From our everyday experience, we know that any transmission of information (i.e. com-munication) must be carried by a physical support: in order to let someone know about something we can say it to him, write him an SMS or a letter about it, etc, i.e. use either acoustic waves, electrons, or paper to carry this information to our friend. This idea can be expressed in the following principle:
No-signalling principle. Any transmission of information must be carried by a physical support leaving the emitter after the message is chosen.
This principle is satisfied by several if not all physical theories, including classical and quantum physics [8, 9]. In fact, the no-signalling principle is tightly related to quantum physics, since it can be seen restricts both the possibility of cloning quantum systems and the possibility of discriminating between quantum states, two peculiarities of quantum physics [10, 11].
In practice, many physical supports are available for communication in nature, like the ones mentioned above, but since the advent of special relativity, it is generally admitted that none of them can carry information faster than light in vacuum. The existence of an upperbound on the speed of all communication led Bell to enunciate the principle oflocal causality:
“The direct causes (and effects) of events are near by, and even the indirect causes (and effects) are no further away than permitted by the velocity of light.”
J. S. Bell [12]
In other words, the speed of light c= 29907920458m
s
, taken as an upper bound on any communication speed, naturally defines the limit between events in space-time which can have a direct causal relation with each other, and events which cannot. More precisely, for every event K the set of events that can be influenced by a decision taken at K is defined by its future light-cone. Similarly, all events that can influence a process taking place atK are contained in its past light-cone (c.f. Figure 1.1a).
1.1.1 Local correlations
It is of particular interest to ask which correlationsP(ab|xy) can be observed, according to the principle of local causality, in the situation of Figure 1.1b in which two measure-ments are performed in a space-like manner. Indeed, in this case, no information can be
Bell tests in bipartite scenarios
space time
past light-cone future light-cone
space a) time
K
b)
A B
a b
Λ1 Λ2 Λ3
Figure 1.1: a) Space-time diagram showing the regions containing events that can influ-ence, or that can be influenced by K according to Bell’s principle of local causality. b) Bipartite Bell experiment in which events creating the outcomes a and b are space-like separated. Apart from the inputs x and y which are chosen atA and B, the outcome a can depend only on the regions Λ1 and Λ2, and bonly on Λ2 and Λ3.
exchanged directly between the two measurement events. Letting the (free) choices of measurement settingx andybe made locally at the time of measurement inAandB, we see that the outcomeais produced by Alice’s measurement device before any information about Bob’s choice of measurementycan reach it. Thus,acannot depend ony. Similarly the production ofb by Bob’s measurement device cannot depend onx.
Still, the measurement processes atAand B can depend on more variables than only xand y. In particular, ais allowed to depend on the whole content of its past light-cone, including Λ2, a region of space-time which can also influence the process creating b, and Λ1 which cannot influence b. Let us thus denote by λ1, λ2, λ3 all variables which belong to the corresponding regions Λ1,Λ2,Λ3 and which are relevant to make predictions about a and b. The most precise prediction of a that can be given prior to measurement and in agreement with the principle of local causality can then be described by a probability distribution of the form PA(a|x, λ1, λ2). Allowing this distribution to also depend on λ3, resulting in a prediction of the form PA(a|x, λ1, λ2, λ3), can only gives a more precise description ofathan a locally causal theory1. Similarly, the distributionPB(b|y, λ1, λ2, λ3) describes predictions aboutbat least as well as any locally causal theory.
Since the processes happening at A and B cannot influence each other, they are independent. The average bipartite correlations produced in this situation must thus be of the form
P(ab|xy) =X
λ
p(λ)PA(a|x, λ)PB(b|y, λ) (1.1.1) whereλ= (λ1, λ2, λ3) and p(λ) is a probability distribution, i.e. p(λ)≥0,P
λp(λ) = 1.
We refer to this decomposition as thelocality condition. Any correlations which can be decomposed in this way are calledlocal, and conversely any correlations which admit no such decomposition are referred to as beingnonlocal.
Note that here the regions Λ1,Λ2,Λ3 in Figure 1.1b extend up to immemorial times and depend on the precise space-time positions of A and B. Bell showed that different regions Λ0 with nicer properties can be chosen in order to reach the same decomposition
1Remember thatxandyare only chosen atAandB, in a way that is independent ofλ1,λ2 andλ3. Allowing a prediction ofato depend onλ3thus still doesn’t allow it to depend ony.
1.1 No-signalling and local causality
(1.1.1). Namely, any region Λ0 that screens off the regions Λi that we considered here, i.e.
that already contains the information from Λ which is relevant to make predictions about aandb [13], is good enough to reach equation (1.1.1).
1.1.2 No-signalling correlations
If no decomposition of the form (1.1.1) exists for some correlationsP(ab|xy), some kind of influence must have taken place between the two measurement events. Yet, this influence might not be available through the correlations to transmit a message. Indeed, users having only access to the variablesa, b, x, ycan only encode a message to be carried from AtoB by the influences in the choice of their inputs xandy. And this message can only be decoded from the observation of the outcomesa and b. Thus, in order to be able to use some correlations to communicate, the statistics of one party’s outcome must depend on the other party’s choice of measurement. In other words they must violate one of the no-signalling conditions:
P(a|xy) =X
b
P(ab|xy) =P(a|x)∀ y P(b|xy) =X
a
P(ab|xy) =P(b|y) ∀ x.
(1.1.2)
Note that these constraints are also sufficient: ifP(b|xy) 6= P(b|x0y) for some x, x0, y, b, then Alice can always send a message to Bob by choosing between x and x0, and re-peating the experiment enough times to allow for Bob to discriminate between these two probabilities. Correlations satisfying the conditions (1.1.2) are calledno-signalling.
Violation of the no-signalling conditions allows for communication, which is very com-mon in nature. However, violation of these constraints between space-like separated events would allow for faster-than-light communication. Assuming that no physical sup-port can carry information faster than light, this would directly contradict the principle of no-signalling.
1.1.3 Geometrical representation
When talking about correlations, it is often useful to represent these probabilities in the vector space obtained by concatenating all components ofP(ab|xy). Let us briefly describe a few sets of correlations in this space.
For concreteness, we consider here the scenario where a, b, x, y = 0,1 can only take binary values. Every conditional probability distributionP(ab|xy) can then be represented as the vector
~
p= (P(00|00), P(10|00), P(00|10), P(10|10), . . . , P(11|11))∈R16 (1.1.3) which belongs to a 16-dimensional vector space. Since probabilities satisfy the normal-ization condition P
abP(ab|xy) = 1 ∀ x, y, the space spanned by the correlation vectors
~
p is in fact only 12-dimensional. Moreover, probabilities are always positive and must thus satisfy the constraints P(ab|xy) ≥ 0 ∀ a, b, x, y. This restricts the set of vectors ~p that correspond to valid correlationsP(ab|xy) within this 12-dimensional space. Since the number of positivity constraints is finite, the set of valid correlation vectors is a polytope (see Appendix A), which we refer to as thepositivity polytope.
Similarly, no-signalling correlations are normalized and positive. Moreover they sat-isfy the no-signalling conditions (1.1.2). These linear conditions define theno-signalling
Bell tests in bipartite scenarios
N S Q
L
Figure 1.2: Schematic representation of the set of local (L), quantum (Q), and no-signalling (N S) correlations. Note the inclusionL⊂Q⊂N S.
subspace, which is of dimension 8 here. The set of all no-signalling correlations is thus the slice of the positivity polytope with this subspace. This is again a polytope (see Appendix A), which is usually called theno-signalling polytope.
The set of local correlations, as defined by (1.1.1), can also be described by a polytope in the space of correlations. Indeed it is known [20] that any local correlation P(ab|xy) can be decomposed as a convex combination
P(ab|xy) =X
µ
p(µ)Pµ(ab|xy), p(µ)≥0, X
µ
p(µ) = 1 (1.1.4) of deterministic local strategiesPµ(ab|xy) = PA(a|x, µ)PB(b|y, µ)∈ {0,1}. On the other hand any convex combination of deterministic local strategies is also local. The set of local correlations thus corresponds to the convex hull of the deterministic local strategies.
Since the number of such strategies is finite, this set is also a polytope, thelocal polytope.
Whereas any inequality satisfied by the local polytope is a validBell inequality, the facets of this polytope aretight Bell inequalities.
Finally, it is also useful to characterize the set of quantum correlations. These corre-lations are all the ones which can be obtained by measuring a quantum stateρwith some local measurement operatorMa|x and Mb|y. They can thus always be written as
P(ab|xy) = tr(Ma|x⊗Mb|yρ). (1.1.5) Whereρ≥0, trρ= 1,Ma|x≥0 P
aMa|x=11,Mb|y ≥0 P
bMb|y =11.
The set of quantum correlations is convex, but admits an infinite number of extremal points. It is thus not a polytope. Nevertheless, it can be efficiently characterized by a hierarchy of semi-definite programs [21, 22]. While quantum correlations can violate Bell inequalities, these correlations always satisfy the no-signalling condition (1.1.2). The boundary of this set thus lies between the two preceding sets as represented in Figure 1.2.
Note that since quantum correlations can be nonlocal, they can require an exchange of influences between the measurement events. However these influences remains out of reach from us because quantum correlations satisfy the no-signalling condition.
1.1.4 Experimental loopholes
Knowing that quantum physics can violate the locality condition (1.1.1) is one thing.
Verifying that nature violates it is another, which requires the observation of a faithful