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1 doi: 1.138/nature5677 An experimental test of non-local realism Simon Gröblacher, 1, Tomasz Paterek, 3, 4 Rainer Kaltenbaek, 1 Časlav Brukner, 1, Marek Żukowski,3, 1 Markus Aspelmeyer, 1, and Anton Zeilinger 1, 1 Faculty of Physics, University of Vienna, Boltzmanngasse 5, A 19 Vienna, Austria Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A 19 Vienna, Austria 3 Institute of Theoretical Physics and Astrophysics, University of Gdansk, ul. Wita Stwosza 57, PL 8-95 Gdansk, Poland 4 The Erwin Schrödinger International Institute for Mathematical Physics (ESI), Boltzmanngasse 9, A 19 Vienna, Austria SUPPLEMENT I: AN EXPLICIT NON-LOCAL HIDDEN-VARIABLE MODEL We construct an explicit non-local model compliant with the introduced assumptions (1)-(3). It perfectly simulates all quantum mechanical predictions for measurements in a plane of the Poincaré sphere. We model the correlation function of the singlet state, E QM a = a b, for which all local averages A and B vanish. In particular, the violation of any b CHSH-type inequality can be explained within the model and, in addition, all perfect correlations state can be recovered. Let us start with a source that emits photons with well-defined polarization. Polarization u is sent to Alice and v to Bob. Alice sets her measuring device to a and Bob to b. The hidden-variable λ [, 1] is carried by both particles and predetermines the individual measurement result as follows: A A( a, u, λ) = { +1 for λ [, λa ], 1 for λ (λ A, 1], with λ A = 1 (1 + u a), (1) where A is the outcome of Alice. This means, whenever λ λ A the result of the measurement A is +1, and for λ > λ A the result is 1. Note that the measurement settings only enter in λ A and are hence independent of the hidden-variable λ of the source. The outcome of Bob is given by B B( a, b, u, v, λ) = { +1 for λ [x1, x ], 1 for λ [, x 1 ) (x, 1], with x 1, x [, 1] arbitrary but x x 1 = 1 (1 + v b). () All non-local dependencies are put on the side of Bob. His measuring device has the information about the setting of Alice, a, and her polarization u. The requirement of the non-local models discussed here is that the local averages performed on the subensemble of definite (but arbitrary) polarizations u and v obey Malus Law, i.e. A u = u a for Alice, and B v = v b for Bob. Indeed, a straight-forward calculation shows that this requirement is fulfilled for both Alice and Bob: A u = λa 1 dλ dλ = λ A 1 = u a, λ A (3) B v = x x 1 x1 1 dλ dλ dλ = (x x 1 ) 1 = v b. x (4) Thus the current construction fulfills one of our aims we recover Malus law. In order to get the correct formula for correlated counts one can fix the value of x 1 and x in the following way: x 1 = 1 4 [1 + u a v b + a b], x = 1 4 [3 + u a + v b + a b]. (5) With this definitions whenever x 1 λ A x the expectation value for measurements on the subensembles reproduces quantum correlations. Simply: x1 λa x 1 A u B v = dλ + dλ dλ + dλ = (λ A x 1 x ) + 1 = a b. (6) x 1 λ A x 1

2 doi: 1.138/nature5677 Therefore, in the next step, one must find the conditions for which both x 1 and x take values from [, 1] and x 1 and x take values from [, 1]. To this end, using definitions (5) one finds that the first condition is equivalent to a set of four inequalities: 1 + v b a b + u a 3 + v b, 3 v b a b + u a 1 v b. (7) Note that the upper bound 3 + v b cannot be exceeded by the middle term, as well as the lower bound 3 v b. Thus, this set of four inequalities is equivalent to a single one: Similarly, the second condition can be reexpressed as: Finally, the validity condition for the model is a conjunction of (8) and (9): a b + u a 1 v b. (8) a b u a 1 + v b. (9) a b ± u a 1 v b. (1) If this relation is not satisfied the model does not recover quantum correlations. Either it becomes inconsistent since x 1 or x leave their range or the necessary relation x 1 λ A x is not satisfied, or both. This is the origin of the incompatibility with general quantum predictions. Nevertheless the model can explain all perfect correlations and the violation of the CHSH inequality. Imagine a source producing pairs with the following property: whenever polarization u is sent to Alice polarization v = u is sent to Bob. Both parties locally observe random polarizations. For Alice, the local average over different polarizations yields A = 1 A u + 1 A u = 1 u a 1 u a =, (11) as it should be for the singlet state. The same result holds for Bob. In this way, we have reproduced the randomness of local measurement outcomes, typical for measurements on entangled states. With the same source, one can explain perfect correlations for measurements along the same basis, i.e. b = ± a. To see how the model works take v = u and b = a, and find that x 1 = 1 [1 + u a] = λ A and x = 1. As it should be, Bob s outcomes are always opposite to Alice s: B B( a, a, u, u, λ) = { +1 for λ [λa, 1], 1 for λ [, λ A ). (1) If in the same subensemble we take b = a we obtain x 1 = and x = λ A, which results in B = A, again in full agreement with quantum mechanics. Note that for these measurement settings, i.e. in order to obtain perfect correlations, condition (1) imposes no additional restrictions since it is always satisfied. For example, if u is sent to Alice and b = a one obtains 1 ± u a 1 u a, which always holds. The same argument applies to the other subensemble and other measurement possibilities b = ± a. Finally, the full predictions of quantum theory are recovered if Alice and Bob restrict their measurements to lie in planes orthogonal to vectors u and v, respectively, i.e. u a = v b =. In this case, condition (1) is satisfied, as a b 1. In general, if condition (1) is satisfied, i.e. for a consistent set of parameters, our model reproduces quantum correlations since they are already reproduced in every subensemble and hence averaging over different polarizations is not necessary: AB = A u B u = A u B u = a b. (13) Therefore every experimental violation of the CHSH inequality can be explained by the presented non-local model.

3 doi: 1.138/nature SUPPLEMENT II: DERIVATION OF THE INEQUALITY Following Leggett [1], one can take a source which distributes pairs of well-polarized photons. Different pairs can have different polarizations. The size of a subensemble in which photons have polarizations u and v is described by the weight function F ( u, v). In every such subensemble individual measurement outcomes are determined by hidden-variables λ. The hidden-variables are distributed according to the distribution ρ u, v (λ). For any dichotomic measurement results, A = ±1 and B = ±1, the following identity holds: 1 + A + B = AB = 1 A B. (14) If the signs of A and B are the same A + B = and A B =, and if A = B then A + B = and A B =. Any kind of non-local dependencies are allowed. Taking the average over the subensemble with definite polarizations gives: 1 + dλρ u, v (λ) A + B = dλρ u, v (λ)ab = 1 dλρ u, v (λ) A B, (15) which in an abbreviated notation, where the averages are denoted by bars, is 1 + A + B = AB = 1 A B. (16) Since the average of the modulus is greater or equal to the modulus of the averages one gets the set of inequalities 1 + A + B AB 1 A B. (17) From now on only the upper bound will be considered, however all steps apply to the lower bound as well. We will discuss the moment where the lower bound becomes equal to the minus upper bound and the modulus appears in the inequality. With the assumption that photons with well defined polarization obey Malus law: the upper bound of Eq. (17) becomes: A = u a, B = v b, (18) AB 1 u a k v b l, (19) where a k and b l are unit vectors associated with the measurement settings of Alice and Bob, respectively. Taking the average over arbitrary polarizations one obtains: E kl 1 sin θ u dθ u dφ u sin θ v dθ v dφ v F (θ u, φ u, θ v, φ v ) u a k v b l, () where all the vectors and the weight function F (θ u, φ u, θ v, φ v ) are written in the spherical coordinate system. We want to stress that the correlation function E kl can be experimentally measured. Let us denote the plane spanned by a k and b l as the xy-plane and the angle relative to the z axis as θ. Thus, vectors a k and b l are parameterized using angles within the xy-plane, φ ak and φ bl, respectively. The scalar products read: and the inequality transforms to: u a k = sin θ u cos(φ ak φ u ), (1) v b l = sin θ v cos(φ bl φ v ), () E kl 1 sin θ u dθ u dφ u sin θ v dθ v dφ v F (θ u, φ u, θ v, φ v ) sin θ u cos(φ ak φ u ) sin θ v cos(φ bl φ v ). The sines of θ u and θ v describe the magnitude of the projection of u and v onto the xy-plane, respectively. These magnitudes can always be decomposed into the sum and the difference of two real numbers: sin θ u = n 1 + n, (3) sin θ v = n 1 n. (4) 3

4 doi: 1.138/nature5677 Note that both n 1 and n are functions of θ u and θ v only. We insert this decomposition into the last inequality, and hence the terms multiplied by n 1 and n read: cos(φ ak φ u ) cos(φ bl φ v ) = sin φ a k + φ bl (φ u + φ v ) cos(φ ak φ u ) + cos(φ bl φ v ) = cos φ a k + φ bl (φ u + φ v ) respectively. We make the following substitution for the measurement angles: and change the integration variables φ u, φ v to ψ, χ: sin (φ a k φ bl ) + φ u φ v, cos φ a k φ bl (φ u φ v ), ξ = φ a k + φ bl, ϕ = φ ak φ bl, (5) ψ = φ u + φ v, χ = φ u φ v. (6) The absolute value of the Jacobian of this transformation equals 1, thus it does not introduce any new factors to the integral. Within this new variables one arrives at: E kl (ξ, ϕ) 1 n cos ϕ χ sin θ u dθ u dψ cos(ξ ψ) n 1 sin ϕ χ sin θ v dθ v dχf (θ u, θ v, ψ, χ) sin(ξ ψ), where in the correlation function E kl (ξ, ϕ) we explicitly state the angles it is dependent on. The expression within the modulus is a linear combination of two harmonic functions of ξ ψ, and therefore is a harmonic function itself. Its amplitude reads n cos ( ϕ χ ) + n 1 sin ( ϕ χ ), and the phase is some fixed real number α: E kl (ξ, ϕ) 1 sin θ u dθ u dψ n cos ( ϕ χ sin θ v dθ v dχf (θ u, θ v, ψ, χ) ) + n 1 sin ( ϕ χ ) cos(ξ ψ + α). (7) In the next step we average both sides of this inequality over the measurement angle ξ = φa k +φ b l. This means an integration over ξ [, π) and multiplying by 1 π. Experimentally one should perform a series of measurements in which the angle between the observables is kept constant, ϕ = const., and the two measurement vectors are rotated in their plane. The integral of the ξ-dependent part of the right-hand side of (7) therefore reads: By denoting the average of the correlation function over the angle ξ as: one can write (7) as dξ π cos(ξ ψ + α) = π. (8) E kl (ϕ) dξ π E kl(ξ, ϕ), (9) E kl (ϕ) 1 4 sin θ u dθ u dψ sin θ v dθ v dχf (θ u, θ v, ψ, χ) π n ϕ χ cos + n ϕ χ 1 sin

5 doi: 1.138/nature5677 Further, the integrand is no longer dependent on ψ, and the ψ integration results in the marginal weight function: The last inequality can be simplified to: F (θ u, θ v, χ) = dψf (θ u, θ v, ψ, χ). (3) E kl (ϕ) 1 4 sin θ u dθ u sin θ v dθ v dχf (θ u, θ v, χ) π n ϕ χ cos + n ϕ χ 1 sin. This inequality is valid for any choice of observables in the plane defined by a k and b l. One can introduce two new observable vectors in this plane and write the inequality for the averaged correlation function E k l (ϕ ) of these new observables. The sum of these two inequalities than is One can use the triangle inequality E kl (ϕ) + E k l (ϕ ) 4 π ( n ϕ χ cos + n 1 sin ϕ χ sin θ u dθ u sin θ v dθ v dχf (θ u, θ v, χ) ) + n cos ϕ χ + n 1 sin ϕ χ. x + y x + y, (31) (x1 + y 1 ) + (x + y ) x 1 + x + y1 + y, (3) for the two-dimensional vectors x = (x 1, x ) and y = (y 1, y ), with components defined by: x 1 = n cos ϕ χ x = n 1 sin ϕ χ This implies that the integrand is bounded from below by: n ϕ χ cos + n ϕ χ 1 sin n ( cos ϕ χ One can further approximate this bound as cos( ϕ χ sin( ϕ χ + cos ϕ χ ) + cos( ϕ χ ) + sin( ϕ χ, y 1 = n cos ϕ χ,, y = n 1 sin ϕ χ. + n cos ϕ χ ) ( + n 1 sin ϕ χ ) sin ϕ ϕ and + n 1 sin ϕ χ + sin ϕ χ. ) ) sin ϕ ϕ. (33) This approximation follows after using the formula for the sine of the difference angle to the right-hand side argument ϕ ϕ = ϕ χ ϕ χ. Namely, sin ϕ ϕ = sin ϕ χ cos ϕ χ cos ϕ χ sin ϕ χ cos ϕ χ + cos ϕ χ sin ϕ χ sin ϕ χ. After these approximations the lower bound equals minus the upper bound, and one can shortly write the modulus: E kl (ϕ) + E k l (ϕ ) 4 π ϕ sin(ϕ ) sin θ u dθ u sin θ v dθ v π dχf (θ u, θ v, χ) n + n

6 doi: 1.138/nature5677 Recall that the numbers n 1 and n are functions of θ u and θ v only. Thus one can perform the integration over χ, which results in yet another marginal weight function: 6 Going back to the magnitudes: F (θ u, θ v ) = dχf (θ u, θ v, χ) (34) E kl (ϕ) + E k l (ϕ ) π ϕ sin(ϕ ) sin θ u dθ u sin θ v dθ v π F (θ u, θ v ) sin θ u + sin θ v. (35) This inequality is valid for any set of four observables in one plane and for any choice of the plane. The bound involves only the angles of vectors u and v relative to the axis orthogonal to the plane of the observables. For a plane orthogonal to the initial one, e.g. the xz-plane, the inequality has a similar form: E mn (ϕ y ) + E m n (ϕ y) π sin ϕ y ϕ y sin θ u dθ u sin θ v dθ v F (θ u, θ v ) sin θ u + sin θ v, (36) where the primed angles θ under the square root are now relative to the y axis (the distribution of vectors is still the same). We add the inequalities for orthogonal observation planes, (35) and (36), choose ϕ = ϕ y = and ϕ = ϕ z = ϕ y to obtain: E kl (ϕ z ) + E k k () + E mn(ϕ y ) + E m m () 4 π sin ϕ ( ) sin θ u dθ u sin θ v dθ v F (θ u, θ v ) sin θ u + sin θ v + sin θ u + sin θ v. On the left-hand side we use the notation ϕ z and ϕ y to stress that the averaged correlations in the moduli are valid for observables from orthogonal planes. We apply the triangle inequality (3) to the expression within the bracket. This time the components of vectors x are y read: x 1 = sin θ u, y 1 = sin θ u, (37) x = sin θ v, y 1 = sin θ v. (38) The integrand is bounded by: sin θ u + sin θ v + sin θ u + sin θ v (sin θ u + sin θ u) + (sin θ v + sin θ v). (39) Now let us consider the term involving vector u only. Since both θ u π and θ u π their sines are always positive, i.e.: (sin θ u + sin θ u) sin θ u + sin θ u. (4) Recall that the angles θ u and θ u (of two spherical coordinate systems) are relative to orthogonal Cartesian axes z and y, respectively. Thus, vector u has the following components in the Cartesian coordinate system: u = (δ, cos θ u, cos θ u ), with δ + cos θ u + cos θ u = 1, (41) The normalization implies that cos θ u + cos θ u 1, which is equivalent to sin θ u + sin θ u 1. (4) The same steps obviously apply to vector v and one can conclude that sin θ u + sin θ v + sin θ u + sin θ v. (43) 6

7 doi: 1.138/nature Since the function F (θ u, θ v ) is normalized, the final inequality reads: E kl (ϕ z ) + E k k () + E mn(ϕ y ) + E m m () 4 4 π sin ϕ. (44) [1] A. J. Leggett, Found. Phys. 33, 1469 (3). 7