the process of , we get

(23)

This can be interpreted as a probability measure of party  receiving the value  from the environment :

(24)

For  to represent a probability measure, the values of  for every deterministic value  must be non-negative and have to sum up to 1:

(25)
(26)

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

(27)

The matrix  fixed to providing the state  to the party  is

(28)

Therefore, the probability of party  observing  is

(29)

and the law of total probability requires

(30)

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

(31)

and the non-negativity condition

(32)

hold. Because a stochastic process can be written as a convex mixture of deterministic operations, it is sufficient to ask for

(33)
(34)

for every operation  from the set  of all deterministic operations. Thanks to linearity, we can straightforwardly extend these requirements to multiple parties, and arrive at Theorems 2 and 3.

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

(35)

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

(36)

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

(37)
(38)

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 quantum framework given by Equations (37) and (38) in the classical limit is equivalent to the description of classical correlations without predefined causal order given by Equations (31) and (32).

Proof.

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

(39)

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.,

(40)

We can rewrite  as

(41)

Analogously, the stochastic matrix  of the party can be translated into the quantum framework and becomes

(42)

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

(43)
(44)

Substituting  with  and  with  yields

(45)
(46)
(47)
(48)

which proves the claim. The multi-party case follows through linearity. ∎

Iii Polytope of classical processes without predefined causal order

iii.1 Polytopes

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.

Here, we derive the polytope of classical processes without predefined causal order. This polytope is represented by the dashed-dotted lines in Figure 2. A projection of the polytope for three parties and binary inputs/outputs onto a plane is given in Figure 6.

Figure 6: Here, we see a projection of the polytope of classical processes without predefined causal order among three parties and with binary inputs/outputs. The circular identity channel  and the circular bit-flip channel  are logically inconsistent; they can be used to reproduce the grandfather’s paradox. The solid lines mark the deterministic-extrema polytope and the dashed-dotted lines mark the additional space of logically consistent processes. Point  is an extremal point of the polytope and is a uniform mixture of the deterministic processes  and . The behavior of this point is shown in Figure 12. Point  is an extremal point of the deterministic-extrema polytope, and is described in Figure 11.

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

(49)

consisting of  variables. The deterministic operations party  can apply are

(50)
(51)

where  produce a constant , respectively, and where the matrix  is the identity and  the negation. The equalities

(52)
(53)
(54)

enforce

(55)

This is shown as follows:

(56)
(57)
(58)
(59)

By eliminating three variables using the total-probability conditions (52), (53), and (54) from above, we get

(60)

with the non-negativity conditions

(61)
(62)

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).

Figure 7: All logically consistent single-party correlations that can be obtained with a feedback channel (see Figure 5) can be simulated without feedback channel.

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

(63)
(64)

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:

(65)
(66)
(67)
(68)
(69)
(70)

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.

(a)

(b)

(c)
Figure 8: (a) Both parties  and  receive a constant each. (b) Party  receives a constant and sends a bit through the identity  or the bit-flip  channel to . (c) Same as (b), where the parties are interchanged.

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

(71)

resulting in a -dimensional polytope with  linear constraints (non-negative probabilities):

(72)

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

(73)

where  is output for input  and  otherwise, imply the total-probability conditions for all remaining deterministic operations of the same dimension, i.e.,

(74)
(75)
(76)
(77)
Proof.

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

(78)

A general deterministic matrix  of the same dimension, where  is mapped to , is expressed as

(79)

On the one hand, using the antecedent above, we have

(80)

with . On the other hand, we can rewrite  as

(81)
(82)
(83)