Relativistic Causality vs. No-Signaling as the limiting paradigm for correlations in physical theories

Relativistic Causality vs. No-Signaling as the limiting paradigm for correlations in physical theories

Abstract

The ubiquitous no-signaling constraints state that the probability distribution of the outputs of any subset of parties is independent of the inputs of the complementary set of parties; here we re-examine these constraints to see how they arise from relativistic causality. We show that while the usual no-signaling constraints are sufficient, in general they are not necessary to ensure that a theory does not violate causality. Depending on the exact space-time coordinates of the measurement events of the parties participating in a Bell experiment, relativistic causality only imposes a subset of the usual no-signaling conditions. We first revisit the derivation of the two-party no-signaling constraints from the viewpoint of relativistic causality and show that the they are both necessary and sufficient to ensure that no causal loops appear. We then consider the three-party Bell scenario and identify a measurement configuration in which a subset of the no-signaling constraints is sufficient to preserve relativistic causality. We proceed to characterize the exact space-time region in the tripartite Bell scenario where this phenomenon occurs. Secondly, we examine the implications of the new relativistic causality conditions for security of device-independent cryptography against an eavesdropper constrained only by the laws of relativity. We show an explicit attack in certain measurement configurations on a family of randomness amplification protocols based on the -party Mermin inequalities that were previously proven secure under the old no-signaling conditions. We also show that security of two-party key distribution protocols can be compromised when the spacetime configuration of the eavesdropper is not constrained. Thirdly, we show how the monogamy of non-local correlations that underpin their use in secrecy can also be broken under the relativistically causal constraints in certain measurement configurations. We then inspect the notion of free-choice in the Bell experiment and propose a definition of free-choice in the multi-party Bell scenario. We re-examine the question whether quantum correlations may admit explanations by finite speed superluminal influences propagating between the spacelike separated parties. Finally, we study the notion of genuine multiparty non-locality in light of the new considerations and propose a new class of causal bilocal models in the three-party scenario. We propose a new Svetlichny-type inequality that is satisfied by the causal bilocal model and show its violation within quantum theory.

I Introduction.

The recent experimental confirmation of the violation of Bell inequalities (1); (2) in systems of electron spins (3), entangled photons (4); (5), etc. has made a compelling case for the “non-locality” of quantum mechanics. Quantum phenomena exhibit correlations between space-like separated measurements that appear to be inconsistent with any local hidden variable explanation. The “spooky action at a distance” of quantum non-locality is now embraced and utilized in fundamental applications such as device-independent cryptography and randomness generation (7); (6); (8) and reductions in communication complexity (16). Moreover, this non-locality has also been used to show that even a tiny amount of free-randomness can be amplified (58); (8) and that extensions of quantum theory which incorporate a particular notion of free-choice cannot have better predictive power than quantum theory itself (71). The quantum non-local correlations are known to be fully compatible with the no-signaling principle, i.e., the space-like separated parties cannot use the non-local correlations to communicate superluminally (67).

Since the proposal of Popescu and Rohrlich (18), it has been realized that non-local correlations might take on a more fundamental aspect. Not only quantum theory but any future theory that might contain the quantum theory as an approximation is now expected to incorporate non-locality as an essential intrinsic feature. This program has led to the formulation of device-independent information-theoretic principles (28); (29) that attempt to derive the set of quantum correlations from amongst all correlations obeying the no-signaling principle. In parallel, cryptographic protocols have been devised based on the input-output statistics in Bell tests such that their proof of security only relies on the no-signaling principle. When one considers such post-quantum cryptography (53); (54), randomness amplification (58); (59); (8); (66), etc. the eavesdropper Eve is assumed to be limited to the preparation of boxes (input-output statistics) obeying a set of constraints collectively referred to as the no-signaling constraints. The general properties of no-signaling theories have been investigated (60) in a related program to formulate an information-theoretic axiomatic framework for quantum theory. On the other hand, quantum theory does not provide a mechanism for the non-local correlations. Several theoretical proposals have been put forward to explain the phenomenon of non-local correlations between quantum particles via superluminal communication between them. These models go beyond quantum mechanics but reproduce the experimental statistical predictions of quantum mechanics, the most famous of these models being the de Broglie-Bohm pilot wave theory (69); (70).

In all relativistic theories, “causality,” is imposed i.e., the requirement that causes must precede effects in all space-time rest frames. Before going further, two remarks are in order here. First, by an effect we mean any possible event, even if it has been affected by other events (causes) indirectly. Second, we shall use as general a correlation point of view as possible, regardless of the physical theory from which the correlations arise. In this context, let us also note that given an arbitrary space-time structure, the question of causal order for any two measures has been formalized using intuitions from optimal transport theory (78). Furthermore it has been proven that in any hyperbolic space-time, casuality of the evolution of measures supported on time-slices is an observer independent concept (79). Both formalisms have also been applied in the study of the evolution of the quantum wavepackets showing in particular the natural consistency of the causality concept with the relativistic continuity equation (80). From the perspective of communication, the requirement of relativistic causality strictly demands that no faster-than-light (FTL) transmission of information takes place between a sender and a receiver. The no-signaling principle being in ubiquitous use (in device-independent cryptography, axiomatic formulations, etc. as explained earlier), a natural question is to explore whether the no-signaling constraints that are currently in use precisely capture the constraints imposed by relativistic causality, i.e., to derive the no-signaling constraints from relativistic causality. In this paper, we investigate this question and find that, surprisingly, the answer is that the multi-party no-signaling principle requires a modification to capture the notion of causality. In particular, the actual no-signaling constraints that one must impose in a multi-party Bell experiment are dictated by the space-time configuration of the measurement events in the experiment. This modification of the no-signaling constraints naturally leads to a number of implications in tasks of device-independent cryptography against relativistic eavesdroppers and to explanations of quantum correlations via finite-speed superluminal influences. This paper is therefore, propaedeutic to a larger project undertaken by the authors to explore the relativistic causality constraints and their implications on the foundations of quantum theory.

The structure of the paper is as follows. We initially establish the setup of the Bell experiment and recall the assumptions in the Bell theorem. We then define the notion of relativistic causality that we use in this paper (and that is commonly accepted, i.e., that there be no causal loops in spacetime) and revisit the derivation of the two-party no-signaling constraints from causality. We then show that, in the multi-party scenario, only a restricted subset of the no-signaling constraints is required to ensure that no causal loops appear. We explicitly identify a region of space-time for the measurement events in a Bell scenario where the usual no-signaling constraints fail. In this regard, we extend a particular framework of “jamming” non-local correlations by Grunhaus, Popescu and Rohrlich in (19) based upon an earlier suggestion of Shimony in (15). We then examine the implications of the restricted subset of no-signaling constraints for device-independent cryptographic tasks against an eavesdropper constrained only by the laws of relativity. We detail explicit attacks on known protocols for randomness amplification based on the GHZ-Mermin inequalities using boxes that obey the new relativistic causality conditions. We show that from that perspective the security theory needs revision, and - in a way - to the some degree collapses. We also explore the implications on some of the known features of no-signaling theories (60), in particular we find that the phenomenon of monogamy of correlations is significantly weakened in the relativistically causal theories and that the monogamy of CHSH inequality violation (32) disappears in certain spacetime configurations. The notions of freedom-of-choice and no-signaling are known to be intimately related (58). We re-examine how the notion of free-choice as proposed by Bell and formalized by Colbeck and Renner (57); (56) can be stated mathematically within the structure of a space-time configuration of measurement events. A breakthrough result in (50) was a claim that any finite superluminal speed explanation of quantum correlations could lead to superluminal signaling and must hence be discarded. We re-examine this question in light of the modified relativistic causality and free-will conditions. Both non-relativistic quantum theory and relativistic quantum field theory (67) are well-known to obey a no superluminal signaling condition, proposals to modify quantum theory by introducing non-linearities have been shown to lead to signaling (40); (42); (41). We end with discussion and open questions concerning feasible mechanisms for the point-to-region superluminal influences.

Ii Preliminaries.

ii.1 Notation.

Let us first establish the setup of a typical Bell experiment. In the Bell scenario denoted , we have space-like separated parties, each of whom chooses from among possible measurement settings and obtains one of possible outcomes. In general, the number of inputs and outputs for each party may vary, but this will not concern us in this paper. The inputs of the -th party will be denoted by random variable (r.v.) taking values in and the outputs of this party will be denoted by r.v taking values in . Accordingly, the conditional probability distribution of the outputs given the inputs will be denoted by

(1)

Following (56); (58), we also consider the notion of a spacetime random variable (SRV), which is a random variable together with a set of spacetime coordinates in some inertial reference frame at which it is generated. A measurement event is thus modeled as an input SRV together with an output SRV . As in typical studies of Bell experiments, here we consider the measurement process as instantaneous, i.e., and share the same spacetime coordinates. Denote a causal order relation between two SRV’s and by if in all inertial reference frames, i.e., is in the future light cone of (so that may cause ). A pair of SRVs is spacelike separated if .

In this paper, we will have occasion to distinguish the specific spacetime location at which correlations between random variables manifest themselves, i.e., the particular spacetime location at which the correlations are registered, from the spacetime locations at which the random variables themselves are generated. Accordingly, we label by the SRV denoting the correlations between the output SRVs and with its associated spacetime location being at the earliest (smallest ) intersection of the future light cones of and in the reference frame .

ii.2 The two-party Bell theorem.

Consider the two-party Bell experiment with two spacelike separated parties Alice and Bob. The inputs of Alice and Bob are denoted by SRV’s and the outputs by . If the measurement process is considered to be instantaneous, and share the same space-time coordinates as do . In the two-party Bell scenario, the results of the experiment are described by the set of conditional probability distributions . Let denote a set of underlying variables describing the state of the system under consideration, these could in general be local or non-local. We have evidently

(2)

In this scenario, the Bell theorem (2) is based on the following assumptions (9); (10):

  1. Outcome Independence: The statistical correlations between the outputs arise from ignorance of the underlying variable

    (3)

    This is evidently true for deterministic models (those that output deterministic answers for inputs ) as well as for stochastic models and is motivated by a notion of realism, i.e., that the measurements merely reveal pre-existing outcomes encoded in .

  2. Parameter-Independence: For each so-called “microstate” , the probability of an outcome on Alice’s side is assumed to be (stochastically) independent of the experimental setting (the parameters of the device) on Bob’s side,

    (4)

    The justification for this assumption comes from special relativity, which imposes that spacelike separated measurements do not influence each other’s underlying outcome probability distributions.

  3. Measurement Independence: The measurement inputs are uncorrelated with the underlying variable

    (5)

    This is also sometimes called the free-choice assumption, i.e., that the inputs are chosen freely, independent of . We elaborate on the notion of freeness in this two-party Bell experiment in Section II.4 (56); (57).

There are other assumptions such as: Fair Sampling, No Backward Causation, Reality being single valued, etc. (10); (11) but these will not concern us in this paper. Substituting Eqs.(3, 2, 5) in Eq.(2), we obtain the description of a local hidden variable model:

(6)

Figure 1 shows the causal structure represented by the two-party Bell experiment represented in terms of a Directed Acyclic Graph (DAG) (72). Remark that the local hidden variable model in Eq.(6) can also be arrived at starting from other postulates such as local causality (12).

ii.3 Two-party no-signaling

From the assumptions of parameter-independence and measurement-independence, one can deduce the no-signaling constraints:

(7)

The no-signaling conditions in this two-party Bell experiment formally state that probability distribution of the outcomes of any party is independent of the input of the other party.

ii.4 Two-party freedom-of-choice

The assumption of measurement-independence or free-choice can be formally expressed in terms of spacetime random variables. In the two-party Bell experiment, one imposes the free-choice constraints

(8)

One formal way to define the freedom-of-choice condition in terms of spacetime random variables was formulated by Colbeck and Renner (CR) in (56). Recall that we denote a causal order relation by if in all inertial reference frames, i.e., is in the future light cone of so that may cause . CR formulate the notion of free-choice (formalising Bell’s notion from (57)) as follows:

Definition 1 ((56), (57)).

A spacetime random variable is said to be free if is uncorrelated with every spacetime random variable such that , i.e., for all such , we have .

In other words, is free if it is uncorrelated with any that it could not have caused, where the requirement for causing is that lies in the future light cone of . Clearly, Eq.(II.4) follows if one adopts this notion of freedom-of-choice, although note that the Def. 1 is strictly stronger than just the conditions (II.4) imposed in the Bell theorem.

ii.5 A sufficient set of multi-party no-signaling conditions

In this paper, our focus is on the no-signaling and free-will constraints above which are intimately related to each other. In particular, we consider the generalization of the no-signaling conditions to the multi-party scenario. The generalized multi-party no-signaling constraints are usually stated as follows (see for example (60)):

(9)

In words, the above constraints state that the outcome distribution of any subset of parties is independent of the inputs of the complementary set of parties (while Eq.(9) imposes this for subsets of parties, one can straightforwardly show that this also implies that the marginal distribution for smaller sized subsets is well-defined (60)). Now, given that as stated earlier the justification of the two-party no-signaling constraint came from the causality constraints of special relativity, the natural question which we investigate in this paper is to what extent the multi-party no-signaling constraints in Eq.(9) are imposed by the causality constraints of special relativity. In other words, while the constraints in Eq.(9) are clearly sufficient to ensure that no superluminal signaling takes place, we derive the set of necessary and sufficient constraints that ensure that no causal loops appear in the theory. We will see that the multi-party no-signaling conditions in fact depend on the space-time coordinates of the measurement events in the Bell experiment and we propose an appropriate modification of these constraints.

Iii Results.

The main results of the paper are as follows.

  1. We derive in Prop. 2 the no-signaling constraints in the two-party Bell experiment from relativistic causality and show that these constraints are both necessary and sufficient in this scenario.

  2. We show that in general, depending on the exact space-time coordinates of the measurement events of the parties participating in a Bell experiment, relativistic causality strictly only imposes a subset of the usual no-signaling conditions. In particular, we show that in the three-party Bell experiment, two sets of relativistic causal constraints are possible: the usual no-signaling conditions and a subset of the no-signaling conditions in Prop. 3 which ensure causality is preserved in certain measurement configurations (such as in Fig. 5).

  3. We geometrically analyze in Prop.5 the exact space-time region in the three-party Bell scenario where the constraints in Prop. 3 are necessary and sufficient to prevent causal loops.

  4. We propose in Definition 7 a rigorous definition of the free-choice constraints in the multi-party Bell experiment and show in Prop. 9 how these are compatible with the relativistic causality conditions.

  5. We examine the implications of the causality constraints on device-independent cryptography against an adversary constrained only by the laws of relativity. In the task of randomness amplification, we demonstrate in Prop. 12 an explicit attack that renders a family of multi-party protocols based on the -party Mermin inequalities insecure, when the measurement events of the honest parties conform to certain spacetime configurations. In the task of key distribution, we show in Prop. 14 that even two-party protocols can be rendered insecure unless assumptions are made concerning the spacetime location of the eavesdropper’s measurement event or shielding of the honest parties to any possible superluminal influences, even those respecting causality.

  6. We then examine properties that were previously considered to be common to all no-signaling theories. In particular, we show in Prop. 18 that the paradigmatic phenomenon of monogamy of correlations violating the CHSH inequality no longer holds in relativistically causal theories in certain measurement configurations.

  7. We re-examine, in Section XI the question whether quantum correlations may admit explanations by finite speed superluminal influences propagating between the spacelike separated parties, by modifying the argument from (50) against such -causal models to show that they would lead to causal loops in certain measurement configurations.

  8. Finally, we investigate the phenomenon of multiparty non-locality in light of the new considerations and propose in Def. 19 a new class of relativistically causal bilocal models in the three-party scenario. We formulate a new Svetlichny-type inequality in Lemma 21 that is satisfied by this class of models and examine its violation within quantum theory.

Iv Relativistic Causality.

In this paper, we work in the regime of special relativity, i.e., flat spacetime with no gravitational fields, this regime is valid within small regions of spacetime where the non-uniformities of any gravitational forces are too small to measure. We consider the causal structure of measurement events occurring at fixed spacetime locations . Within this regime, the relativistic causality constraint we consider is simply that :

  • No causal loops occur, where a causal loop is a sequence of events, in which one event is among the causes of another event, which in turn is among the causes of the first event.

Causality implies that for two causally related events taking place at two spatially separated points, the cause always occurs before the effect, and this sequence cannot be changed by any choice of a frame of reference. Relativistic Causality is a consequence of prohibiting faster-than-light transmission of information from one spacetime location to another space-like separated location in any inertial frame of reference. A violation of this condition would lead to the well-known “grandfather-paradoxes”. In other words, if an effect that occurs at a space-time location in an inertial reference frame precedes its cause that occurs at space-time location in (i.e., ) then an observer at may in turn affect the cause at and lead to a paradox. An explicit example of a closed causal loop is shown in the proof of Prop. 2.

A few remarks are in order. Firstly, note that faster-than-light propagation in one privileged frame of reference alone does not lead to causal loops. Secondly, note that the “principle of causality”, i.e., the invariance of the temporal sequence of causally related events is also directly related to the macroscopic notion of arrow of time from the second law of thermodynamics (38). Finally, we remark that in the General Theory of Relativity, the field equations allow for solutions in the form of closed timelike curves, and there has been much debate over these, with proposals such as the chronology protection conjecture and a self-consistency principle (47); (48) suggested to prevent time-travel paradoxes.

V Deriving No-Signaling constraints from Relativistic Causality.

v.1 Derivation of the two-party no-signaling constraints.

Let us first revisit the derivation of the no-signaling constraint from relativistic causality in the typical two-party Bell experiment. In any run of the Bell experiment, Alice freely chooses at spacetime location in , her measurement input and obtains (instantaneously) the output . Similarly, Bob who is at a space-like separated location, in the same run, freely chooses at his input and obtains an output . The requirement of space-like separation, i.e., , ensures that the measurement events fall outside each other’s light cone. The conditional probability distributions of outputs given the inputs that they obtain in the Bell experiment jointly constitute a “box” .

The causality constraint is the requirement that faster-than-light information transmission from Alice to Bob or Bob to Alice is forbidden, i.e., Alice and Bob cannot use their local measurements to signal to one another. Note that here, Alice and Bob choose their measurement inputs freely, i.e., we assume the free-choice constraints in Eq.(II.4).

Figure 1: The causal structure of the two-party Bell experiment represented by a directed acyclic graph. The inputs and outputs of the two parties are denoted by and respectively. The inputs are chosen freely so no arrows are pointed towards . The correlations between the outputs may be explained as caused by a shared random variable which may be local or non-local depending on the theory under consideration. The input of each party is prevented from influencing the output distribution of the other party due to the spacelike separation between them.
Proposition 2.

In the two-party Bell experiment, the usual no-signaling constraints

(10)

are necessary and sufficient to ensure that relativistic causality is not violated.

Proof.

Suppose by contradiction that one of the constraints in Eq.(2) was violated, for definiteness, let us suppose that the marginal distribution of Bob’s outputs depended upon the input of Alice, i.e., suppose

(11)

Note that by assumption, Alice chooses her measurement freely, i.e.,

(12)

so suppose that in some specific run Alice chooses the input at spacetime location . Since Alice and Bob’s measurement events are spacelike separated, Eq.(11) would imply physically that a superluminal influence propagated from Alice’s system to Bob’s system informing Bob of Alice’s input . Explicitly, consider that Alice and Bob share many copies of a system obeying (11), when Alice chooses input on all her subsystems, Bob guesses with a probability strictly larger than uniform by inspecting his local statistics . This is illustrated in Fig.2 where in a particular frame of reference labeled by axes (for concreteness, this can be taken to be the laboratory frame at which Alice and Bob’s systems are at rest), the superluminal influence is shown to be instantaneous, i.e., the signal travels parallel to the time axis and reaches Bob’s system at spacetime location . Now, consider the inertial reference frame labeled by of Charlie and Dave who are moving uniformly at some speed relative to . At the space-time location Charlie and Bob’s world-lines intersect, suppose Bob informs Charlie of the value at this point. Charlie immediately transmits this information via the same superluminal mechanism to Dave. In the frame , this superluminal influence again travels parallel to the time axis . The information about thus reaches Dave at spacetime location which is in the causal past of as shown in Fig. 2. Alice has thus managed to transmit the information about to her causal past. Dave may then transmit by a sub-luminal signal that reaches Alice at and Alice could now decide freely to not measure and measure instead. Therefore, we see that Eqs.(11) and (12) have resulted in a causal loop, which would lead to grandfather-style paradoxes. In order to prevent causal loops and preserve the notion of relativistic causality while still keeping the notion of free will in Eq.(12), we therefore impose the no-signaling condition

(13)

Analogous reasoning to prevent a closed causal loop starting from Bob’s measurement gives the other no-signaling condition

(14)

We have therefore derived the necessity of the two-party no-signaling constraints from the requirement that there be no closed causal loops. That these constraints are also sufficient is clear, since these ensure that the causal structure in Fig. 1 is maintained. Note that while the outputs can be correlated with each other, this correlation is attributed to the underlying (local or non-local) hidden variable rather than due to any superluminal influence propagating from one party to another.    

Figure 2: An explicit violation of causality occurs when the two-party no-signaling condition in Eq.(2) is violated as in Eq.(11). In the figure, we consider the violation in terms of Bob’s output distribution depending on Alice’s input. Since Alice’s input was chosen freely at , this dependence implies a superluminal signal containing the information was transmitted from to at , here the signal is shown to travel at infinite speed, i.e. parallel to the -axis. The world-lines of Charlie and Dave who move at uniform velocity relative to Alice and Bob are also shown. At , Charlie’s world line interesects Bob who informs Charlie of . Charlie then uses the same superluminal signal to send to Dave at , this signal travels parallel to the -axis as shown. Note that is in the causal past of Alice, so Dave can send via a subluminal signal to Alice. Alice can then freely chose to not measure and measure instead. Thus, we have a closed causal loop resulting in a violation of causality.
Figure 3: In an inertial reference frame the two (spatially separated) events and appear to occur simultaneously, i.e., . In the inertial reference frame , event occurs before , i.e., . In another inertial reference frame on the other hand, event occurs before , i.e., . To maintain causality, we must have that the output distribution at must be independent of the input at , and similarly must be independent of the input at .

This situation is illustrated in the space-time diagram of the measurement process in Fig. 3. As seen in the figure, while causal relationships are manifestly Lorentz invariant, the specific time sequence of events changes in different inertial frames of reference, in particular spacelike separated events have no absolute time ordering between them. In the reference frame, the measurement events of Alice and Bob occur simultaneously. In the inertial reference frame, Bob’s measurement precedes that of Alice (), while in the reference frame, Alice’s measurement precedes that of Bob (). We see that the imposition of the constraints in Eq.(2) prevents any causal loop (and resulting “grandfather paradoxes”), or in other words relativistic causality is guaranteed by the two-party no-signaling principle. Note that the “freedom-of-choice” condition that Alice and Bob are allowed to choose their measurements freely (using a private random number generator, for example) is necessary in the argument above to identify the choice of measurement by Alice as cause and observation by Bob of his output distribution as effect. Furthermore, remark that it is essential to consider different inertial reference frames to make the argument, no causality violation would occur if faster-than-light propagation occurred in only one reference frame.

As shown by Eberhard in (67), the no-signaling requirements are satisfied in both non-relativistic quantum theory and in the relativistic quantum field theory. In quantum field theory, this requirement would be ensured by the vanishing of commutators of field operators and representing Alice and Bob’s observables respectively, i.e., (67). In the non-relativistic quantum theory that is usually used to analyze Bell experiments, the description of the Alice-Bob composite system by a density operator in the tensor product space , the description of quantum operations by local Kraus operators acting on the respective Hilbert spaces and the partial trace rule ensure that the local statistics only depend on the reduced density matrices of the respective party, and no superluminal communication even using entangled states is possible.

v.2 Modification of the three-party no-signaling constraints to the relativistic causality constraints.

The causal structure of the three-party Bell experiment is shown in Fig. 4. Alice’s spacetime random variables corresponding to her input and output are generated at spacetime location in inerial reference frame , similarly Bob’s input-output are generated at and Charlie’s input-output are generated at . We now investigate the question:

  • What are the necessary and sufficient conditions in the three-party Bell scenario that ensure that no causal loops appear?

We shall see that the answer depends upon the exact spacetime locations of the measurement events in the Bell experiment.

Proposition 3.

Consider the three-party measurement configuration shown in Fig. 5, where in an inertial frame , the spacetime locations of the measurement events , and are such that the intersection of the future light cones of and is contained within the future light cone of . The necessary and sufficient constraints to ensure that no causality violation occurs in this configuration are given by

Proof.

The constraints in Eq.(3) guarantee that the marginal distributions , , and are well-defined. Firstly, notice that the fact that the marginal is also well-defined for all is guaranteed as a consequence of these constraints by the relation

where we used the first and second equalities from Eq.(3) successively. The constraint that each party’s marginal distribution is well-defined is necessary in order not to violate causality by the two-party result from Prop. 2, i.e., if either Alice’s or Charlie’s output statistics exhibited a dependence on Bob’s input ( or ) then by the Prop. 2, Bob could signal his input to his causal past and a causal loop would result.

We now move to the two-party distributions. Here, compared to the usual no-signaling constraints in Eq.(9), the constraints in Eq.(3) only ensure that the A-B joint distribution and the B-C joint distribution are well-defined independent of the remaining party’s input. The fact that this is necessary can be seen as follows. In the measurement configuration in Fig. 5, the spacetime random variable corresponding to the correlations between the outcomes of and manifests itself at the intersection of the future light cones of and , and in particular, can be verified outside the future light cone of . Any dependence of the joint distribution on the input at can then result in a causal loop, following an analogous reasoning to the two-party result in Prop. 2. Similarly, we see that it is necessary for to be well-defined independent of the input at .

We now move to see the sufficiency of the four constraints in Eq.(3). To see this, we note that as opposed to the usual sufficient no-signaling constraints (9), these constraints do not ensure that the marginal is well-defined. In other words, the joint output distribution of Alice and Charlie can explicitly depend upon Bob’s input as even though the marginal distribution of each party does not depend on . This implies that a superluminal influence propagated from Bob’s system to the joint system of Alice and Charlie altering their joint distribution while keeping their marginal distributions unaffected. The spacetime random variable corresponding to the correlations between the outputs of and only manifests itself at the intersection of the future light cones of and . Briefly, when this intersection is contained within the future light cone of , is timelike separated from and hence any influence of by the choice of input at does not lead to signaling. More formally, the argument is stated as follows.

We want that the choice of input at does not signal to any space-time location via the change of correlations . Now, two possibilities (attempts to signal to via ) arise.

  1. Alice at and Charlie at transmit their outputs by a light signal at speed to . In this case, must be contained in the intersection of the future light cones of and . Now, the crucial property of the measurement configuration of Fig. 5 is that the intersection of the future light cones of and is contained within the future light cone of . Therefore, is timelike separated from . Formally, the sum of the time taken for a superluminal influence (at any arbitrary speed ) to move from to and the time taken for a light signal (at speed ) to move from to is less than the time taken for a light signal at speed to travel from to directly. A similar condition holds for the influence traveling via . Mathematically, these constraints are captured by the following equations

    (17)

    where denotes the time taken for a signal at speed to travel from to , denotes the time taken for a light signal (at speed ) to travel from to , and so on. The fact that is always in the future light cone of ensures that no superluminal transmission of information takes place.

  2. Alternatively, Alice (or Charlie or both) transmits her output via a subsequent measurement and a similar superluminal influence as Bob did, i.e., Alice superluminally influences the measurement events at two spacelike separated locations and (with the property that the intersection of the future light cones of and is contained within the future light cone of ) in such a way as to change the joint distribution while retaining the marginal distributions and . If and subsequently send a light signal, then as in the previous argument, the output at is available only in a location in the future light cone of . The intersection of the future light cones of and is directly seen to be contained within the future light cone of and which is in turn contained within the future light cone of , so that the information about the outputs at and is again only available at a timelike separation from . Similarly, if and subsequently send a superluminal influence as Bob did, then we repeat the argument to see that any concentration of information only occurs within the future light cone of .

We therefore deduce that in measurement configurations such as in Fig. 5, the constraints in Eq.(3) are necessary and sufficient as opposed to the usual three-party no-signaling constraints from Eq.(9) which while being sufficient are not strictly necessary in order to prevent a violation of causality.

Example 4.

As a simple illustrative example of the above, consider a box with binary inputs and binary outputs for the three parties of the form

(18)

This box deterministically outputs for both inputs and has marginals of the form

Here is a local box that returns correlated outcomes for any input with uniform marginals for (such a local box is generated by shared randomness between and ), and is the Popescu-Rohrlich box (18) that returns outcomes satisfying also with uniform marginals and . We see that such a box explicitly satisfies the relativistic causality constraints from Prop. 3 since the local marginals , and are well-defined (independent of the inputs of the other party), and the two-party marginals and are given explicitly as

(19)

which are also well-defined, independent of the input of the remaining party.

This relativistically causal box maximally saturates the three-party Mermin-type expression (even beyond the customary promise on the input set ), since we have for that and for that . However, despite this maximal violation, notably in this box, the outputs are always deterministic.

Figure 4: The causal structure of the three-party Bell experiment. The outputs are correlated via a common . The inputs are chosen freely according to a notion of free-choice (see Section VIII). The input cannot signal to change the distribution of a remote party’s output or , analogously for inputs and . On the other hand, depending upon the spacetime configuration of the measurement events, the input may influence the correlations between remote parties’ outputs without violating causality. In particular, in the measurement configuration shown in Fig. 5, the input may influence the correlations .
Figure 5: A particular spacetime configuration of measurement events in the three-party Bell experiment. The spacetime locations of Alice, Bob and Charlie’s measurement events in some inertial reference frame labeled by axes are given by , and respectively. The correlations between the outputs of Alice and Bob is denoted by a spacetime random variable which manifests itself (is checked by Alice and Bob) at a spacetime location within the intersection of the future light cones of Alice and Bob. Similarly, the correlations between the outputs of the other pairs of parties is denoted by and . The crucial property of this measurement configuration is that the intersection of the future light cones of and is contained within the future light cone of .

Having seen that in certain measurement configurations, the set of constraints in Eq.(3) as opposed to the full set of no-signaling constraints is already necessary and sufficient to ensure no violation of causality, we now identify the spacetime configurations where this occurs. To do this, we consider the situation where the superluminal influence propagates at some fixed speed in an inertial reference frame . We slightly alter the notation and consider the situation from a cryptographic perspective with Bob replaced by Eve with input-output and Alice-Charlie replaced by Alice-Bob . Thus, we fix the spacetime coordinates and of Alice and Bob, and investigate from which spacetime location Eve is able to influence the correlations without affecting the marginal distributions and , i.e., in which measurement configurations (3) is necessary and sufficient to prevent the violation of causality. Accordingly, our result is then the following Proposition 5 whose proof is in the Supplemental Material.

Proposition 5.

Consider measurement events with corresponding spacetime coordinates , in a chosen inertial reference frame . Then a measurement event can superluminally influence the correlations between and at speed without violating causality in I if and only if its space coordinate satisfies

(20)

for any circle with as a chord and having angle as the angle in the corresponding minor segment, where and , and if its time coordinate satisfies

(21)

Outside of the spacetime region in Proposition 5, the usual no-signaling constraints from (9) turn out to be necessary and sufficient to ensure causality is preserved. Figure 4 shows the causal structure represented by the three-party Bell experiment represented in terms of a Directed Acyclic Graph (DAG) (72). We can now consider the implications of the spacetime structure on the causal relationship, namely we can consider the study of Bayesian networks of spacetime random variables (SRVs) in light of the considerations of this section, a study which we pursue in future work. Recall that a Bayesian network, or probabilistic directed acyclic graphical model is a probabilistic graphical model (a type of statistical model) that represents a set of random variables and their conditional dependencies via a directed acyclic graph (DAG). Now, in the DAG that represents a causal structure in the three-party scenario, an additional ingredient of a new type of edge representing a causal link between the spacetime variable and the effects would be added depending on the spacetime location of the measurement events. This edge represents the causal link that does not influence the marginal distributions of and themselves but changes the joint distribution of . The additional ingredient when considering spacetime variables is that the random variable representing the correlations between and is only created in the future light cone of provided that the intersection of the future light cones of and is contained within the future light cone of . The causal network corresponding to a particular spacetime arrangement of measurement events would incorporate the additional edges as possible influences that do not lead to superluminal signaling and respect causality.

v.3 Multi-party Relativistic Causality.

We now extend to the general -party scenario the considerations of the previous subsections. In the mutli-party case, different subsets of the usual no-signaling constraints are sufficient to preserve causality, depending on the spacetime locations of the measurement events of the parties. Here, we focus on an extension in the simplest -dimensional scenario, when the parties are arranged in a line in some reference frame.

Consider parties with respective fixed laboratory space coordinates in frame . Let us identify the possible space-time region from which a party can influence the correlations between the systems of the parties. Clearly, the time coordinates of the parties at which the influence traveling at in frame can be felt are given by

(22)

By the results of the previous section, we know that to influence two-party correlations for parties and where , must belong to

(23)

All points in the intersection of over all pairs , i.e., <