As mentioned before, we shall estimate bounds on the MS polynomials for two classical communication models, namely the grouping and thebroadcasting models.
In the two models, thenparties have access to shared randomness and are allowed to communicate their inputs to some subset of the other parties. Given the information available to them, each party then produces a local output. The two models depend on a parameter m (or k =n−m) which quantifies the extent of multipartite non-locality.
• Grouping: Then parties are grouped into m subsets. Within each group, the parties are free to collaborate and communicate to each other, but are not allowed to do so between distinct groups.
Figure 6.1: Thegrouping (left) and thebroadcasting (right) models forn= 4 parties.
Grouping: Within each group, every party can communicate its input to any other party, as indicated by the arrows. a) If all parties join into one group (m = 1), they can achieve any correlation. b) andc) If they split into m= 2 groups they can realize some non-local correlations but not all; our tests do not distinguish between these two ways of forming 2 groups. d) If they are all separated (m =n), they can only reproduce local correlations.
Broadcasting: The broadcasting parties communicate their input to all other parties (small arrows). e) Only one party broadcasts its input; the n parties can realize some non-local correlations but not all. f) If k= 3 = n−1 parties broadcast their input, they can also achieve any correlation (recall that we restrict our analysis to correlation functions).
• Broadcasting: Out of the n parties, k = n −m of them can broadcast their input to all other parties. The remainingmparties cannot communicate their input to any other party.
The two models are illustrated in Figure 6.1, for the case n= 4.
In the measurement scenario that we consider in this work, restricted to correla-tion funccorrela-tions, a model withk =n−mbroadcasting parties is more powerful than a communication model withmdisjoint groups. Indeed, consider the grouping model;
since in the correlation coefficients ha1. . . ani it is only the product of all outputs that matters and not each output individually, we can assume that within each of the m groups, all parties send their input to one singled-out party, who can then produce the correct output for the entire group. The m parties thus singled-out (one for each group) do not need to send any input, but receive inputs from other parties. This situation clearly involves less communication than the broadcasting model.
Yet, we find that the bounds on the MS expressions are identical for both models.
Let us indeed state our main result:
Theorem. For both the grouping and the broadcasting models,
|Snm| ≤2(n−m)/2. (6.5)
Moreover this bound is tight, i.e., for each model there exists a strategy that yields
|Snm| = 2(n−m)/2 (in the case of the grouping model, this is true for any possible grouping of the n parties into m groups).
We refer to [H] for the proof. Note that a third model, the restrained-subset model, was also identified in [H]; this is the strongest model that satisfies the theo-rem. However, it is less natural than the two considered here.
The fact that the same bounds hold for the two (or the three) models, and for any possible grouping of the n parties into m groups, is not trivial. Actually, it is possible to construct inequalities that distinguish between these models. It thus appears to be a special property of the MS expressions. Note also that the fact that the above bounds are tight for the two (or three) models implies that the bounds for any intermediary model, in which for instance two parties join to form a group and another one broadcasts its input, can be readily computed; in this case it would correspond to k= 2 or m=n−2.
Suppose that one observes a violation of the inequality |Snm| ≤2(n−m)/2. One can then conclude that in order to reproduce the corresponding nonlocal correlations in the framework of our communication models, the n parties need to join together and form strictly fewer thanmgroups, or strictly more thank =n−mparties must broadcast their input. Thus, the above bounds on Snm allow us to give a bound on the multipartite character of the observed nonlocal correlations (an upper bound on m, or a lower bound on k).
6.3 Multipartite non-locality of quantum states.
Here we discuss the violation of the inequalities (6.5) for n-partite GHZ-like and W states.
Partially entangled GHZ states are defined as
|GHZθi= cosθ|00...0i+ sinθ|11...1i. (6.6) It was conjectured in [78] that when the values of Mn exceed the local bounds, their maximal violations are given by 2(n−1)/2sin 2θ (see FIG. 2 in [H]). Numerical optimizations induce us to conjecture that the same holds for Mn+. For maximally entangled GHZ states (θ =π/4), this is known to be the maximal violation allowed by quantum mechanics. Upon comparison with the bound (6.5), we conclude that all n-partite GHZ states with θ > π/8 are maximally non-local according to our criterion (i.e., all parties must be grouped together or n−1 parties must broadcast their input to reproduce their correlations). Less entangled GHZ states, on the other hand, cannot be simulated if the parties are separated in more than m−1 groups
or if fewer than k+ 1 =n−m+ 1 parties broadcast their inputs whenever θ > θc with
sin 2θc = 2−m−12 . (6.7)
Interestingly, θc is the same for alln.
Consider now the W state
|Wni= 1
√n(|10. . .0i+|01. . .0i+. . .+|0. . .01i).
Numerical optimizations suggest that the maximal values of the MS polynomials for these states are upper-bounded by a small constant for all n (see FIG. 3 in [H]). A semi-analytical argument [H] leads us to believe that the asymptotic maximal values of the MS polynomials are indeed
|M∞| ≃1.62, |M∞+|= 2 r2
e ≃1.72. (6.8)
SinceSnn−1 =Mn+>1 forn≥3, letting one party broacast his input is not sufficient to reproduce the correlations of the W state. However, we cannot reach the same conclusion for k = n−m ≥ 2 broadcasting parties since the criterion (6.5) is not violated in this case. Similarly, for the grouping model it is not sufficient that two parties join together (so that thenparties formn−1 groups) to simulate a W state, but we find no violation if they are separated in less than n−1 groups.
6.4 Conclusion
We have thus shown how one can quantify multipartite non-locality, by comparing quantum multipartite correlations to correlations that can be obtained in commu-nication models such as the grouping or the broadcasting scenarios.
A series of tests to evaluate bounds on multipartite non-locality has been in-troduced. While GHZ states exhibit a strong form of multipartite non-locality ac-cording to our criterion, we have found that W states violate our inequalities only for small values of k, giving a small lower bound for its non-locality. As our ap-proach only gives bounds, it could be that other inequalities would be more adapted to estimate the non-locality of W states. Or, it might also be that there exists a communication model with only a few parties broadcasting their inputs or join-ing together, which is sufficient to reproduce the correlations of W states. Findjoin-ing which one of these possibilities is the correct one is an interesting problem for future research.
Finally, let us stress that the criterions we presented here can be tested exper-imentally. It would thus be worth (re-)considering experiments on multipartite non-locality in view of our results.
Chapter 7
Can one see entanglement ?
This is a tautology: quantum correlations appear in quantum systems. Now, the quantum world is often thought of as the world of the infinitely small, the invisible.
A world that reasonably seems to be unaccessible to our direct experience.
Is it really so ? Couldn’t we experience quantum correlations more directly than through the whole machinery usually involved in quantum experiments? Couldn’t we for instance replace man-made detectors in Bell-type experiments by human eyes, and see directly the entangled photon pairs?
Interestingly, the answer to that intriguing question appears to be optimistic:
the human eye is an extraordinary light-sensitive detector, and quantum optics experiments with human eye detectors seem indeed realistic! We could show first that the main characteristic of the human eye, namely a detection threshold, is no limitation to demonstrate entanglement [I]. We have then refined our analysis [J], and showed, more precisely, that naked human eyes could be used in Bell-type experiments to detect multi-photon states obtained by cloning single-photon qubits via simulated emission.
7.1 Threshold detectors can in principle detect entanglement
Studies on the photon detection characteristics of the human eye (see [79] and ref-erences therein) show that it is characterized by a threshold number of incident photons, below which no neural signal is sent to the brain. This is in contrast to standard man-made detectors, whose response to very low intensities is usually linear. The first question one might ask is whether this characteristics could be problematic with a view to demonstrating entanglement or non-locality.
We thus considered in a preliminary theoretical study [I] the very simple toy model of a detector with a perfect threshold occurring atNth photons: the detector does not detect fewer thanNth incoming photons, but it always clicks when at least Nth photons arrive. We could calculate the violation obtained for the CHSH Bell inequality [65], first when exactly N =Nth independent pairs of entangled photons arrive on the detectors. Surprisingly, although the probabilities of detection can
become very low, a violation of the Bell inequality can still be observed, without post-selection, for any value of the threshold Nth. However, the resistance to noise of the violation decreases when Nth gets larger.
Interestingly, when more than exactly Nthare sent to the detectors, or even when the number of photons follow a poissonian distribution, non-locality (and therefore entanglement) can still be demonstrated with perfect threshold detectors, without any further assumption.
Of course, the response function of the eye is not a perfect step function, with a perfect threshold at Nth photons, but is a typical S-shaped curve rather. We have found that, for smooth thresholds, the demonstration of quantum non-locality in the strict sense is compromised. It is still possible however to demonstrate entanglement with N independent pairs in the same state, by performing post-selection. We refer to [I] for a detailed discussion.
It is interesting at this point to establish a link with the study presented in [K]. In that paper, the violation of the CHSH inequality was studied in multi-pair scenarios, for both independent and indistinguishable multi-pairs, and for different detection schemes, by invoking voting strategies (e.g., majority, 3/4 or unanimous votes). This is closely related to the threshold detection scheme considered here: for instance, if N =Nth as in the first case above, the threshold detection corresponds to an unanimous vote.
Let us just mention that the study of [K] provides an interesting insight into the complex relation between entanglement and non-locality. In particular, although independent pairs contain less entanglement than indistinguishable photons (for the same number of photon pairs), they allow a larger violation of CHSH, and the violation is more resistant to noise. In this sense, independent pairs are more non-local than indistinguishable photons.