CCausal and causally separable processes
Ognyan Oreshkov and Christina Giarmatzi , QuIC, Ecole Polytechnique de Bruxelles, C.P. 165, Universit´e Libre de Bruxelles, 1050 Brussels, Belgium Centre for Engineered Quantum Systems, Centre for Quantum Computer and Communication Technology,School of Mathematics and Physics, University of Queensland, Brisbane, Queensland 4072, Australia
The idea that events are equipped with a partial causal order is central to our understanding of physics in thetested regimes: given two pointlike events A and B , either A is in the causal past of B , B is in the causal pastof A , or A and B are space-like separated. Operationally, the meaning of these order relations corresponds toconstraints on the possible correlations between experiments performed in the vicinities of the respective events:if A is in the causal past of B , an experimenter at A could signal to an experimenter at B but not the other wayaround, while if A and B are space-like separated, no signaling is possible in either direction. In the contextof a concrete physical theory, the correlations compatible with a given causal configuration may obey furtherconstraints. For instance, space-like correlations in quantum mechanics arise from local measurements on jointquantum states, while time-like correlations are established via quantum channels. Similarly to other variables,however, the causal order of a set of events could be random, and little is understood about the constraints thatcausality implies in this case. A main di ffi culty concerns the fact that the order of events can now generallydepend on the operations performed at the locations of these events, since, for instance, an operation at A couldinfluence the order in which B and C occur in A’s future. So far, no formal theory of causality compatible withsuch dynamical causal order has been developed. Apart from being of fundamental interest in the context of in-ferring causal relations, such a theory is imperative for understanding recent suggestions that the causal order ofevents in quantum mechanics can be indefinite. Here, we develop such a theory in the general multipartite case.Starting from a background-independent definition of causality, we derive an iteratively formulated canonicaldecomposition of multipartite causal correlations. For a fixed number of settings and outcomes for each party,these correlations form a polytope whose facets define causal inequalities. The case of quantum correlations inthis paradigm is captured by the process matrix formalism. We investigate the link between causality and theclosely related notion of causal separability of quantum processes, which we here define rigorously in analogywith the link between Bell locality and separability of quantum states. We show that causality and causal sepa-rability are not equivalent in general by giving an example of a physically admissible tripartite quantum processthat is causal but not causally separable. We also show that there are causally separable quantum processes thatbecome non-causal if extended by supplying the parties with entangled ancillas. This motivates the concepts ofextensibly causal and extensibly causally separable (ECS) processes, for which the respective property remainsinvariant under extension. We characterize the class of ECS quantum processes in the tripartite case via simpleconditions on the form of the process matrix. We show that the processes realizable by classically controlledquantum circuits are ECS and conjecture that the reverse also holds. I. INTRODUCTION
The possibility for dynamical and indefinite causal structures in quantum theory and more general probabilistic theories hasrecently attracted a great deal of interest, both from a foundational point of view and in the context of quantum informationprocessing [1–23]. Motivated by the long standing search for a theory of quantum gravity, where the causal structure is expectedto be dynamical as in General Relativity but fundamentally probabilistic in nature, as well as by the exploration of novel quantumarchitectures beyond the standard circuit model, operational ways of thinking about causal order in a probabilistic setting haveprovided new perspectives on quantum mechanics, its possible applications, and routes for potential extensions.A general framework for the study of correlations between local experiments without the assumption of a predefined causalorder between them was proposed in Ref. [4]. In this so called process framework , each experiment is associated with aninput and an output system between which an experimenter can perform di ff erent operations, but no specific assumption aboutthe existence of a causal structure in which the experiments are embedded is made. When the experiments take place at fixedlocations in a background space-time in circumstances defined without post-selection, the causal structure of space-time imposessignaling constraints on the correlations between the experiments. For example, there can be signaling from one experiment toanother only if the former takes place in the past light cone of the latter, but no signaling between space-like separated locationsor from the future to the past is possible. In Ref. [4], it was shown that if the local operations are described by quantummechanics, it is possible to conceive correlations that are incompatible with any underlying causal structure. Such correlationsallow two parties, Alice and Bob, to establish correlations that violate a causal inequality , which is impossible if their operationstake place in a causal order, even if that order is random. A similar possibility was subsequently shown to exist in a multipartitesetting even when the local operations are purely classical [9], which in the bipartite case is not possible [4]. It is not known atpresent whether such joint processes could have a physical realization without post-selection, that is, whether one could preparea setup that leads to correlations violating causal inequalities between separate experimenters who locally experience the validityof standard quantum mechanics. a r X i v : . [ qu a n t - ph ] S e p Another peculiar e ff ect that seems at odds with causality, which has a physical realization without post-selection, arises whenlocal quantum operations are applied in an order that depends on the value of a variable prepared in a quantum superposition[3, 5, 6, 10, 17], a technique known as ‘quantum switch’ [3]. This approach allows achieving certain tasks that are impossibleif the quantum operations are applied in a definite causal order. In contrast to the violation of a causal inequality, however, thisconclusion depends on the assumed description of the local operations and is theory-dependent.So far, the analysis of these e ff ects has relied on semi-rigorous considerations about what it means for a process to be compati-ble with ‘definite causal order’. A fully rigorous argument requires such considerations to be rooted in a clear notion of causality,which, however, in this background-independent setting has been lacking. Such a notion is expected to have a universal expres-sion which can be applied in the context of any number of parties, but how to formulate it turns out to be a nontrivial problem.Simple considerations in the multipartite case show that the causal order of a set of local experiments should most generally beconsidered to be a random variable that can depend on the settings of these experiments. The latter possibility cannot be excludedsince compatibly with our intuition of causality we can conceive of scenarios in which the setting in a given local experimentcan influence the order in which other experiments take place in the future. In other words, causality should be expressed as arule that constrains the joint conditional probabilities for the events in the local experiments and the causal order between them,allowing for the possibility that causal configurations unfold as a result of events in the past. A formal theory of such dynamicalcausal order is essential not only for understanding the subject of indefinite causal order in quantum mechanics or more generaltheories, but also for the problem of inferring causal structure beyond the classic paradigm of underlying deterministic variablesand static causal relations [24].In this paper, we develop rigorous theory-independent and theory-dependent notions of causality in the process frameworkand characterize the structure and relations between the corresponding classes of processes they define. Section II is devoted tothe theory-independent perspective, which contains our core result. We formalize the process framework in theory-independentterms and propose a definition of causality which allows for the possibility of dynamical causal order. We develop a number ofconcepts, such as multipartite signaling, reduced and conditional processes, and derive necessary and su ffi cient conditions fora process to be causal, which are expressed in the form of an iteratively defined canonical decomposition of the probabilitiesin the process. This decomposition can be understood as describing a causal ‘unraveling’ of the events in the experiment in asequence, showing that the proposed notion of causality yields the structure expected from intuition. Apart from being logicallynon-trivial, this result has important conceptual implications – it presents us with an understanding of causal order as a randomfunction on random events rather than the ordering of underlying locations in which events happen. This perspective is in thespirit of the idea of background independence in general relativity, according to which there are no underlying locations, butonly events and the relations between them. In Section III, we focus on the quantum process framework, where we developdi ff erent theory-dependent notions of causality, which in principle have analogues in more general process theories too. Specifi-cally, we investigate several possible generalizations of the bipartite notion of causal separability, which was previously definedheuristically in the bipartite case by postulating a particular form of the quantum process matrix [4]. We show that this formcan be understood as arising from the canonical decomposition of causal processes under the condition that each process in thisdecomposition is a valid quantum process. We define the multipartite concept based on this principle. We show that the sets ofcausal and causally separable processes are not equivalent in the multipartite case, by giving an explicit example of a class ofprocesses that are causal but not causally separable. This example is based on the ‘quantum switch’ technique discussed earlier.We also show that, surprisingly, there exist causally separable (and hence causal) quantum processes that become non-causal ifextended by supplying the parties with an entangled input ancilla. This example of ‘activation of non-causality’ is constructedbased on a suitable modification of the non-causal process matrix of Ref. [4]. This observation motivates the concepts of ex-tensibly causal and extensibly causally separable (ECS) processes, for which the respective property remains invariant underextension with arbitrary input ancillas. We derive a characterization of the class of ECS quantum processes in the tripartitecase in terms of simple conditions on the form of the process matrix, which generalize the known form of bipartite causallyseparable process matrices. In the bipartite case, causal separability and extensible causal separability are equivalent, hence theclass of ECS processes can be regarded as another possible multipartite generalization of the previously known bipartite concept.Finally, we consider the class of processes realizable by classically controlled quantum circuits, which we show is inside theclass of ECS processes. These, too, are equivalent to the causally separable processes in the bipartite case and provide a possiblemultipartite generalization based on a di ff erent principle. We conjecture that the processes that can be obtained by classicallycontrolled quantum circuits are equivalent to the ECS processes, and hence are described by process matrices obeying the simpleconditions we have derived. We provide arguments in favor of this conjecture based on analysis in the tripartite case. In SectionIV, we summarize our results and discuss future research directions. II. THE PROCESS FRAMEWORKA. General processes
The process framework introduced in Ref. [4] describes probabilities for the outcomes of local experiments associated withdi ff erent parties, Alice, Bob, Charlie, etc., performed in abstract circumstances defined without assuming the existence of aglobal causal order between the experiments, but only a local order of the events in each of them. Each local experiment canbe thought of as performed inside an isolated laboratory, where, at a given instant, an input system is received in the laboratory,it is subject to some operation that yields one of a set of possible outcomes, and, at a given later instant, an output systemis sent out of the laboratory. The input and output systems are assumed to provide the only means of information exchangebetween events in the laboratory and any events in the rest of the whole experiment. The framework in Ref. [4] was developedfor the case where the local experiments are described by standard quantum mechanics, under a set of specific assumptions.These assumptions are that the joint probabilities of the outcomes of the local experiments are non-contextual functions of thetransformations (described by completely positive (CP) maps) associated with the local outcomes, and that the local experimentscan be extended to act on ancillas prepared in any joint quantum state.There is a straightforward way in which an analogous theory can be formulated starting from any generalized operationalprobabilistic theory that has a formulation in the circuit framework [25–28] following the construction in Ref. [4]. Indeed, theconcepts of transformation and state are defined for any such theory, and so is the idea of a composite system that is employed inthe notion of adding an ancilla. [Note that the representation of the quantum process framework [4] in terms of process matrices (see Section III) is built around the Choi-Jamiołkowski isomorphism [29, 30], which may not be available for arbitrary theories,but this concerns the representation of the framework.] However, the above assumptions underlying the extension from a circuittheory to a process theory, albeit arguably natural, are by no means mandatory. For example, one can conceive of extensions ofquantum theory in which the joint probability distributions are contextual, but nevertheless for each single party the marginalprobabilities are non-contextual and consistent with standard quantum mechanics. One can also conceive of theories in whichthe allowed non-signaling ancillary resources are not quantum states, although they give valid non-contextual probabilities forthe outcomes of any combination of local quantum measurements [32]. It is therefore of interest to formulate a general processframework in operational terms without additional assumptions about how that framework may be related to theories expressedin the circuit framework. This is important also for the question of understanding the concept of causal inequality introduced inRef. [4], which tests the compatibility of a process with an underlying causal structure in theory-independent terms.To this end, we will describe each local experiment, say that of Alice, by two variables – a setting s A , and an outcome o As forthat setting. What these variables are supposed to correspond to in practice will be discussed below. The possible settings fora given local experiment are assumed to belong to some set S A , and the outcomes for each value s A of the setting to a set O As .Since we can formally extend the possible outcomes for each setting with fictitious outcomes that never occur, without loss ofgenerality we can assume that the sets O As are identical for all s A ∈ S A , i.e., O As ≡ O A . A particular event in Alice’s laboratory isthus described by a pair of variables ( s A , o A ) ∈ S A × O A . An operation is a collection of possible events { ( s A , o A ) } o A ∈ O A for a fixedvalue of s A ∈ S A . The very occurrence of the local experiments, as well as the circumstances in which they take place, wouldbe conditional on some variable that we will denote by w A , B , C , ··· , which belongs to some set Ω A , B , C , ··· of possible such variables.What the variable w A , B , C , ··· is supposed to correspond to in practice will also be discussed below. Definition II.1. (Process):
Mathematically, we define a process W A , B , ··· for a set of local experiments (or parties) S = { A , B , · · · } as the collection of conditional probabilities W A , B , ··· ≡ { P ( o A , o B , ... | s A , s B , ..., w A , B , ··· ) } , (1) o X ∈ O X , s X ∈ S X , X ∈ S , for a given value of w A , B , ··· ∈ Ω A , B , ··· . Definition II.2. (Trivial process):
For the purposes of expressing more succinctly certain conditions later, it is convenient toallow the set of local experiments S = { A , B , · · · } in the definition to be the empty set {} as a special case. In that case, thecorresponding process will be referred to as the trivial process . We define it to consist of a single probability – that for the trivialoutcome given the trivial setting – which is equal to 1.In abstract terms, a theory in the process framework is specified by listing the di ff erent types of input and output systems,all possible settings and outcomes between input and output systems of specific types, all possible variables w A , B , ··· for whichwe have a valid occurrence of a set of local experiments S = { A , B , · · · } , and the corresponding processes (1). Similarly tooperational probabilistic theories in the circuit framework [25–28], it is understood that equivalence classes of the variables s X , o X , and w A , B ,... , with regard to the probabilities (1) are taken, and these variables are identified with their equivalence classes.But what are these variables supposed to describe in practice? In Refs. [15, 16], it was argued that there are two main ideas thatunderlie the concept of operation in the standard circuit framework for operational probabilistic theories [25–28]. The first one,termed the closed-box assumption , is the idea that the input and output systems of an operation are the only means of informationexchange responsible for the correlations between the outcomes of that operation and the outcomes of other operations in theglobal experiment. The second idea, termed the no-post-selection criterion , which makes sense assuming a predefined notion oftemporal ordering as in the standard circuit formulation, is that the variable that defines an operation, or the setting s X , can beknown with certainty before the time of interaction with the input system unconditionally on any events in the future.Since no predefined global time is assumed in our picture, the latter condition will be imagined to hold only with respect tothe local temporal sequence of events observed by each experimenter. Furthermore, we will assume that the variable w A , B , ··· thatdefines the global setup in which the individual experiments take place is also obtained without post-selection. We can makesense of this idea by imagining that the variable is associated with an event that fits within each of the local temporal frames ofthe experimenters and is such that it occurs before any of them receives the input system. We will call processes that describeexperiments of this kind pre-selected processes . (For a generalization that admits post-selection, see Ref. [16]).For the rest of this paper, we will consider only pre-selected processes. We will drop the explicit specification ‘pre-selected’for brevity, and will refer to them simply as processes, unless we want to explicitly emphasize the assumption of pre-selection.We will also drop the explicit specification of the variable w A , B , ··· on which the joint experiment is conditioned, and we willsimply write W A , B , ··· ≡ { p ( o A , o B , ... | s A , s B , ... ) } , keeping in mind that every process describes circumstances defined by such avariable and hence all probabilities we consider are implicitly conditional on such a variable. B. Causal processes
In the circuit framework for operational probabilistic theories, causality is defined as the property that the probability distri-bution over the outcomes of a given operation in a circuit do not depend on what operations take place in the absolute future orabsolute elsewhere [33] of that operation as defined by the strict partial order (SPO) of the circuit composition [26, 27]. Morespecifically, every circuit describes a set of operations taking place at the vertices of a directed acyclic graph, whose directededges (the circuit ‘wires’) correspond to systems that go from one operation to another. Such a graph defines a SPO on the op-erations in a circuit (a precise definition of SPO is given below) – one operation is in the absolute past of another (equivalently,the latter is in the absolute future of the former) if there exists a directed path from the former to the latter through the graph.If there is no directed path connecting two operations, we say that one is the absolute elsewhere of the other. If we imaginethat there is a local experiment taking place at every vertex of such a graph, the property of causality says that the probabilitiesfor the outcomes of local experiments that are in the causal past or causal elsewhere of a given local experiment cannot dependon the setting of that experiment. A circuit theory that obeys this condition, such as standard quantum theory, is called causal,and for such a theory the SPO defined by the circuit composition can be interpreted as causal order [26, 27]. This interpretationcorresponds to the intuitive idea that, if the setting of a local experiment is regarded as up to the ‘free choice’ of an experimenter,then any correlations between that setting and other variables must indicate a causal influence of the setting on those variables.From this perspective, causality can be understood as the condition that a variable can influence only variables in its immediatelocation or in its absolute future.In the process framework, we do not assume the existence of a given circuit in which the local experiments are embedded.Thus, there is no natural SPO with respect to which to define causality. Nevertheless, we may ask whether the probabilitiesdescribed by a given process are compatible with the existence of a SPO with respect to which causality is satisfied. Howto formulate this precisely, however, is not immediately clear because the process framework can describe situations in whichthe SPO may be random. For instance, it can describe the correlations between local experiments that can be embedded indi ff erent circuits according to some probability distribution. Clearly, if the SPO between the local experiments is random, itmust be the case that conditionally on that SPO taking any particular value, the probabilities of the outcomes of the partiesgiven their settings must obey the above notion of causality. This condition, however, is not su ffi cient to capture the idea ofcausality. For example, consider the local experiments of two parties, Alice and Bob, which are embedded at random in one oftwo possible causal circuits where they occur in di ff erent orders. The probabilities for all events and the specific circuit could besuch that, conditionally on any particular circuit being realized, the joint probabilities of the outcomes of the parties given theirsettings obey the above notion of causality, but nevertheless the setting of Alice could be correlated with the circuit in whichher experiment is embedded, and thereby with the SPO on the two local experiments. Intuitively, such a situation should be inconflict with causality, because if Alice’s setting could not influence events that occur in the past, it should not influence whetheror not Bob performs an operation in the past. The circuit notion of causality cannot be used to define such an independence fromthe past, because there the past is defined assuming a fixed circuit. This indicates that we need a more general notion of causalitythat imposes constraints on how the SPO on the local experiments can depend on the parties’ settings. A simple possibilityis to require that the SPO on the local experiments must be independent of the parties’ setting. This condition, however, istoo restrictive, because, compatibly with the idea of causality, we can conceive of scenarios where the setting of a given partyinfluences the order in which other parties perform their experiments in that party’s absolute future. Thus, a more sophisticateddefinition of causality is needed for the process framework. We next develop such a definition.First, let us review the properties of SPO and introduce some terminology. A SPO on a nonempty set of local elements S = { A , B , C , · · · } is a binary relation ≺ which satisfies the following conditions: (1) irreflexivity – not A ≺ A ; (2) transitivity – if A ≺ B and B ≺ C , then A ≺ C ; (3) anti-symmetry – if A ≺ B , then not B ≺ A . When two local experiments A and B satisfy A ≺ B (equivalently, B (cid:31) A ), we will say that A is in the absolute past of B , or that B is in the absolute future of A [33]. It will beconvenient to introduce the notation A (cid:14) B (equivalently, B (cid:15) A ), which means A (cid:44) B and not A ≺ B , that is, A and B are di ff erent and A is not in the absolute past of B (equivalently, B is not in the absolute future of A ). We will also introduce the notation A (cid:14)(cid:15) B ,which means A (cid:14) B and A (cid:15) B , that is, A and B are di ff erent and A is neither in the absolute past nor in the absolute future of B (and hence, B is neither in the absolute past nor in the absolute future of A ). In the case when A (cid:14)(cid:15) B , we will say that A and B are absolutely independent , or that A is in the absolute elsewhere [33] of B (and similarly, B is in the absolute elsewhere of A ). A prototypical example of these relations is the causal order between the points in a Minkowski space-time – the absolutepast / future of a given point corresponds to the points in the past / future light-cone of this point, excluding the point itself, whilethe absolute elsewhere consists of the points that are space-like separated from the point.Note that if a set of elements S = { A , B , · · · } is equipped with a SPO, the elements X and Y in any pair ( X , Y ) ∈ S × S arerelated by X ≺ Y , X (cid:31) Y , X (cid:14)(cid:15) Y , or X = Y . The SPO on the set S = { A , B , · · · } is equivalently described by the list of respectiverelations for each such pair, which we will denote by κ ( A , B , · · · ). (This list obviously must respect the properties of SPO listedabove.) Since for pairs ( X , X ) of identical elements this relation is trivially X = X , when we explicitly describe κ ( A , B , · · · ), wewill only list the pairwise relations for all pairs of distinct elements of the set (if any). Note that this description is generallyredundant due to the transitivity of SPO. If we are given the pairwise relations for a set S = { A , B , · · · } , we have, in particular,pairwise relations for any nonempty subset S (cid:48) = { X , Y , · · · } ⊂ S , i.e., a SPO κ ( A , B , · · · ) on S implies a SPO κ ( X , Y , · · · ) on S (cid:48) ⊂ S , S (cid:48) (cid:44) {} .As discussed above, the SPO κ ( A , B , · · · ) on a set of local experiments S = { A , B , · · · } in terms of which causality wouldbe defined can most generally be random and correlated with the events in these experiments. The notion of causality wouldimpose constraints on the possible correlations. We want these constraints to formalize the following intuition about causality: The choice of setting in a local experiment cannot a ff ect the occurrence of events in the absolute past or absolute elsewhereof that experiment, nor the SPO on such events and the experiment in question. Since a process is defined by the conditional probabilities for the outcomes of the local experiments given their settings anddoes not assume the existence of probabilities for the settings, we will formulate the above constraint at the level of probabilitiesconditional on the settings. We define this as follows.
Definition II.3. (Causal process):
A process W A , B , ··· ≡ { p ( o A , o B , · · · | s A , s B , · · · ) } for a nonempty set of local experiments S = { A , B , · · · } is called causal if and only if there exists a probability distribution p ( κ ( A , B , · · · ) , o A , o B , · · · | s A , s B , · · · ), (cid:80) κ ( A , B , ··· ) p ( κ ( A , B , · · · ) , o A , o B , · · · | s A , s B , · · · ) = p ( o A , o B , · · · | s A , s B , · · · ), where the random variable κ ( A , B , · · · ) takes valuesin the possible SPOs on S = { A , B , · · · } , such that for every local experiment, e.g. A , every subset X = { X , Y , · · · } of the rest ofthe local experiments, and every SPO κ ( A , X , Y , · · · ) ≡ κ ( A , X ) on the local experiment in question and that subset, we have p ( κ ( A , X ) , A (cid:14) X , o X | s A , s B , · · · ) = p ( κ ( A , X ) , A (cid:14) X , o X | s B , · · · ) . (2)Here, o X denotes collectively the outcomes of all local expriments in X , and A (cid:14) X denotes the condition that all these localexperiments are in the causal past or causal elsewhere of A (i.e., A (cid:14) X , A (cid:14) Y , · · · , for all X , Y , · · · ∈ X ). [The probability p ( κ ( A , X ) , A (cid:14) X , o X | s A , s B , · · · ) is understood obtained from p ( κ ( A , B , · · · ) , o A , o B , · · · | s A , s B , · · · ) by summing over all cases inwhich κ ( A , B , · · · ) is compatible with κ ( A , X ) and A (cid:14) X (obviously, if κ ( A , X ) itself is not compatible with A (cid:14) X , the respectiveprobability is zero) and over all possible outcomes of the local experiments in the complement of X ]. Remark.
A monopartite process is trivially causal.For a process W A , B , ··· that is causal, the binary relation ≺ of the SPO κ ( A , B , · · · ) can be interpreted as causal order . Inthat case, we will use the terms ‘ causal past ’, ‘ causal future ’, ‘ causal elsewhere ’ and ‘ causally independent ’ in the place of‘absolute past’, ‘absolute future’, ‘absolute elsewhere’ and ‘absolutely independent’, respectively. We will also refer to the listof pairwise relations κ ( A , B , · · · ) as the causal configuration of the local experiments (in the case of a monopartite process, thecausal configuration is trivial).Our goal next is to understand the structure of causal processes that arises from this definition and show that it correspondsexactly to what one expects from intuition. C. Fixed-order causal processes, (no) signaling, reduced and conditional processes
Before we consider the case of general causal processes, it will be instructive to investigate the special case of causal processesfor which the causal configuration of the local experiments is fixed. As we will show, the constraints on such processes can be ex-pressed via the concept of signaling, which we develop below. We also introduce several related concepts that will be of use later.
Definition II.4. (Fixed-order causal process):
A process W A , B , ··· ≡ { p ( o A , o B , · · · | s A , s B , · · · ) } is called fixed-order causal ifit is compatible with a deterministic causal configuration, i.e., if it satisfies condition (2) for a SPO κ ( A , B , · · · ) that takes aparticular value κ ( A , B , · · · ) = κ ∗ ( A , B , · · · ) with unit probability for all possible settings of the parties: p ( κ ( A , B , · · · ) , o A , o B , · · · | s A , s B , · · · ) = , i ff κ ( A , B , · · · ) (cid:44) κ ∗ ( A , B , · · · ) , ∀ s A ∈ S A , ∀ s B ∈ S B , · · · , ∀ o A ∈ O A , ∀ o B ∈ O B , · · · . (3)Since our definition of causal process implies that the setting of a local experiment cannot be correlated with the outcomesof local experiments that are in the absolute past or absolute elsewhere of that experiment, one may expect that for any fixedcausal configuration of the local experiments, causality would impose constraints on the possibility for signaling between them,similarly to the case in the circuit framework. In the case of two experiments, signaling can be defined as follows: Definition II.5. (Bipartite signaling):
We say that there is no signaling from Alice ( A ) to Bob ( B ) in a bipartite process W A , B if and only if the probabilities of the process satisfy p ( o B | s B , s A ) ≡ (cid:88) o A ∈ O A p ( o A , o B | s B , s A ) = p ( o B | s B ) , (4) ∀ s A ∈ S A , s B ∈ S B , o B ∈ O B , i.e., the marginal probabilities for the outcomes of Bob are independent of the setting of Alice for any possible setting of Bob.Equivalently, we say that there is signaling from Alice to Bob in the process W A , B if and only if this condition is not satisfied.For a fixed-order causal process W A , B , where one of the relations A ≺ B , B ≺ A , or A (cid:14)(cid:15) B holds with unit probability for allsettings of the parties, we can see that signaling is possible from one experiment to the other only if the former is in the causal pastof the latter, which agrees with the notion of causality in the circuit framework [26, 27]. Indeed, assume for example that B ≺ A ,i.e., p ( κ ( A , B ) = B ≺ A | s A , s B ) = ∀ s A ∈ S A , ∀ s B ∈ S B (and hence p ( κ ( A , B ) = A ≺ B | s A , s B ) = p ( κ ( A , B ) = A (cid:14)(cid:15) B | s A , s B ) = ∀ s A ∈ S A , ∀ s B ∈ S B ). Then, we have p ( o B | s A , s B ) = p ( A ≺ B , o B | s A , s B ) + p ( B ≺ A , o B | s A , s B ) + p ( A (cid:14)(cid:15) B , o B | s A , s B ) = p ( A (cid:14) B , o B | s A , s B ) = p ( A (cid:14) B , o B | s B ) = p ( o B | s B ) , ∀ s A ∈ S A , s B ∈ S B , o B ∈ O B , (5)i.e., there is no signaling from Alice to Bob. In a similar way, we see that if A ≺ B , there is no signaling from Bob to Alice, whileif A (cid:14)(cid:15) B , there is no signaling from Alice to Bob and no signaling from Bob to Alice.In the case of more than two local experiments, the relevant generalization of the above notion of signaling may not beimmediately obvious. Notice that if a given bipartite process W A , B involves no signaling between A and B , such a process isin principle compatible with the causal configuration A (cid:14)(cid:15) B (in fact, it is compatible with any causal configuration of the twoparties). However, in the case of processes for more than two local experiments, even if there is lack of signaling between anypair of experiments for all possible settings of the rest of the experiments, the process may not be compatible with a causalconfiguration in which all experiments are causally independent.To see this, consider three local experiments performed by Alice, Bob, and Charlie, where each party’s input and outputsystems are classical bits, and each party is allowed to perform any classical stochastic operation from the input bit to the outputbit. Let the experiments of Bob and Charlie be causally independent, and let Alice’s experiment be in the absolute future ofBob’s experiment, but in the absolute elsewhere of Charlie’s experiment (i.e., the causal configuration of the three parties is AB C a) CBA b) BA C c) FIG. 1: Certain types of multipatite signaling correlations do not involve bipartite signaling and do not imply the existence of acausal connection between any particular pairs of channels. The example discussed in the text could arise from any of themechanisms sketched here.[ B ≺ A , A (cid:14)(cid:15) C , B (cid:14)(cid:15) C ]). Imagine that Charlie receives his input system in one of the two possible states 0 or 1 with probability1 /
2, and depending on that state, Alice and Bob are in one of the following two scenarios. In the first scenario (say, whenCharlie receives 0), Bob receives a random bit as an input system, his output bit is sent unaltered into the input system of Alice,and Alice’s output bit is discarded. In the second scenario (when Charlie receives 1), Bob again receives a random input bit, butthis time his output bit is flipped before sending it into Alice’s input, and Alice’s output bit is again discarded. In both cases,the output system of Charlie is discarded. Clearly, the described situation can be realized in agreement with a fixed causalconfiguration of the parties – all we need to do is supply Bob with a random bit and correlate the channel from Bob to Alicewith the input system of Charlie, discarding the outcomes of Alice and Charlie. The mechanism realizing this is sketched inFig. 1a. Note that the tripartite process corresponding to this scenario would involve no signaling from Bob to Alice in spite ofthe existence of a channel from Bob to Alice. This is the case irrespectively of what operation Charlie performs. Obviously,there can be no signaling from Alice to Bob either, since Alice operates in the future of Bob, nor can there be signaling betweenAlice and Charlie, or between Bob and Charlie, since Charlie is causally independent of both Alice and Bob. Thus, we haveno signaling between any pair of parties, no matter what the setting of the third party is. Yet, the possible correlations betweenthe parties cannot be realized if all parties are causally independent because if Alice and Charlie measure their input bits andcollect the results of their measurements, they can infer the bit sent out by Bob, which is impossible if all parties are causallyindependent. We might say that in this case we have signaling from Bob to Alice and Charlie together. But intuitively, given thedescribed scenario, this signaling should be from Bob to Alice only, since there is no channel connecting Bob’s output system toCharlie’s input. However, the latter conclusion is based on knowledge about the mechanism by means of which the correlationsare established, or about the causal configuration of the parties, and does not follow solely from the correlations between them.Indeed, the tripartite joint probabilities for the outlined scenario are symmetric with respect to interchanging the roles of Aliceand Charlie, and thus they could arise from a di ff erent mechanism in a situation where Alice is causally independent of bothBob and Charlie, and Charlie is in the causal future of Bob (Fig. 1b). They could also arise from a channel from Bob to bothAlice and Charlie (Fig. 1c) which transforms Bob’s output bit into either correlated or anti-correlated random input bits forAlice and Charlie. We therefore see that, at the level of the joint probabilities for the parties’ experiments, there is no wayof distinguishing between these di ff erent mechanism of information transmission, and hence no way of giving a definitionof signaling among a proper subset of the parties that unambiguously captures the existence of such a mechanism. We can,however, give an unambiguous definition of lack of signaling between two complementary subsets of the parties (Fig. 2), aswell as an associated notion of multipartite signaling, generalizing the bipartite case. Definition II.6. (Multipartite signaling):
Consider an n -partite process W , ··· , n for a set of local experiments S = { , · · · , n } , n = , , · · · (in the case of n =
0, this is understood as the empty set, and correspondingly the process is the trivial process). Let A = { , · · · , k } and B = { k + , · · · , n } , 0 ≤ k ≤ n , be two complementary subsets of the experiments, A ∪ B = S , A ∩ B = {} (for simplicity, we take them to be the first k and the next n − k experiments, which can always be ensured by relabeling). Wesay that there is no signaling from the subset A to the complementary subset B in the process W , ··· , n if and only if p ( o k + , · · · , o n | s , · · · , s n ) ≡ p ( o k + , · · · , o n | s k + , · · · , s n ) , (6) ∀ s j ∈ S j , o j ∈ O j , j = , · · · , n . Equivalently, we say that there is signaling from (1 or · · · or k ) to ( k + · · · or n ) if and only if this condition is not satisfied. ( o , s ) ... ( o k , s k ) A B ( o k +1 , s k +1 ) ... ( o n , s n ) FIG. 2: Pictorial representation of the definition of multipartite signaling.
Remark.
There is no signaling from or to the empty subset.Note that this definition only says whether there is signaling from one or more local experiments from a given subset to oneor more local experiments from the complementary subset, but in the general case it does not identify pairs of experimentsbetween which there is signaling. In the case of two experiments, the definition reduces to the notion of bipartite signalingdefined earlier.
Definition II.7. (Non-signaling process):
A process W , ··· , n for a set of local experiments S = { , · · · , n } , n = , , · · · , iscalled non-signaling if and only if there is no signaling from A to B for any pair of complementary subsets A and B of S . Remark.
Monopartite processes and the trivial process are non-signaling.From the definition of causal process, one easily obtains the following relation between the existence of multipartite signalingamong the local experiments described by a given process and the causal configuration of these experiments.
Proposition II.1.
In an n -partite fixed-order process W , ··· , n , n ≥
1, compatible with a deterministic causal configuration κ ∗ (1 , · · · , n ), there can be signaling from (1 or · · · or k ) to ( k + · · · or n ), only if at least one of { , · · · , k } is in the absolutepast of at least one of { k + , · · · , n } according to κ ∗ (1 , · · · , n ).It turns out that we can formulate necessary and su ffi cient conditions for a process to be fixed-order causal, which areexpressed entirely in terms of the condition stated in Proposition II.1 applied to di ff erent subsets of the experiments. Toformulate the conditions precisely, we will need to introduce the concept of reduced process . Definition II.8. (Reduced process):
Consider an n -partite process W , ··· , n , n ≥
0, for a set of local experiments S = { , · · · n } .Let A = { , · · · , k } and B = { k + , · · · , n } , 0 ≤ k < n , be two complementary subsets of the experiments (specified up torelabeling), such that there is no signaling from B to A . This means that p ( o , · · · , o k | s , · · · , s n ) = p ( o , · · · , o k | s , · · · , s k ) , (7) ∀ s j ∈ S j , o j ∈ O j , j = , · · · n , i.e., we have well defined conditional probabilities p ( o , · · · , o k | s , · · · , s k ) for the experiments in A . The collection of theseprobabilities will be called reduced process for A and will be denoted by W A ≡ W , ··· , k .Note that if a multipartite process is a valid pre-selected process, any of its reduced processes is also a valid pre-selectedprocess because it is defined conditionally on the same pre-selected event. Note also that a general multipartite process need notadmit any reduced processes apart from the trivial process and itself, since it may involve signaling from every proper subset ofthe local experiments to its complementary subset.Before we state the conditions for a process to be fixed-order causal, we introduce another concept that will be needed later. Definition II.9. (Conditional process):
Consider an n -partite process W , ··· , n , n ≥
0, for a set of local experiments S = { , · · · , n } . Let A = { , · · · , k } and B = { k + , · · · , n } , 0 ≤ k < n , be two complementary subsets of the experiments (specifiedup to relabeling), such that there is no signaling from B to A (and hence we can define a reduced process W A ≡ W , ··· , k ). Foreach fixed event ( s , o , · · · s k , o k ) in A for which p ( o , · · · , o k | s , · · · , s k ) (cid:44)
0, consider the collection of conditional probabilities { p ( o k + , · · · , o n | s k + , · · · , s n , s , o , · · · , s k , o k ) } . These can be thought of as an ( n − k )-partite process for B dependent on the event( s , o , · · · , s k , o k ) in A . The collection of these processes for all values of ( s , o , · · · , s k , o k ) for which p ( o , · · · , o k | s , · · · , s k ) (cid:44) conditional process and will be denoted by W B|A ≡ W k + , ··· , n | , ··· , k . The relation between the whole process andthe reduced and conditional processes can be written in the compact form W A , B ≡ W , ··· , n = W k + , ··· , n | , ··· , k ◦ W , ··· , k ≡ W B|A ◦ W A , (8)where the product ◦ between W B|A and W A denotes multiplication of the respective probabilities of these processes, whendefined, for the same value of the event in A : p ( o , · · · , o n | s , · · · , s n ) = p ( o k + , · · · , o n | s k + , · · · , s n , s , o , · · · , s k , o k ) p ( o , · · · , o k | s , · · · , s k ) , (9)for p ( o , · · · , o k | s , · · · , s k ) (cid:44)
0, and p ( o , · · · , o n | s , · · · , s n ) = , (10)for p ( o , · · · , o k | s , · · · , s k ) = Proposition II.2.
A process W , ··· , n for a set of local experiments S = { , · · · , n } , n ≥
1, is compatible with a determinis-tic causal configuration κ ∗ (1 , · · · , n ) of these experiments (and is thereby fixed-order causal) if and only if, for the assumedcausal configuration, Proposition II.1 holds for the full process and all of its reduced processes for all bipartitions of the localexperiments into two complementary subsets. The Proof S1 is given in the Appendix.We next turn to general causal processes, beginning with the bipartite case. D. Bipartite causal processes
Consider a process W A , B describing the local experiments of two parties, Alice and Bob. If the process is causal, there existprobabilities p ( A ≺ B | s A , s B ), p ( B ≺ A | s A , s B ), p ( A (cid:14)(cid:15) B | s A , s B ), with p ( A ≺ B | s A , s B ) + p ( B ≺ A | s A , s B ) + p ( A (cid:14)(cid:15) B | s A , s B ) =
1. We cantherefore write the joint probabilities of the process in the form p ( o A , o B | s A , s B ) = p ( A ≺ B | s A , s B ) p ( o A , o B | s A , s B , A ≺ B ) + p ( B ≺ A | s A , s B ) p ( o A , o B | s A , s B , B ≺ A ) + p ( A (cid:14)(cid:15) B | s A , s B ) p ( o A , o B | s A , s B , A (cid:14)(cid:15) B ) , (11)where each of the probability distributions p ( o A , o B | s A , s B , A ≺ B ), p ( o A , o B | s A , s B , B ≺ A ), and p ( o A , o B | s A , s B , A (cid:14)(cid:15) B ), is definedassuming that p ( A ≺ B | s A , s B ) (cid:44) p ( B ≺ A | s A , s B ) (cid:44)
0, and p ( A (cid:14)(cid:15) B | s A , s B ) (cid:44)
0, respectively, otherwise that term is absent from theexpansion. The definition of causality (2) implies that p ( A ≺ B | s A , s B ) = p ( A ≺ B | s A ), p ( B ≺ A | s A , s B ) = p ( B ≺ A | s B ), p ( A (cid:14)(cid:15) B | s A , s B ) = p ( A (cid:14)(cid:15) B ). Since the sum of these probabilities must be unity, we obtain p ( A ≺ B | s A ) = p ( A ≺ B ), p ( B ≺ A | s B ) = p ( B ≺ A ), i.e., thecausal configuration of the local experiments is independent of the parties’ settings. Thus, the probabilities of a bipartite causalprocess W A , Bc have the form p ( o A , o B | s A , s B ) = p ( A ≺ B ) p ( o A , o B | s A , s B , A ≺ B ) + p ( B ≺ A ) p ( o A , o B | s A , s B , B ≺ A ) + p ( A (cid:14)(cid:15) B ) p ( o A , o B | s A , s B , A (cid:14)(cid:15) B ) , (12)where the probability distributions p ( o A , o B | s A , s B , A ≺ B ) ≡ p ( A ≺ B , o A , o B | s A , s B ) / p ( A ≺ B ), p ( o A , o B | s A , s B , B ≺ A ) ≡ p ( B ≺ A , o A , o B | s A , s B ) / p ( B ≺ A ), and p ( o A , o B | s A , s B , A (cid:14)(cid:15) B ) ≡ p ( A (cid:14)(cid:15) B , o A , o B | s A , s B ) / p ( A (cid:14)(cid:15) B ), whenever defined, describe pro-cesses, which we will denote by W A ≺ B , W B ≺ A , and W A (cid:14)(cid:15) B , respectively. (Note that we can imagine that the causal configura-tion κ ( A , B ) taking values A ≺ B , B ≺ A , or A (cid:14)(cid:15) B , is associated with an event in the past of both A and B , i.e., the processes W A ≺ B ,0 W B ≺ A , and W A (cid:14)(cid:15) B , can be thought of as proper pre-selected processes.) The assumption of causality imposes conditions onthese processes too. Specifically, it can be seen that each of them must obey a no-signaling constraint compatible with theconcrete causal configuration it is conditioned on: the first one must involve no signaling from Bob to Alice, p ( o A | s A , s B , A ≺ B ) = p ( o A | s A , A ≺ B ); the second one must involve no signaling from Alice to Bob, p ( o B | s A , s B , B ≺ A ) = p ( o B | s B , B ≺ A ); and the thirdone must involve no signaling in either direction, p ( o A | s A , s B , A (cid:14)(cid:15) B ) = p ( o A | s A , A (cid:14)(cid:15) B ), p ( o B | s A , s B , A (cid:14)(cid:15) B ) = p ( o B | s B , A (cid:14)(cid:15) B ),i.e., these are fixed-order causal processes. In a compact form, we can write W A , Bc = p ( A ≺ B ) W A ≺ B + p ( B ≺ A ) W B ≺ A + p ( A (cid:14)(cid:15) B ) W A (cid:14)(cid:15) B , (13)i.e., a bipartite causal process has the form of a probabilistic mixture of processes that are compatible with the di ff erent mutuallyexclusive causal configurations of the parties (and correspondingly involve only one-way signaling in the respective direction,or no signaling). This form is not only necessary but also su ffi cient for a process to be causal because it explicitly gives a jointprobability distribution p ( κ ( A , B ) , o A , o B | s A , s B ) = p ( κ ( A , B )) p ( o A , o B | s A , s B , κ ( A , B )) that obeys the condition for causality (2)when each conditional distribution p ( o A , o B | s A , s B , κ ( A , B )) obeys the no-signaling constraints compatible with κ ( A , B ). Indeed,we have p ( A (cid:14) B , o B | s A , s B ) = p ( B ≺ A , o B | s A , s B ) + p ( A (cid:14)(cid:15) B , o B | s A , s B ) = p ( B ≺ A ) p ( o B | s A , s B , B ≺ A ) + p ( A (cid:14)(cid:15) B ) p ( o B | s A , s B , A (cid:14)(cid:15) B ) = p ( B ≺ A ) p ( o B | s B , B ≺ A ) + p ( A (cid:14)(cid:15) B ) p ( o B | s B , A (cid:14)(cid:15) B ) = p ( A (cid:14) B , o B | s B ) , (14)and similarly p ( B (cid:14) A , o A | s A , s B ) = p ( B (cid:14) A , o A | s A ).Since the non-signaling probabilities p ( o A , o B | s A , s B , A (cid:14)(cid:15) B ) are compatible with the one-way signaling constraints for thecases A ≺ B or B ≺ A , we can also write the probabilities (12) in the non-unique form p ( o A , o B | s A , s B ) = p ( w A (cid:14) B ) p ( o A , o B | s A , s B , w A (cid:14) B ) + p ( w B (cid:14) A ) p ( o A , o B | s A , s B , w B (cid:14) A ) , (15)where w A (cid:14) B and w B (cid:14) A are two mutually exclusive variables for which the experiments of Alice and Bob respect the relations A (cid:14) B and B (cid:14) A , respectively, with the probabilities of these variables satisfying p ( w A (cid:14) B ) + p ( w B (cid:14) A ) =
1. In a compact form, thiscan be written W A , Bc = q W A (cid:14) B + (1 − q ) W B (cid:14) A , ≤ q ≤ , (16)where W Y (cid:14) X is a process that involves no signaling from Y to X , i.e., W Y (cid:14) X = W Y | X ◦ W X . (17)The constraint (16) (equivalently, (15)) provides a means of testing whether a given bipartite process theory is compatible withcausal order. For every fixed number of settings and fixed number of outcomes for each party, the joint probabilities satisfyingEq. (15) form a convex polytope, which is the convex hull of the polytope of probabilities that involve no signaling from Aliceto Bob, and the polytope of probabilities that involve no signaling from Bob to Alice [34]. The non-trivial facets of this ‘causalpolytope’ define bipartite causal inequalities, similar to the one in Ref. [4], whose violation by a given process theory indicatesthat the theory is not compatible with causal order. Note that a causal inequality does not need to be a facet of the causal polytope– it may correspond to an external plane. For instance, the causal inequality of Ref. [4], which concerns the case where oneparty has a binary input and a binary output while the other one has a quaternary input and a binary output, is not a facet ofthe respective causal polytope [21]. One way of seeing this is to note that the derivation of the inequality in Ref. [4] onlyused certain consequences of the requirement that the causal configuration of the parties must be independent of the parties’settings, but not the full requirement. The bipartite causal polytope for binary inputs and binary outputs has been characterizedby Branciard [34] (see Ref. [21]). E. The tripartite and n -partite causal processes In the case of more than two parties, causal processes need not have the simple form of probabilistic mixtures of fixed-order causal processes with probability weights that are independent of the parties’ settings. This is because, consistently withcausality, we have the possibility that the causal configuration of a subset of the local experiments may depend on the settingsof other local experiments in their past. For example, imagine that we have a tripartite experiment where the input and outputsystems of each party correspond to the internal (e.g., spin) degrees of freedom of a particle that enters the respective laboratoryat a given instant and leaves it at a given later instant. The time at which each party receives her / his particle is determined bysome predefined mechanism, which also governs any exchange of information taking place outside of the parties’ laboratories.1(Note that in order for the internal degrees of freedom of the particle to constitute the only means of information exchangebetween each local experiment and the rest of the experiment, the experiment should be so designed that no communication viathe times of input or output of the parties is possible. For example, each party may be restricted not to possess any common timereference frame with the rest of the experiment and to perform her / his operation during a fixed time interval with a stopwatch.)In such a case, if Charlie receives a particle first, the operation that he applies on the system could a ff ect the order in whichAlice and Bob receive their particles afterwards, since we can conceive of a mechanism that selects di ff erent future scenarios forthat order conditionally on the outcome of a measurement performed on the internal degrees of freedom of the particle comingout of Charlie’s laboratory. This can result in the di ff erent scenarios depicted in Fig. 3. By construction, the outlined setup iscompatible with the condition that the setting of each local experiment can be chosen independently of events in the causal pastand causal elsewhere of that experiment, as well as of the causal configuration of such events and the experiment in question, soit would be associated with a valid causal process. AB C C CAB AB
FIG. 3: In a causal setup where Charlie performs his experiment in the causal past of both Alice and Bob, the causalconfiguration of Alice and Bob may depend on the setting of Charlie.Clearly, the dependence of the causal configuration of the parties on the parties’ settings cannot be arbitrary, because it mustagree with causality. To formulate the constraints on this dependence, we will need to introduce some more terminology.For any fixed causal configuration κ (1 , · · · , n ) of the local experiments S = { , · · · , n } , there are local experiments that arein no-one else’s causal future. The full set of such local experiments, { i , j , · · · } ⊂ { , · · · , n } , will be referred to as the localexperiments that are first , or as the first consecutive set and will be denoted by [ i , j , · · · ] I . Next, if the first consecutive setdoes not include all of the local experiments, there are local experiments whose causal past contains local experiments from[ i , j , · · · ] I and only from [ i , j , · · · ] I . The full set of these will be referred to as the local experiments that are second , or as the second consecutive set , and will be denoted by [ k , l , · · · ] II . Then, if the first and second consecutive sets do not include all localexperiments, there are local experiments whose causal past contains local experiments from both sets [ i , j , · · · ] I and [ k , l , · · · ] II and only from those sets. The full set of these will be referred to as the local experiments that are third , or as the third consecutiveset , and will be denoted by [ p , q , · · · ] III , and so on.The following proposition will play a central role in our derivation of the form of multipartite causal processes.
Proposition II.3.
Consider a causal process for S = { , · · · , n } , n ≥
1, with an associated joint probability distribution p ( κ (1 , · · · , n ) , o , · · · , o n | s , · · · , s n ), where κ (1 , · · · , n ) are the causal configurations of the local experiments. The probabil-ity for the first K consecutive sets to consist of specific local experiments, [1 I , · · · , n I ] I , · · · , [1 K , · · · , n K ] K , these experimentsto have a specific causal configuration κ (1 I , · · · , n K ), the experiments in the first K − I consecutive sets to have a specific setof outcomes o I , · · · , o n K − I , and a given (possibly empty) subset { K , · · · , g K } ⊂ { K , · · · , n K } of the local experiments in the K th set (given up to relabeling) to have specific outcomes o K , · · · , o g K , can depend non-trivially only on the settings of the localexperiments indicated in the first K − I consecutive sets and the subset { K , · · · , g K } , p ( κ (1 I , · · · , n K ) , [1 I , · · · , n I ] I , · · · , [1 K , · · · , n K ] K , o I , · · · , o g K | s , · · · , s n ) = p ( κ (1 I , · · · , n K ) , [1 I , · · · , n I ] I , · · · , [1 K , · · · , n K ] K , o I , · · · , o g K | s I , · · · , s g K ) , (18)where we define the 0 th set as the empty set. The Proof S2 is given in the Appendix.An important consequence of Proposition II.3 is that the probability for a given set of local experiments to be first is indepen-dent of the settings of all parties (this is the case of K = { K , · · · , g K } being empty). For example, consider thedi ff erent causal configurations of three parties – Alice ( A ), Bob ( B ), and Charlie ( C ) – which are compatible with [ C ] I (Fig. 3).Each of the individual configurations has a probability that may depend on the setting of Charlie, but the overall probability forCharlie to be first, i.e., for any one of these configurations to be realized (which is the sum of the probabilities for the individualconfigurations), is independent of the settings of all parties, including Charlie. This independence of the first consecutive set onthe settings of all parties will play a key role in our characterization of the structure of multipartite causal processes. We will firstdevelop the characterization for the case of three parties in order to illustrate the underlying principle, and then we will extend itto the general multipartite case.2 Groups of tripartite causal configurations whose probabilities are independent of the parties’ settings,defined by the set of parties that are first[ A ] I : [ A ≺ B , A ≺ C , B ≺ C ] or [ A ≺ B , A ≺ C , C ≺ B ] or [ A ≺ B , A ≺ C , B (cid:14)(cid:15) C ][ B ] I : [ B ≺ A , B ≺ C , A ≺ C ] or [ B ≺ A , B ≺ C , C ≺ A ] or [ B ≺ A , B ≺ C , A (cid:14)(cid:15) C ][ C ] I : [ C ≺ A , C ≺ B , A ≺ B ] or [ C ≺ A , C ≺ B , B ≺ A ] or [ C ≺ A , C ≺ B , A (cid:14)(cid:15) B ][ A , B ] I : [ A (cid:14)(cid:15) B , A ≺ C , B (cid:14)(cid:15) C ] or [ A (cid:14)(cid:15) B , A (cid:14)(cid:15) C , B ≺ C ] or [ A (cid:14)(cid:15) B , A ≺ C , B ≺ C ][ A , C ] I : [ A (cid:14)(cid:15) C , A ≺ B , B (cid:14)(cid:15) C ] or [ A (cid:14)(cid:15) C , A (cid:14)(cid:15) B , C ≺ B ] or [ A (cid:14)(cid:15) C , A ≺ B , C ≺ B ][ B , C ] I : [ B (cid:14)(cid:15) C , B ≺ A , C (cid:14)(cid:15) A ] or [ B (cid:14)(cid:15) C , B (cid:14)(cid:15) A , C ≺ A ] or [ B (cid:14)(cid:15) C , B ≺ A , C ≺ A ][ A , B , C ] I : [ A (cid:14)(cid:15) B , B (cid:14)(cid:15) C , A (cid:14)(cid:15) C ] TABLE I: The mutually exclusive groups of tripartite causal configurationsThe groups of tripartite causal configurations compatible with the di ff erent possibilities for the first consecutive set of partiesare listed in Table I. In terms of these possibilities, the probabilities of a tripartite causal process can be written p ( o A , o B , o C | s A , s B , s C ) = p ([ A ] I ) p ( o A , o B , o C | s A , s B , s C , [ A ] I ) + p ([ B ] I ) p ( o A , o B , o C | s A , s B , s C , [ B ] I ) + p ([ C ] I ) p ( o A , o B , o C | s A , s B , s C , [ C ] I ) + p ([ A , B ] I ) p ( o A , o B , o C | s A , s B , s C , [ A , B ] I ) + p ([ A , C ] I ) p ( o A , o B , o C | s A , s B , s C , [ A , C ] I ) + p ([ B , C ] I ) p ( o A , o B , o C | s A , s B , s C , [ B , C ] I ) + p ([ A , B , C ] I ) p ( o A , o B , o C | s A , s B , s C , [ A , B , C ] I ) , (19)where p ([ A ] I ) + p ([ B ] I ) + p ([ C ] I ) + p ([ A , B ] I ) + p ([ A , C ] I ) + p ([ B , C ] I ) + p ([ A , B , C ] I ) = , (20)and the probability distributions p ( o A , ... | s A , ..., [ · · · ] I ) for a given [ · · · ] I , defined whenever p ([ · · · ] I ) (cid:44)
0, describe processeswhich we will denote by W [ ··· ] I . (Note that we can imagine that the variable [ · · · ] I is associated with an event in the past of alllocal experiments, i.e., these can be thought of as a proper pre-selected process.)In a compact form, Eq. (19) can be written W A , B , Cc = p ([ A ] I ) W [ A ] I + p ([ B ] I ) W [ B ] I + p ([ C ] I ) W [ C ] I + p ([ A , B ] I ) W [ A , B ] I + p ([ A , C ] I ) W [ A , C ] I + p ([ B , C ] I ) W [ B , C ] I + p ([ A , B , C ] I ) W [ A , B , C ] I , (21)i.e., the overall process is a mixture of processes defined conditionally on the di ff erent scenarios [ · · · ] I . The processes W [ ··· ] I cannot be arbitrary but must be compatible with causality, the conditions for which we derive next.Consider the case in which one party is first, say [ C ] I (Fig. 3). There are three distinct causal configurations compatible withthis case, in which A ≺ B , B ≺ A , or A (cid:14)(cid:15) B (Table I). We can expand p ( o A , o B , o C | s A , s B , s C , [ C ] I ) conditionally on these configura-tions as follows: p ( o A , o B , o C | s A , s B , s C , [ C ] I ) = p ( o C | s A , s B , s C , [ C ] I ) × [ p ( A ≺ B | s A , s B , s C , o C , [ C ] I ) p ( o A , o B | s A , s B , s C , o C , A ≺ B , [ C ] I ) + p ( B ≺ A | s A , s B , s C , o C , [ C ] I ) p ( o A , o B | s A , s B , s C , o C , B ≺ A , [ C ] I ) + p ( A (cid:14)(cid:15) B | s A , s B , s C , o C , [ C ] I ) p ( o A , o B | s A , s B , s C , o C , A (cid:14)(cid:15) B , [ C ] I )] , (22)where p ( o A , o B | s A , s B , s C , o C , κ ( A , B ) , [ C ] I ) is defined when p ( κ ( A , B ) | s A , s B , s C , o C , [ C ] I ) (cid:44)
0, and p ( A ≺ B | s A , s B , s C , o C , [ C ] I ) + p ( B ≺ A | s A , s B , s C , o C , [ C ] I ) + p ( A (cid:14)(cid:15) B | s A , s B , s C , o C , [ C ] I ) = . (23)3From Proposition II.3, we have that p ( o C | s A , s B , s C , [ C ] I ) ≡ p ([ C ] I , o C | s A , s B , s C ) / p ([ C ] I ) = p ([ C ] I , o C | s C ) / p ([ C ] I ) = p ( o C | s C , [ C ] I ) . Similarly, we have p ( A ≺ B | s A , s B , s C , o C , [ C ] I ) = p ( A ≺ B | s A , s C , o C , [ C ] I ) , p ( B ≺ A | s A , s B , s C , o C , [ C ] I ) = p ( B ≺ A | s B , s C , o C , [ C ] I ) , p ( A (cid:14)(cid:15) B | s A , s B , s C , o C , [ C ] I ) = p ( A (cid:14)(cid:15) B | s C , o C , [ C ] I ) , (24)which together with Eq. (23) implies p ( A ≺ B | s A , s B , s C , o C , [ C ] I ) = p ( A ≺ B | s C , o C , [ C ] I ) , p ( B ≺ A | s A , s B , s C , o C , [ C ] I ) = p ( B ≺ A | s C , o C , [ C ] I ) , p ( A (cid:14)(cid:15) B | s A , s B , s C , o C , [ C ] I ) = p ( A (cid:14)(cid:15) B | s C , o C , [ C ] I ) . (25)Substituting this in Eq. (22), we obtain p ( o A , o B , o C | s A , s B , s C , [ C ] I ) = p ( o C | s C , [ C ] I ) × [ p ( A ≺ B | s C , o C , [ C ] I ) p ( o A , o B | s A , s B , s C , o C , A ≺ B , [ C ] I ) + p ( B ≺ A | s C , o C , [ C ] I ) p ( o A , o B | s A , s B , s C , o C , B ≺ A , [ C ] I ) + p ( A (cid:14)(cid:15) B | s C , o C , [ C ] I ) p ( o A , o B | s A , s B , s C , o C , A (cid:14)(cid:15) B , [ C ] I )] , (26)with p ( A ≺ B | s C , o C , [ C ] I ) + p ( B ≺ A | s C , o C , [ C ] I ) + p ( A (cid:14)(cid:15) B | s C , o C , [ C ] I ) = , (27)where the probability distributions p ( o A , o B | s A , s B , s C , o C , A ≺ B , [ C ] I ), p ( o A , o B | s A , s B , s C , o C , B ≺ A , [ C ] I ), and p ( o A , o B | s A , s B , s C , o C , A (cid:14)(cid:15) B , [ C ] I ) describe bipartite processes for Alice and Bob for every fixed value of ( s C , o C ). Theassumption of causality implies conditions for these processes too. They must respect the no-signaling constraints imposed bythe causal configuration κ ( A , B ) they are conditioned on – the first one must involve no signaling from Bob to Alice, the secondone must involve no signaling from Alice to Bob, and the third one must involve no signaling between Alice and Bob in eitherdirection. This follows from the fact that p ( o A , o B | s A , s B , s C , o C , κ ( A , B ) , [ C ] I ) = p ([ C ] I , κ ( A , B ) , o A , o B , o C | s A , s B , s C ) p ([ C ] I ) p ( o C | s C , [ C ] I ) p ( κ ( A , B ) | s C , o C , [ C ] I ) , (28)and the observation that since only the numerator on the right-hand side depends on s A , o A , s B , and o B , the respective no-signalingconstraints on the quantity on the left-hand side follow from the requirement that the numerator is compatible with Eq. (2).Notice that the probabilities p ( o C | s C , [ C ] I ) in Eq. (26) define a reduced monopartite process for Charlie, W C , while theprobabilities enclosed by the square brackets define a conditional bipartite process W A , B | Cc , which is causal (indicated by thesubscript c ) for every fixed ( s C , o C ). In a compact form, this can be written W [ C ] I = W A , B | Cc ◦ W C . (29)The form (29) is necessary for a causal process for which all causal configurations that have non-zero probabilities respect [ C ] I (in that case, a causal process of the general form (21) reduces to the term W [ C ] I ). It is also su ffi cient, because this form providesan explicit joint probability distribution p [ C ] I ( κ ( A , B , C ) , o A , o B , o C | s A , s B , s C ) – equal to p ([ C ] I , κ ( A , B ) , o A , o B , o C | s A , s B , s C ) = p ( o C | s C , [ C ] I ) p ( κ ( A , B ) | s C , o C , [ C ] I ) p ( o A , o B | s A , s B , s C , o C , κ ( A , B ) , [ C ] I ) when κ ( A , B , C ) is compatible with [ C ] I , and to zerootherwise – for which condition (2) is satisfied with respect to every party. Indeed, condition (2) is satisfied with respectto C since the probability for any party being in the causal past or causal elsewhere of C is zero. It is also satisfied withrespect to A (similarly for B ) since the no-signaling constraints respected by p ( o A , o B | s A , s B , s C , o C , κ ( A , B ) , [ C ] I ) guarantee that p [ C ] I ( κ ( A , B , C ) , A (cid:14) B , A (cid:14) C , o B , o C | s A , s B , s C ) = p [ C ] I ( κ ( A , B , C ) , A (cid:14) B , A (cid:14) C , o B , o C | s B , s C ). The necessary and su ffi cient conditionsfor a causal process compatible with [ A ] I and [ B ] I are analogous.Let us now consider the case where two parties are first, say [ B , C ] I . The possible causal configurations in this case (Table I)are depicted in Fig. 4. Similarly to the previous case, using the assumption of causality, we can expand the probabilities p ( o A , ... | s A , ..., [ B , C ] I ) conditionally on the di ff erent configurations as follows: p ( o A , o B , o C | s A , s B , s C , [ B , C ] I ) = p ( o B , o C | s B , s C , [ B , C ] I ) × [ p ( B ≺ A , C (cid:14)(cid:15) A | s B , o B , s C , o C , [ B , C ] I ) p ( o A | s A , s B , o B , s C , o C , B ≺ A , C (cid:14)(cid:15) A , [ B , C ] I ) + p ( B (cid:14)(cid:15) A , C ≺ A | s B , o B , s C , o C , [ B , C ] I ) p ( o A | s A , s B , o B , s C , o C , B (cid:14)(cid:15) A , C ≺ A , [ B , C ] I ) + p ( B ≺ A , C ≺ A | s B , o B , s C , o C , [ B , C ] I ) p ( o A | s A , s B , o B , s C , o C , B ≺ A , C ≺ A , [ B , C ] I )] , (30)4with p ( B ≺ A , C (cid:14)(cid:15) A | s B , o B , s C , o C , [ B , C ] I ) + ( B (cid:14)(cid:15) A , C ≺ A | s B , o B , s C , o C , [ B , C ] I ) + p ( B ≺ A , C ≺ A | s B , o B , s C , o C , [ B , C ] I ) = , (31)where the probabilities p ( o B , o C | s B , s C , [ B , C ] I ) in Eq. (30) define a reduced bipartite process that involves no signaling between B and C , and the probabilities in the square brackets describe a conditional process for A . The fact that there is no signalingbetween B and C in the first process follows easily from Proposition II.3.It turns out that the decomposition over di ff erent causal configurations does not yield any nontrivial conditions on the proba-bilities of the conditional process enclosed in the square brackets, i.e., the simpler form p ( o A , o B , o C | s A , s B , s C , [ B , C ] I ) = p ( o B , o C | s B , s C , [ B , C ] I ) p ( o A | s A , s B , o B , s C , o C , [ B , C ] I ) (32)is both necessary and su ffi cient for a valid W [ B , C ] I . Necessity is obvious since Eq. (30) implies Eq. (32). Su ffi ciency followsfrom the fact that the right-hand side of Eq. (32) is compatible with the particular case p ( B ≺ A , C ≺ A | s B , o B , s C , o C , [ B , C ] I ) = p ( o A , o B , o C | s A , s B , s C , [ B , C ] I ) imposed by κ ( A , B , C ) are that there isno signaling from Alice to Bob and Charlie, and no signaling between Bob and Charlie in their reduced bipartite process. Theseare clearly guaranteed by Eq. (32) when the reduced process { p ( o B , o C | s B , s C , [ B , C ] I ) } involves no signaling between Bob andCharlie. Therefore, similarly to Eq. (29), we can write Eq. (32) in the compact form W [ B , C ] I = W A | B , C ◦ W B , Cns , (33)where W B , Cns is a non-signaling bipartite process for Bob and Charlie, and W A | BC is a monopartite process for Alice conditionalon the events in the laboratories of Bob and Charlie. C C CA A ABBB CABAA BB CC CAB C ABA B C AB C AB C
FIG. 4: The three possible tripartite causal configurations included in the group where B and C are first. From left to right:[ B (cid:14)(cid:15) C and B ≺ A and C (cid:14)(cid:15) A ], [ B (cid:14)(cid:15) C and B ≺ A and C ≺ A ], [ B (cid:14)(cid:15) C and B (cid:14)(cid:15) A and C ≺ A ].Finally, in the case where all of the parties are first, we only have the constraint that W [ A , B , C ] I = W A , B , Cns (34)is a tripartite non-signaling process. Again, this follows from Proposition II.3.Therefore, we have obtained that a tripartite causal process W A , B , Cc must have the form W A , B , Cc = p ([ A ] I ) W B , C | Ac ◦ W A + p ([ B ] I ) W A , C | Bc ◦ W B + p ([ C ] I ) W A , B | Cc ◦ W C + p ([ A , B ] I ) W C | A , B ◦ W A , Bns + p ([ A , C ] I ) W B | A , C ◦ W A , Cns + p ([ B , C ] I ) W A | B , C ◦ W B , Cns + p ([ A , B , C ] I ) W A , B , Cns , (35)with suitable probability weights p ([ A ] I ), p ([ B ] I ), p ([ C ] I ), p ([ A , B ] I ), p ([ A , C ] I ), p ([ B , C ] I ), and p ([ A , B , C ] I ). Thisform is also su ffi cient for a tripartite process to be causal because it explicitly gives a probability distribution p ( κ ( A , B , C ) , o A , o B , o C | s A , s B , s C ) = (cid:80) [ ··· ] I p ([ · · · ] I ) p ( κ ( A , B , C ) , o A , o B , o C | s A , s B , s C , [ · · · ] I ) that satisfies Eq. (2). Indeed, wehave seen that each of the distributions p ( κ ( A , B , C ) , o A , o B , o C | s A , s B , s C , [ · · · ] I ) in this convex mixture is an extension of a causalprocess { p ( o A , o B , o C | s A , s B , s C , [ · · · ] I ) } , and hence it satisfies Eq. (2). Since the weights p ([ · · · ] I ) in the mixture are independentof s A , s B , and s C , and Eq. (2) is linear in p ( κ ( A , B , C ) , o A , o B , o C | s A , s B , s C , [ · · · ] I ), the equation is satisfied by the whole mixturetoo.Condition (35) can be further simplified by noticing that the processes corresponding to the cases in which two or three partiesare first have forms compatible with cases in which only a single party is first. For instance, W [ B , C ] I satisfies the necessary andsu ffi cient conditions for a valid W [ B ] I or a valid W [ C ] I , while W [ A , B , C ] I satisfies the necessary and su ffi cient conditions for anyof W [ A ] I , W [ B ] I , or W [ C ] I . The compatibility of W [ B , C ] I with [ C ] I , for example, can be seen from the fact that Eq. (32) (or5Eq. (33)) is compatible with the case [ C ] I in which C ≺ B ≺ A , since the only constraints in that case are that Alice cannot signal toBob and Charlie, and that Bob cannot signal to Charlie, which are satisfied by the probabilities in Eq. (32). Similarly, W [ B , C ] I is compatible with [ C ] I . A process W [ A , B , C ] I is compatible with any causal configuration since it does not involve signalingbetween any of the parties. These observations suggest that we can group (in a generally non-unique way) the terms in theprobabilistic mixture (21) so as to obtain a mixture of three processes W A , B , Cc = p ( w ( B , C ) (cid:14) A ) W ( B , C ) (cid:14) A + p ( w ( A , C ) (cid:14) B ) W ( A , C ) (cid:14) B + p ( w ( A , B ) (cid:14) C ) W ( A , B ) (cid:14) C , (36)where w ( B , C ) (cid:14) A , w ( A , C ) (cid:14) B , and w ( A , B ) (cid:14) C , are some mutually exclusive variables whose probabilities satisfy p ( w ( B , C ) (cid:14) A ) + p ( w ( A , C ) (cid:14) B ) + p ( w ( A , B ) (cid:14) C ) =
1, such that conditionally on these variables, the causal configuration of the parties belongs toone of the groups compatible with ( B , C ) (cid:14) A (meaning B (cid:14) A ∧ C (cid:14) A ), ( A , C ) (cid:14) B , and ( A , B ) (cid:14) C , respectively, while the processes W ( B , C ) (cid:14) A , W ( A , C ) (cid:14) B , and W ( A , B ) (cid:14) C , satisfy the most general causal constraints compatible with these groups. For instance, con-ditionally on w ( B , C ) (cid:14) A , the causal configurations of the parties may belong to any of the groups defined by [ A ] I , [ A , B ] I , [ A , C ] I ,and [ A , B , C ] I . The process W ( B , C ) (cid:14) A would itself be a probabilistic mixture of processes compatible with these groups, whichmost generally satisfy the constraints satisfied by W [ A ] I . That is, W ( B , C ) (cid:14) A = W B , C | Ac ◦ W A , (37) W ( A , C ) (cid:14) B = W A , C | Bc ◦ W B , (38) W ( A , B ) (cid:14) C = W A , B | Cc ◦ W C . (39)Obviously, the existence of a convex decomposition (36) is both necessary and su ffi cient for a tripartite process to be causal,since any process of the form (35) can be written in the form (36), while Eq. (36) is a special case of Eq. (35).As in the bipartite case, for any fixed number of settings and fixed number of outcomes for each party, the constraint (36) pro-vides a means of testing whether the corresponding tripartite probabilities are compatible with causality. The set of probabilitiesthat satisfy Eq. (36) is the convex hull of the probabilities compatible with causal configurations in which ( B , C ) (cid:14) A , ( A , C ) (cid:14) B ,and ( A , B ) (cid:14) C . One can see that the latter form polytopes, since the constraints imposed by causality in each of these cases arelinear. For example, in the case of ( A , B ) (cid:14) C , we have the constraint that p ( o A , o B , o C | s A , s B , s C , w ( A , B ) (cid:14) C ) involve no signalingfrom Alice and Bob to Charlie, and that for every ( s C , o C ), the resultant process between Alice and Bob is causal. The firstrequirement corresponds to a set of linear constraints. The second requirement corresponds to the condition that for every fixed( s C , o C ), the probabilities p ( o A , o B | s A , s B , s C , o C , w ( A , B ) (cid:14) C ) ≡ p ( o A , o B , o C | s A , s B , s C , w ( A , B ) (cid:14) C ) / p ( o C | s C , w ( A , B ) (cid:14) C ) are the proba-bilities describing a causal process for Alice and Bob, which themselves belong to a polytope and hence respect a set of linearinequalities. Plugging these probabilities in the respective inequalities and multiplying both sides by p ( o C | s C , w ( A , B ) (cid:14) C ) wouldyield a set of linear inequalities for p ( o A , o B , o C | s A , s B , s C , w ( A , B ) (cid:14) C ). Therefore, these probabilities also form a polytope, and sodo the probabilities of the form (36). The nontrivial facets of the polytope of probabilities (36) would define tripartite causalinequalities, whose violation indicates incompatibility with causal order. Examples of tripartite causal inequalities for binary in-puts and outputs can be found in Refs. [7, 9] (we have not investigated whether these are facets of the respective causal polytope).The extension of the conditions for causality of a process to the case of n parties can be defined iteratively. The followingtheorem provides the generalization of condition (35): Theorem II.1.
A process for a set of parties S = { , · · · , n } , n ≥
1, is causal if and only if it can be written in the form W S c = (cid:88) X⊂S , X (cid:44) {} p X W S\X|X c ◦ W X ns , (40)where the sum is over all nonempty subsets X of the local experiments S , p X are suitable probability weights (which canbe interpreted as the probability for X to be first, p X = p ([ X ] I )), S\X denotes the relative complement of X in S , W X ns is anon-signaling reduced process for X , and the conditional process W S\X|X c is either the trivial process (when X = S ) or other-wise can be written in the same form (40) for every given value of the possible events in X . The Proof S3 is given in the Appendix.As in the bipartite and tripartite cases, we can simplify the conditions for an n -partite process to be causal by noticing that theconstraints on a process compatible with a given set of k (1 ≤ k ≤ n ) parties being first are compatible with the constraints on aprocess compatible with the case in which only a single one of the k parties is first. Therefore, by an argument analogous to theone in the tripartite case, we obtain the following alternative formulation of the conditions.6 Theorem II.2. (Canonical causal decomposition):
A causal process for n parties is one that can be written in the (generallynon-unique) form W , ··· , nc = n (cid:88) i = q i W (1 , ··· , i − , i + , ··· , n ) (cid:14) i , q i ≥ , ∀ i , n (cid:88) i = q i = , (41)with W (1 , ··· , i − , i + , ··· , n ) (cid:14) i = W , ··· , i − , i + , ··· , n | ic ◦ W i , (42)where the ( n − W , ··· , i − , i + , ··· , n | ic is either trivial (when n =
1) or has the form (41) for every valueof the event in i .The weights q i in Eq. (42) can be thought of as the probabilities q i ≡ p ( w (1 , ··· , i − , i + , ··· , n ) (cid:14) i ) for a mutually exclusive set of vari-ables w (1 , ··· , i − , i + , ··· , n ) (cid:14) i for which the causal configurations of the parties belong to a group such that (1 , · · · , i − , i + , · · · , n ) (cid:14) i .Theorem II.2 (alternatively Theorem II.1) gives iteratively formulated necessary and su ffi cient conditions for a process to becausal in the general multipartite case. It can be understood as describing an ‘unraveling’ of the di ff erent possible sequences ofoperations in steps: first, the party that is first and his / her monopartite process are selected at random based on some probabilitydistribution; next, the party that is second and his / her monopartite process are selected at random from some probability distribu-tion that most generally can depend on the first party’s setting and outcome; next, the party that is third and his / her monopartiteprocess are selected from some probability distribution that most generally can depend on the settings and outcomes of the firsttwo parties, and so on. We refer to this intuitive decomposition as the canonical causal decomposition of a causal process.By an argument analogous to the one in the tripartite case, one easily sees from Theorem II.2 that for any fixed number ofsettings and outcomes for each party, the causal probabilities for n parties form a polytope, provided that the causal probabilitiesfor ( n −
1) parties form a polytope. By induction, this implies a polytope structure for the general multipartite case. The nontrivialfacets of such a polytope define causal inequalites. Examples of n -partite causal inequalities, where n = k +
1, for binary inputsand outputs can been found in Refs. [7, 9]. It would be interesting to check if these inequalities are facets of the respective causalpolytope.
III. THE QUANTUM PROCESS FRAMEWORKA. General quantum processes
The quantum process framework introduced in Ref. [4] is a particular theory within the general operational framework forpre-selected processes discussed in the previous section. It is based on a set of assumptions about the local operations of theparties and the joint probabilities for their outcomes, which we review next.The first main assumption is that of local quantum mechanics [4], which says that each local experiment is described as instandard quantum mechanics. Specifically, let X and X denote the input and output systems of a local experiment X . It isassumed that these systems are associated with Hilbert spaces H X and H X of dimensions dim H X = d X and dim H X = d X ,respectively. The set of operations that can be performed between the input and output systems is the set of standard quantumoperations (or quantum instruments [35]). A quantum operation has a set of outcomes labeled by j = , . . . , n . Each outcomeinduces a specific transformation from the input to the output, which corresponds to a completely positive (CP) map M Xj : L ( H X ) → L ( H X ), where L ( H ) is the space of linear operators over the (finite-dimensional) Hilbert space H . The action ofeach M Xj on an operator σ ∈ L ( H X ) can be written in the Kraus form [36] M Xj ( σ ) = (cid:80) mk = E jk σ E † jk , m = d X d X , where theKraus operators E jk : H X → H X satisfy (cid:80) mk = E † jk E jk ≤ X , ∀ j . The set of CP maps (cid:110) M Xj (cid:111) nj = corresponding to all possibleoutcomes of a quantum operation has the property that (cid:80) nj = M Xj is CP and trace-preserving (CPTP), which is equivalent to thecondition (cid:80) nj = (cid:80) mk = E † jk E jk = X .The second main assumption is that the joint probabilities for the outcomes of the operations of a set of parties, Alice, Bob,Charlie, · · · , is a non-contextual function of the local CP maps, p ( i , j , k , · · · |{M Ai } , {M Bj } , {M Ck } · · · ) = ω ( M Ai , M Bj , M Ck , · · · ) . (43)The requirement that local procedures agree with standard quantum mechanics implies that the function ω should be linear inthe local CP maps [4].7Such a linear function can be written in a convenient form by expressing each local CP map as a positive semidefiniteoperator using a version of the Choi-Jamiołkowsky (CJ) isomorphism [29, 30]. In this isomorphism, the CJ operator M A A i ∈L ( H A ⊗ H A ) corresponding to a linear map M Ai : L ( H A ) → L ( H A ) is defined as M A A i : = (cid:2) I ⊗ M i ( | φ + (cid:105)(cid:104) φ + | ) (cid:3) T , where | φ + (cid:105) = (cid:80) d A j = | j j (cid:105) ∈ H A ⊗ H A is a (not normalized) maximally entangled state on two copies of H A , the set of states {| j (cid:105)} d A j = is an orthonormal basis of H A , I is the identity map, and T denotes matrix transposition in the basis {| j (cid:105)} d A j = of H A and somespecific basis of H A . The CJ operator defined in this way does not depend on the choice of basis of H A , but does dependon the choice of basis of H A [31]. For the purposes of the present paper, the latter basis can be an arbitrary fixed basis. Wenote, however, that within the time-symmetric generalization of the framework developed in Ref. [16], this basis has a nontrivialphysical significance related to the transformation of time reversal. Specifically, in that formulation, the Hilbert space H A onwhich the CJ operator is defined is not interpreted as the original output Hilbert space of the CP map, but a time-reversed copyof it. In this paper, we will not be concerned with that formulation, but will simply regard the CJ representation of CP maps,defined for an arbitrary choice of basis, as a mathematical convenience. Using the CJ representation, the joint probabilities (43)can be written in the form p ( i , j , k , · · · |{M Ai } , {M Bj } , {M Ck } , · · · ) = Tr (cid:104) W A A B B C C ··· (cid:16) M A A i ⊗ M B B j ⊗ M C C k ⊗ · · · (cid:17)(cid:105) , (44)where W A A B B C C ··· ∈ L ( H A ⊗ H A ⊗ H B ⊗ H B ⊗ H C ⊗ H C ⊗ · · · ).The last main assumption behind the quantum process framework is that the local operations of the parties can be extendedto act on input ancillas A (cid:48) , B (cid:48) , C (cid:48) , · · · , that are allowed to be prepared in an arbitrary quantum state ρ A (cid:48) B (cid:48) C (cid:48) ··· , ρ A (cid:48) B (cid:48) C (cid:48) ··· ≥ ρ A (cid:48) B (cid:48) C (cid:48) ··· =
1. Upon such an extension, the original operator W A A B B C C ··· becomes W A A B B C C ··· ⊗ ρ A (cid:48) B (cid:48) C (cid:48) ··· [4]. Therequirement that the probabilities are non-negative for any combination of local CP maps M Ai , M Bj , M Ck , · · · , on the extendedsystems A = A A (cid:48) A , B = B B (cid:48) B , C = C C (cid:48) C , · · · , implies that [4] W A A B B C C ··· ≥ . (45)In addition, since the probabilities should sum up to 1 for a complete set of local outcomes, we have the condition thatTr (cid:104) W A A B B C C ··· (cid:16) M A A ⊗ M B B ⊗ M C C ⊗ · · · (cid:17)(cid:105) = , (46) ∀ M A A , M B B , M C C , · · · ≥ , Tr A M A A = A , Tr B M B B = B , Tr C M C C = C , · · · , where Tr X denotes partial trace over X . Here, we have used the fact that a linear map M X is CPTP if and only if its CJ operatorsatisfies M X X ≥ X M X X = X . An operator W A A B B C C ··· that satisfies conditions (45) and (46) is called a processmatrix [4]. Knowing the process matrix, by Eq. (44) we have the probabilities for the outcomes of any combination of localoperations of the parties, i.e., the process matrix provides a complete description of a process. (Here, the set S X of possiblesettings of a given party is the set of quantum operations with the respective input and output systems.)The process matrix can be expanded in a Hilbert-Schmidt basis of orthogonal matrices on the Hilbert spaces of the inputand output systems of the parties, which is helpful in analyzing di ff erent properties of the correlations that the process allows.A Hilbert-Schmidt basis of L ( H X ) is given by a set of Hermitian operators { σ X µ } d X − µ = , with σ X = X , Tr σ X µ σ X ν = d X δ µν , andTr σ Xj = j = , ..., d X −
1. In such a basis, a process matrix can be written W A A B B C C ··· = (cid:88) i , j , k , l , m , n ··· w i jklmn ··· σ A i ⊗ σ A j ⊗ σ B k ⊗ σ B l ⊗ σ C m ⊗ σ C n ⊗ · · · , (47) w i jklmn ··· ∈ R , ∀ i , j , k , l , m , n , · · · . It turns out that many properties of process matrices can be formulated entirely as statements about the nonzero terms in theabove expansion [4]. For this purpose, it is convenient to introduce the following terminology. Non-zero terms proportional to σ A i ⊗ rest ( i ≥
1) will be called terms of type A , non-zero terms proportional to σ A i ⊗ σ B j ⊗ rest ( i , j ≥
1) will be called termsof type A B , etc. Every process matrix also contains a non-zero term proportional to the identity operator on all systems. Thisterm will be referred to as of type
1, or as the identity term .In Ref. [4], it was shown that, in the bipartite case, an operator W A A B B satisfies condition (46) if and only if it contains atmost terms from the following types: A , B , A B , A B , A A B , A B B . This rule also includes the monopartite case,which is obtained when the input and output systems of one of the parties is trivial (the one-dimensional Hilbert space C ).Specifically, a monopartite operator W A A satisfies condition (46) if and only if it contains at most terms of type A . Thetypes of allowed terms can be generalized to the n -partite case as follows.8TABLE II: The types of terms allowed in a tripartite process matrix W A A B B C C C B C B B C B C B C C B B C A C A B C A B A B C A B C A B C C A B B C A A C A C A C C A B A B C A B C A B C C A B A B C A B C A B C C A B B A B B C A B B C A B B C C A A C A A B C A A B A A B C A A B C A A B C C A A B B C W A A B B C C ( A , B ) (cid:14) C compatible with ( A , B ) (cid:14) C . C B B C B C B C C A B A B C A B C C A A C A C A C C A B A B C A B C C A B A B C A B C A B C C A B B A B B C A B B C C A A B A A B C A A B C C A B C A B C A A B C A B B C Proposition III.1.
An operator of the form (47) satisfies condition (46) if and only if in addition to the identity term it containsat most terms in which there is a nontrivial σ operator on X and a trivial one (the identity operator) on X for some party X ∈ { A , B , C , · · · } .In the Appendix, we present Proof S4 of the above proposition for the case of three parties and the general case followsaccordingly. From the analysis in Proof S4 we see that a general operator W A A B B C C can contain up to 64 types ofterms. The condition for normalization of probabilities (46) narrows the types of terms to the 38 types listed in Table II. Thepositive semidefiniteness condition (45) does not limit any further the allowed types of terms, because one can conceive of apositive semidefinite matrix containing nonzero terms of any chosen type (this can be ensured by taking the nontrivial σ termswith non-zero coe ffi cients of su ffi ciently small magnitude relative to the weight of the identity term which is always fixed).Thus, an operator W A A B B C C is a valid tripartite process matrix, i.e., it satisfies conditions (45) and (46), if and only if itsatisfies condition (45) and contains only terms of the types listed in Table II, where the identity term comes with the weight w = d A d B d C . In a similar way, one proves the allowed types of terms in the general n -partite case. (For an alternativeformulation of the conditions for an operator to be a valid process matrix, see Ref. [19].)The types of terms that appear in the expansion of a process matrix are closely related to the signaling between the partiesthat the process allows. For example, a bipartite process involves signaling from Bob to Alice if and only if the process matrixcontains terms of type A B or A B B [4]. To state the condition for (no) signaling in the multipartite case, it is convenient tointroduce the following terminology (see also Ref. [19]). Consider a Hilbert-Schmidt term σ A i ⊗ σ A j ⊗ σ B k ⊗ σ B l ⊗ σ C m ⊗ σ C n ⊗· · · as in Eq. (47). The restriction of this term onto, say, subsystems B C C · · · is defined as σ B l ⊗ σ C m ⊗ σ C n ⊗ · · · . Proposition III.2. An n -partite process matrix for a set of parties { , · · · , n } does not permit signaling from, say, (1 and 2 and · · · and k ) to ( k + k + · · · and n ) if an only if it contains only terms whose restriction onto 1 · · · k k are of theallowed types for a process matrix on { , · · · , k } as described in Proposition III.1. The Proof S5 is given in the Appendix.As an example, a tripartite quantum process that is causal and compatible with a situation in which Charlie is first (Fig. 3)should involve no signaling from Alice and Bob to Charlie, and hence it can only contain the types of terms listed in Table III.These constraints on the allowed types of terms imposed by causal order will turn out to play an important role in the character-ization of the so-called causally separable quantum processes, which we define in the next subsection.9 B. Causally separable quantum processes
Given that quantum processes have a simple description in terms of process matrices, it is natural to ask whether the propertyof causality can also be expressed in terms of simple conditions on these matrices. Consider a bipartite quantum process forAlice and Bob, and assume that it is a fixed-order process compatible with the causal configuration A ≺ B . In that case, as arguedearlier, the only constraint imposed by causal order is that the process should involve no signaling from Bob to Alice. As pointedout in the previous subsection, there can be signaling from Bob to Alice if and only if the process matrix W A A B B containsterms of type A B or A B B . Therefore, a process matrix is compatible with A ≺ B if and only if none of these types of termsappear in its expansion. This means that such a process matrix has the form W A ≺ B = W A A B ⊗ B , (48)where W A A B ≥ W A A B = d A ) contains at most terms of type A , B , A B , A B , A A B . (This is equivalent tosaying that W A A B is a valid process matrix for the case where Bob has a trivial output system, H B = C .)Similarly, in the case where A (cid:14)(cid:15) B , the process matrix has the form W A (cid:14)(cid:15) B = W A B ⊗ A B , (49)where W A B ≥
0, Tr W A B =
1. Such a process is realized in a situation in which Alice and Bob receive input systems in a jointquantum state with a density matrix W A B , and their output systems are discarded.We can unify these two conditions to write down the form of a process matrix compatible with B (cid:14) A , which is identical to (48), W B (cid:14) A = W A A B ⊗ B , (50)where W A A B is a valid process matrix for the case where H B = C .As shown in Ref. [37] within a di ff erent framework, all process matrices of the type (50) can be realized by embedding theexperiments of Alice and Bob in a quantum circuit, so that Bob’s experiment does not precede Alice’s experiment in the orderof the circuit composition. Most generally, this corresponds to providing Alice with an input system that is entangled with anancilla, then sending Alice’s output together with the ancilla through a quantum channel into Bob’s input, and then discardingBob’s output. Such a process is referred to as quantum ‘channel with memory’.As we have seen earlier, a bipartite causal process is one that can be written in the form (16), where W A (cid:14) B and W B (cid:14) A aretwo processes compatible with A (cid:14) B and B (cid:14) A , respectively. It is then tempting to conjecture that the class of causal quantumprocesses might be those whose process matrices can be written in the form W A A B B = q W A (cid:14) B + (1 − q ) W B (cid:14) A , ≤ q ≤ , (51)where W A (cid:14) B and W B (cid:14) A have the form defined in Eq. (50). Certainly, since the probabilities for the outcomes in the quantumprocess framework are linear functions of the process matrix, a process matrix of the form (51) describes a causal process.However, the condition for a process to be causal (Eq. (16)) does not imply that W A (cid:14) B and W B (cid:14) A in the convex decompositionof the process should themselves be quantum process; only their convex mixture needs to be. While it is conceivable that thestructure of quantum processes might imply the form (51) (indeed, this has been shown to hold for a limited class of bipartitequantum processes [14]), there is no obvious reason to expect this to hold in the general case. In fact, we will see that the naturalgeneralization of condition (51) to the multipartite case is not equivalent to the condition that a process is causal (the same holdsalso for other possible generalizations that we will discuss later). Very recently, the same was shown to hold also in the bipartitecase, by Feix, Ara´ujo, and Brukner [39].A bipartite quantum process that admits the decomposition (51) was called causally separable [4]. One way to think of the re-lation between causal and causally separable quantum processes is in analogy with the relation between Bell-local and separable(non-entangled) quantum states. Given an arbitrary multipartite quantum state with a density matrix ρ AB ··· , the probabilities forthe outcomes of a set of local POVM measurements { M Ai } i ∈ O A , { M Bj } j ∈ O B , · · · ( (cid:80) i ∈ O A M Ai = A , (cid:80) j ∈ O B M Bj = A , · · · ) are givenby p ( i , j , · · · |{ M Ai } i ∈ O A , { M Bj } j ∈ O B , · · · ) = Tr( ρ AB ··· M Ai ⊗ M Bj ⊗ · · · ) . (52)A Bell-local state is one for which the joint probabilities for the outcomes of any combination of local measurements admitsa local hidden variable description (and hence such a state cannot be used to violate any Bell inequality [40]), i.e., the jointdistribution can be written as a probabilistic mixture of factorizing local distributions, p ( o A , o B , · · · | s A , s B , · · · , ρ AB ) = (cid:88) λ p ( λ ) p ( o A | s A , λ ) p ( o B | s B , λ ) · · · , (53)0where λ is some variable with a probability distribution p ( λ ), s A , s B , · · · are the local measurement settings (each correspondingto a specific local POVM measurement { M Ai } i ∈ O A , { M Bj } j ∈ O B , · · · ), and o A , o B , · · · are their outcomes (corresponding to i , j , · · · in the expression (52)). A separable quantum state is one for which each of the local distributions p ( o A | s A , λ ), p ( o B | s B , λ ), · · · inEq. (52) itself can be thought of as arising from the respective local measurement being applied on a local quantum state, whichmeans that the density matrix of the state can be written ρ AB ··· = (cid:88) λ p ( λ ) ρ A ( λ ) ⊗ ρ B ( λ ) ⊗ · · · . (54)A separable quantum state is clearly Bell local, but the reverse is known not to be true [41]. The relation between causal (16)and causally separable (51) bipartite quantum processes can be seen in an analogous way – a causally separable process is onefor which the processes into which we decompose the process are themselves valid quantum processes.Here, we propose to extend the notion of causal separability to the multipartite case based on this analogy. Definition III.1. (Causally separable quantum process):
A quantum process is called causally separable if and only if it canbe decomposed in the canonical form given by Theorem II.2, with the additional condition that each process on the right-handside of Eq. (41) is a quantum process. (Note that since the canonical form is defined iteratively, the latter is understood to holdfor all conditional processes in this definition.)By a direct analogy, causally separable processes can be defined for any theory formulated in the process framework, but herewe will be interested specifically in quantum processes. The process matrix of a causally separable quantum process will becalled a causally separable process matrix.
C. Non-equivalence between causal and causally separable multipartite processes: a tripartite example
We now give an example of a tripartite quantum process that is causal but causally non-separable, which demonstrates thatthese two concepts are not equivalent, at least in the case of more that two parties. A similar conclusion based on the sameexample has been obtained independently by Costa and is presented in Ref. [19].The example is inspired by the idea of superposition of causally ordered quantum circuits by means of the so-called quantumswitch technique [3], where the order of two black-box quantum operations is made to depend on the value of a quantum controlbit prepared in superposition of the two possible logical values. Each of the input and output systems of Alice and Bob in ourexample will be assumed to be a two-dimensional (qubit) system. We can imagine that this is the spin degree of freedom of aspin- particle, which enters each laboratory, interacts with the devices inside, and leaves. The particle could be prepared soas to go in superposition along two di ff erent possible paths – along one path, it goes first through Alice’s laboratory and thenthrough Bob’s, whereas along the other path it goes first through Bob’s laboratory and then through Alice’s. For simplicity, wecan imagine that the experiment is arranged in such a way that the particle would always go through Bob’s laboratory at a fixedtime, but depending on the value of the control bit, it would go through Alice’s laboratory before or after that. It is assumedthat independently of the time at which the system may go through Alice’ laboratory in a given run, Alice would apply the sameoperation on it. To understand the e ff ect of such a setup, consider first the case in which Alice and Bob each apply a unitaryoperation on the system, U A and U B , respectively. Let us denote the Hilbert space of the control qubit (path degree of freedom)by H c , and that of the system (spin degree of freedom) by H s . If | (cid:105) c corresponds to the path in which Alice is before Boband | (cid:105) c to the path in which Bob is before Alice, if we initially prepare the particle in the state, say, ρ csin = | Ψ (cid:105)(cid:104) Ψ | csin , where | Ψ (cid:105) csin = ( α | (cid:105) c + β | (cid:105) c ) | ψ (cid:105) s , at the end it will be in the state ρ csf i = | Ψ (cid:105)(cid:104) Ψ | csf i , where | Ψ (cid:105) csf i = α | (cid:105) c U sB U sA | ψ (cid:105) s + β | (cid:105) c U sA U sB | ψ (cid:105) s .Now, if a third party, Charlie, performs an operation on the joint system H c ⊗ H s subsequently, he can distinguish this situationfrom a situation in which the order between the operations of Alice and Bob is conditioned on a classical bit (e.g., modeledby the initial state of the control qubit being in a ‘classical’ mixture of the two possible values, | α | | (cid:105)(cid:104) | s + | β | | (cid:105)(cid:104) | s , insteadof a coherent superposition) by performing a suitable measurement. In fact, it was shown in Ref. [6] that by exploiting sucha coherent strategy, Charlie can perfectly distinguish between a pair of unitaries U A and U B that commute or anticommute byusing each of the unitaries only once, which is impossible if the order of the unitaries is conditioned on a classical bit. Anexperimental demonstration of this e ff ect was recently reported in Ref. [17].In the general case, the operations of Alice and Bob need not be unitary and may have di ff erent possible outcomes. Everysuch operation, however, can be seen as the result of a joint unitary on the input system and a local ancilla, such that the outcomeremains stored on the local ancilla in a particular basis. Similarly, any local ‘choice’ of operation can be modeled by a largerunitary on all systems involved plus a local ancilla that carries the ‘choice’ variable. Thus, we can have Alice and Bob performgeneral operations in this setup by purifying their local operations to unitaries and deferring the reading of their outcomes to theend of the whole experiment. (Note that in order not to destroy the superposition, the whole experiments needs to be performed1coherently, which may be unrealistic for local operations performed by macroscopic devices, but is in principle compatible withstandard quantum mechanics.)In our example, we will take α = β = √ , and we will assume, as described above, that Charlie can operate on both the pathand spin degrees of freedom of the particle after it has interacted with Alice and Bob. In other words, Charlie’s input systemwill be four dimensional, and we will formally decompose it into two qubit subsystems, H C = H C c ⊗ H C s , where H C c and H C s correspond to the path and spin degrees of freedom, respectively. Since Charlie operates last, we do not need to introducea non-trivial output system for him, i.e., his output system will be assumed one-dimensional. The process matrix relating thelocal experiment of Alice, Bob, and Charlie in this setup can easily be obtained by describing the experiment in the form of acircuit in which Alice’s operation is represented by two controlled operations at two possible times, such that one of them wouldact nontrivially depending on the state of the control qubit (left diagram on Fig. 5). Using the CJ representation of the channelsconnecting the di ff erent boxes, we obtain W A A B B C C = | W (cid:105)(cid:104) W | A A B B C C , (55)where | W (cid:105) A A B B C C = ( | (cid:105) C c | ψ (cid:105) A | Φ + (cid:105) A B | Φ + (cid:105) B C s + | (cid:105) C c | ψ (cid:105) B | Φ + (cid:105) B A | Φ + (cid:105) A C s ) / √ , (56)with | Φ + (cid:105) = | (cid:105) + | (cid:105) . It can be verified that W A A B B C C contains only allowed terms. This process matrix is a rank-oneprojector, and hence it cannot be written as a convex mixture of di ff erent process matrices. Therefore, if it is causally separable,it must be of one of the types W ( A , B ) (cid:14) C , W ( B , C ) (cid:14) A , or W ( A , C ) (cid:14) B . But each of these types of process matrices should permit nosignaling from two of the parties to the third one (e.g., in the first case there can be no signaling from Alice and Bob to Charlie).However, the above process matrix permits signaling to any of the parties from some of the other parties. Indeed, to see that therecan be signaling from Alice and Bob to Charlie, imagine that Alice and Bob choose to perform the unitary operations U A and U B . In this case, Charlie will receive the state [ | (cid:105) C c ( U B U A | ψ (cid:105) ) C s + | (cid:105) C c ( U A U B | ψ (cid:105) ) C s ] / √
2, which can be di ff erent for di ff erentchoices of the unitaries of Alice and Bob, and can therefore yield di ff erent probabilities for the outcomes of some measurementof Charlie. To see that we can have signaling from Alice to Bob or vice versa, notice first that there can be no signaling fromCharlie to Alice and Bob (Charlie has a trivial output system). This means that we have a well-defined reduced process for Aliceand Bob, whose process matrix is W A A B B =
12 ( | ψ (cid:105)(cid:104) ψ | A ⊗ | Φ + (cid:105)(cid:104) Φ + | A B ⊗ B + | ψ (cid:105)(cid:104) ψ | B ⊗ | Φ + (cid:105)(cid:104) Φ + | B A ⊗ A ) . (57)This is a causally separable bipartite process matrix that can be interpreted as describing an equally weighted probabilisticmixture of two fixed-order processes – the first one describes a situation in which the input state | ψ (cid:105) is sent into Alice’s input,her output is sent into Bob’s input through the identity channel, and Bob’s output is discarded; the second one describes theanalogous situation with the roles of Alice and Bob interchanged. Clearly, since in the first situation there is an ideal channelfrom Alice to Bob, there can be signaling from Alice to Bob in this process (even if imperfect on average), and similarly fromBob to Alice. Therefore, the process matrix given by Eqs. (55) and (56) is not causally separable.The fact that the process is causal follows immediately from the fact that the reduced process for Alice and Bob is causallyseparable (and hence also causal). Specifically, we have W AB = W B (cid:14) A + W A (cid:14) B = W B | A [ A ] I ◦ W A [ A ] I + W A | B [ B ] I ◦ W B [ B ] I .But the tripartite process is simply W ABC = W C | AB ◦ W AB = W C | AB ◦ W B | A [ A ] I ◦ W A [ A ] I + W C | AB ◦ W A | B [ B ] I ◦ W B [ B ] I , whichis the form of a causal process. This observation suggests how the probabilities of Alice, Bob, and Charlie can be simulatedwithout using a quantum switch, if we allow the parties to have larger input and output systems. Since the reduced probabilitiesof Alice and Bob can be realized by conditioning their order on a classical random bit, all that is needed in order for the tripartiteprocess to be reproduced in this way is for Charlie to receive the information about the settings and outcomes of Alice and Bobso as to produce the necessary p ( o C | s A , o A , s B , o B , s C ). Therefore, if in addition to the qubit system that goes between Alice andBob there is another (possibly infinite-dimensional) system on which each party writes down his / her setting and outcome (rightdiagram on Fig. 5), and this system at the end enters Charlie’s laboratory (or, alternatively, the state on Charlie’s original inputsystem is prepared based on this information), the process can be simulated using classically random causal configurations.By a similar argument we can construct a large class of multipartite processes that are causal but not causally separable.Consider a situation in which the order of all but one of the parties is conditioned on the state of a control system prepared insuperposition, and subsequently all systems on which these parties have operated together with the control system are sent intothe input of the last party. If all systems were initially prepared in a pure state and all channels are unitary ones, the processmatrix will have rank 1, and unless the process is fixed-order causal, it cannot be causally separable. Yet, it will be causalbecause the reduced process for all parties except for the last one will be causally separable (and hence causal) due to the factthat when we trace out the control system, the process for these parties would be a classical probabilistic mixture of fixed-orderprocesses. Since the full process is obtained by multiplying the conditional process of the last party with the reduced process ofthe previous ones, the full process is causal. It can be simulated using classical control of the order of the parties by allowinglarger input and output systems by which the settings and outcomes of all other parties are made available to the last one.2 AAB t system C AB t C | i + | ip | i or | i system A FIG. 5: The left diagram illustrates the circuit with quantum control. The right diagram illustrates a simulation of the samecorrelations with a classically controlled circuit using input and output systems of larger dimensions.
D. Non-causality can be activated by shared entanglement
We now show another peculiar property of the concepts of causality and causal separability of quantum processes. One of thekey assumptions in the derivation of the quantum process matrix framework is that every process can be extended by supplyingthe parties with ancillary input systems in an arbitrary quantum state, yielding another valid process. Intuitively, since a jointinput state is a non-signaling process that is compatible with any causal configuration, one may expect that by adding such astate to a causal quantum process would yield again a causal process. We now show that this is not the case. We refer to thise ff ect as activation of non-causality.We give a particular example of a tripartite causal quantum process matrix, constructed on the basis of the bipartite processmatrix presented in Ref. [4], W A A B B =
14 ( A A B B + √ σ A z σ B x σ B z + √ σ A z σ B z ) , (58)which itself can violate a causal inequality and is hence non-causal (see Ref. [4]). Here, the input and output systems of Aliceand Bob are two-level systems. In our tripartite construction, the input and output systems of Alice and Bob are also two-levelsystems, and we add Charlie, who has a trivial input system and a two-level output system. In terms of the Pauli matrices σ x , σ y , σ z , the process matrix we consider has the form W A A B B C =
14 ( A A B B C + √ σ A z σ B z σ B z σ C x + √ σ A z σ B z σ C z ) . (59)The fact that this is a valid process matrix follows from the fact that it has the right normalization, contains only allowed σ terms, and is positive semidefinite. The latter is easy to see by noticing that relative to the {| (cid:105) , | (cid:105)} basis of system B (this is theeigenbasis of σ z corresponding to eigenvalues + −
1, respectively), the process matrix can be written W A A B B C = | (cid:105)(cid:104) | B ⊗
14 ( A A B C + √ σ A z σ B z σ C x + √ σ A z σ C z ) + | (cid:105)(cid:104) | B ⊗
14 ( A A B C − √ σ A z σ B z σ C x − √ σ A z σ C z ) . (60)Now, the operator ( A A B C + √ σ A z σ B z σ C x + √ σ A z σ C z ) is identical to that in Eq. (58) except that we have the system C in the place of B , and this operator has been shown to be positive semidefinite. The operator ( A A B C − √ σ A z σ B z σ C x − √ σ A z σ C z ) di ff ers only by the fact that the nontrivial σ terms come with a minus sign, and can be obtained from the first operatorby a unitary transformation (e.g., one that takes σ C x to − σ C x and σ C z to − σ C z , such as σ C y ).To see that this process matrix describes a causally separable process, note that it permits no signaling from Alice and Bobto Charlie, i.e., it can be formally written as W A , B , C = W A , B | C ◦ W C . But conditionally on any event in Charlie’s laboratory,3which is most generally described by some CP map with CJ operator M C ≥
0, Alice and Bob are left with a bipartite processwith process matrix W A A B B M C = Tr C [( M C ⊗ A A B B ) W A A B B C ] / Tr[ M C ] . (61)This process matrix is obviously a linear combination of the identity and terms containing only σ z operators on di ff erent sub-systems, i.e., it is diagonal in a given local basis (the {| (cid:105) , | (cid:105)} basis for each subsystem). It was shown in Ref. [4] that allsuch bipartite process matrices are causally separable (though we remark that the same was shown not to hold for multipartiteprocesses [9]).Imagine now that we supply Bob and Charlie with the entangled input state | Φ + (cid:105)(cid:104) Φ + | C (cid:48) B (cid:48) , which yields the new process W A A B B (cid:48) B C (cid:48) C = W A A B B C ⊗ | Φ + (cid:105)(cid:104) Φ + | C (cid:48) B (cid:48) . (62)If Charlie performs the identity unitary channel from C (cid:48) to C in his laboratory, which is described by M C (cid:48) C = | Φ + (cid:105)(cid:104) Φ + | C (cid:48) C ,Alice and Bob are left with the bipartite process W A A B B (cid:48) B =
14 ( A A B B (cid:48) B + √ σ A z σ B z σ B (cid:48) x σ B z + √ σ A z σ B z σ B (cid:48) z ) . (63)This can be easily seen from the fact that taking the partial trace of W A A B B (cid:48) B C (cid:48) C with the operator | Φ + (cid:105)(cid:104) Φ + | C (cid:48) C is formallyidentical (up to a normalization) to a local projection in a quantum-state teleportation protocol [38], which amounts to ‘teleport-ing’ the part of the matrix on C onto B (cid:48) . (Note that the standard notion of teleportation is defined for quantum states and notprocess matrices, and the protocol requires a correcting operation on the receiver’s side since a projection of the kind above,which does not require correction, cannot be accomplished deterministically [38]). The process matrix (63) is similar to (58),except that the local operators on B in the non-trivial sigma terms in Eq. (58) are now on B (cid:48) , and there is a σ z operator on B in each such term. This process matrix is non-causal, because it allows Alice and Bob to obtain any correlations that they couldobtain using the non-causal process matrix (58). This can be done as follows. Alice always performs the same operations thatshe would perform with the process matrix (58). Bob performs a measurement on system B in the {| (cid:105) , | (cid:105)} basis. If he obtainsthe outcome | (cid:105) , then it is as if Alice and Bob share the process matrix (58) with B (cid:48) in the place of B . He will then apply anyoperation from B (cid:48) to B that he would apply from B to B with the process matrix (58), which yields the same joint probabilitiesfor Alice and Bob as those with the process matrix (58). If Bob obtains the outcome | (cid:105) for his measurement on B , then it isas if Alice and Bob share the same process matrix as (58) with B (cid:48) in the place of B but with a minus sign in front of each ofthe two nontrivial σ terms. This process matrix is equivalent to the previous one under a change of basis by the unitary σ B (cid:48) y .Therefore, Bob can simply apply from B (cid:48) to B the same operations he would apply from B to B with the process matrix (58)but transformed by the unitary transformation σ B (cid:48) y . Again, this yields the same joint probabilities for Alice and Bob as with theprocess matrix (58). In particular, Alice and Bob can use this strategy to violate the causal inequality described in Ref. [4]. Theprocess matrix (63) is thus non-causal, and so is the tripartite process matrix (62).It is not known at present whether non-causal processes can be realized in agreement with the known laws of quantummechanics without resorting to post-selection. We have seen in the previous subsection that we can realize causally non-separableprocesses, which are nevertheless causal. Here, we see that certain causal processes can become non-causal when supplied withshared entanglement. The ability to extend a process with shared entanglement seems natural to expect for any experimentallyrealizable process. From this perspective, this result suggests that either non-causal processes may be possible, or that there mayexist causally separable processes, as defined above, that cannot be realized in practice. E. Extensibly causal and extensibly causally separable quantum processes
The fact that according to our definition of causal separability there exist causal processes that may be activated to non-causalones by shared entanglement naturally suggests the definition of the following classes of processes that do not have thiscounterintuitive property.
Definition III.2. (Extensibly causal quantum process):
A quantum process that is causal and remains causal under extensionwith input systems in an arbitrary joint quantum state is called extensibly causal.
Definition III.3. (Extensibly causally separable (ECS) quantum process):
A quantum process that is causally separable andremains causally separable under extension with input systems in an arbitrary joint quantum state is called extensibly causallyseparable (ECS).4The process matrices of these types of processes will also be referred to as extensibly causal and ECS process matrices,respectively.
Note.
These definitions can be formulated analogously for more general process theories that permit composite local systems.Do these classes of processes correspond to something easy to describe in practice, and are they di ff erent at all? It isimmediate to see the following facts. Observation 1:
All bipartite causally separable processes are ECS. This is because, if we add an arbitrary joint input ancillato a process matrix of the form (51), we again obtain a process matrix of the same form. Therefore, the notion of extensiblecausal separability can be seen as another possible multipartite extension of the bipartite notion of causal separability, which,however, is linked in a less direct way to the theory-independent notion of causality.
Observation 2:
Extensibly causal and ECS processes are not equivalent in general. Indeed, the causally non-separabletripartite process (55) based on the quantum switch is also extensibly causal (our proof that it is causal applies also if the partiesshare entangled input ancillas).
Comment:
Recently, Feix, Ara´ujo, and Brukner gave an example of a bipartite quantum process that is causal but notextensibly causal [39], proving that causality and extensible causality are di ff erent in the bipartite case too. While in thetripartite case we have seen that extensible causality is also di ff erent from causal separability, it is currently an open problemwhether the same holds in the bipartite case.In the next subsection, we derive a characterization o f the tripartite ECS processes in terms of conditions on the form of theprocess matrix which generalize the conditions in the bipartite case (Eqs. (50), (51)). F. Structure of tripartite ECS process matrices
Recalling the definition of causally separable process, let us first state an obvious consequence of this definition for thestructure of causally separable (though not necessarily ECS) process matrices. Since the probabilities of a quantum process arelinear in the process matrix, the requirement that a causally separable process decomposes as in Theorem II.2 where all processeson the right-hand side of Eq. (41) are valid quantum processes means that a causally separable process matrix is one that can bewritten in the form W ··· n n cs = n (cid:88) i = q i W (1 , ··· , i − , i + , ··· , n ) (cid:14) i , ≤ q i , ∀ i , n (cid:88) i = q i = , (64)where W (1 , ··· , i − , i + , ··· , n ) (cid:14) i is a process matrix which describes a process W (1 , ··· , i − , i + , ··· , n ) (cid:14) i with the property W (1 , ··· , i − , i + , ··· , n ) (cid:14) i = W , ··· , i − , i + , ··· , n | ics ◦ W i , (65)where for n > W , ··· , i − , i + , ··· , n | ics is a causally separable process for every value of the event in i , and for n = W (1 , ··· , i − , i + , ··· , n ) (cid:14) i is a quantum process that permits no signalingfrom the rest of the parties to i guarantees that both the reduced and the conditional process on the right-hand side of Eq. (65)are valid quantum processes (this can be seen from the (no) signaling condition in Proposition III.2).In the case of two parties, we have seen that the process matrices W A (cid:14) B , whose processes obey W A (cid:14) B = W A | Bcs ◦W B (note thatany monopartite process is trivially causally separable and ECS), are those that can be written in the form W A (cid:14) B = W B B A ⊗ A ,and the general form of bipartite causally separable process matrices is (51). As noted already, this is also the general form ofthe bipartite ECS process matrices. Our goal is to obtain a similar conditionfor triparite ECS processes.First, let us consider a process of the form W ( A , B ) (cid:14) C = W A , B | Ccs ◦ W C , where W C is a monopartite quantum process and W A , B | Ccs is a bipartite conditional process which is causally separable for each possible event in C . Since in particular thereshould be no signaling from Alice and Bob to Charlie in such a process, its process matrix, which we will denote W A A B B C C ( A , B ) (cid:14) C ,can at most contain the types of terms listed in Table III. These are the terms that do not permit signaling from Alice and Bob toCharlie according to Proposition III.2.We will first obtain necessary and su ffi cient conditions for such a process to be ECS. Note that we have not proven yet thata general ECS process matrix should have the form (64) where each of the terms W (1 , ··· , i − , i + , ··· , n ) (cid:14) i is itself ECS. This will beshown later.5Every event in Charlie’s laboratory is described by some CP map with CJ operator M C C ≥
0, Tr M C C ≤ d C . Conditionallyon such an event, Alice and Bob are left with the process matrix W A A B B M C C = Tr C C [ W A A B B C C ( A , B ) (cid:14) C ( A A B B ⊗ M C C )] / p ( M C C ) , (66)where p ( M C C ) is the probability for the event M C C to occur in Carlie’s laboratory (given the appropriate setting), whichis independent of the operations performed by Alice and Bob since the process involves no signaling from Alice and Bob toCharlie. More specifically, p ( M C C ) = Tr[ W C C M C C ] , (67)where W C C = Tr A A B B [ W A A B B C C ( A , B ) (cid:14) C ( A A B B d A d B ⊗ C C )] (68)is the reduced process of Charlie. The requirement that the conditional process for Alice and Bob is causally separable meansthat for all M C C , W A A B B M C C = q M C C W A (cid:14) BM C C + (1 − q M C C ) W B (cid:14) AM C C , (69)where W A (cid:14) BM C C and W B (cid:14) AM C C are valid quantum processes compatible with A (cid:14) B and B (cid:14) A , respectively, and q M C C ∈ [0 ,
1] (allobjects generally depend on M C C ). For convenience, we will write this simply in the form W A A B B M C C = A ⊗ ˜ W A B B M C C + B ⊗ ˜ W A A B M C C , (70)where ˜ W A B B M C C ≥ W A A B M C C ≥
0, and the whole operator is a valid process matrix, i.e., it contains only allowed terms and isproperly normalized.A su ffi cient condition for this to hold is that W A A B B C C ( A , B ) (cid:14) C = A ⊗ ˜ W A B B C C + B ⊗ ˜ W A A B C C , (71)where ˜ W A B B C C ≥ W A A B C C ≥ W A B B C C ≥ W A A B C C ≥ C , but these terms have tocancel in the sum.) Indeed, we haveTr C C [ W A A B B C C ( A , B ) (cid:14) C ( A A B B ⊗ M C C )] / p ( M C C ) = W A A B B M C C = A ⊗ ˜ W A B B M C C + B ⊗ ˜ W A A B M C C , ∀ M C C ≥ , (72)where ˜ W A B B M C C = Tr C C [ ˜ W A B B C C ( A B B ⊗ M C C )] / p ( M C C ) ≥ , (73)˜ W A A B M C C = Tr C C [ ˜ W A A B C C ( A A B ⊗ M C C )] / p ( M C C ) ≥ , (74)and it is easy to see that since W A A B B C C ( A , B ) (cid:14) C contains only the types of terms listed in Table III, W A A B B M C C can only containallowed terms.It is immediate to see that this condition is su ffi cient also for the process matrix W A A B B C C ( A , B ) (cid:14) C to be ECS. This is because if W A A B B C C has the above properties, any extension W A A B B C C ⊗ ρ A (cid:48) B (cid:48) C (cid:48) , where ρ A (cid:48) B (cid:48) C (cid:48) is a density matrix, also has theseproperties.We now show that the form (71) is also a necessary condition for an ECS process matrix compatible with ( A , B ) (cid:14) C , whichwe will denote by W A A B B C C ecs ;( A , B ) (cid:14) C . The proof makes use of the ‘teleportation’ technique that we used in showing the activationof non-causality. Imagine that we supply Alice and Charlie respectively with ancillary systems A (cid:48) and C (cid:48) of dimension d C d C each, which are prepared in the maximally entangled state | φ + (cid:105)(cid:104) φ + | A (cid:48) C (cid:48) / ( d C d C ), where | φ + (cid:105) = (cid:80) d C d C i = | i (cid:105) A (cid:48) | i (cid:105) C (cid:48) . Conditionallyon Charlie performing a suitable operation and obtaining an outcome with CP map M C C C (cid:48) ∝ | φ + (cid:105)(cid:104) φ + | ( C C ) C (cid:48) , Alice and Bobwill be left sharing a process matrix which, up to a normalization factor, has an identical form to that of W A A B B C C ecs ;( A , B ) (cid:14) C but with6 A (cid:48) in the place of C C . The requirement that this is a causally separable bipartite process matrix means that W A A B B C C ecs ;( A , B ) (cid:14) C mustbe of the form (71).So far, we have only obtained necessary and su ffi cient conditions for an ECS process matrix W A A B B C C ecs ;( A , B ) (cid:14) C compatible with( A , B ) (cid:14) C (and similarly for permutations of A , B , C ). We next prove the general case. Proposition III.3.
Every tripartite ECS process matrix can be written in the form W A A B B C C ecs = q W A A B B C C ecs ;( A , B ) (cid:14) C + q W A A B B C C ecs ;( A , C ) (cid:14) B + q W A A B B C C ecs ;( B , C ) (cid:14) A , q i ≥ , ∀ i = , , , (cid:88) i = q i = , (75)where W A A B B C C ecs ;( A , B ) (cid:14) C contains only terms from table III and has the form (71), and analogously for W A A B B C C ecs ;( A , C ) (cid:14) B and W A A B B C C ecs ;( B , C ) (cid:14) A by permutation. The Proof S6 is given in the Appendix.The extension of this form to an arbitrary number of parties is left for future investigation. G. Processes realizable by classically controlled quantum circuits
Bipartite ECS processes have a clear experimental realization. This raises the question of whether multipartite ECS processescan also be realized in practice, and if so, whether they correspond to a natural class of experimental procedures. (Note that in thebipartite case, ECS processes are equivalent to causally separable processes, but we have already seen that there are multipartitecausally separable processes that can become non-causal under extension with entangled ancillas, and these do not have a knownexperimental realization.) Here, we will show that a particular class of processes which can be realized in practice, referred toas classically controlled quantum circuits, belong to the class of ECS processes, which is the smallest class of causal quantumprocesses that we have considered so far. Based on certain considerations, we furthermore conjecture that all ECS processes canbe realized in this way (this is certainly true in the bipartite case).The idea of a classically controlled circuit can be thought of as falling within the paradigm of quantum lambda calculus withclassical control [42, 43]. If we regard the local experiments of the parties as black-box operations, we may think that theyare called, only once each, as part of a computation where at every time step a quantum operation is applied on some part ofa quantum register depending on a classical protocol that may use as a variable the outcomes of past operations. If black-boxoperations are involved in such a computation, their outcomes cannot be directly used (they remain ‘inside the box’ until theend), but the order of subsequent operations of the circuit may nevertheless depend indirectly on the event inside such a blackbox, since it can be decided based on a measurement on the output system.More concretely, we define such a process to have the following general realization. We begin with some su ffi ciently largequantum system (or ‘register’) in a given quantum state. We perform a quantum operation on it and conditionally on the outcomeof that operation we determine which party will be first, which subsystem of the register will be his / her input system, and whatoperation will be applied after the black box of that party, all according to some specified rule. We apply the black-box operationof the first party on the decided subsystem, perform the decided operation after it, and depending on its outcome and the outcomeof our first operation decide which party will be second, and so on. This continues until all parties are called (by definition, theprotocol is such that each party is called exactly once). This model can be formalized in di ff erent equivalent ways, which maybe suitable for di ff erent purposes, and we will consider some simplifications below when we discuss a tripartite example. Thefact that this model gives rise to valid quantum processes can be seen from the fact that if we formally write the operation insideeach box and calculate the joint probabilities for the outcomes of all boxes using the standard rules of quantum mechanics forall possible outcomes of the protocol, we see that they are linear and non-contextual functions of the respective CP maps of theparties. The same holds if we introduce ancillary systems prepared in an arbitrary state and consider extended operations of theparties that act on parts of them.In the case of only two parties, we know that any (extensibly) causally separable process can be implemented in this way,since it most generally corresponds to embedding at random the local experiments of Alice and Bob into one of two possiblefixed circuits, which can be chosen conditionally on the outcome of a measurement on some state at the very beginning. Sinceafter the first party is chosen there is only one possible choice for the second party, no measurement after the first party is needed.Reversely, any bipartite process that we may obtain via this model has the form of an ECS process. Fist notice that the process isindependent of the operation applied after the last party. Also, the outcome of any operation after the first party can be ignoredsince there is only one choice for the last party, i.e., that operation can be assumed deterministic. Finally, the outcomes of theoperation before the first party can be grouped into two coarse-grained outcomes such that conditionally on one of them thefirst party is Alice and on the other one it is Bob. But since after the outcome of that operation and before the input of the firstparty the quantum register is in some particular quantum state, the rest of the experiment simply corresponds to a deterministic7circuit in which Alice and Bob are embedded in a particular order. Therefore, the process realized by such a procedure is just aprobabilistic mixture of the processes of two fixed-order circuits, which is the claimed form.In the case of more than two parties, the equivalence between the two concepts is less obvious, but we can easily arguethat all processes obtained by classically controlled circuits are ECS. First, it is clear that depending on the outcome of the firstmeasurement (which has a probability independent of any future operations and therefore of the settings of the parties), there willbe one party that is first and hence the subsequent process that results from the protocol can involve no signaling from the rest ofthe parties to that first party. Therefore, the subsequent process has a well-defined reduced process for the first party. Taking intoaccount all possible outcomes of the first measurement, the whole process will be just a probabilistic mixture of processes ofthis kind where one party is first, which is Eq. (64). But conditionally on the outcome of the first party, the procedure for the restof the parties looks analogously, so Eq. (65) holds too, i.e., the process is causally separable. Including ancillas onto which theoperations of the parties can be extended does not change anything in this argument. Therefore, every process realizable with aclassically controlled quantum circuit is ECS.We conjecture that the reverse also holds. We provide some partial considerations that support this conjecture, based onanalysis of the restrictions on the allowed terms in processes realized by classically controlled quantum circuits in the tripartitecase. We will focus on the question of implementing by a classically controlled quantum circuit an ECS process matrix ofthe type W ( A , B ) (cid:14) Cecs , which has the form (71). Implementability of a matrix of this kind is both necessary and su ffi cient forthe implementability of a general tripartite ECS process matrix as described in Proposition III.3, since by using a suitablemeasurement at the beginning we can select with the right probability which of the three process matrices in the mixture on theright-hand side of Eq. (75) to realize subsequently. CAB AB t C C A A B B A B M ,CP M ,CP A B N ,CP T P N ,CP T P C A B / ⇢ FIG. 6: Realization of an ECS process compatible with ( A , B ) (cid:14) C by a classically controlled quantum circuit.The protocol begins most generally with some quantum system prepared in a state ρ . After Charlie operates on some sub-system, we apply some operation based on whose outcome we determine who is second, on what subsystem he / she would act,and what operation will be applied after that. Note that without loss of generality we may assume that there is a pre-specifiedsubsystem on which the second party will operate since any subsystem of the same dimension can be mapped onto the desig-nated subsystem by a unitary transformation that can be absorbed as part of the definition of the present operation. Also, withoutloss of generality we may assume that this operation has only two outcomes, since we can group the outcomes into those forwhich Alice will be next, and those for which Bob will be next, and any conditioning of the operation following the next partyon the fine-grained outcome within each group can be equivalently done by a single future operation acting on a larger systemthat includes some subsystem on which the classical information about the outcome at this step is copied (still something thatwe can include as part of the definition of the operation at this step). Since there is only a single possibility for the last party,the operation after the second party can be regarded as a deterministic operation (or a CPTP map) from all systems to the inputof the last party. We leave the possibility that this last operation may be defined conditionally on the first outcome rather thanabsorb the conditioning on that outcome into a larger operation, in order to avoid complications arising from the fact that thedi ff erent parties may have input and output systems of di ff erent dimensions. The outlined procedure is sketched in Fig. 6, wherethe two possible sequences of transformations arising from the two possible outcomes of our first operation are depicted in blueand green, respectively. The two CP maps corresponding to the outcomes of the operation after Charlie must sum up to a CPTPmap, since they correspond to the two possible outcomes of a standard quantum operation.Each of the two possible developments (blue and green) of this protocol is a non-deterministic linear supermap [44] fromthe local CP maps of the parties into the real numbers, the result of which equals the probability for the particular sequence ofevents. This can be written in a similar form as the formula for the probabilities of the outcomes of the parties in a valid process,8except that in the place of the process matrix we would have an operator ˜ W A A B B C C i ≥
0, where i = ,
2, labels the particulardevelopment, which generally would not be a valid process matrix. However, ˜ W A A B B C C + ˜ W A A B B C C = W A A B B C C cs ;( A , B ) (cid:14) C would be a valid process matrix realized through this classically controlled quantum circuit.Consider now just one of the two possible developments, say, the blue one, in which Alice is second and Bob is last (labeled by1). One can see that since Bob is last and his output system is discarded, we have ˜ W A A B B C C = B ⊗ ˜ W A A B C C (similarly,in the other case we have ˜ W A A B B C C = A ⊗ ˜ W A B B C C ). Notice that if the transformation N , CPTP after Alice was notrequired to be CPTP but could be any CP map N , CP , for a suitable choice of the initial state ρ and of the CP maps M , CP and N , CP we could realize any ˜ W A A B C C ≥
0. This is simply because we can choose the density operator ρ C C (cid:48) proportional to˜ W A A B C C , where the part of ˜ W A A B C C on A A B C is stored on C (cid:48) , and we can ‘teleport’ this part of the operator onto itsdesired subsystem by using CP maps M , CP and N , CP that have CJ operators proportional to projectors on maximally entangledstates as needed to realize the ‘teleportation’ (the traces of these CP maps can be chosen to ensure the overall trace of theresultant operator ˜ W A A B C C ). However, the restriction that the transformation after Alice is trace-preserving, N , CPTP , placesconstraints on what kind of ˜ W A A B C C can be obtained. Indeed, the CJ operator of N , CPTP cannot contain terms of type A , A (cid:48) A , and A (cid:48) . Considering the calculation of ˜ W A A B C C based on the CJ operators of ρ , M , CP and N , CP , we see that thelack of these types of terms in N , CPTP implies the lack of any term with a nontrivial σ on A in ˜ W A A B C C . This is the onlyconstraint on the possible types of terms in ˜ W A A B C C . The possible types of terms are exactly those allowed in the operator˜ W A A B C C in Eq. (71). Similarly, we see that the allowed terms in ˜ W A B B C C (Bob second, Alice last) are the same as thosein ˜ W A B B C C in Eq. (71). These are the terms allowed in a process matrix compatible with Charlie being first, except thatboth ˜ W A A B C C and ˜ W A B B C C may contain terms of type C and C C . The fact that these terms should cancel in the sum B ⊗ ˜ W A A B C C + A ⊗ ˜ W A B B C C = W A A B B C C cs ;( A , B ) (cid:14) C follows from the fact that this is a valid ECS process, and can be seento be ensured by the requirement that M , CP + M , CP is CPTP.The only restriction on the operators B ⊗ ˜ W A A B C C and A ⊗ ˜ W A B B C C imposed by this model, apart from their positive-semidefiniteness and the normalization of their sum, seems to be the absence of the forbidden terms in each of them, as well asof the forbidden terms in their sum. If this is indeed the case, then any ECS process could be realized by a suitable classicallycontrolled quantum circuit. A strictly rigorous proof requires showing that apart from the lack of these forbidden terms, there canbe no other hidden constraints on the pair of operators B ⊗ ˜ W A A B C C and A ⊗ ˜ W A B B C C (which, of course, are guaranteedto be properly normalized). One way of doing it could be by exhibiting an explicit constructive procedure for implementing anygiven ECS process, which would be of additional interest on its own right. We leave this question, and the multipartite case, forfuture investigation. IV. CONCLUSION
In this paper, we proposed a rigorous definition of causality in the process framework [4], which takes into account the fact thatthe causal order between a set of local experiments may in general be random and correlated with the settings of some of them.We derived the structure of causal processes permitting such ‘dynamical’ causal order in the general multipartite case, which iscaptured by an iteratively formulated canonical form expressed in terms of reduced and conditional processes. The canonicalform can be interpreted as an unraveling of the process into a sequence of local experiments, which agrees with the conditionthat the order and outcomes of the experiments prior to a given step is independent of the settings of future experiments. Weshowed that for any fixed number of settings and outcomes for each party, the probabilities of a causal processes form a polytope,referred to as the causal polytope. The facets of this polytope define causal inequalities, whose violation by a given process canbe interpreted as demonstrating the non-existence of causal order between the local experiments.We investigated this concept and the related concept of causal separability in the quantum process theory introduced in Ref. [4],whose properties were detailed here in the multipartite case. We proposed a definition of causal separability, which reduces tothe one for the case of two parties [4], based on the canonical form of causal processes. Specifically, a causally separablequantum process was defined as a causal quantum process that has a causal decomposition such that the di ff erent processesappearing in this decomposition are themselves valid quantum processes. We showed that the set of causally separable quantumprocesses is strictly within the set of causal quantum processes, by exhibiting an example of a tripartite process that is causal butnot causally separable. Very recently, the same was shown to hold also in the bipartite case [39]. We also gave an example ofa causally separable (and hence also causal) process that becomes non-causal when extended by supplying the parties with anentangled ancillary state. Based on this observation, we proposed two extended notions of causality and causal separability calledextensible causality and extensible causal separability, which require preservation of the respective property under extending theprocess with entangled input ancillas. Although they are di ff erent in the general case, the sets of causally separable and ECSprocesses are equivalent in the bipartite case. We showed that the sets of extensibly causal and causally separable processesare di ff erent in general via the same tripartite example that we used to show that causal and causally separable processes aredi ff erent. At present we do not know if the same separation holds in the bipartite case. However, it was recently shown that9 multipartite quantum processes causal ECS classically controlled q. circuits causally separable extensibly causal a) Multipartite case. bipartite quantum processes causal causally separable = extensibly causally separable (ECS) = classically controlled quantum circuits extensibly causal b) Bipartite case. FIG. 7: A Venn diagrammatic sketch of our present knowledge of the di ff erent sets of quantum processes that we haveintroduced, in the general multipartite case and in the bipartite case. The white segments are non-empty. The gray segments aresets for which at present we do not know if they are empty or not.causal and extensibly causal processes are di ff erent in the bipartite case, similarly to the multipartite case [39].Finally, we derived a simple characterization of the ECS quantum processes in the tripartite case in terms of conditions on theform of their process matrices, which extends the conditions for (extensibly) causally separable process matrices in the bipartitecase. We conjectured that the set of ECS processes is equivalent to the processes that can be obtained within the paradigm ofclassically controlled quantum circuits and provided evidence for this based on analysis of the restrictions that this paradigmimposes on the tripartite process matrices it can create. The ECS processes and the processes obtainable by classically controlledquantum circuits are equivalent in the bipartite case.Our present understanding of the relation between all these di ff erent classes of quantum processes is illustrated for the generalmultipartite case and for the bipartite case in Fig. 7a and Fig. 7b, respectively. An obvious open problem is whether the graysegments in these figures are empty or not.Another problem of fundamental importance is to understand the class of quantum processes that are physically admissiblein agreement with the known laws of quantum mechanics, and where this class stands with respect to all of the above classes.Are the processes that can be realized by classically controlled quantum circuits all the physically admissible causally separableprocesses? Where does the class of quantum-controlled quantum circuits stand? At present, this is the most general operationallyfeasible paradigm that we are aware of and all known processes realizable through it seem to be extensibly causal. Could theclass of extensibly causal processes be equivalent to quantum-controlled quantum circuits? And most intriguingly, are therephysically admissible non-causal processes?The implications of our results are not limited to the subject of indefinite causal order in quantum mechanics. They canbe useful also for the problem of inferring causal structure [24], both in classical and quantum theory [45]. The subject ofcausal inference concerns many disciplines, from philosophy and machine learning to sociology and medicine. Our formulationof a background-independent operational notion of causality that admits dynamical causal relations opens the road to a moregeneral paradigm for causal inference than the one assuming deterministic underlying variables and static causal relations [24].The decomposition of causal processes derived here implies constraints on the possible causal orders compatible with givensetting-outcome correlations, which can serve as a basis for developing more sophisticated causal inference tools. Acknowledgments
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Proof S1. Proposition II.2 . The ‘only if’ part is contained in the very Proposition II.1. To prove the ‘if’ part, take an arbitraryexperiment, say, 1. Let { , · · · , k } , up to relabeling, be the set of local experiments that are in the causal past or causal elsewhereof 1, and { k + , · · · , n } be the set of local experiments that are in the causal future of 1. Since the causal configuration of thelocal experiments is assumed fixed, the condition for the process to be causal reduces to the requirement that for every such 1,we have p ( o , · · · , o k | s , s , · · · , s n ) = p ( o , · · · , o n | s , · · · , s n ). But from the transitivity and anti-symmetry of causal order itfollows that none of the experiments { , · · · , k } is in the causal future of any of the experiments { k + , · · · , n } . This impliesthat we have a reduced k -partite process for { , · · · , k } , i.e., p ( o , · · · , o k | s , s , · · · , s n ) = p ( o , · · · , o k | s , · · · , s k ). The desiredcondition then follows from Proposition II.1 applied to the k -partite process. Proof S2. Proposition II.3 . First, observe that the property (18) holds for the case where the specified K consecutive setsexhaust all local experiments { , · · · , n } . This is because, in this case, each of the local experiments in the K th consecutive set iscausally preceded by or causally independent from every other local experiment. Hence, the definition of causality (2) directlyimplies the desired relation. The general case follows by induction from this special case and the following Lemma. Lemma S1.
Let the property (18) hold for K = K (cid:48) + I, where K (cid:48) ≥
1. Then it also holds for K = K (cid:48) . Proof.
Observe that p ( κ (1 I , · · · , n K (cid:48) ) , [1 I , · · · , n I ] I , · · · , [1 K (cid:48) , · · · , n K (cid:48) ] K (cid:48) , o I , · · · , o g K (cid:48) | s , · · · , s n ) (S1) = (cid:80) [1 K (cid:48) + I , ··· , n K (cid:48) + I ] K (cid:48) + I p ( κ (1 I , · · · , n K (cid:48) ) , [1 I , · · · , n I ] I , · · · , [1 K (cid:48) , · · · , n K (cid:48) ] K (cid:48) , [1 K (cid:48) + I , · · · , n K (cid:48) + I ] K (cid:48) + I , o I , · · · , o g K (cid:48) | s , · · · , s n ) , where the sum on the right-hand side is over all sets of local experiments that can be the (K (cid:48) + I) th set when the first K (cid:48) consecutivesets are the specified ones. If Eq. (18) holds for K = K (cid:48) + I, all terms in the sum can depend non-trivially only on the settings ofthe parties in the first K (cid:48) consecutive sets, and hence the same must hold for the quantity on the left-hand side: p ( κ (1 I , · · · , n K (cid:48) ) , [1 I , · · · , n I ] I , · · · , [1 K (cid:48) , · · · , n K (cid:48) ] K (cid:48) , o I , · · · , o g K (cid:48) | s , · · · , s n ) (S2) = p ( κ (1 I , · · · , n K (cid:48) ) , [1 I , · · · , n I ] I , · · · , [1 K (cid:48) , · · · , n K (cid:48) ] K (cid:48) , o I , · · · , o g K (cid:48) | s I , · · · , s n K (cid:48) ) . What remains to be shown is that this probability cannot depend on the settings s ( g + K (cid:48) , · · · , s n K (cid:48) .Note that here we cannot apply straightforwardly the causality condition (2) as we did in the case when the first K (cid:48) consec-utive sets were assumed to contain all local experiments. This is because for a particular causal configuration κ ∗ (1 I , · · · , n K (cid:48) )compatible with [1 I , · · · , n I ] I , · · · , [1 K (cid:48) , · · · , n K (cid:48) ] K (cid:48) , it is generally not the case that p ( κ ∗ (1 I , · · · , n K (cid:48) ) , [1 I , · · · , n I ] I , · · · , [1 K (cid:48) , · · · , n K (cid:48) ] K (cid:48) , o I , · · · , o g K (cid:48) | s , · · · , s n ) = p ( κ ∗ (1 I , · · · , n K (cid:48) ) , o I , · · · , o g K (cid:48) | s , · · · , s n ) . (S3)Indeed, in order for the first K (cid:48) consecutive sets to be the specified ones, it is necessary and su ffi cient that: 1) the local experimentsin the specified K (cid:48) consecutive sets have a causal configuration compatible with these sets, and 2) each of the local experimentsthat are not in the specified K (cid:48) consecutive sets is in the causal future of at least one of the local experiments in the K (cid:48) th consecutive set. [In the case where the K (cid:48) sets were assumed to contain all local experiments, only condition 1) was relevantand hence the equality (S3) held.] Consider a particular causal configuration κ ∗ (1 I , · · · , n K (cid:48) ) compatible with [1 I , · · · , n I ] I , · · · ,[1 K (cid:48) , · · · , n K (cid:48) ] K (cid:48) (when the causal configuration κ (1 I , · · · , n K (cid:48) ) in the probability on the left-hand side of Eq. (18) is not compatiblewith the specified consecutive sets, that probability is trivially zero). Let us denote by 1 rest , · · · , l rest , l = n − (cid:80) K (cid:48) K = I (cid:80) n K m = m , therest of the local experiments, i.e, those that do not belong to the assumed first K (cid:48) consecutive sets. We have p ( κ ∗ (1 I , · · · , n K (cid:48) ) , [1 I , · · · , n I ] I , · · · , [1 K (cid:48) , · · · , n K (cid:48) ] K (cid:48) , o I , · · · , o g K (cid:48) | s , · · · , s n ) = p ( κ ∗ (1 I , · · · , n K (cid:48) ) , (1 K (cid:48) ≺ rest ∨ · · · ∨ n K (cid:48) ≺ rest ) , · · · , (1 K (cid:48) ≺ l rest ∨ · · · ∨ n K (cid:48) ≺ l rest ) , o I , · · · , o g K (cid:48) | s , · · · , s n ) . (S4)We will show that the probability on the right-hand side can be written as a linear combination of probabilities for which thecondition of causality (2) straightforwardly implies independence of s ( g + K (cid:48) , · · · , s n K (cid:48) .To this end, we write p ( κ ∗ (1 I , · · · , n K (cid:48) ) , (1 K (cid:48) ≺ rest ∨ · · · ∨ n K (cid:48) ≺ rest ) , · · · , (1 K (cid:48) ≺ l rest ∨ · · · ∨ n K (cid:48) ≺ l rest ) , o I , · · · , o g K (cid:48) | s , · · · , s n ) = p ( κ ∗ (1 I , · · · , n K (cid:48) ) , o I , · · · , o g K (cid:48) | s , · · · , s n ) − p ( κ ∗ (1 I , · · · , n K (cid:48) ) , ( K (cid:48) (cid:14) rest ) ∨ · · · ∨ ( K (cid:48) (cid:14) l rest ) , o I , · · · , o g K (cid:48) | s , · · · , s n ) , K (cid:48) = { K (cid:48) , · · · , n K (cid:48) } , and K (cid:48) (cid:14) l means 1 K (cid:48) (cid:14) l ∧ · · · ∧ n K (cid:48) (cid:14) l .By the definition of causality, the term p ( κ ∗ (1 I , · · · , n K (cid:48) ) , o I , · · · , o g K (cid:48) | s , · · · , s n ) on the right-hand side is in-dependent of s ( g + K (cid:48) , · · · , s n K (cid:48) . We need to prove that the second term is also independent of s ( g + K (cid:48) , · · · , s n K (cid:48) .Observe that the proposition “( K (cid:48) (cid:14) rest ) ∨ · · · ∨ ( K (cid:48) (cid:14) l rest )” is true when a proposition of the following kind istrue: for some nonempty subset of { rest , · · · , l rest } , say, { rest , · · · , r rest } , 1 ≤ r ≤ l , define the proposition“( K (cid:48) (cid:14) rest ) ∧ · · · ∧ ( K (cid:48) (cid:14) r rest ) ∧ ¬ [( K (cid:48) (cid:14) ( r + rest ) ∨ · · · ∨ ( K (cid:48) (cid:14) l rest )]”. The di ff erent nonempty subsets of { rest , · · · , l rest } yielddi ff erent such propositions that describe a complete set of mutually exclusive scenarios for which “( K (cid:48) (cid:14) rest ) ∨ · · · ∨ ( K (cid:48) (cid:14) l rest )”is true. Therefore, the probability p ( κ ∗ (1 I , · · · , n K (cid:48) ) , ( K (cid:48) (cid:14) rest ) ∨ · · · ∨ ( K (cid:48) (cid:14) l rest ) , o I , · · · , o g K (cid:48) | s , · · · , s n ) is a sum of probabili-ties of the form p ( κ ∗ (1 I , · · · , n K (cid:48) ) , ( K (cid:48) (cid:14) rest ) , · · · , ( K (cid:48) (cid:14) r rest ) , ¬ [( K (cid:48) (cid:14) ( r + rest ) ∨ · · · ∨ ( K (cid:48) (cid:14) l rest )] , o I , · · · , o g K (cid:48) | s , · · · , s n ),up to relabeling of 1 rest , · · · , l rest , where 1 ≤ r ≤ l . But every such probability can be further written as p ( κ ∗ (1 I , · · · , n K (cid:48) ) , ( K (cid:48) (cid:14) rest ) , · · · , ( K (cid:48) (cid:14) r rest ) , o I , · · · , o g K (cid:48) | s , · · · , s n ) − p ( κ ∗ (1 I , · · · , n K (cid:48) ) , ( K (cid:48) (cid:14) rest ) , · · · , ( K (cid:48) (cid:14) r rest ) , ( K (cid:48) (cid:14) ( r + rest ) ∨ · · · ∨ ( K (cid:48) (cid:14) l rest ) , o I , · · · , o g K (cid:48) | s , · · · , s n ). By the definition of causality, the first of these terms is independent of s ( g + K (cid:48) , · · · , s n K (cid:48) . Considering again the di ff erent realizations of “( K (cid:48) (cid:14) ( r + rest ) ∨ · · · ∨ ( K (cid:48) (cid:14) l rest )” by propositions of theform “( K (cid:48) (cid:14) ( r + rest ) ∧ · · · ∧ ( K (cid:48) (cid:14) ( r + q ) rest ) ∧ ¬ [( K (cid:48) (cid:14) ( r + q + rest ) ∨ · · · ∨ ( K (cid:48) (cid:14) l rest )]” for 1 ≤ q ≤ l − r (up to relabelingof the local experiments), the probability in the second term can again be written as a sum of probabilities of the form p ( κ ∗ (1 I , · · · , n K (cid:48) ) , ( K (cid:48) (cid:14) rest ) , · · · , ( K (cid:48) (cid:14) r rest ) , ¬ [( K (cid:48) (cid:14) ( r + rest ) ∨ · · · ∨ ( K (cid:48) (cid:14) l rest )] , o I , · · · , o g K (cid:48) | s , · · · , s n ), where now r is strictlylarger than the one in the previous step. We can continue this for every new term until we reach r = l . In this way, the probability p ( κ ∗ (1 I , · · · , n K (cid:48) ) , [1 I , · · · , n I ] I , · · · , [1 K (cid:48) , · · · , n K (cid:48) ] K (cid:48) , o I , · · · , o g K (cid:48) | s , · · · , s n ) is decomposed entirely into a linear combination ofprobabilities of the form p ( κ ∗ (1 I , · · · , n K (cid:48) ) , ( K (cid:48) (cid:14) rest ) , · · · , ( K (cid:48) (cid:14) r rest ) , o I , · · · , o g K (cid:48) | s , · · · , s n ), 1 ≤ r ≤ l , which by the definitionof causality are independent of s ( g + K (cid:48) , · · · , s n K (cid:48) . This completes the proof of Lemma S1. Proof S3. Theorem II.1 . The necessity of the form (40) follows from Proposition II.3. Indeed, let s M and o M denote thecollection of settings and outcomes, respectively, of the local experiments in a subset M ⊂ S . In terms of the set of parties X that are first, the probabilities of a causal process W S c can most generally be expanded p S c ( o S | s S ) = (cid:88) X⊂S , X (cid:44) {} p ([ X ] I | s S ) p ( o X | s S , [ X ] I ) p ( o S\X | s S\X , s X , o X , [ X ] I ) . (S5)But, as noted earlier, Proposition II.3 implies that p ([ X ] I | s S ) = p ([ X ] I ), and that p ( o X | s S , [ X ] I ) = p ([ X ] I , o X | s S ) / p ([ X ] I | s S ) = p ([ X ] I , o X | s X ) / p ([ X ] I ) = p X ns ( o X | s X , [ X ] I ) are the probabilities of a non-signaling process for X . We therefore have W S c = (cid:88) X⊂S , X (cid:44) {} p X W S\X|X ◦ W X ns , (S6)where p X = p ([ X ] I ). Next, if X (cid:44) S , we can similarly expand the probabilities of the process W S\X|X in terms of the set ofparties Y that are second: p ( o S\X | s S\X , s X , o X , [ X ] I ) = (cid:88) Y⊂S , Y (cid:44) {} p ([ Y ] II | s S , o X , [ X ] I ) p ( o Y | s S , o X , [ X ] I , [ Y ] II ) × p ( o S\ ( X∪Y ) | s S , o X , o Y , [ X ] I , [ Y ] II ) . (S7)Again, from Proposition II.3 we have that p ([ Y ] II | s S , o X , [ X ] I ) = p ([ X ] I , [ Y ] II , o X | s S ) / p ([ X ] I , o X | s S ) = p ([ X ] I , [ Y ] II , o X | s X ) / p ([ X ] I , o X | s X ) = p ([ Y ] II | s X , o X , [ X ] I ). Similarly, p ( o Y | s S , o X , [ X ] I , [ Y ] II ) = p ([ X ] I , [ Y ] II , o X , o Y | s S ) / p ([ X ] I , [ Y ] II , o X | s S ) = p ([ X ] I , [ Y ] II , o X , o Y | s X , s Y ) / p ([ X ] I , [ Y ] II , o X | s X ) = p Y ns ( o Y | s Y , s X , o X , [ X ] I , [ Y ] II ) are the probabilities of a non-signaling process for Y for each value of ( s X , o X ). In otherwords, we obtain that for each value of the events ( s X , o X ) in X , W S\X|X has the form (S6). The argument is completelyanalogous for the next conditional process that appears, W S\ ( X∪Y ) |X∪Y , which, if nontrivial, can be expanded in terms of thedi ff erent possibilities for the third consecutive set, and so on. This can be continued until we reach the last consecutive set inevery possible grouping of the parties into consecutive sets, which proves the necessity of the form (40).To prove su ffi ciency, we will show that if every process of the form (40) is causal for 1 ≤ n ≤ n (cid:48) , then the same must hold for n = n (cid:48) +
1. The general case then follows by induction from this and the fact that a monopartite ( n =
1) process, which has theform (40), is causal. Let an n (cid:48) -partite process have the form (40), i.e., its probabilities can be written p ( κ ( S ) , o S | s S ) = (cid:88) X⊂S , X (cid:44) {} p X p ns ( o X | s X ) p ( o S\X | s S\X , s X , o X ) , (S8)where the probabilities p ( o S\X | s S\X , s X , o X ) describe a conditional process W S\X|X c , which, if non-trivial, has an analogous formfor every possible value of ( s X , o X ). Such a conditional process is therefore causal for every possible value of ( s X , o X ) according4to our assumption. This means that there exists a probability distribution p ( κ ( S\X ) , o S\X | s S\X , s X , o X ), where κ ( S\X ) is thecausal configurations of the experiments in
S\X , such that (cid:80) κ ( S\X ) p ( κ ( S\X ) , o S\X | s S\X , s X , o X ) = p ( o S\X | s S\X , s X , o X ), whichfor every fixed ( s X , o X ) obeys the causality condition (2). We want to show that there exists a distribution p ( κ ( S ) , o S | s S ), where κ ( S ) is the causal configuration of all experiments S , such that (cid:80) κ ( S ) p ( κ ( S ) , o S | s S ) = p ( o S | s S ), which also obeys the causalitycondition (2). The following distribution will be shown to satisfy these desiderata: p ( κ ( S ) , o S | s S ) = p X p ns ( o X | s X ) p ( κ ( S\X ) , o S\X | s S\X , s X , o X )for κ ( S ) = [ κ ( S\X ); i ≺ j , ∀ i ∈ X , ∀ j ∈ S\X ; i (cid:14)(cid:15) j , ∀ i , j ∈ X ], p ( κ ( S ) , o S | s S ) = κ ( S ) (cid:44) [ κ ( S\X ); i ≺ j , ∀ i ∈ X , ∀ j ∈ S\X ; i (cid:14)(cid:15) j , ∀ i , j ∈ X ].According to this distribution, p X = p ([ X ] I ), and the causal configuration of all local experiments for [ X ] I is always such that each of the local experiments in X is in the causal past of all local experiments in S\X ,while the probability for the causal configuration and outcomes of
S\X given the events in X and the set-tings in S\X is p ( κ ( S\X ) , o S\X | s S\X , s X , o X ). The distribution p ( κ ( S ) , o S | s S ) has the correct marginal p ( o S | s S )by construction. To show that it satisfies condition (2), we will show that p ( κ ( S ) , o S | s S , [ X ] I ), which equals p ns ( o X | s X ) p ( κ ( S\X ) , o S\X | s S\X , s X , o X ) for κ ( S ) = [ κ ( S\X ); i ≺ j , ∀ i ∈ X , ∀ j ∈ S\X ; i (cid:14)(cid:15) j , ∀ i , j ∈ X ] and vanishes oth-erwise, satisfies this condition. The fact that the whole mixture p ( κ ( S ) , o S | s S ) = (cid:80) X⊂S , X (cid:44) {} p ([ X ] I ) p ( κ ( S ) , o S | s S , [ X ] I )satisfies it then follows from the linearity of the condition. Consider a given local experiment l ∈ S\X . Let X (cid:48) ⊂ X and Y (cid:48) ⊂ ( S\X ) \ l . We have p ( κ ( X (cid:48) , Y (cid:48) , l ) , l (cid:14) X (cid:48) , l (cid:14) Y (cid:48) , o X (cid:48) , o Y (cid:48) | s S ) = p ( κ ( X (cid:48) , Y (cid:48) , l ) , l (cid:14) Y (cid:48) , o X (cid:48) , o Y (cid:48) | s S ) = p ns ( o X (cid:48) | s X (cid:48) ) p ( κ ( Y (cid:48) , l ) , l (cid:14) Y (cid:48) , o Y (cid:48) | s l , s Y (cid:48) , s X , o X ) for κ ( X (cid:48) , Y (cid:48) , l ) = [ κ ( Y (cid:48) , l ); i ≺ j , ∀ i ∈ X (cid:48) , ∀ j ∈ Y (cid:48) ∪ l ; i (cid:14)(cid:15) j , ∀ i , j ∈ X (cid:48) ],and p ( κ ( X (cid:48) , Y (cid:48) , l ) , l (cid:14) X (cid:48) , l (cid:14) Y (cid:48) , o X (cid:48) , o Y (cid:48) | s S ) = p ( κ ( S\X ) , o S\X | s S\X , s X , o X ) satisfiescondition (2), it follows that p ( κ ( Y (cid:48) , l ) , l (cid:14) Y (cid:48) , o Y (cid:48) | s l , s Y (cid:48) , s X , o X ) = p ( κ ( Y (cid:48) , l ) , l (cid:14) Y (cid:48) , o Y (cid:48) | s Y (cid:48) , s X , o X ). This proves that p ( κ ( X (cid:48) , Y (cid:48) , l ) , l (cid:14) X (cid:48) , l (cid:14) Y (cid:48) , o X (cid:48) , o Y (cid:48) | s S ) = p ( κ ( X (cid:48) , Y (cid:48) , l ) , l (cid:14) X (cid:48) , l (cid:14) Y (cid:48) , o X (cid:48) , o Y (cid:48) | s S\ l ), which is condition (2). Similarly, if we take l ∈ X , consider two arbitrary subsets X (cid:48) ⊂ X\ l , Y (cid:48) ⊂ S\X . When Y (cid:48) (cid:44) {} , we have p ( κ ( X (cid:48) , Y (cid:48) , l ) , l (cid:14) X (cid:48) , l (cid:14) Y (cid:48) , o X (cid:48) , o Y (cid:48) | s S ) = Y (cid:48) = {} , we have p ( κ ( X (cid:48) , Y (cid:48) , l ) , l (cid:14) X (cid:48) , l (cid:14) Y (cid:48) , o X (cid:48) , o Y (cid:48) | s S ) = p ( κ ( X (cid:48) , l ) , l (cid:14) X (cid:48) , o X (cid:48) | s S ) = p ns ( o X (cid:48) | s X ) = p ns ( o X (cid:48) | s X (cid:48) ), whichagain proves that p ( κ ( X (cid:48) , Y (cid:48) , l ) , l (cid:14) X (cid:48) , l (cid:14) Y (cid:48) , o X (cid:48) , o Y (cid:48) | s S ) = p ( κ ( X (cid:48) , Y (cid:48) , l ) , l (cid:14) X (cid:48) , l (cid:14) Y (cid:48) , o X (cid:48) , o Y (cid:48) | s S\ l ), i.e., we have seen thatcondition (2) is satisfied for every l . This completes the proof of Theorem II.1. Proof S4. Proposition III.1 . The proof follows the idea of the proof for the bipartite case in Ref. [S4]. Here, we detail it forthe case of three parties. The n -partite follows analogously.Expanding the CJ operator of a local CP map in the Hilbert-Schmidt basis, M X X = (cid:80) µν r µν σ X µ σ X ν , r µν ∈ R , we observe thatthe trace-preserving condition Tr X M X X = X is equivalent to the requirement r = d X , r i = i >
0. Thus, CJ operatorscorresponding to CPTP maps are positive semidefinite operators of the form M X X = d X + (cid:88) i > a i σ X i + (cid:88) i , j > t i j σ X i σ X j , a i , t i j ∈ R . (S10)It turns out that condition (46) can be equivalently imposed only for operators M X X of the form (S10) without the constraint M X X ≥
0. Clearly, an operator W A A B B C C ··· that satisfies Eq. (46) for all operators M X X of the form (S10) satisfiesEq. (46) for positive semidefinite operators M X X of this form in particular. The converse follows from the fact that anyoperator M X X of the form (S10) can be written as a real linear combination of positive semidefinite operators of the form(S10): M X X = (cid:80) i α i M X X i , where M X X i ≥ i , and (cid:80) i α i = α i ∈ R , ∀ i . We will use this fact to recastcondition Eq. (46) as a statement about the types of non-zero terms in the Hilbert-Schmidt expansion of W A A B B C C ··· .In the case of three parties, the expansion of W A A B B C C reads W A A B B C C = (cid:88) i , j , k , l , m , n w i jklmn σ A i σ A j σ B k σ B l σ C m σ C n , (S11) w i jklmn ∈ R , ∀ i , j , k , l , m , n . (S12)Let us fix M A A = A A d A and M B B = B B d B , and consider an arbitrary M C C of the form (S10). Condition (46) becomes1 d A d B d C Tr[ W A A B B C C ( A A ⊗ B B ⊗ ( C C + (cid:88) n > c n σ C + (cid:88) mn > p mn σ C m σ C n ))] = , (S13)5TABLE S1: The types of terms that are forbidden in a tripartite process matrix W A A B B C C . C C C B B C B C C B B B B C B B C C A A C A C C A B A B C A B C C A B B A B B C A B B C C A A A A C A A C C A A B A A B C A A B C C A A B B A A B B C A A B B C C which, using the expansion of the process matrix, becomes d A d B d C ( w + (cid:88) n > w n c n + (cid:88) mn > w mn p mn ) = , (S14) ∀ c n , p mn ∈ R . This implies w = d A d B d C and w n = w mn = ∀ m , n > M A A = A A d A and M C C = C C d C , and considering an arbitrary M B B of the form (S10), we obtain w l = w kl = k , l >
0, while by fixing M B B = B B d B and M C C = C C d C , and considering an arbitrary M A A ofthe form (S10), we obtain w j = w i j = i , j > M A A = A A d A , and we use the previously obtained constraints, we obtain w l n = w kl n = w lmn = w klmn = ffi cients can be shown to vanish by suitably choosing the parameters in M B B and M C C in orderto select only the term with that coe ffi cient). Then, if we fix M B B = B B d B , we obtain w j n = w j mn = w i j n = w i j mn = M C C = C C d C , we obtain w j l = w jkl = w i j l = w i jkl = M A A , M B B , and M C C , of the form (S10). Using the constraints obtainedfrom the special cases above, we obtain w j l n = w j lmn = w jkl n = w jklmn = w i j l n = w i j lmn = w i jkl n = w i jklmn =
0. Thus,we have shown that all coe ffi cients w i jklmn , except for w , that may appear in the result of taking the trace of W A A B B C C with a general combination of M A A , M B B , M C C of the form (S10), must vanish. This is also a su ffi cient condition for thenormalization condition (46) to hold. All these forbidden terms for a process matrix are listed in Table (S1). Proof S5. Proposition III.2 .Explicitly, by the definition of (no) signaling (6) and the expression for the probabilities of a process in terms of the processmatrix (44), there is no signaling from (1 and 2 and · · · and k ) to ( k + k + · · · and n ) if and only if p ( o , · · · , o k + |{M o } , · · · , {M no n } ) ≡ (cid:88) o , ··· , o k Tr (cid:104) W ··· n n (cid:16) M o ⊗ · · · ⊗ M n n o n (cid:17)(cid:105) ≡ Tr (cid:20) W ··· n n (cid:18) M ⊗ · · · ⊗ M k k ⊗ M ( k + ( k + o k + ⊗ · · · ⊗ M n n o n (cid:19)(cid:21) = Tr (cid:104) W ( k + ( k + ··· n n (cid:16) M ( k + ( k + o k + ⊗ · · · ⊗ M n n o n (cid:17)(cid:105) ≡ p ( o k + , · · · , o k + |{M k + o k + } , · · · , {M no n } ) , for all local quantum operations {M o } , · · · , {M no n } , where M X i X i = (cid:80) o Xi M X i X i o Xi , ∀ i . Here, the operator W ( k + ( k + ··· n n is givenby W ( k + ( k + ··· n n = Tr ··· k k W ··· n n d · · · d k , (S15)which is obtained for the case where M i i = i i / d i , ∀ i = , · · · , k . This condition is equivalent to the condition thatTr ··· k k (cid:20) W ··· n n (cid:18) M ⊗ · · · ⊗ M k k ⊗ ( k + ( k + ··· (cid:19)(cid:21) = W ( k + ( k + ··· n n , ∀ M , · · · , M k k , (S16)where M , · · · , M k k are the CJ operators of CPTP maps (this is because any linear operator V ( k + ( k + ··· n n is fully deter-mined by the values of Tr (cid:104) V ( k + ( k + ··· n n (cid:16) M ( k + ( k + ⊗ · · · ⊗ M n n (cid:17)(cid:105) for all possible M ( k + ( k + ≥ , · · · , M n n ≥ ff erent types of terms in satisfying or violating condition (S16), consider the representation of W ··· n n as a linear combination of Hilbert-Schmidt terms of di ff erent types and the contribution that each such term makes to the quantityon the left-hand side of Eq. (S16). Assume that W ··· n n contains only terms of the types stated in Proposition III.2. The iden-tity term is such a term. When the identity term is partially traced with any combination of local CPTP maps M , · · · , M k k ,it yields exactly the right-hand side of Eq. (S16). From the rest of the terms that satisfy the condition in Proposition III.2, wecan distinguish two types. The first type are those that have a nontrivial σ operator on i and i for some i = , · · · , k .They yield zero when partially traced with any combination of local CPTP maps M , · · · , M k k , since a CPTP map M i i doesnot contain terms of type i (which is necessary to get a non-trivial partial trace with the term in question). The second type ofterms are those that do not have any nontrivial σ operator on any of the systems i and i , i = , · · · , k , and hence, when partiallytraced with any combination of local CPTP maps M , · · · , M k k , only the W ··· n n contains only the types of termsstated in Proposition III.2.To prove the reverse, assume that W ··· n n contains at least one term whose restriction onto 1 · · · k k is not a valid term fora process matrix for { , · · · , k } . Every such term has the form O α ⊗ σ α ⊗· · ·⊗ O α m ⊗ σ α m ⊗ α m + α m + ⊗· · ·⊗ α k α k ⊗ Q ( k + ( k + ··· n n ,where α i , i = , · · · , k , are di ff erent numbers from 1 to k , 1 ≤ m ≤ k , O α i is either the identity or some nontrivial σ operatoron α i , σ α i is a nontrivial σ operator on α i , and Q ( k + ( k + ··· n n is a non-zero operator on ( k + ( k + · · · n n , which isproportional to a tensor product of nontrivial σ operators and ff erent subsystems, such that the whole term is anallowed term for a process matrix. We want to show that if such a term is present in the process matrix, Eq. (S16) can beviolated for a specific choice of the local CPTP maps M , · · · , M k k . Out of all such terms, consider one for which m hasthe smallest value (there may be more than one of these). Consider the following choice of local CPTP maps constructed basedon this term: for i = m + , · · · , k , choose M α i α i = d α i α i α i , and for j = , · · · , m , choose M α j α j = d α j ( α j α j + (cid:15) α j O α j ⊗ σ α j ),where (cid:15) α j > α j α j + (cid:15) α j O α j ⊗ σ α j ) ≥ ffi ciently small non-zero (cid:15) α j ). Considerthe Hilbert-Schmidt expansion of the tensor product M ⊗ · · · ⊗ M k k . From this expansion, only the identity term and theterm proportional to O α ⊗ σ α ⊗ · · · ⊗ O α m ⊗ σ α m will survive when we plug M ⊗ · · · ⊗ M k k in the expression on the left-handside of Eq. (S16). This is because in order for any other term to survive, it would be necessary that W ··· n n contains a term ofa form similar to O α ⊗ σ α ⊗ · · · ⊗ O α m ⊗ σ α m ⊗ α m + α m + ⊗ · · · ⊗ α k α k ⊗ Q ( k + ( k + ··· n n but with a smaller value of m than theone we have chosen, which contradicts the assumption that we have chosen the smallest value. Plugging M ⊗ · · · ⊗ M k k inthe expression on the left-hand side of Eq. (S16) therefore yields W ( k + ( k + ··· n n + (cid:15) Q ( k + ( k + ··· n n for some (cid:15) ≥