Chained Bell inequality

On top of these inequalities, we also tested theN-settings chained Bell inequality, which can be written as

I^N =E₁₁+E₁₂+E₂₂+. . .+E_{N N} −E_N1 ≤2(N−1) =I_L^N. (1.4.3) The values obtained experimentally are reported for N ≤ 6 in table 1.2. The chained inequality has a number of applications which we mention below.

N I_L I^exp I^tom I^exp−I_L

p_noise (%) (σ units)

2 2 2.731 ±0.015 2.683 49 27

3 4 4.907 ±0.019 4.925 48 18

4 6 7.018 ±0.023 6.999 44 15

5 8 8.969 ±0.026 8.996 37 11

6 10 10.91 ±0.028 10.954 33 8

Table 1.2: Measurement of the chained inequalities withN settings per side.

Randomness certified by the no-signalling principle

An interesting property of the chained Bell inequality is that the marginal probabilities P(a|x) and P(b|y) tend to 1/2 as the violation of the inequality increases. This allows one to certify that the outcomes produced by measuring the quantum system must be truly random, in the sense that no algorithm can possibly predict the measurement out-comes [31].

More precisely, the amount of true randomness that could be extracted from the ex-perimental results found by Alice can be evaluated by finding the largest marginal prob-abilityP^∗(a|x) which is compatible with the measured Bell inequality violationI^exp. We

Bell tests in bipartite scenarios

2Nï2 2Nï3/2 2Nï1 2Nï1/2 2N

0.5 0.6 0.7 0.8 0.9 1

I^N P*(a|x)

Quantum bound

Noïsignaling bound for all N

N=4N=5 N=6 N=2 N=3

Figure 1.8: Maximum marginal probability compatible with a violation of theN-settings chained inequality. The bound implied by the no-signalling principle is identical for all N.

performed this optimization over the set of quantum correlations as well as among all no-signalling correlations. The result is shown in Figure 1.8 together with the experimentally achieved valuesI^exp.

The strongest bound imposed by the no-signalling principle is P^∗(a|x) = 0.7455± 0.0057, achieved for the inequality with N = 4 settings. This allows one in principle to extractH_min(a|x) =−log₂P^∗(a|x) = 0.41±0.01 random bits per run.

EPR2 local part

Another property of the chained inequality is that its maximum quantum value I_Q^N = 2Ncos _2N^π

, achievable by measuring a singlet state [32], approaches the no-signalling boundI_{N S}^N = 2N as the number of settingsN increases. This allows one to conclude that the singlet state has no local part in the sense of EPR2 [33].

Indeed, if a fractionp_L of the measured pairs would behave locally during an experi-mental evaluation of the chained inequality (1.4.3) yielding the valueI^exp, the following equation would hold:

I^exp =p_LI_L^N + (1−p_L)I_{N L}^N , (1.4.4) where I_L^N is the value ofI^N achieved with the local pairs of particles, and I_{N L}^N a value of the same expression achieved on the rest of the particles. Since the following bounds hold: I_L^N ≤2(N−1) andI_{N L}^N ≤I_{N S}^N = 2N, the local partp_Lof the measured states must be bounded by

pL≤p^max_L =N −I^exp

2 . (1.4.5)

For I^exp = I_Q^N, we find p^max_L = N 1−cos _2N^π N→∞

−→ 0. Thus, for every number of settingsN, testing the chained inequality can provide an upperbound on the local content of the state measured which eventually converges to 0.

In our case, the best bound on pL is found for N = 4 settings, yielding p^max_L = 0.491±0.012. While recent work could demonstrate an even lower value [34], this simple

1.4 Experimental violation of Bell inequalities with a commercial source of entanglement

experiment already shows that at least half of the photon pairs produced by the source are nonlocal.

1.4.4 Conclusion

In this experiment, we relied on the fair sampling assumption because the single pho-tons detectors were not efficient enough to close the detection loophole. Moreover the detection events were not space-like separated. Yet, this experiment shows that a simple demonstration of several interesting results of quantum information theory is nowadays possible with modest equipment.

Bell tests in bipartite scenarios

Chapter 2

Nonlocality with three and more parties

In the precedent chapter, we focused on Bell-type experiments involving two parties only.

While this is the simplest case, and indeed the most often discussed one, the idea of local correlations can be extended straightforwardly to multipartite scenarios involving more parties.

Labeling the (output,input) of a third party Charly by (c,z), the locality condition (1.1.1) generalises to:

P(abc|xyz) =X

λ

p(λ)P(a|x, λ)P(b|y, λ)P(c|z, λ), (2.0.1) and similarly for more parties. Tripartite correlations P(abc|xyz) are then referred to as nonlocal if and only if they cannot be decomposed as (2.0.1).

2.1 Defining genuine multipartite nonlocality

Just like entanglement can have more forms in a multipartite scenario than in the bipar-tite case [35], it is easy to realize that the definition (2.0.1) does not capture the whole potential of nonlocality in a tripartite scenario. Consider indeed some bipartite nonlocal correlationsP_AB(ab|xy) and arbitrary statistics for Charly P_C(c|z). The product of the two distributionsP(abc|xyz) = PAB(ab|xy)PC(c|z) violates (2.0.1) and is thus nonlocal.

However it is clear that Charly plays no role in the nonlocality of these correlations. These correlations are thus not genuinely three-way nonlocal.

This observation was first made by Svetlichny in 1987 [36], who proposed an inequality capable of certifying (if violated) that correlations cannot be explained by a mechanism involving fewer than 3 parties. This is the Svetlichny inequality

S =E111+E112+E121−E122+E211−E212−E221−E222 ≤4 (2.1.1) withE_xyz =P₁

a,b,c=0(−1)^a+b+cP(abc|xyz), which is satisfied by all tripartite correlations of the form

P(abc|xyz) =X

λ

p1(λ)P_AB(ab|xy, λ)P_C(c|z, λ)

+X

λ

p2(λ)PAC(ac|xz, λ)PB(b|y, λ) +X

λ

p3(λ)PBC(bc|yz, λ)PA(a|x, λ)

(2.1.2)

Nonlocality with three and more parties

withp_i ≥0 andP

λp₁(λ) +p₂(λ) +p₃(λ) = 1.

If being unable to decompose some tripartite correlations P(abc|xyz) in the form of (2.1.2) is sufficient to conclude that none of the parties was separated from the other ones in the process that created these correlations, it was pointed out recently that this condition is not always necessary (see paper [N] and [37]).

To understand why this is the case, let us consider the situation in which the three measurement events producing a, b, and c, are not simultaneous but follow an order:

Alice measures first, then Bob, and finally Charly (A < B < C). If decompositions of the form (2.1.2) exist for the observed correlations we might want to conclude that these correlations can be reproduced by some interaction between pairs of parties. Yet, this is not possible if every such decomposition happens to containP_AB terms that are signalling from B to A, i.e. such that P

bP_AB(ab|xy, λ) depends on y. Indeed, in the considered configuration (A < B < C), y can always be chosen freely aftera. The distribution of a thus cannot depend ony.

It thus seems important, from a physical point of view, to consider decompositions (2.1.2) that are compatible with the situation in which the correlations are produced.

In order to conclude something about the nature of correlations that is independent of the situation in which they appear, we suggest to require a consistent decomposition (2.1.2) to exist for all possible measurement situations. Thus, we say that correlations areSvetlichny-sequential iff they can be decomposed as

P(abc|xyz) =X

λ

p₁(λ)P^TAB(ab|xy, λ)P_C(c|z, λ)

+X

λ

p₂(λ)P^TAC(ac|xz, λ)P_B(b|y, λ) +X

λ

p₃(λ)P^TBC(bc|yz, λ)P_A(a|x, λ) (2.1.3) for every possible ordering of the measurements. Non-Svetlichny-sequential correlations are then called genuinely tripartite nonlocal. For correlations that are not genuinely tripartite nonlocal in this sense, a biseparable model cannot be constructed coherently for all possible ordering of the measurements. HereP^TAB(ab|xy, λ) depends on the order of measurement between Alice and Bob. Namely, P^TAB(ab|xy, λ) = P^A<B(ab|xy, λ) = P(a|x, λ)P(b|y, axλ) if Alice measures first, and P^TAB(ab|xy, λ) = P^B<A(ab|xy, λ) = P(b|y, λ)P(a|x, byλ) if Bob measures first.

Note that the problem we just mentioned can also be avoided by requiring all terms ap-pearing in (2.1.2) to be no-signalling. Namely, if all bipartite correlations termsP_AB(ab|xy), P_AC(ac|xz) andP_BC(bc|yz) satisfy the no-signalling constraints (1.1.2) in equation (2.1.2), then it is always possible to reproduce these correlations with communication between only two parties, independently of the order in which the parties perform their measure-ments. We call correlations having such a decomposition Svetlichny-no-signalling. One can show that the condition obtained by requiring terms in (2.1.2) to be no-signalling is strictly stronger than the one given by equation (2.1.3) (which is also strictly stronger than (2.1.2) without the no-signalling requirement). Some correlations can thus be repro-duced with communication between two parties only, and in a way that is consistent with all possible orders of the measurement, even though they are not Svetlichny-no-signalling (see paper [N] for more information about that).

To conclude, note that the three definitions we just discussed generalize straightfor-wardly to scenarios involving an arbitrary number of parties. Moreover, they all reduce to the locality condition (1.1.1) in the bipartite case. This is possibly one reason why details about the time ordering of the measurement events is usually not discussed when talking about nonlocal correlations. Yet, a precise account of the order in which measurements