Table Of ContentEinstein–Podolsky–Rosen correlations of vector bosons
Pawe l Caban, Jakub Rembielin´ski, and Marta W lodarczyk
∗ † ‡
Department of Theoretical Physics, University of Lodz
Pomorska 149/153, 90-236 L ´od´z, Poland
(Dated: February 2, 2008)
WecalculatethejointprobabilitiesandthecorrelationfunctioninEinstein–Podolsky–Rosentype
experiments with a massive vector boson in the framework of quantum field theory. We report on
the strange behavior of the correlation function (and the probabilities) – the correlation function,
which in the relativistic case still depends on the particle momenta, for some fixed configurations
has local extrema. We also show that relativistic spin-1 particles violate some Bell inequalities
more than nonrelativistic ones and that thedegree of violation of theBell inequality is momentum
8 dependent.
0
0 PACSnumbers: 03.65Ta,03.65Ud
2
n
I. INTRODUCTION affects the violation of the Bell-type inequalities. Our
a
analysisshowsthatrelativisticvectorbosonsviolateBell
J
inequalities stronger than nonrelativistic spin-1 particles
1 Different aspects ofquantum informationtheory [1]in
andthatthedegreeofviolationofBellinequalitydepends
2 the relativistic context have been discussed in many pa-
pers [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, on the particle momentum.
] In Sec. II we establish notation and recall basic
h 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
facts concerningthe massivespin-1 representationof the
p 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], mostly for
Poincar´e group and quantum spin-1 boson field. In
- massive particles. Photons have only been discussed in
t Sec. III we define one- and two-particle states which
n a few papers [5, 6, 7, 15, 27, 33, 34, 35, 39, 40]. Most
transformcovariantlywith respectto the Lorentzgroup.
a of these works were performed in the framework of rela-
u Inthenextsectionwediscussthespinoperator. Sec.Vis
tivistic quantum mechanics. However, for the discussion
q devotedtotheexplicitcalculationoftheprobabilitiesand
ofrelativisticcovariancethemostappropriateframework
[ correlationfunctionforthebosonpairinthescalarstate.
is the quantum field theory (QFT) approach. Recently
In Sec. VI we discuss our correlation function and prob-
1 we have discussed the Einstein–Podolsky–Rosen (EPR)
v correlationfunctionforapairofspin-1 massiveparticles abilities . Sec.VII is devotedto the analysisofBell-type
0 2 inequalitiesforspin1particlesintherelativisticcontext.
in the QFT framework [9]. In the present paper we con-
0 The last section contains our concluding remarks.
sider a pair of spin-1 massive particles in this framework
2 In the paper we use the natural units ~ = c = 1 and
3 and calculate the correlation function in EPR-type ex- the metric tensor ηµν =diag(1, 1, 1, 1).
. periments for such a pair in a covariantscalar state. We − − −
1
also calculate the probabilities of the definite outcomes
0
of spin projections measurements performed by two ob-
8
II. PRELIMINARIES
0 servers – Alice and Bob.
: We observe very surprisingbehavior of the correlation
v
For the readers convenience we recall the basic facts
i function (as well as the probabilities). In the center-of-
X and formulas concerning the spin one representation of
mass frame for the definite configuration of the parti-
r cles momenta and directions of the spin projection mea- the Poincar´egroup and quantum vector boson field.
a
surements the correlation function still depends on the
value ofthe particle momentum. It alsoappearsthatfor
A. Massive representations of the Poincar´e group
some configurations this dependence is not monotonic.
In other words, for fixed spin measurement directions
and particle momenta directions, the correlation func- Let us denote by the carrier space of the irre-
H
tion (and probabilities) can have an extremum for some ducible massive representation of the Poincar´egroup. It
finite value of the particle momentum. As far as we are is spanned by the four-momentum operator eigenvectors
aware this is the first time, that such behavior of the k,σ
| i
correlation function has been reported.
This strange behavior of the correlation function also Pˆµ k,σ =kµ k,σ , (1)
| i | i
k2 = m2, with m denoting the mass of the particle, and
σ its spin component along z axis. We use the following
∗Electronicaddress: [email protected] Lorentz-covariantnormalization
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected] k,σ k′,σ′ =2k0δ3(k k′)δσσ′. (2)
h | i −
2
The vectors k,σ can be generated from standard vec- where dµ(k)=Θ(k0)δ((k0)2 ω2) d3k is the Lorentz-
tor k˜,σ , wh|ere ik˜ = m(1,0,0,0) is the four-momentum invariant measure, ω =√k2−+mk2,≡a2ω(kk) and a (k) are
| i k †σ σ
of the particle in its rest frame. We have k,σ = creation and annihilation operators of the particle with
| i
U(Lk)k˜,σ , where Lorentz boost Lk is defined by re- four-momentumkandspincomponentalongz-axisequal
| i
lations k =Lkk˜, Lk˜ =11. The explicit form of Lk is to σ. They fulfill canonical commutation relations
k0 kT [a†σ(k),a†σ′(k′)] = [aσ(k),aσ′(k′)]=0, (10a)
Lk = mmk 11+mmk(m⊗+kTk0) !, (3) [aσ(k),a†σ′(k′)] = 2k0δ(k−k′)δσσ′. (10b)
The field satisfies Klein-Gordon equation and Lorentz
where k0 =√m2+k2. transversality condition, which imply
By means of Wigner procedure we get
m2 =k2, k eµ (k)=0. (11)
µ σ
U(Λ)k,σ = s (R(Λ,k))Λk,λ , (4)
| i Dλσ | i The one-particle states |k,σi := a†σ(k)|0i transform ac-
cording to (4) provided that
where the Wigner rotation R(Λ,k) is defined as
cRo(rΛre,lka)tio=nsLo−Λfk1sΛpLink.onBeepcaaurtsieclewse, ianrethgeoisnegquteol awnealwyzilel U(Λ)a†σ(k)U†(Λ) = Dλσ(R(Λ,k))a†λ(Λk), (12a)
focus on the representation 1(R(Λ,k)) (R). There U(Λ)aσ(k)U†(Λ) = Dλ∗σ(R(Λ,k))aλ(Λk). (12b)
D ≡ D
exists such unitary matrix V that every matrix (R) is Here 0 denotes Poincar´einvariantvacuumwith 00 =
related to R by D 1; a (|ki)0 = 0. Equations (8,12) imply the Wehin|bierg
σ
condition|sifor amplitudes eµ (k)
(R)=VRV . (5) σ
†
D eµ (Λk)=Λµeν (k) (R(Λ,k)) . (13)
(R) are generated by Si, i=1,2,3, σ ν λ D σλ
D From Eq. (13) we have
0 1 0 0 1 0 e(k)=L e(k˜), (14)
− k
S1 = 1 1 0 1 , S2 = i 1 0 1 ,
√2 √2 − where L is given by (3) and we used the fact that
0 1 0 0 1 0 k
R(Lk,k˜) = 11. Therefore, to find the explicit form of
1 0 0 eµ (k) it is enough to determine eµ (k˜). From Eq. (11)
σ σ
S3 = 0 0 0 , (6) we get
0 0 1
− 0 0 0
[eµ (k˜)]= , (15)
σ
(see e.g. [44]), that is (R)=eiφS. Taking into account e˜ !
the form of generatorDs of the rotations R i.e. [Ii] =
jk where e˜is a 3 3 matrix. Now, from the Weinberg con-
−iǫijk, we can easily determine the explicit form of ma- dition (13) for×pure rotations and by means of Eq. (5)
trix V
and Schur’s Lemma we find
1 i 0 e˜=VT, (16)
1 −
V = 0 0 √2 . (7)
√2 where explicit form of V is given by (7). Finally from
1 i 0
(14) we have
kT
B. Vector field e(k)= 11+mmk(m⊗+kTk0) !VT. (17)
Under Lorentz groupaction the vector boson field op- Equations (16,17) imply
erator ϕˆµ(x) transforms according to
e µ (k)e (k)= δ , (18a)
∗σ µλ − σλ
U(Λ)ϕˆµ(x)U (Λ)=(Λ 1)µϕˆν(Λx). (8)
† − ν
eµ (k)e (k)= (VVT) , (18b)
The field operator has the standard momentum expan- σ µλ − σλ
sion e µ (k)eν (k)= ηµν + kµkν, (18c)
∗σ σ − m2
ϕˆµ(x)=(2π)−3/2 dµ(k) eikxeµσ(k)a†σ(k) 0 0 1
−
σ=X0,±1Z (cid:2) where e(k)VVT =e∗(k), and VVT = 0 1 0 .
+e ikxe µ (k)a (k) , (9)
− ∗σ σ 1 0 0
−
(cid:3)
3
III. COVARIANT STATES There are also two independent tensor states, the sym-
metric traceless
A. One-particle covariant states
Ψµν = 1(δµ δν +δµδν 1ηµνη )(α,k),(β,p) ,
| symi 2 α β β α− 2 αβ | i
(30)
Inthe discussionofLorentz-covarianceitisconvenient
and the antisymmetric one
to use states
(µ,k) =eµ (k)k,σ , (19) Ψµν = 1(δµ δν δµδν )(α,k),(β,p) . (31)
| i σ | i | asymi 2 α β − β α | i
which transform covariantly In the sequel we will analyze correlations in the scalar
state (28).
U(Λ)(µ,k) =(Λ 1)µ (ν,Λk) . (20)
| i − ν| i
They are normalized as follows [c.f. (2)]
IV. SPIN OPERATOR
(µ,k)(ν,p) =2k0δ(k p)e µ(k)eν (p). (21)
h | i − ∗σ σ
When we want to calculate explicitly correlationfunc-
Arbitraryone-particlestatecanbeexpandedinthestan-
tions, we need to introduce the spin operator for rela-
dard basis k,σ as well as in the covariant one (19)
| i tivistic massive particles. Several possibilities have been
discussed in the literature (see e.g. [8, 9, 10, 11, 24, 28,
ψ = dµ(k)ψ (k)k,σ = dµ(k)Ψ (k)(µ,k) ,
σ µ 36, 37, 38, 40, 41]). We choose the operator
| i | i | i
Z Z
(22)
where 1 Pˆ
Sˆ = Wˆ +Wˆ0 , (32)
Ψ (k)eµ (k)=ψ (k). (23) m Pˆ0+m!
µ σ σ
which is the most appropriate [8, 38, 45]. Here
B. Two-particle covariant states
Wˆµ = 1ǫµνγδPˆ Jˆ (33)
2 ν γδ
In analogyto (19) we candefine covariantbasis in the
two-particle sector of the Fock space is the Pauli-Lubanski four-vector, Pˆν is the four-
momentum operator, Jˆ denote the generators of the
µν
|(µ,k),(ν,p)i=eµσ(k)eνλ(p)|(k,σ),(p,λ)i, (24) Lorentz group such that U(Λ) = exp(iωµνJˆµν), and we
assumeǫ0123 =1. Consequentlythe spinoperatorSˆ acts
where |(k,σ),(p,λ)i=a†σ(k)a†λ(p)|0i. The most general on one-particle states according to
two-particle state has the form
Sˆ k,σ =S k,λ , (34)
λσ
| i | i
Ψ = dµ(k)dµ(p)ψ (k,p)(k,σ),(p,λ)
σλ
| i Z | i where Si are defined by (6). In the Fock space Sˆ takes
dµ(k)dµ(p)Ψ (k,p)(µ,k),(ν,p) . (25) standard form
µν
≡ | i
Z
One can see that Sˆ = dµ(k)a†(k)Sa(k), (35)
Z
Ψ (k,p)eµ (k)eν (p)=ψ (k,p). (26)
µν σ λ σλ where the column matrix
a(k) = (a (k),a (k),a (k))T. From Eqs. (18a,19) we
Moreover it holds +1 0 1
get −
Ψµν(k,p)=Ψνµ(p,k), (27a)
kµΨµν(k,p)=Ψµν(k,p)pν =0. (27b) Sˆ|(α,k)i=− e(k)STe†(k)η αβ|(β,k)i. (36)
(cid:2) (cid:3)
We can now define two-particle states transforming ac- In real experiments detectors register only particles
cording to irreducible representations of Lorentz group. whose momenta belong to some definite region Ω in mo-
Thescalarstatedescribingparticleswithsharpmomenta mentum space. Therefore we need the operator which
is defined as acts similar to (34) on particles with four-momenta be-
longing to Ω and yields 0 in all other cases. Such an
Ψ =η (µ,k),(ν,p) . (28)
s µν operator has the following form
| i | i
In terms of (24) it takes the form
Sˆ = dµ(k)a (k)Sa(k). (37)
Ω †
|Ψsi=ηµνeµσ(k)eνλ(p)|(kσ),(p,λ)i. (29) ZΩ
4
V. PROBABILITIES AND THE CORRELATION In Eq. (40) Πˆi , i = 0,1,2, denotes a projector on the
Ω
FUNCTION subspace of two-particle states, in which exactly i parti-
cles have momenta from Ω. From Eqs. (40, 41) we find
Let us consider two distant observers, Alice and Bob,
in the same inertial frame, sharing a pair of bosons in Πˆ1 =2Nˆ Nˆ2, (42)
scalar state Ψs defined by (29). Ω Ω− Ω
| i
Now let Alice measure spin component of her boson
in direction a and Bob spin component of his boson in and
direction b, where a = b = 1. Their observables are
(a Sˆ ) and (b Sˆ |),|resp|ec|tively, where (ω Sˆ ) is de-
· A · B · Ω Πˆ1 (k,λ),(p,σ)
fined by (37) with Ω equal A and B, and ω equal to a Ω| i
and b, respectively. We assume that A B = . Now = χ (k)+χ (p) 2χ (k)χ (p) (k,λ),(p,σ) . (43)
Ω Ω Ω Ω
∩ ∅ − | i
we would like to explicitly calculate probabilities P of
σλ Therefore from (38) and (43) we finally get
(cid:2) (cid:3)
obtainingparticularoutcomesσ andλbyAliceandBob,
respectively (σ and λ can take values 1 and 0). Let us
±
first notice, that from Eqs. (24), (34), and (37) we have (ω Sˆ )Π1 (k,λ),(p,σ)
· Ω Ω| i
(ω SˆΩ)(k,λ),(p,σ) =ω· χΩ(k) 1−χΩ(p) Sλ′λδσ′σ
· | i
=ω· χΩ(k)Sλ′λδσ′σ+χΩ(p)Sσ′σδλ′λ |(k,λ′),(p,σ′)i, +χΩ(p) 1−nχΩ(k) S(cid:2)σ′σδλ′λ |((cid:3)k,λ′),(p,σ′)i. (44)
(38)
(cid:2) (cid:3) (cid:2) (cid:3) o
where the characteristic function χΩ(q) is defined in a By definition the observable ωSˆΩΠˆ1Ω measures the spin
standard way component of one particle in the direction ω, therefore
its spectral decomposition is
1 when q Ω,
χΩ(q)= ∈ (39)
( 0 when q ∈/ Ω. (ω·SˆΩ)Πˆ1Ω =1·Πˆ+Ωω−1·Πˆ−Ωω+0·Πˆ0Ωω, (45)
However,in EPR-typeexperiments we take into account
only such measurements in which Alice and Bobregister where the projectorsΠˆ±Ωω and Πˆ0Ωω correspondto eigen-
oneparticleeach. Thereforeweareactuallyinterestedin values 1 and 0, respectively. Simple calculation gives
spectral decomposition of observable (ω Sˆ )Πˆ1, where ±
· Ω Ω
Πˆ1 is a projector (in the two-particle sector of the Fock
Ω
sspitaucaet)ioonninthwehsiuchbspexaaccetloyf osntaetepsarctoicrrleeshpaosndminogmteonttuhme Πˆ±Ωω = 12(ω·SˆΩ) (ω·SˆΩ)±11 Πˆ1Ω, (46a)
from the region Ω. To find the explicit form of the Πˆ1Ω Πˆ0Ωω = 11−(ω·Sˆ(cid:2)Ω)2 Πˆ1Ω. (cid:3) (46b)
we use the particle number operator Nˆ answering the
Ω (cid:2) (cid:3)
question how many particles have momentum from Ω.
Now we can find explicitly the probabilities P men-
In the two-particle sector of the Fock space we have the σλ
tioned above in the state (29).
obvious spectral decomposition of Nˆ
Ω
NˆΩ =0·Πˆ0Ω+1·Πˆ1Ω+2·Πˆ2Ω, (40) P = hΨs|ΠˆσAaΠˆλBb|Ψsi. (47)
σλ
Ψ Ψ
and in the basis (24) h s| si
Nˆ (k,λ),(p,σ) = χ (k)+χ (p) (k,λ),(p,σ) . (41) From Eqs. (38, 43, 44, 46) we find
Ω Ω Ω
| i | i
(cid:2) (cid:3)
Πˆ±Ωω|Ψsi= 12ηµνeµλ(k)eνσ(p) (ω·S)2±ω·S λ′λδσ′σχΩ(k) 1−χΩ(p)
n(cid:2) (cid:3)+ (ω·S)2±(cid:2)ω·S δ′δδλ′(cid:3)λχΩ(p) 1−χΩ(k) |(k,λ′),(p,σ′)i, (48a)
(cid:2) (cid:3) (cid:2) (cid:3)o
Πˆ0Ωω|Ψsi=ηµνeµλ(k)eνσ(p) δλ′λδσ′σ χΩ(k)+χΩ(p) 2−(ω·S)2λ′λδσ′σχΩ(k) 1−χΩ(p)
n (cid:2) (cid:3) −(ω·S)2σ′σδλ′λχΩ(p(cid:2)) 1−χΩ(k(cid:3)) |(k,λ′),(p,σ′)i. (48b)
(cid:2) (cid:3)o
5
Let us assume, that Alice can measure only the bosons Eqs. (51). Therefore in our case (s = 1) the correlation
with four-momentum k and Bob those with four- function takes the following form
momentum p, i.e.
C (k,p)=P +P P P , (54)
ab ++ + +
χ (p)=χ (k)=0, (49) −−− − − −
A B
which in notation (52) reads
and
1
C (k,p)= Tr N(k,a)ηN(p,b)η . (55)
χA(k)=χB(p)=1. (50) ab −2+ (kp)2 { }
m4
After a little algebra we find
Of course the above correlation function can be also
1 found by means of standard formula
P = Tr M(k,a)ηM(p,b)η
±± −4(cid:2)N2(+k,(akmp)4)η2N(cid:3)(p{,b)η}, (51a) Cab(k,p)= hΨs|(a·hSˆΨAs)|(Ψbs·iSˆB)|Ψsi. (56)
1
P±∓ = 4 2+ (kp)2 Tr{M(k,a)ηM(p,b)η After some calculation we get
m4
+(cid:2)N(k,a)ηN(cid:3)(p,b)η}, (51b) Cab(k,p)
1
P = Tr T(k,a)ηM(p,b)η , (51c) 2 [a (k p)][b (k p)]
0± 2 2+ (kmp4)2 { } = 2+ (kmp4)2n−a·b− m·2(m×+k0)(·m+×p0)
1
P = (cid:2) (cid:3)Tr M(k,a)ηT(p,b)η , (51d) (a p)(b k) (a b)(p k)
±0 2 2+ (kmp4)2 { } − · · m−2 · ·
1 (a p)(b p) p2(a b) (a k)(b k) k2(a b)
P = (cid:2) Tr(cid:3) T(k,a)ηT(p,b)η , (51e) + · · − · + · · − ·
00 2+ (kp)2 { } m(m+p0) m(m+k0)
m4
(k p)(a p)(b k) (k p)2(a b)
where we have introduced the following notation + · · · − · · . (57)
m2(m+k0)(m+p0)
o
N(q,ω)αβ ≡ e∗αλ(q)(ω·S)λσeβσ(q), (52a) In the next sectionwe will analyze behavior of the prob-
M(q,ω)αβ ≡ e∗αλ(q)(ω·S)2λσeβσ(q), (52b) abilities and the correlationfunction in the CMF frame.
T(q,ω)αβ ≡ e∗αλ(q) δλσ−(ω·S)2λσ eβσ(q),(52c)
VI. PROBABILITIES AND CORRELATION
(for explicit formof matrice(cid:2)s N, M, T see A(cid:3)pp. A.). All FUNCTION IN CMF FRAME
the above probabilities add up to 1.
Using Eqs. (51) and (A1) one can easily find explicit In the CMF frame p= k and probabilities (51) take
form of the probabilities for arbitrary a, b, k and p. the form −
However,theresultingformulasappeartoberatherlong
andwe do not put them here. In the nextsectionwe are P = 1 (1+2x)2 2(1+2x)(a b)
going to limit ourselves to the simpler case when Alice ±± 4[2+(1+2x)2] − ·
and Bob are at rest with respect to the center of mass +4x(a n)(bn n) 4x(x+1) (a n)2+(b n)2
· · − · ·
frame (CMF) of the boson pair. 2
+ a b+2x(a n)(b n) ,(cid:2) (58a(cid:3))
In EPR-type experiments we usually analyze the spin · · ·
correlation function defined as P = (cid:2) 1 (1+2x)2+(cid:3)2(o1+2x)(a b)
±∓ 4[2+(1+2x)2] ·
s
4x(a n)(bn n) 4x(x+1) (a n)2+(b n)2
C = λσP , (53)
ab λσ − · · − · ·
2
σ,Xλ=−s + a·b+2x(a·n)(b·n) ,(cid:2) (58b(cid:3))
where σ, λ denote spin projections on the directions a P = (cid:2) 1 1+4x(1+(cid:3)x)o(a n)2
andb,respectively,andPλσ isthejointprobabilityofob- 0± 2[2+(1+2x)2] ·
tainingresultsσ,λ. Letusnoticethatcaseswhenσ orλ n 2
a b+2x(a n)(b n) , (58c)
equal0donotcontributetothecorrelationfunction(53). − · · ·
In principle one could define the ”normalized” correla- P = (cid:2) 1 1+4x(1+(cid:3)x)o(b n)2
tion function as C [Eq. (53)] divided by P . ±0 2[2+(1+2x)2] ·
However we preferatbo deal with the function (σ5,3λ)6=0whλicσh a b+2xn(a n)(b n) 2 , (58d)
P − · · ·
contains more information. The ”normalized” correla-
tion function can be also easily calculated by means of P00 = 2+(cid:2)(1+12x)2 a·b+2x(a·n(cid:3))(ob·n) 2, (58e)
(cid:2) (cid:3)
6
0.4 0.7
0.35
0.6
n
o
bability 0 00.2..235 PP++−+==PP−−−+ elationfuncti 00..45
pro corr 0.3
0.15
0.2
0.1
0.1
0.05
0
0 P00 0xm 2 4 6 8 10
0 2 4 6 8 10 x
x
FIG. 2: The plot shows dependence of correlation function
FIG. 1: The plot shows dependence of probabilities Pσλ in Cab inCMFframeonxfora b= 1,a n=b n=0. The
pCrMobFabfirlaitmiees oPn++xafnord aP−·b− =hav−e1m, aax·imnu=mbeq·unal=to03./8Tfhoer function has maximum equal·to1/√−2 for·xm=(·√2−1)/2.
x=1/2. Probabilities P0± and P±0 vanish.
0.2
0.18
0.16 P++=P
twhheerceorxrel=ati(cid:16)o|nmk|f(cid:17)u2n,ctnio=n r|ekkd|u.cFesurttohermore in this frame probability 00 ..011.241 PP0+±−==PP±−−0−+
0.08
0.06
0.04 P00
C (k, k)=
ab
− 0.02
2
(1+2x)(a b)+2x(a n)(b n) (59) 0
2+(1+2x)2 − · · · 0 2 4 6 8 10
(cid:2) (cid:3) x
FIG. 3: The plot shows dependence of probabilities Pσλ in
CMF frame on x for a b = 1/2, a n = b n = 1/2. The
· − · ·
For a given configuration of directions a, b and n the probabilities P++, P−−, P0± and P±0 have maxima for x =
probabilities and the correlation function depend on the 1/2 while probabilities P00, P−+ and P−− have minima for
value of the three-momentum of the particles. What is x=1 and x=1/3, respectively.
very unexpected, for some configurations the probabili-
ties and the correlation function have local extrema. It
Ultra-relativistic limit
suggests that for some values of momenta Bell inequali-
tiesmaybeviolatedstronger. We discussthis possibility
In the ultra-relativistic limit the probabilities take the
in the next section.
form
Configurations can be found where the correlation P =P = 1(1 (a n)2)(1 (b n)2), (60a)
±± ±∓ 4 − · − ·
fwuhnicletioonthaenrdpsroombaeboilfitthieesparroebmaboinliottieosnhicav(seeleocFailgesx.t1r,e2m)a., P0± = (a·2n)2(1−(b·n)2), (60b)
Citioensfiagrueramtoionnostocnanicaalnsodbseucfohucnodn,fiwghueraretiaolnlsthwehperreobaallbiol-f P±0 = (b·2n)2(1−(a·n)2), (60c)
P = (a n)2(b n)2, (60d)
the probabilities and the correlation function have local 00 · ·
extrema (see Figs. 3, 4). and the correlation function vanishes
C (k, k)=0, (61)
ab
Finally let us consider the ultra-relativistic (x ) −
−→∞
andnon-relativistic(x 0)limits offormulas(58)and whichmeansthatforultrafastparticlesthereisnocorre-
−→
(59). lation between outcomes of measurements performed by
7
P++
P+- 1
0 .2 0.8
P-+ 1 P00 0.6 1
P-- 0.1 0.4
0.5 0.2 0.5
0 0
--11 0 Öb×Ön× --11 0 Öb×Ön×
--00..55 --00..55
ÖÖ××ÖÖ××00 -0.5 ÖÖ××ÖÖ××00 -0.5
aann 00..55 aann 00..55
-1 -1
1 1
0.5 0.5
0.4 0.4
P0± 0.3 1 P±0 0.3 1
0.2 0.2
0.1 0.5 0.1 0.5
0 0
--11 0 Öb×Ön× --11 0 Öb×Ön×
--00..55 --00..55
ÖÖ××ÖÖ××00 -0.5 ÖÖ××ÖÖ××00 -0.5
aann 00..55 aann 00..55
-1 -1
1 1
FIG. 5: Dependenceof probabilities (60) on a nand b nin theultra-relativistic limit.
· ·
Alice and Bob. One can notice that in this limit none of
0.4 the probabilities(60)dependonrelativeconfigurationof
directions a and b but only on their configuration with
0.35
respecttodirectionofthemomentumn. Fig.5illustrates
n
ctio 0.3 the dependence of probabilities (60) on scalar products
n a n and b n.
u
f 0.25 · ·
n
o
ati 0.2
el
corr 0.15
0.1
0.05
0
0 xm 2 4 6 8 10
x
FIG.4: Theplotshowsdependenceofthecorrelationfunction
Cab in CMF frame on x for a b= 1/2, a n=b n=1/2.
· − · ·
Ithasmaximumequalto(√19 1)/8forxm =(√19 2)/6.
− −
Non-relativistic limit and the correlation function reads
Now in non-relativistic limit the probabilities are
C = 2a b. (63)
P = 1 (1 (a b))2, (62a) ab −3 ·
±± 12 − ·
P = 1 (1+(a b))2, (62b)
±∓ 12 ·
P =P = 1(1 (a b)2), (62c)
0± ±0 6 − ·
P = 1(a b)2, (62d)
00 3 ·
8
Letus note that inthis limit probabilitiesand the corre-
lationfunctiondo notdepend onthe momentumk. One 1.05
canalsoeasilycheckthatinthiscasetheyarethesameas
calculated in the framework of non-relativistic quantum 1
mechanics in the singlet state
0.95
Ψ = 1 (1 1 0 0 + 1 1 ), (64)
| i √3 | i|− i−| i| i |− i| i 0.9
where 1 , 0 and 1 are states with spin component
| i | i |− i 0.85
along z-axis equal to 1, 0 and 1, respectively.
−
0.8
VII. BELL-TYPE INEQUALITIES
0.75
Thespin-1systemhasthreedegreesoffreedom,which 0.7
0 0.2 0.4 0.6 0.8 1
makes the full analysis of Bell inequalities much more x
difficult and subtle (see e.g. [46, 47, 48]). In the present
paper we will show that at least for some Bell-type in- FIG. 6: The plot shows dependence of the left side of the
equalities its violation strongly depends on the particle inequality (72) on x. The plotted function has maximum
momenta. Moreover we discuss inequality which is sat- valueequal to 3√2/4 for x=(√2 1)/2.
−
isfied for the nonrelativistic correlation function but is
violated in the relativistic case.
in the theory which fulfills the assumptions of local re-
For spin-1 particles the most commonly discussed
2 alism. This inequality, similar to the CHSH one, is not
Bell-type inequality is the Clauser-Horne-Shimony-Holt
violated in the nonrelativistic quantum mechanics. In-
(CHSH) inequality [50]:
deed, inserting (63) into (68) we get the inequality
C C + C +C 2 (65)
ab ad cb cd
| − | | |≤ 2
(a b+b c+c a) 1 (69)
In (65) Cab denotes the correlationfunction of spin pro- − 3 · · · ≤
jections on the directions a and b. One can easily check
which is equivalent to
that (65) is also valid for spin-1 particles. (see e.g. [51]).
The nonrelativistic correlationfunction (63) does notvi- 1
3 (a+b+c)2 1. (70)
olatethe inequality(65). Indeed, inserting(63)into(65)
3 − ≤
we get
(cid:2) (cid:3)
The left side of (70) is largest when a+b+c = 0. In
a b a d + c b+c d 3. (66) this case (70) is of course fulfilled. Therefore nonrela-
| · − · | | · · |≤
tivisticquantummechanicsdoesnotviolatetheBell-type
The largestvalue of the left side of (66) is equal to 2√2,
inequality (68).
therefore (66) holds in all configurations. In the rela-
However, we show that the inequality (68) can be vi-
tivistic framework, inserting (59) into (65) we get the
olated in the relativistic framework. Inserting (59) into
following inequality:
inequality (68) we get
1
(1+2x) (a b) (a d) 2
2+(1+2x)2 · − · (1+2x)(a b+b c+c a)
n(cid:12) 2+(1+2x)2 − · · ·
2x ((cid:12)(cid:12)a n)(b (cid:2)n) (a n)(d n(cid:3)) n
− · · − · · +2x (a n)(b n)+(b n)(c n)+(c n)(a n) 1.
(cid:12) · · · · · · ≤
+(cid:2) (1+2x) (c·b)+(c·d) (cid:3)(cid:12)(cid:12) (cid:2) (cid:3)o (71)
(cid:12)
2x (c(cid:12) n)(b n(cid:2))+(c n)(d n)(cid:3) 1. (67) Intheconfigurationa+b+c=0,a n=b n=c n=0
− (cid:12)· · · · ≤ · · ·
(71) takes the form
Our numerical(cid:2)simulations show that the(cid:3)(cid:12)(cid:12)loargest value
of left side of (67) is equal to 1. Therefo(cid:12)re the CHSH 3(1+2x)
1, (72)
inequality is not violated in the relativistic framework, 2+(1+2x)2 ≤
either.
and one can easily check that this inequality is violated
Thereforeforspin1particleswehavetoconsiderother
for 0 < x < 1/2. (Let us note that the value x = 0
Bell-type inequalities.
corresponds to the nonrelativistic limit for which the in-
AccordingtoMermin’spaper[49],inEPR-typeexper-
equalityis notviolated). The dependence ofthe leftside
iments with a pair of spin-1 particles in the singlet state
of the inequality (72) is shown on the Fig. 6.
the following inequality has to be satisfied
In the paper [49] another Bell-type inequality, which
C +C +C 1, (68) isviolatedinthenonrelativisticcase,isconsidered. This
ab bc ca
≤
9
1.2
1.15
1.15 1.1
1.1 1.05
1.05 1
1 x=0 0.95
x=1/6 0.9
0.95
0.85
0.9
0.8
0.85
0.75
0.8
π/6 π/4 π/3 π/2 0.7
θ[rad] 0 0.2 0.4 0.6 0.8 1
x
FIG. 7: The plot shows dependence of the left side of the
FIG. 8: The plot shows dependence of the left side of the
inequality (75) on θ. The value x = 0 corresponds to the
inequality (75) on x for θ = 2π/3 corresponding to the con-
nonrelativisticcaseTheplottedfunctionhasmaximumvalue
figurationa+b+c=0. Theplottedfunctionhasmaximum
equal to 3√2/4 for x=(√2 1)/2.
− valueequal to 9/8 for x=1/6.
inequality contains not only a correlation function but
VIII. CONCLUSIONS
alsotheaveragevalueofthedifferenceofspinprojections
measured by Alice and Bob and has the following form:
We have discussed joint probabilities and the corre-
λ σ P (a,b) C +C . (73) lation function of two relativistic vector bosons in the
λσ ac bc
| − | ≥
λ,σ framework of quantum field theory. We have classified
X
two-particlecovariantstates and defined the observables
WehavecalculatedtheprobabilitiesP (a,b)[Eqs.(58)]
λσ correspondingtodetectorsmeasuringthespinofthepar-
therefore we can analyze the inequality (73) also in the
ticles with momenta belonging to a given region of mo-
relativistic framework. Inserting (58) and (63) into (73)
mentum space. Using this formalism we have explicitly
we obtain the inequality
calculated the correlation function and the probabilities
in the scalar state. We observed strange behavior of
2
(1+2x)(a b+b c+c a) thecorrelationfunctionandtheprobabilities. Itappears
2+(1+2x)2 − · · ·
that in the CMF frame for the definite configuration of
n
+2x (a n)(b n)+(b n)(c n)+(c n)(a n) the particles momenta and directions of the spin projec-
· · · · · ·
(cid:2) + 1 (a b)+2x(a n)(b n) 2 1. (cid:3)(74) tionmeasurements,thecorrelationfunctionstilldepends
onthevalueoftheparticlesmomenta. Recallthatfortwo
2 · · · ≤
(cid:2) (cid:3) o fermions the correlation function in CMF frame in the
This inequality is stronger than (71). Let us analyze
singletstatedoesnotdependonmomentum[9]. Further-
the inequality (74) in the configuration considered in
more, in the bosonic case for fixed spin measurement di-
[49], that is let us assume that a, b, c are coplanar and
rections, the correlation function (and the probabilities)
a b = cos(π 2θ), a c = b c = cos(π/2+θ). More-
can have extrema for some finite values of the particles
· − · ·
over let us assume that a n = b n = c n = 0. In this
momenta. ThisaffectsthedegreeofviolationofBell-type
· · ·
configuration (74) takes the following form inequalities. We have discussed the Bell-type inequality
(68) which is fulfilled in the nonrelativistic limit but is
2(1+2x) 2sinθ+cos(2θ) +cos2(2θ)
1. (75) violated in some finite region of the particles momenta.
2+(1+2x)2 ≤
(cid:2) (cid:3) We havealsoshownthat Bell-type inequality (73) which
is violated for nonrelativistic spin-1 particles in the rela-
Wehaveshownthedependence oftheleftsideof(75)on
tivistic case is violated more in some finite region of the
θ for two chosen vales of x: x = 0 corresponding to the
particles momenta.
nonrelativistic case and x=1/6 on Fig. 7. We show the
dependence of the left side of (75) on x in Fig. (8). We
havechosenθ =2π/3correspondingtotheconfiguration
a+b+c=0 considered earlier. Acknowledgments
Summarizing, our analysis shows that relativistic vec-
tor bosons violate Bell inequalities more than nonrela- This work has been supported by the University of
tivistic spin-1 particles and that the degree of violation Lodz grantandpartiallysupportedby the EuropeanSo-
oftheBellinequalitydependsontheparticlemomentum. cial Fund and Budget of State implemented under the
10
Integrated Regional Operational Programme. Project:
GRRI-D.
APPENDIX A: EXPLICIT FORM OF MATRICES
Nαβ, Mαβ AND Tαβ
The explicit form of matrices (52) is
0 i (q ω)T
Nαβ(q,ω)=−mi (q×ω) −iǫijkωk+imq⊗(q××ωm)(Tm−+(qq0×)ω)⊗qT ij , (A1a)
h i
Mαβ(q,ω)
= q2−(mω2·q)2 qmT qm0 − m(ω(m·+q)q20) −ωTωm·q , (A1b)
mq qm0 − m((ωm·+q)q20) −ωωm·q 11−ω⊗ωT− m(ωm·+qq(cid:2)0) ω⊗qT+q(cid:3)⊗ωT + 1− ((mω+·qq0))22 q⊗mq2T !
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
Tαβ(q,ω)= (ωm·q2)2 ωm·q ωT+ m(ω(m·q+)qq0T) . (A1c)
ωm·q ω+ m(ω(m·+q)qq0) ω⊗ωT+ m(ωm·+qq0) ω⊗(cid:2)qT+q⊗ωT (cid:3)+ m2((ωm·+q)q20)2q⊗qT !
(cid:2) (cid:3) (cid:2) (cid:3)
[1] M. A. Nielsen and I. L. Chuang, Quantum Computation SocietyforOpticalEngineering,Bellingham,WA,1997),
andQuantumInformation (CambridgeUniversityPress, vol. 3076 of Proceedings of SPIE, pp. 141–145, arXiv:
Cambridge, 2000). quant-ph/0205187.
[2] D. Ahn, H. J. Lee, and S. W. Hwang, Rela- [13] M. Czachor, Phys. Rev. Lett. 94, 078901 (2005), arXiv:
tivistic entanglement of quantum states and nonlocal- quant-ph/0312040.
ity of Einstein–Podolsky–Rosen(EPR) paradox, arXiv: [14] R. M. Gingrich and C. Adami, Phys. Rev. Lett. 89,
quant-ph/0207018 (2002). 270402 (2002), arXiv: quant-ph/0205179.
[3] D.Ahn,H.J.Lee,S.W.Hwang,andM.S.Kim,Isquan- [15] R.M.Gingrich,A.J.Bergou,andC.Adami,Phys.Rev.
tum entanglement invariant in special relativity?, arXiv: A 68, 042102 (2003), arXiv: quant-ph/0302095.
quant-ph/0304119 (2003). [16] C. Gonera, P. Kosin´ski, and P. Ma´slanka, Phys. Rev. A
[4] D.Ahn,H.J.Lee,Y.H.Moon,andS.W.Hwang,Phys. 70, 034102 (2004), arXiv: quant-ph/0310132.
Rev.A 67, 012103 (2003), arXiv: quant-ph/0209164. [17] N.L.Harshman,Phys.Rev.A71,022312(2005),arXiv:
[5] P. M. Alsing and G. J. Milburn, Quantum In- quant-ph/0409204.
formation and Computation 2, 487 (2002), arXiv: [18] S. He, S. Shao, and H. Zhang, J. Phys. A: Math. Gen.
quant-ph/0203051. 40, F857 (2007), arXiv: quant-ph/0701233.
[6] S.D.BartlettandD.R.Terno,Phys.Rev.A71,012302 [19] S. He, S. Shao, and H. Zhang, Quantum Entangled En-
(2005), arXiv: quant-ph/0403014. tropy: Helicity versus Spin, arXiv: quant-ph/0702028
[7] P. Caban and J. Rembielin´ski, Phys. Rev. A 68, 042107 (2007).
(2003), arXiv: quant-ph/0304120. [20] T. F. Jordan, A. Shaji, and E. C. G. Sudarshan, Phys.
[8] P. Caban and J. Rembielin´ski, Phys. Rev. A 72, 012103 Rev. A 73, 032104 (2006), arXiv: quant-ph/0511067.
(2005), arXiv: quant-ph/0507056. [21] T.F.Jordan,A.Shaji,andE.C.G.Sudarshan,Lorentz
[9] P. Caban and J. Rembielin´ski, Phys. Rev. A 74, 042103 transformations that entangle spins and entangle mo-
(2006), arXiv: quant-ph/0612131. menta, arXiv: quant-ph/0608061 (2006).
[10] M. Czachor and M. Wilczewski, Phys. Rev. A 68, [22] P. Kosin´ski and P. Ma´slanka, Depurification by Lorentz
010302(R) (2003), arXiv: quant-ph/0303077. boosts, arXiv: quant-ph/0310145 (2003).
[11] M. Czachor, Phys. Rev. A 55, 72 (1997), arXiv: [23] W. T. Kim and E. J. Son, Phys. Rev. A 71, 014102
quant-ph/9609022. (2005), arXiv: quant-ph/0408127.
[12] M.Czachor,inPhotonicQuantum Computing,editedby [24] H.LiandJ.Du,Phys.Rev.A68,022108 (2003),arXiv:
S.P.HolatingandA.R.Pirich(SPIE-TheInternational quant-ph/0211159.
