The relevance of random choice in tests of Bell inequalities with atomic qubits
aa r X i v : . [ qu a n t - ph ] J a n The relevance of random choice in tests of Bellinequalities with two atomic qubits
Emilio SantosDepartamento de F´ısica. Universidad de Cantabria. Santander. SpainNovember 4, 2018
Abstract
It is pointed out that a loophole exists in experimental tests of Bellinequality using atomic qubits, due to possible errors in the rotationangles of the atomic states. A sufficient condition is derived for closingthe loophole.PACS numbers: 03.65.Ud, 03.67.Mn, 37.10.Ty, 42.50.Xa
A recent experiment, by a group of Maryland, has measured the corre-lation between the quantum states of two Yb + ions separated by a distanceof about 1 meter[1]. The authors claim that the experiment is relevant be-cause it violates a CHSH[2] (Bell) inequality, modulo the locality loophole,closing the detection loophole. In my opinion that assertion does not makefull justice to the relevance of the experiment. The truth is that it is thefirst experiment which has tested a genuine Bell inequality. Actually theresults of previous experiments, in particular those involving optical photonpairs[3], did not test any genuine Bell inequality, that is an inequality whichis a necessary condition for the existence of local hidden variables (LHV)models. The inequalities tested in those experiments should not be qualifiedas Bell´s because their derivation involves additional assumptions. Conse-quently their violation refutes only restricted families of LHV models, namelythose fulfilling the additional assumption. ( For details see[4].)The aim of the present letter is to point out the existence of a loopholein the Maryland experiment[1], or more generally in Bell tests with atomicqubits, in addition to the locality loophole. Blocking that loophole will be1traightforward using random choice of the measurements, as is explainedbelow.In general I will consider experiments where a pair of atoms (or ions)is prepared in an entangled state. Then Alice performs a rotation of thestate of her atom by an angle θ a and, after a short time, she may detectfluorescence of the atom illuminated by an appropriate laser. Similarly Bobperforms a rotation of his atom by an angle θ b and, after that, he maydetect fluorescence too. I shall label p ++ ( θ a , θ b ) the probability of coincidencedetection and p −− ( θ a , θ b ) the probability that neither Alice nor Bob detectfluorescence. Similarly p − + ( θ a , θ b ) ( p + − ( θ a , θ b )) will be the probability thatonly Bob (Alice) detects fluorescence. In the Maryland experiment[1] (seetheir eq.(6)), a function E ( θ a , θ b ) is defined by E ( θ a , θ b ) = p ++ ( θ a , θ b ) + p −− ( θ a , θ b ) − p + − ( θ a , θ b ) − p − + ( θ a , θ b ) . (1)Then the authors define a parameter S by S = | E ( θ a , θ b ) + E ( θ ′ a , θ b ) | + | E ( θ a , θ ′ b ) − E ( θ ′ a , θ ′ b ) | , (2)and claim that the CHSH[2] inequality S ≤ E ( θ a , θ b ) = p ( θ a , θ b ) + p ( θ a + π, θ b + π ) − p ( θ a , θ b + π ) − p ( θ a + π, θ b ) , (3)where they label p ( θ a , θ b ) the quantity which I have labeled p ++ ( θ a , θ b ) . Definition eq.(3) , in place of eq.(1) , rests upon assuming the equalities p − + ( θ a , θ b ) = p ( θ a + π, θ b ) , p + − ( θ a , θ b ) = p ( θ a + π, θ b ) ,p −− ( θ a , θ b ) = p ( θ a + π, θ b + π ) , which are true according to quantum mechanics, but may not be true in LHVtheories. In any case the authors measured E ( θ a , θ b ) as defined in eq.(1)[5].In order to show that there is a loophole in the experiment, in additionto the locality loophole, I begin remembering that, according to Bell[6], aLHV model will contain a set of hidden variables, λ, a positive normalizeddensity function, ρ ( λ ) , and two functions M a ( λ, θ a ) , M b ( λ, θ b ), θ a and θ b being parameters which may be controlled by Alice and Bob respectively.The latter functions fulfil M a ( λ, θ a ) , M b ( λ, θ b ) ∈ { , } . (4)2n the Maryland experiment[1] the parameters θ a and θ b are angles definingthe quantum states of the two ions. The probability, p ++ ( θ a , θ b ) , that thecoincidence measurement of two dichotomic variables, in two distant regions,gives a positive answer for both variables should be obtained in the LHVmodel by means of the integral p ++ ( θ a , θ b ) = Z ρ ( λ ) M a ( λ, θ a ) M b ( λ, θ b ) dλ. (5)Similarly the probability, p + − ( θ a , θ b ) , that Alice gets the answer “yes” andBob the answer “no” is given by p + − ( θ a , θ b ) = Z ρ ( λ ) M a ( λ, θ a ) [1 − M b ( λ, θ b )] dλ, (6)and analogous expressions for p − + and p −− . A LHV model for an atomic experiment may be obtained by choosing ρ ( λ ) = 12 π , λ ∈ [0 , π ] , M a ( λ, θ a ) = Θ (cid:16) π − | λ − θ a | (cid:17) ,M b ( λ, θ b ) = Θ (cid:16) π − | λ − θ b − π | (cid:17) , mod (2 π ) , (7)where Θ ( x ) = 1 if x >
0, Θ ( x ) = 0 if x <
0. It is easy to see, taking eqs.(5)and (6) into account, that model predictions are (assuming θ a , θ b ∈ [0 , π ]) p ++ ( θ a , θ b ) = p −− ( θ a , θ b ) = | θ a − θ b | π ,p + − ( θ a , θ b ) = p − + ( θ a , θ b ) = 12 − | θ a − θ b | π . (8)Hence I get E ( θ a , θ b ) = 2 π | θ a − θ b | − , (9)and it is not difficult to show that, for any choice of the angles θ a , θ b , θ ′ a , θ ′ b , the model predicts S ≤ S given by eq.(2) . Now let us assume that the experiment is performed so that Alice andBob start measuring the quantity E ( θ a , θ b ) in a sequence of runs of theexperiment. After that they measure E ( θ a , θ ′ b ) in another sequence, thenthey measure E ( θ ′ a , θ b ) and, finally, they measure E ( θ ′ a , θ ′ b ) . Let α be theerror in the rotation performed by Bob on his atom in the first sequence of3uns, so that the rotation angle is θ b + α rather than θ b in the measurementof E ( θ a , θ b ) . Similarly I shall assume that the rotation angles are θ ′ b + β, θ b + γ and θ ′ b + δ in the measurements of E ( θ a , θ ′ b ), E ( θ ′ a , θ b ) and E ( θ ′ a , θ ′ b ) , respectively. For simplicity I will assume that no error appears in Alicerotations. The errors are considered small, specifically | α | , | β | , | γ | , | δ | < π/ . I shall prove that, taking into account the errors in the measurement of theangles, the LHV model prediction for the parameter S , eq.(2) may apparentlyviolate the CHSH[2] inequality S ≤ . To do that let us choose, as in theMaryland experiment[1], θ a = π , θ b = π , θ ′ a = 0 , θ ′ b = 3 π . (10)The values predicted by the LHV model for the relevant quantities are E ( θ a , θ b + α ) = − . − απ , E ( θ a , θ ′ b + β ) = − . βπ ,E ( θ ′ a , θ b + γ ) = − . γπ , E ( θ ′ a , θ ′ b + δ ) = 0 . δπ . (11)Then the parameter actually measured in the experiment is S ′ = | E ( θ a , θ b + α ) + E ( θ ′ a , θ b + γ ) | + | E ( θ a , θ ′ b + β ) − E ( θ ′ a , θ ′ b + δ ) | , (12)and the LHV prediction for that parameter is S ′ == 2 + 2 π ( α − β − γ + δ ) , which may violate the inequality S ′ ≤ α, β, γ and δ. In particular the results of the Maryland experiment[1] arereproduced by choosing2 α/π = 0 . , β/π = − . , γ/π = − . , δ/π = − . . The errors in the angles are of order 7 o or less. It is plausible that errorsas high as these may appear in experiments with atomic qubits but not inoptical photon experiments. I stress that no violation of a Bell inequalityby a LHV model is produced. Actually the parameter S ′ of eq.(12) is not aCHSH parameter as defined in eq.(2) . In the following I shall prove that the loophole may be closed by randomchoice of the angles to be measured. To begin with, it is easy to see that the4HV model predictions do not violate the inequality S ′ ≤ θ b is the same in all measurementsof that angle, and similarly for θ ′ b . In fact the inequality is fulfilled if α = β and γ = δ, as may be seen by looking at eq.(12) . In the following I derivea sufficient condition for the fulfillement of the inequality, S ′ ≤ , for theactually measurable quantity S ′ , by the predictions of any LHV model.Let us assume that there is a (normalized) probability distribution, f a ( x ) , for the errors when Alice rotates her atom by an angle θ a and another dis-tribution, f ′ a ( y ) , when she rotates her atom by an angle θ ′ a . Similarly I shallassume that there are similar disitribuions f b ( u ) and f ′ b ( v ) for the errors inthe rotations, by Bob, of the angles θ b and θ ′ b . I shall show that a sufficientcondition for the inequality S ′ ≤ S ′ will be obtained from probabilities defined as follows (compare with eqs.(5)and (6)) p ++ ( θ a , θ b ) = Z ρ ( λ ) M a ( λ, θ a + x ) M b ( λ, θ b + u ) dλf a ( x ) dxf b ( u ) du, (13) p + − ( θ a , θ b ) = Z ρ ( λ ) M a ( λ, θ a + x ) [1 − M b ( λ, θ b + u )] dλf a ( x ) dxf b ( u ) du, and similarly for the other quantities p ij with i, j = + , − . Now we may definenew quantities Q a ( λ, a ) = Z M a ( λ, θ a + x ) f a ( x ) dx, (14) Q b ( λ, b ) = Z M b ( λ, θ b + u ) f b ( u ) du,Q a ( λ, a ′ ) = Z M a ( λ, θ ′ a + y ) f ′ a ( y ) dy,Q b ( λ, b ′ ) = Z M b ( λ, θ ′ b + v ) f ′ b ( v ) dv, which fulfil the conditions (compare with eqs.(4))0 ≤ Q a ( λ, a ) , Q a ( λ, a ′ ) , Q b ( λ, b ) , Q b ( λ, b ′ ) ≤ . (15)5he consequence is that we may obtain a new LHV model for the experimentwith the quantities Q, eqs.(15) , in place of the quantities M , eqs.(4) . Theexistence of the model implies the fulfillement of the inequality S ′ ≤ . From our proof it is rather obvious that the essential condition required toblock the loophole is that the probability distribution of errors made by Bobare independent of what rotation is performed by Alice in the partner atom,and similarly the errors made by Alice should be independent of the rotationperformed by Bob. A simple method to insure that independence is thatAlice makes at random the choice whether to rotate her atom by the angle θ a or by the angle θ ′ a , and similarly Bob. That is, after every preparation ofthe entangled state of the atom, Alice should make a random choice (withequal probabilities) between the rotation angles θ a and θ ′ a and similarly Bobshould make a random choice, independently of Alice , between θ b and θ ′ b . References [1] D. N. Matsukevich, P. Maunz, D.L. Moehring, S. Olmschenk and C.Monroe,
Phys. Rev. Lett. , 150404 (2008).[2] J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt,
Phys. Rev. Lett. , 880 (1969).[3] M. Genovese, Phys. Reports , 319 (2005).[4] E. Santos, Found. Phys. , 1643 (2004).[5] D. N. Matsukevich, private communication.[6] J. S. Bell, Physics1