As shown by Leggett himself [11], his model cannot reproduce the correlations of the singlet state, and this incompatibility with quantum predictions can be experi-mentally tested. The proof of incompatibility relies on the violation of what we shall call Leggett-type inequalities: these criteria say that, under Leggett’s assumptions (3.7–3.8), a measurable quantity L necessarily satisfies L ≤ Lmax, while quantum theory predicts that L > Lmax can be observed for suitable measurements.
The first experimental test of Leggett’s model was presented in 2007 by Gr¨oblacher et al.[59]. However, in order to reach a conclusion regarding the falsification of the model, they needed an additional assumption. Indeed, Leggett’s original inequali-ties, and in particular the version used in [59], suppose that data are collected from infinitely many measurement settings. To circumvent this problem, Gr¨oblacher et al. assumed the rotation invariancy of the system being measured, and then claimed a violation of their inequality.
It was soon realized how one could do without this unnecessary assumption:
inspired by the way that inequality was derived, we could indeed come up with a finite version of it, directly testable [E] (see also [60] for a similar derivation). This allowed us to demonstrate the first really conclusive experimental falsification of Leggett’s model.
Here we shall derive a new Leggett-type inequality [F], in a much simpler way than it was done for the first inequalities that had been published and tested [11, 59, 60],[E]. Our inequality has been experimentally tested, and allowed a clear refutation of Leggett’s model.
3.2.1 General inequality for non-signaling models
In order for the decomposition (3.1) to be a valid mixture of correlations, all distri-butions Pλ should be non-negative. When writing Pλ(~a,~b) in the form of eq. (3.2), one can see that the non-negativity implies the general constraints
−1 +|MλA(~a) +MλB(~b)| ≤Cλ(~a,~b)≤1− |MλA(~a)−MλB(~b)| (3.9) or equivalently, in a more compact form:
|MλA(~a)±MλB(~b)| ≤1±Cλ(~a,~b) . (3.10) Constraints on the marginals MλA orMλB thus imply constraints on the correlation coefficients Cλ, and vice versa.
Let us now consider one measurement setting ~a for Alice and two measurement settings~b,~b′ for Bob, and let’s combine the previous inequalities (3.10) that we get for (~a,~b) and (~a,~b′). Using the triangle inequality, one gets
|Cλ(~a,~b)±Cλ(~a,~b′)| ≤2− |MλB(~b)∓MλB(~b′)|. (3.11) These constraints must hold for all probability distributions Pλ. After inte-gration over the λ’s, one gets, for the averaged correlation coefficients C(~a,~b) = R dλρ(λ)Cλ(~a,~b):
|C(~a,~b)±C(~a,~b′)| ≤2− Z
dλρ(λ) |MλB(~b)∓MλB(~b′)|. (3.12)
3.2.2 Derivation of a simple Leggett-type inequality
Now we derive an inequality satisfied by Leggett’s model, which can be experimen-tally tested. Inequality (3.12) implies, for the particular form of eq. (3.8) for Bob’s marginals:
|C(~a,~b) +C(~a,~b′)| ≤2− Z
dλρ(λ)|~v·(~b−~b′)| . (3.13) Let’s consider three triplets of settings1 (~ai,~bi,~b′i), with the same angleϕbetween all pairs (~bi,~b′i), and such that~bi−~b′i = 2 sinϕ2~ei, where{~e1, ~e2, ~e3}form an orthogonal basis (see Figure 3.1). After combining the three corresponding inequalities (3.13), using the fact that P3
i=1|~v·~ei| ≥ 1 and the normalization R
dλρ(λ) = 1, we finally get the Leggett-type inequality
1 3
3
X
i=1
|C(~ai,~bi) +C(~ai,~b′i)| ≡ L3(ϕ) ≤ 2− 2 3|sinϕ
2| . (3.14)
1Note that other choices of measurement settings can lead to other Leggett-type inequalities, which might be more robust, but would bring more settings into play [F].
Figure 3.1: Alice’s and Bob’s settings~ai,~biand~b′i used to test inequality (3.14). The three directions~ei of~bi−~b′i (thin dotted arrows) must be orthogonal.
The correlation coefficients C(~a,~b) can be experimentally estimated; for the sin-glet state, quantum mechanics predicts CΨ−(~a,~b) = −~a·~b. Thus, when~ai is judi-ciously chosen to be along the direction of~bi+~b′i, the quantum mechanical prediction for L3(ϕ) is
LΨ−(ϕ) = 2|cosϕ
2| , (3.15)
which violates inequality (3.14) for a large range of values ϕ. More specifically, for a pure singlet state, the violation occurs for |ϕ| < 4 arctan13 ≃ 73.7◦, and the maximal violation is obtained for |ϕ|= 2 arctan13 ≃36.9◦. In the case of imperfect interference visibility V ( ˜LΨ−(ϕ) = 2V|cosϕ2|), a violation can still be observed as long asV > Vth(3) =q
1−(13)2 = 2√32 ≃94.3%.
An important feature of Leggett-type inequalities compared to Bell inequalities is that the bound is not a fixed number, independent of quantum physics. Instead, the bound in Leggett-type inequalities, like the model itself, depends on the quantum measurements that are performed. Consequently all experimental data aiming at disproving Leggett’s model should present evidence that the settings used in the experiment have been properly adjusted.
3.2.3 Experimental falsification of Leggett’s model
We tested our Leggett-type inequality (3.14) with pairs of polarization-entangled photons obtained via a non-collinear type-II parametric down conversion process in a Barium-beta-borate crystal.
We chose the following Bloch vectors, as shown on Figure 3.1:
~a1 =~x , ~b1,~b′1 =³ cosϕ
2,±sinϕ 2,0´
,
~a2 =~y , ~b2,~b′2 =³
0,cosϕ
2,±sinϕ 2
´
, (3.16)
~a3 =~z , ~b3,~b′3 =³
±sinϕ
2,0,cosϕ 2
´,
1.4 1.5 1.6 1.7 1.8 1.9 2
−90° −60° −30° 0 30° 60° 90°
L
3upper bound for Leggett’s model
QM
(pure singlet)
experiment
ϕ
Figure 3.2: Experimental values for L3 over a range of separation angles ϕ (points with error bars) violate the bound given by Legget’s model (solid line), and follow qualitatively the expected value for the quantum mechanical prediction (dashed line).
so that
~bi−~b′i = 2 sinϕ
2~ei, with (~e1, ~e2, ~e3) = (~y, ~z, ~x). (3.17) The x axis in this notation corresponds to ±45◦ linear, the y axis to circular, and the z axis to horizontal/vertical polarization, the latter coinciding with the natural basis of the parametric down conversion process in the nonlinear optical crystal.
For given measurement settings, we estimated the correlation coefficientsC(~a,~b), and computed the values of L3(ϕ) over a larger range ofϕ with an integration time of T = 15 s per setting. The results are shown on Figure 3.2. The variation of L3 with ϕ is compatible with the quantum mechanical prediction for a singlet with residual colored noise and an orientation uncertainty of the quarter wave plate of 0.2◦ (recall that in Leggett-type inequalities, it is important to adjust the settings accurately). The largest violations of (3.14) are found for ϕ =±25◦ with 40.6 and 38.1 standard deviations, respectively.
Clearly, the experimental data agree with quantum mechanics and falsify Leggett’s model.