correlations are said to be not directly accessible if they require perfect synchronization between some measurements. In this case at least part of these correlations involve si-multaneous measurements and are thus eveninaccessible in principle.
A v-causal model is then said to be quantum if every time its correlations are easily accessible they are also in agreement with the quantum prediction. v-causal models which are not quantum can in principle be detected experimentally, whereas quantum v-causal models are experimentally indistinguishable from quantum physics without extraordinary synchronization capabilities.
Even though both easily accessible and hardly accessible correlations are in principle measurable, we would like to say something aboutv-causal models independently of their typical speedv based only on the measurement of easily accessible correlations.
5.1.2 Influences without communication?
As mentioned in chapter 1, the fact that faster-than-light influences be needed in order to reproduce some correlations does not necessarily allow these correlations to be used to signal faster than light. Rather, a violation of the no-signalling conditions (1.1.2) is needed to allow correlations to be used for communication. The superluminal influences of a v-causal model can thus remain hidden from observers having only access to the produced correlations if these correlations are no-signalling. In particular, as long as the correlations produced agree with the quantum predictions, they are no-signalling and thus cannot be used to communicate.
Since all easily accessible correlations produced by a quantum v-causal model are quantum, simultaneous measurements must be considered in order to allow quantum v-causal models with arbitrary speedv to produce correlations diverging from the quantum prediction.
It was suggested in [93, 94] that the correlations predicted by av-causal model in this situation could become signalling and allow for faster-than-light communication. Here, we investigate this question in more detail, and show that it is indeed possible to communicate faster than light in a v-causal world in which all easily accessible correlations are in agreement with quantum physics.
Note that a first example of situation in which av-causal model was shown to allow for faster than light communication was put forward recently in [95]. However, this example requires the observation of supra-quantum correlations in order to conclude. It thus doesn’t apply to quantumv-causal models, and is not likely to lead to an experimental application.
We present below a general approach which allows one to test if the existence of signalling correlations can be deduced from the knowledge of potentially accessible cor-relations. We then examine whether such a test can be expected to be conclusive if correlations observed experimentally are assumed to be the ones predicted by quantum theory.
5.2 The hidden influence polytope
Following the above discussion, we consider a space-time configuration in which some measurements are simultaneous in order to open the possibility for quantum v-causal models to produce non-quantum correlations. We then examine whether the correlations
Finite-speed hidden influences
time
space A
D
B C
C0 B0
Figure 5.2: In the four-partite Bell-type experiment characterized by the space-time or-deringR= (A < D <(B ∼C)), no influence can be exchanged between Bob and Charly.
However, if Charly delays his measurement, he can allow the configuration to recover a complete order T1 = (A < D < B < C0). Similarly, Bob can delay his measurement in order to obtain the orderT2= (A < D < C < B0). .
produced by the model in this configuration can remain no-signalling or not2.
For definiteness, let us consider here the 4-partite space-time configurationR= (A <
D < (B ∼ C)) shown in Figure 5.2. A v-causal model in this situation must produce BC correlations that are local, even after conditioning on what happened at A and D (see paper [M] for more details). The correlations P(bc|yz, axdw) must thus satisfy all bipartite Bell inequalities X
bcyz
βbcyzi P(bc|yz, axdw)≤β0i (5.2.1) where{(β0i, βibcyz)}i denote the coefficients of all Bell inequalities that are relevant given the number of inputs and outputs of each party.
On the other hand, the correlationsP(abcd|xyzw) are no-signalling if and only if they satisfy the 4-partite no-signalling conditions:
X
a
P(abcd|xyzw) =P(bcd|yzw) , X
b
P(abcd|xyzw) =P(acd|xzw) X
c
P(abcd|xyzw) =P(abd|xyw) , X
d
P(abcd|xyzw) =P(abc|xyz). (5.2.2) No-signalling correlations produced by a v-causal model in the R configuration must thus satisfy both condition (5.2.1) and (5.2.2). Since these form a finite set of linear conditions, they define a polytope in the space of correlations (c.f. Appendix A). We refer to this polytope as thehidden influence polytope associated toR.
In order to test whether a v-causal model satisfies the above conditions, we need to know which correlations the model produces in theRconfiguration. But sinceBandCare measured simultaneously inR, the correlationsP(abcd|xyzw) are not directly accessible:
their observation requires perfect synchronization between some of the measurements.
Still, given the properties of v-causal models, one can show that some parts of the 4-partite distributionP(abcd|xyzw) can be deduced indirectly.
2Note that signalling could be activated in cases where the model only produces no-signalling corre-lations as well. Indeed, if a marginal probability distribution can have different (no-signalling) values depending on the time chosen by some other party to perform his measurement, in the fashion of [93, 94], this change in the correlation can allow to guess the time of measurement chosen by a distant party.
However we don’t consider this possibility here.
5.2 The hidden influence polytope
To see this, consider that Charly, in the experiment of Figure 5.2, could perform his measurement at C as planed initially or choose to delay it to C0 (or even to never do it). In any case, since he can in principle make his choice outside of the past v-cone of A, B and D, his choice cannot affect what happens at A, B and D. Thus, the ABD marginal produced by the model must be the same in the R configuration as in the T1 = (A < D < B < C0) configuration. Since correlations in the T1 configuration are easily accessible, the ABD marginal in the R configuration can be determined through measurements inT1.
Similarly, one can show that the ACD marginal in the R configuration must equal that in the T2 = (A < D < C < B0) configuration. It is thus also easily accessible, and both the ABD and the ACD marginal in the R configuration can in principle be known. TheBCmarginal is however clearly inaccessible experimentally since it explicitely requires measurements to be performed simultaneously. These two 3-partite marginals thus constitute the maximum amount of information that one can hope to infer about the R configuration.
In order to reach a conclusion without making assumptions on the value of the un-known marginals, we project the hidden influence polytope onto the subspace spanned by theABD andACDmarginals. This allows to deduce the conditions that are satisfied by thev-causal model in the situation R, in terms of the known marginals only.
Note that whenever an inequality satisfied by this projected polytope is violated, one of the two conditions (5.2.1) or (5.2.2) must be violated as well. Since (5.2.1) cannot be violated in the R configuration, by the definition of v-causality, the model must violate condition (5.2.2) in this configuration, i.e. produce signalling correlations.
Using techniques described in Appendix A, we could find several inequalities of the projected hidden influence polytope in configurationRwhen all parties use binary inputs and outputs. We present one of them below.
5.2.1 Quantum violation and faster-than-light communication
The following inequality is satisfied by all no-signalling correlations produced by av-causal model in theR configuration (c.f. Figure 5.2):
S=−3hA0i − hB0i − hB1i − hC0i −3hD0i
− hA1B0i − hA1B1i+hA0C0i + 2hA1C0i+hA0D0i+hB0D1i
− hB1D1i − hC0D0i −2hC1D1i
+hA0B0D0i+hA0B0D1i+hA0B1D0i
− hA0B1D1i − hA1B0D0i − hA1B1D0i +hA0C0D0i+ 2hA1C0D0i −2hA0C1D1i
≤7,
(5.2.3)
wherehAxi=PA(0|x)−PA(1|x), hAxByi=P
ab(−1)a+bPAB(ab|xy) and so on.
Recall that by construction this inequality only involves correlations that are easily accessible through some experiment. A quantumv-causal model can thus reproduce any value ofS that is achievable with quantum correlations.
Interestingly, this inequality can be violated by measuring a 4-qubit state (c.f. paper [M]). We can thus deduce that the corresponding quantumv-causal model must produce signalling correlations in theR configuration.
Finite-speed hidden influences
D0 time
space A0
B C
D A
Figure 5.3: In the configuration of Figure 5.2, letting the partiesB and C broadcast (at light-speed) their measurement results allows one to evaluate the marginal correlations BCDat the pointD0, which lies outside of the future light-cone ofA(shaded area). If this marginal depends on Alice’s input, it can thus be used for superluminal communication from A to D0. Similarly, if the ABC marginal correlations depend on the measurement wmade at D, superluminal communication is possible fromD to the pointA0.
Thanks to the geometry of this configuration, any signalling obtained in the correla-tions can be used to communicate faster than light as soon asv > c. Indeed, by definition ofv-causal models, signalling can neither happen fromB toACD, nor from C toABD, which lie in the past of C. It must thus happen either from A to BCD or from D to ABC. In both cases, this signalling in the correlations can be used to send signals faster than light (see Figure 5.3).
Finite-speed v-causal models for quantum correlations can thus be used to communi-cate faster than light.