Influence of detector motion in Bell inequalities with entangled fermions
aa r X i v : . [ qu a n t - ph ] F e b Influence of detector motion in Bell inequalities with entangled fermions
Andr´e G. S. Landulfo and George E. A. Matsas
Instituto de F´ısica Te´orica, Universidade Estadual Paulista,Rua Pamplona 145, 01405-900, S˜ao Paulo, S˜ao Paulo, Brazil (Dated: November 3, 2018)We investigate how relativity influences the spin correlation of entangled fermions measured bymoving detectors. In particular, we show that the Clauser-Horne-Shimony-Holt Bell inequality isnot violated by quantum mechanics when the left and right spin detectors move fast enough.
PACS numbers: 03.65.Ud, 03.30.+p
The discovery of the Bell inequalities can be consid-ered one of the most important physics landmarks of the20th century [1]. It allows us to probe the essence ofquantum theory by distinguishing it from local hiddenvariable theories. The genesis of this achievement canbe traced back to the Einstein-Podolsky-Rosen discus-sion about the completeness of quantum mechanics [2].Presently the Clauser-Horne-Shimony-Holt (CHSH) Bellinequality [3] has been shown to be violated 30 standarddeviations [4], which strongly supports quantum mechan-ics. In order to contribute to the intense present debateon the interplay between relativity and quantum mechan-ics (see e.g. Refs. [5, 6, 7, 8, 9, 10]), we investigate herehow the former influences the spin correlation of entan-gled fermions measured by moving detectors. In par-ticular we show that the CHSH Bell inequality can besatisfied rather than violated by quantum mechanics ifthe left and right spin detectors are set in fast enoughrelativistic motion. We adopt natural units ~ = c = 1.Let us assume a system composed of two spin-1/2 par-ticles A and B with mass m and zero total spin angularmomentum. Each particle spin is measured along somearbitrary direction defined on the y ⊥ z plane. The dis-tance between the planes along the x axis is large enoughto make both measurements causally disconnected. Thisis well known that local hidden variable theories satisfythe CHSH Bell inequality | E ( a , b )+ E ( a , b )+ E ( a , b ) − E ( a , b ) | ≤ , (1)where a i ( i = 1 ,
2) are two arbitrary unit vectors con-tained in the y ⊥ z plane along which the spin s A of parti-cle A is measured, and analogously for the two arbitraryunit vectors b j ( j = 1 ,
2) and spin s B of particle B . Here E ( a i , b j ) ≡ lim N →∞ N N X n =1 ( a i · s A )( b j · s B ) (2)is the spin correlation function obtained after an arbi-trarily large number N of experiments is performed, and a i · s A and b j · s B assume ± / x axis. For this purpose let us begin considering a quantum system composed of twospin-1/2 particles. The corresponding normalized statecan be written as [6, 11] (see also Ref. [12]) | ψ i = X s A ,s B Z d p A d p B ψ s A s B ( p A , p B ) | s A , p A i| s B , p B i , (3)where X s A ,s B Z d p A d p B | ψ s A s B ( p A , p B ) | = 1 , (4) h s ′ X , p ′ X | s X , p X i = δ s ′ X s X δ ( p ′ X − p X ) , (5)and X = A, B distinguishes between both particles. Tak-ing P µX and S X ≡ s X ⊗ I as the four-momentum andWigner spin operators, respectively, where s X is half ofthe Pauli matrices, we have P µX | s X , p X i = p µX | s X , p X i S zX | s X , p X i = s X | s X , p X i with p X = ( p p X + m , p X ) and s X = ± /
2. Let usnow assume that the two-particle system is prepared ina singlet state (see, e.g., Refs. [9, 13] for the two-spinornotation used below) ψ ( p A , p B ) = 1 √ (cid:20)(cid:18) f k A ( p A )0 (cid:19) ⊗ (cid:18) f k B ( p B ) (cid:19) − (cid:18) f k A ( p A ) (cid:19) ⊗ (cid:18) f k B ( p B )0 (cid:19)(cid:21) (6)from which we read ψ s A s B ( p A , p B ) = 1 √ f k A ( p A ) f k B ( p B ) × ( δ s A / δ s B − / − δ s A − / δ s B / ) . (7)We describe particles A and B by Gaussian packets: f k X ( p X ) = π − / w − / e − ( p X − k X ) / (2 w ) , w ∈ R + andassume that they move away from the origin in oppositedirections at the same rate along the x axis as defined inthe laboratory frame: k A = − k B = ( | k | , , A and B havevelocities v d A = ( v d A , ,
0) and v d B = ( v d B , , | ψ i in their proper frames trans-formed through an unitary transformation [11, 14]: | ψ i → | ψ ′ i = U (Λ d A ) ⊗ U (Λ d B ) | ψ i , (8)where U (Λ d X ) | s X , p X i = [(Λ d X p X ) /p X ] / × X s ′ X D s ′ X s X (Λ d X , p X ) | s ′ X , Λ d X p X i . (9)The Wigner rotation can be written in matrix form as D (Λ d X , p X ) = ( p X + m ) σ cosh( α d X / p X + m )((Λ d X p X ) + m )] / + ( p xX σ + iǫ xij p iX σ j ) sinh( α d X / p X + m )((Λ d X p X ) + m )] / , (10)where σ and σ i , i = x, y, z , are the usual 2 × d X = cosh α d X sinh α d X α d X cosh α d X with α d X ≡ − tanh − v d X . By using Eqs. (3) and (9) inEq. (8), we obtain | ψ ′ i = X s X ,s ′ X Z d p A d p B (Λ − d A p A ) p A ! (Λ − d B p B ) p B ! × D s ′ A s A (Λ d A , Λ − d A p A ) D s ′ B s B (Λ d B , Λ − d B p A ) × ψ s A s B (Λ − p A , Λ − p B ) | s ′ A , p A i| s ′ B , p B i , (11)where we have performed the change of variable p X → Λ − d X p X and we recall that d p X /p is a relativistic invari-ant. By using Eqs. (7) and (10) in Eq. (11), we can write ψ ′ s A s B ( p A , p B ), which appears in | ψ ′ i = X s A ,s B Z d p A d p B ψ ′ s A s B ( p A , p B ) | s A , p A i| s B , p B i , (12)using the two-spinor notation: ψ ′ ( p A , p B ) = 1 √ (cid:20)(cid:18) a ( p A ) a ( p A ) (cid:19) ⊗ (cid:18) b ( p B ) b ( p B ) (cid:19) − (cid:18) − a ( p A ) a ( p A ) (cid:19) ⊗ (cid:18) b ( p B ) − b ( p B ) (cid:19)(cid:21) . (13)This is the wave function on which the detectors willeffectively act to measure the particle spin. Here a ( p A ) = K A f k A ( q A )[ C A ( q A + m ) + S A ( q xA + iq yA )] ,a ( p A ) = K A f k A ( q A ) S A q zA ,b ( p B ) = − K B f k B ( q B ) S B q zB ,b ( p B ) = K B f k B ( q B )[ C B ( q B + m ) + S B ( q xB − iq yB )] , where K X ≡ ( q X /p X ) / / [( q X + m )( p X + m )] / ,q X ≡ Λ − d X p X ,C X ≡ cosh( α d X / ,S X ≡ sinh( α d X / . Next we trace out the momenta degrees of freedom sincethe detectors do only measure spin. As a result, we ob-tain the following reduced density matrix: τ ′ = Z d p A d p B ψ ′ ( p A , p B ) ψ ′† ( p A , p B )= ( ρ ⊗ ρ ′ − ρ ⊗ ρ ′ − ρ ⊗ ρ ′ + ρ ⊗ ρ ′ ) / ρ ⊗ ρ ′ = (cid:18) − V V (cid:19) ⊗ (cid:18) W
00 1 − W (cid:19) ,ρ ⊗ ρ ′ = (cid:18) − V − V (cid:19) ⊗ (cid:18) − W − W (cid:19) ,ρ ⊗ ρ ′ = (cid:18) − V − V (cid:19) ⊗ (cid:18) − W − W (cid:19) ,ρ ⊗ ρ ′ = (cid:18) V
00 1 − V (cid:19) ⊗ (cid:18) − W W (cid:19) and V ( α d A ) = sinh (cid:16) α d A (cid:17)Z d q A | f k A ( q A ) | q zA ( q A + m )( p A + m ) , (15) W ( α d B ) = sinh (cid:16) α d B (cid:17)Z d q B | f k B ( q B ) | q zB ( q B + m )( p B + m ) , (16)where we have used that d p X /p X = d q X /q X . Now let ususe our previous results to investigate Eq. (1). In quan-tum mechanical terms, the left-hand side of this equationcan be expressed as | E ( a , b )+ E ( a , b )+ E ( a , b ) − E ( a , b ) | = |h C i τ ′ | (17)where h C i τ ′ = tr( τ ′ C ) and C = ( σ · a ) ⊗ [ σ · ( b + b )] + ( σ · a ) ⊗ [ σ · ( b − b )] . By using that h ( σ · u ) ⊗ ( σ · v ) i τ ′ = − (1 − V )(1 − W ) u · v , (18)where u , v = a , a , b , b , we cast Eq. (17) as |h C i τ ′ | = |h ( σ · a ) ⊗ ( σ · b ) i τ ′ + h ( σ · a ) ⊗ ( σ · b ) i τ ′ + h ( σ · a ) ⊗ ( σ · b ) i τ ′ − h ( σ · a ) ⊗ ( σ · b ) i τ ′ | and finally as |h C i τ ′ | = (1 − V )(1 − W ) |h C i τ ′ | (19) π π π π π π π π π π π φ F ( φ ) α = 0.00 α = 0.85 α = 1.39 α = 2.50 α → ∞ π π π π π π π π π π π φ F ( φ ) α = 0.00 α = 2.70 α = 3.12 α = 4.5 α → ∞ FIG. 1: F ( φ ) as given in Eq. (22) is plotted as a function of φ with ˜ w = 4 for different values of α . The top and bottom plotsassume | ˜ k | = 0 .
01 and | ˜ k | = 100, respectively. For α = 0 theusual Bell inequality result is recovered. For α > ∼ .
39, and α > ∼ .
12, we have that F ( φ ) < with |h C i τ ′ | = | a · b + a · b + a · b − a · b | . Eq. (19) is our key formula. Note that when the detectorsare at rest α d A = α d B = 0, we recover the usual result: |h C i τ ′ | = |h C i τ ′ | , i.e. the nontriviality introduced bythe detector motion is isolated in the (1 − V )(1 − W )multiplicative factor. By defining α i , β i ( i = 1 ,
2) as theangles between a i , b i and the x axis, respectively, we get |h C i τ ′ | = | cos( α − β ) + cos( α − β )+ cos( α − β ) − cos( α − β ) | . For our purposes, this is sufficient to take the sim-pler case where a = b . By assuming this and φ ≡ cos − ( a · a ) = cos − ( b · b ) , we obtain |h C i τ ′ | a = b = (1 − V )(1 − W ) | φ − cos(2 φ ) | . Let us, now, focus on the case where both detectorsboost away from each other with the same absolute ra-pidity [15]: α d A = − α d B = −| α | , i.e. v d A = − v d B = π π π π π π π π π π π φ F ( φ ) w → w = 0.52 w = 0.87 w = 3.00 w = 30.0 π π π π π π π π π π π φ F ( φ ) w → w = 0.24 w = 0.37 w = 1.00 w = 30.0 FIG. 2: F ( φ ) as given in Eq. (22) is plotted as a function of φ assuming α → ∞ for different values of the ˜ w width. Thetop and bottom plots assume again | ˜ k | = 0 .
01 and | ˜ k | = 100,respectively. For ˜ w = 0 the usual Bell inequality result isrecovered, while for ˜ w > ∼ .
87 and ˜ w > ∼ .
37 we have that F ( φ ) < tanh | α | . (For | ˜ k | ≪ F ( φ ) ≡ |h C i τ ′ | α dA = − α dB a = b . Then, from Eqs. (15)-(16) we have V ( −| α | ) = W ( | α | )= sinh ( | α | / √ π ˜ w Z ∞−∞ d ˜ q x Z ∞ d ˜ q r G (˜ q x , ˜ q r ) (20)where we have used cylindrical coordinates with q x asthe symmetry axis and G (˜ q x , ˜ q r ) = (˜ q r ) exp [ − ((˜ q x − | ˜ k | ) + (˜ q r ) ) / ˜ w ](˜ q + 1)(˜ q cosh | α | − ˜ q x sinh | α | + 1) (21)with ˜ q r = q r /m , ˜ q x = q x /m , ˜ q = p (˜ q x ) + (˜ q r ) + 1, | ˜ k | = | k | /m and ˜ w = w/m . Then, we finally obtain thesimple expression F ( φ ) = F | φ − cos(2 φ ) | , (22)where F = [1 − V ( −| α | )] . For very narrow wave pack-ets in the momentum space, i.e. ˜ w ≪
1, this is easyto analytically solve the integral in Eq. (20) for particlesmoving slow enough, ˜ k ≈
0, and cast Eq. (22) as F ( φ ) | ˜ k ≈ w ≪ = (cid:18) − ˜ w | α | (cid:19) | φ − cos(2 φ ) | . (23)Clearly for ˜ w →
0, we recover the standard Bell inequal-ity result irrespective of the detector velocities, i.e. thenontriviality driven by the detector motion in Eq. (23)is not present when the entangled particles are describedby momenta eigenstates [8, 10]. This is so because onlywhen particles are described by wave packets, | ψ i (whichis a pure state according to observers lying at rest in thelaboratory) looks like as a mixed state for the movingdetectors once they ignore the momenta degrees of free-dom [6]. (The corresponding “missing information” getshidden in the traced out momenta.)In Fig. 1 we plot F ( φ ) for different detector velocities,i.e. | α | ’s, assuming a wave packet with ˜ w = 4. Theplots on the top and at the bottom take | ˜ k | = 0 .
01 and | ˜ k | = 100, respectively. We note that the standard Bellinequality result is recovered for α = 0 but this is not sowhen the detectors move. Indeed, F ( φ ) decreases as thedetector velocities increase. For α > ∼ .
39, and α > ∼ . F ( φ ) < φ range for the | ˜ k | = 0 .
01 and | ˜ k | = 100 cases, respectively, i.e. for these α intervals the Bell inequality is not violated for every φ .Our numerical integration was cross checked against theMonte Carlo method and we have verified that it repro-duces the analytic value given by Eq. (23) up to 1 partin 10 . Next we analyze how the packet width influencesin our results. In Fig. 2 we plot F ( φ ) for different ˜ w values when the detectors have ultra relativistic veloci-ties: α → ∞ . Again, we have assumed | ˜ k | = 0 .
01 and | ˜ k | = 100 for the top and bottom graphs, respectively.We see that for ˜ w → w increases, F ( φ )decreases. This reflects that the nontriviality associatedwith the detector motion is not manifest when the en-tangled particles are described by momenta eigenstates.For ˜ w > ∼ .
87 and ˜ w > ∼ .
37 we have that F ( φ ) < φ range for the top and bottom plots, respec-tively. Some experimental effort to verify the influence ofthe detector motion in the Bell inequalities using photonscan be found in the literature [5]. The natural general-ization of our results for massless spin-1 particles wouldrequire the entanglement of horizontal/vertical polarizedphotons, which seems distinct from the one consideredin Ref. [5]. This makes both results difficult to compare. Furthermore the replacement of fermions by photons isnot quite straightforward [7]. It would be interesting toverify our results in laboratory since this would also bean indirect test for all the underlying theoretical frame-work. Although conceptually the required experimentalapparatus would be quite simple, this is not obvious tothe present authors how difficult would be its realizationin practice.Modern physics is dominated by quantum mechanicsand relativity. This is fair to say that the Bell inequali-ties probe one of the deepest aspects of quantum mechan-ics. Our analysis shows that the detector state of motionis crucial as one investigates the spin correlation of en-tangled fermions in the context of the Bell inequalitiesonce one assumes the realistic physical situation wherethe particles of the entangled system are described bywave packets rather than by momentum eigenstates.We are grateful to A. O. Pereira for computationalassistance. A.L. and G.M. acknowledge full and partialsupport from Funda¸c˜ao de Amparo `a Pesquisa do Estadode S˜ao Paulo, respectively. G.M. also acknowledges par-tial support from Conselho Nacional de DesenvolvimentoCient´ıfico e Tecnol´ogico. [1] J. S. Bell, Physics , 195 (1964).[2] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. ,777 (1935).[3] J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt,Phys. Rev. Lett. , 880 (1969).[4] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter and A.Zeilinger, Phys. Rev. Lett. , 5039 (1998).[5] A. Stefanov, H. Zbinden, N. Gisin and A. Suarez, Phys.Rev. Lett. , 120404 (2002).[6] A. Peres, P. F. Scudo and D. R. Terno, Phys. Rev. Lett. , 230402 (2002).[7] A. Peres and D. R. Terno, J. Mod. Optics , 1165(2003).[8] H. Terashima and M. Ueda, Quantum Inf. Comput. ,224 (2003).[9] A. Peres and D. R. Terno, Rev. Mod. Phys. , 93 (2004).[10] W. T. Kim and E. J. Son, Phys. Rev. A , 014102(2005).[11] F. R. Halpern, Special Relativity and Quantum Mechan-ics , (Prentice-Hall, Englewood Cliffs, NJ, 1968).[12] We are adopting Halpern [11] and Peres [6] convention,where the relativistic factor p p + m is introduced fur-ther in the Wigner transformation law rather than in thenormalization relation.[13] N. N. Bogolubov, A. A. Logunov and I. T. Todorov, In-troduction to Axiomatic Quantum Field Theory (W. A.Benjamin, Massachusetts, 1975).[14] S. Weinberg,