Table Of ContentDynamics of momentum entanglement in lowest-order QED
L. Lamata,1,2, J. Le´on,1, and E. Solano2,3,
∗ † ‡
1Instituto de Matem´aticas y F´ısica Fundamental, CSIC, Serrano 113-bis, 28006 Madrid, Spain
2Max-Planck-Institut fu¨r Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany
3Secci´on F´ısica, Departamento de Ciencias, Pontificia Universidad Cat´olica del Peru´, Apartado Postal 1761, Lima, Peru
Westudythedynamicsofmomentumentanglementgeneratedinthelowest-orderQEDinteraction
between two massive spin-1/2 charged particles, which grows in timeas thetwo fermions exchange
virtualphotons. Weobservethatthedegreeofgeneratedentanglementbetweeninteractingparticles
with initial well-defined momentum can be infinite. We explain this divergence in the context of
entanglement theory for continuous variables, and show how to circumvent this apparent paradox.
Finally, we discuss two different possibilities of transforming momentum into spin entanglement,
through dynamical operations or through Lorentz boosts.
6
0 PACSnumbers: 12.20.-m,03.67.Mn,03.65.Ud
0
2
n I. INTRODUCTION nonrelativisticlimit,thedynamicsofentanglementinpo-
a sition/momentum of two electrons which interact with
J each other and with a nucleus via a smoothed Coulomb
Entanglement is the physical property that
5 potential. They found that the associated quantum cor-
Schr¨odinger described as Not one but the charac-
2 relationsmanifestatendencytoincreaseasafunctionof
teristic trait of quantum mechanics. It has played a
fundamental role in the study of the completeness of the interaction time.
2
In this paper, we study to the lowest order in QED
v quantum mechanics [1, 2]. Nowadays, in quantum
the interaction of a pair of identical, charged, massive
1 information theory, entanglement is considered as a
2 physical resource, equivalent in many aspects to the spin-1/2particles,andhowthis interactionincreasesthe
0 role energy played in classical and quantum mechanics. entanglement in the particle momenta as a function of
9 time. We chose to work at lowest order, where en-
Most applications in this novel field, like quantum
0 tanglement already appears full-fledged, precisely for its
teleportation [3], quantum communication [4], quantum
5
simplicity. In particular this allows to set aside neatly
0 cryptography [5], and some algorithms of quantum
other intricacies of QED, whose influence on entangle-
/ computation are carried out by using this intriguing
h ment should be subject of separate analysis. Here, the
quantum property [6]. A thorough study of entangle-
p generation of entanglement is a consequence of a con-
ment in quantum information theory would demand a
-
t naturalclassificationbetweendiscrete[7]andcontinuous servation law: the total relativistic four-momentum is
n preserved in the system evolution. This will also be the
variables [8].
a
caseinanyinteractionverifyingthisconservationlaw,as
u In the last few years two apparently different fields,
occurs in closed multipartite systems, while allowing the
q entanglementandrelativity,haveexperiencedintensere-
: change in the individual momentum of each component.
v searchinaneffortfor treatingthem ina commonframe-
In the asymptotic limit, the infinite spacetime intervals
i work [9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Most of those
X involved in the S-matrix result in the generation of an
works investigated the Lorentz covariance of entangle-
r ment through purely kinematic considerations, and only infinite amount of entanglement for interacting particles
a with well-defined momentum. QED is a place where in-
a few of them studied ab initio the entanglement dy-
finities can be avoided, and this will be also true, even
namics. For example, in the context of Quantum Elec-
though for other physical reasons, in the case of diver-
trodynamics (QED), Pachos and Solano [14] considered
gences appearing in momentum entanglement, a distinc-
the generation and degree of entanglement of spin cor-
tive feature of continuous variables [8]. We will also dis-
relations in the scattering process of a pair of massive
cuss two different possibilities of establishing transfer of
spin-1/2 charged particles, for an initially pure product
entanglement between momentum and spin degrees of
state, in the low-energy limit and to the lowest order in
freedom in the collective two-particle system: through
QED.ManoukianandYongram[18]computedthe effect
dynamical operations or Lorentz boosts.
of spin polarization on correlations in a similar model,
InSec. II,weanalizeatlowestorderandatfinitetime
but also for the case of two photons created after e+e
−
the generation of momentum entanglement between two
annihilation, analyzing the violation of Bell’s inequality
electrons. In Sec. III, we calculate the Schmidt decom-
[2]. In an earlier work, Grobe et al. [19] studied, in the
position of the amplitude of a pair of spin-1/2 particles,
showing the growth of momentum entanglement as they
interact via QED. We obtain also analytic approxima-
tions of the Schmidt modes, both, in momentum and
∗Electronicaddress: lamata@imaff.cfmac.csic.es
†Electronicaddress: leon@imaff.cfmac.csic.es configuration spaces. In Appendix A, we include some
‡Electronicaddress: [email protected] notations and definitions related to entanglementtheory
2
for discrete and continuous variables. In Appendix B, a 1 a 1
we address the possibilities of transferring entanglement -
between momenta and spins via dynamical action, with
LocalOperationsandClassicalCommunication(LOCC),
or via kinematical action, with Lorentz transformations. b 2 b 2
t u
II. TWO ELECTRON GREEN FUNCTION IN
PERTURBATION THEORY FIG. 1: Feynmandiagrams for theQED interaction between
twoelectrons(secondorder). Theminussigndenotesthean-
To address the properties of entanglement of a two tisymmetryoftheamplitudeassociatedtothefermionstatis-
electron system one needs the amplitude (wave func- tics.
tion) ψ(x ,x ) of the system, an object with 16 spinor
1 2
components dependent on the configuration space vari- insteadofK(0)inthelefthandsideoftheaboveequation.
ables x , x of both particles. The wave functions were
1 2 The corresponding process is shown in Fig. 1. To deal
studied perturbatively by Bethe and Salpeter [20] and
with this case we choose t = t = t, t = t = t and
1 2 3 4
theirevolutionequationwasalsogivenbyGell-Mannand −
introduce the new variables
Low [21]. The wave function development is closely re-
1
lated to the two particle Green function, t = (t +t ), t ( t,t), (5)
+ 5 6 +
2 ∈ −
K(1,2;3,4) = (Ψ ,T(ψ(x )ψ(x )ψ¯(x )ψ¯(x ))Ψ ) (1) 1
0 1 2 3 4 0
t = (t t ), t ( (t t ),t t ), (6)
5 6 + +
− 2 − − ∈ − −| | −| |
which describes (in the Heisenberg picture) the sym-
metrized probability amplitude for one electron to pro- in Eq. (3), yielding
ceed from the event x to the event x while the other
3 1 K˜(2)(1,2;a,b;t)=
proceeds from x to x . If u (3) describes the electron
4 2 ps
at 3 and up′s′(4) the one at 4, then 2ie2 j1µajµ2b δ(3)(p +p p p )
(2π)4√2E 2E 2E 2E 1 2− a− b
1 2 a b
ψ(x1,x2) = Z dσµ(3)dσν(4)K(1,2;3,4)γ(µ3)γ(ν4) × ∞(k0)2 (pdk0 p )2+iǫ tdt+e−i(Ea+Eb−E1−E2)t+
× ups(3)up′s′(4), (2) Z−∞ − a− 1 Z−t
will be their correlated amplitude at 1, 2. In the free t−|t+| dt e−i(E1−E2+Eb−Ea+2k0)t− 1 2 , (7)
case this is just ups(1)up′s′(2) , but the interaction will ×Z−(t−|t+|) − −{ ↔ }
produce a reshuffling of momenta and spins that may
where K˜(2)(1,2;a,b;t) is a shorthand notation for what
lead to entanglement. The two body Green function K
corresponds to (4) at second order, and jµ = u¯ γµu .
is precisely what we need for analysing the dynamical kl k l
After some straightforwardcalculations, we obtain
generation of entanglement between both electrons.
Perturbatively [20], e2 δ(3)(p +p p p )
K˜(2)(1,2;a,b;t)= 1 2− a− b
4π3 √2E 2E 2E 2E
K(1,2;3,4)=S (1,3)S (2,4) e2 d4x d4x S (1,5) 1 2 a b
F F − Z 5 6 F ×{j1µajµ2b[St(t)+Υt(t)]−j2µajµ1b[Su(t)+Υu(t)]},
S (2,6)γµ D (5,6)γ S (5,3)S (6,4) (8)
× F (5) F (6)µ F F
+ 1 2 (3) with,
···−{ ↔ }
wherealltheobjectsappearingintheexpansionarethose i
S (t) =
toeframfsreoenfitehledrtihgheotrhya.nWdesimdeayofctahlilsKex(np)retsositohne.sTuchceeysswivilel t E1−E2+2Eb−Ea 2−(pa−p1)2
describethetransferofpropertiesbetweenbothparticles (cid:0) sin[(E1+E2(cid:1)−Ea−Eb)t], (9)
duetotheinteraction. Thisreshufflingvanishesatlowest × E +E E E
1 2 a b
− −
order,which gives just free propagationforwardin time: 1
Υ (t)=
t p p
a 1
d3x d3x d3x d3x u (1)u (2)K(0)(1,2;3,4) | − |
1 2 3 4 †p1s1 †p2s2 i 1 + 1 Σsin(Σt)
Z ×γ(03)upasa(3)γ(04)upbsb(4)=θ(t1−t3)θ(t2−t4) ×(cid:26) (cid:20)µ1(Σ2−µ2) 1 ν(Σ2−ν2)(cid:21)
δ δ δ(3)(p p )δ(3)(p p ) (4) + cos(Σt)
× s1sa s2sb 1− a 2− b − Σ2 µ2 Σ2 ν2
(cid:20) − − (cid:21)
whereups(x) =(2π)−3/2(m/E)1/2exp(−ipx)us(p). The + 1 e iµt+ 1 e iνt , (10)
first effects of the interaction appear when putting K(2) Σ2 µ2 − Σ2 ν2 −
(cid:20) − − (cid:21)(cid:27)
3
Σ = E +E E E , (11) III. TWO ELECTRON ENTANGLEMENT
1 2 a b
− −
GENERATION AT LOWEST ORDER
µ = ∆+2p p , (12)
a 1
| − |
ν = ∆+2p p , (13)
a 1
− | − | Theelectronsatx , x willbegenericallydescribedby
3 4
∆ = E E +E E , (14)
1− 2 b− a an amplitude F
and
ψ (x ,x )= d3p d3p F(p ,s ;p ,s )
F 3 4 a b a a b b
S (t) S (t),Υ (t) Υ (t), sXa,sbZ Z
u ↔ t 1 u2 ↔ t (15) ×upa,sa(x3)upb,sb(x4) (17)
↔
thatshouldbe normalizabletoallowforaphysicalinter-
S are the only contributions that remain asymptoti- pretation, i.e.,
t,u
cally (t ) leading to the standard scattering am-
→ ∞
plitude, while Υt,u vanish in this limit. We recall that d3pa d3pb F(pa,sa;pb,sb)2 =1. (18)
these are weak limits: no matter how large its modulus, | |
the expression in Eq. (10) will vanish weakly due to its sXa,sbZ Z
fast oscillatory behavior. On the other hand, the sinc ForseparablestateswhereF(a;b)=f (p ,s )f (p ,s ),
a a a b b b
function in Eq. (9) enforces energy conservation via f and f could be Gaussian amplitudes g centered
a b
around a certain fixed momentum p0 and a certain spin
sin[(E +E E E )t] component s0,
1 2 a b
lim − − =πδ(E +E E E ).
1 2 a b
Tt→h∞is limiEt1sh+owEs2a−lsEoath−aEt bthe entanglement i−n en−er(g1i6e)s g(p,s)= ( δπsσs0)3/2e−(p−p0)2/σ2,
2
increases with time [22], see Appendix A, reaching its
p
maximum (infinite) value when t for particles with which in the limit of vanishing widths give the standard
initialwell-definedmomentaande→ner∞gy. Thisresultisin- -well defined- momentum state δss0δ(3)(p p0).
−
dependent of the chosen scattering configuration. Exact In the absence of interactions,a separable initial state
energyconservationatlargetimes,unitedtoasharpmo- will continue to be separable forever. However, interac-
mentumdistributionoftheinitialstates,wouldnaturally tions destroy this simple picture due to the effect of U.
resultintoahighdegreeofentanglement. Thebetterde- Clearly, the final state
fined the initial momentum of each electron, the larger
the asymptotic entanglement. The physical explanation F(2)(p ,s ;p ,s ;t)= d3p d3p K˜(2)(1,2;a,b;t)
1 1 2 2 a b
to this unbounded growth is the following: the parti-
sXa,sbZ Z
cles with welldefinedmomentum (unphysicalstates)are
F(p ,s ;p ,s ) (19)
a a b b
spreadoverallspace,andthustheirinteractionisubiqui- ×
tous,withtheconsequentunboundeddegreeofgenerated
can not be factorized.
entanglement. Thisisvalidforeveryexperimentalsetup,
In the rest of this section we analyze the final state
except for those pathological cases where the amplitude F(2)(p ,s ;p ,s ;t) in Eq. (19) to show how the vari-
1 1 2 2
cancelsout, due to somesymmetry. Inthe followingsec-
ables p and p get entangled by the interaction. We
1 2
tion,andforillustrativepurposes,wewillsingleoutthese
consider the nonrelativistic regime in which all interven-
two possibilities.
ing momenta and widths p,σ m, so the characteristic
i)Thecaseofanunboundeddegreeofattainableentan- times t under consideration a≪re appreciable. We single
glementduetoanincidentelectronwithwelldefinedmo- outtheparticularcaseofaprojectilefermionascattered
mentum. We consider,with no lossofgenerality,a fuzzy offafuzzytargetfermionbcenteredaroundp0 =0. Asa
b
distribution in momentum of the second initial electron, furthersimplification,weconsiderthe projectilemomen-
for simplicity purposes. tum sharply distributed around p0 (σ p0) so that
a a ≪ a
ii) Basically the same setup as in (i) but with a spe- the initial state can be approximated by
cificspinconfiguration,whichleadstocancellationofthe
δ
anmopalsiytmudpetoatticlaerngteantgimleemsednutegetonesryamtiomne.try, and thus to F(a;b)≈δsas0aδ(3)(pa−p0a)( πsσbs0b)3/2e−(pb−p0b)2/σb2.
2 b
On the other hand, for finite times, nothing prevents (20)
p
a sizeable contribution from Eq. (10). In fact, in the Our kinematical configuration would acquire complete
limiting case where t 1 is large compared to the ener- generality should we introduce a finite momenta p0 for
− b
gies relevant in the problem, it may give the dominant the initial electron b. The reference system would be in
contributionto entanglement. Whether the contribution thiscasemidwaybetweenthelab. systemandthec.o.m.
from Υ (t) and Υ (t) is relevant, or not, depends on the system. In short, the choice p0 = 0 will not affect the
t u b
particular case considered. qualitative properties of entanglement generation.
4
where Σ˜ = p0a(p + q), µ˜ = √2 p2+ p0a 2, and
2m σ
r
p t˜ p0aσt. In the following, we analyze d(cid:16)iffer(cid:17)ent spe-
1 ≡ 2m
cific spin configurations in the non-relativistic limit with
p0a p0=0 π/4 the help of Eq. (23). We consider an incident particle
b
π/4 energy of around 1 eV≪ m (p0a = 1 KeV), and a mo-
mentum spreading σ one order of magnitude less than
p2 p0a. We make this choice of p0a and σ to obtain longer
interaction times, of femtoseconds (t = 2mt˜). Thus the
p0σ
a
parameter values we consider in the subsequent analysis
are p0/m = 0.002 and σ/m = 0.0002. We consider the
a
initial spin state for particles a and b as
FIG. 2: Experimental setup considered in the calculations.
s0s0 = , (24)
| a bi |↑↓i
along an arbitrary direction that will serve to mea-
Wewillworkinthelabframe,whereparticlebshowsa
fuzzy momentum distribution around p0 = 0, and focus sure spin components in all the calculation. The phys-
b
in the kinematical situation in which the final state mo- ical results we are interested in do not depend on
menta satisfy p p = 0 and also p p0 = 1/√2p p0, this choice of direction. The QED interaction, in the
1· 2 α· a α a non-relativistic regime considered, at lowest order, is
α=1,2(seeFig. 2). Thischoicenotonlyavoidsforward
a Coulomb interaction that does not change the spins
scattering divergencies but also simplifies the expression
of the fermions. In fact, (jµ j ) 4m2δ δ ,
of the amplitude in Eq. (19), due to the chosen angles. 1a µ2b ≃ s0as1 s0bs2
For sure, the qualitative conclusions would also hold in (j1µbjµ2a) ≃ 4m2δs0bs1δs0as2. Given the initial spin states
other frames, like the center-of-mass one. We obtain of Eq. (24), depending on whether the channel is t or u,
the possible final spin states are
E1+E2−Ea−Eb|ppba==pp10a+p2−p0a = s s = , (25)
p0 | 1 2it |↑↓i
√2am(p1+p2−√2p0a)+O((p0a/m)3p0a), |s1s2iu = |↓↑i. (26)
(p1+p2−p0a)2 = (p1−p0a/√2)2 + (p2−p0a/√2)2, Duetothefactthattheconsideredfermionsareidentical,
σ2 σ2 σ2 theresultingamplitudeafterapplyingtheSchmidtproce-
(p p )2 =(p p0/√2)2+(p0)2/2, dureisasuperpositionofSlaterdeterminants[23,24,25].
1− a 1− a a WheneverthisdecompositioncontainsjustoneSlaterde-
(p p )2 =(p p0/√2)2+(p0)2/2. (21)
2− a 2− a a terminant(Slater numberequalto1)the state is noten-
tangled: itscorrelationsarejustduetothestatisticsand
Here, boldface characters represent trivectors, otherwise
are not useful for the applications because they do not
they represent their associated norms. We perform now
contain any additional physical information. If the am-
the following change of variables,
plitude contains more than one determinant, the state is
p 1 p0 q 1 p0 entangled. Splitting the amplitude in the corresponding
= p a , = p a , (22)
1 2 ones for the t and u channels, we have
√2 σ − √2 √2 σ − √2
(cid:18) (cid:19) (cid:18) (cid:19)
turning the amplitude in Eq. (19) into
F(2)(p, ;q, ;t)
t
sin[(p+q)t˜] ↑ ↓ ∝
F(2)(p,s1;q,s2;t)∝ Σ˜ sin[(pΣ˜+q)t˜]p2+1p0a 2e−p2/2e−q2/2
(jµ j )sa=s0a (jµj )sa=s0a σ
1a µ2b sb=s0b 1b µ2a sb=s0b e p2/2e q2/2 1 1 (cid:16) (cid:17)
× p2+ p0a 2 − q2+ p0a 2 − − + µ˜/2 −µ˜(Σ˜2 µ˜2) Σ˜sin[(p+q)t˜]
σ σ (cid:26) −
+ (j1µajµ(cid:16)2b)ssba(cid:17)==ss0b0a (cid:16)1 (cid:17) Σ˜sin[(p+q)t˜] − Σ˜2 i µ˜2 cos[(p+q)t˜]−e−i2pm0aµ˜t˜
µ˜/2 −µ˜(Σ˜2 µ˜2) − (cid:18) (cid:19)(cid:27)
(cid:26) − e p2/2e q2/2, (27)
− −
− Σ˜2 i µ˜2 cos[(p+q)t˜]−e−i2pm0aµ˜t˜ with ×
− (cid:18) (cid:19)(cid:27)
p,1 q,2 e−p2/2e−q2/2, (23) F(2)(p,↓;q,↑;t)u ↔F(2)(p,↑;q,↓;t)t,
− { ↔ }! p q. (28)
↔
5
Intheinfinitetimelimitthesincfunctionconvergesto t˜= 1 t˜= 2
δ(p+q), which is a distribution with infinite entangle- 1 1
ment[22]. Thepresenceofthesincfunctionisduetothe
finite time interval of integration in Eq. (7). This kind
of behavior can be interpreted as a time diffraction phe- n n
λ λ
nomenon [26]. It has direct analogy with the diffraction
of electromagnetic waves that go through a single slit of
width 2L comparable to the wavelength λ. The analogy
iscompleteifoneidentifiest˜withLandp+qwith2π/λ. 0 1 2 n 3 4 5 0 1 2 n 3 4 5
In Fig. 3, we plot the modulus of Eq. (27) versus p, q, t˜= 3 t˜= 4
attimest˜=1,2,3,4. Thisgraphicshowstheprogressive
1 1
clustering of the amplitude around the curve q = p,
due to the function sin[(p+q)t˜]. This is a clear sig−nal
p+q
of the growth in time of the momentum entanglement.
n n
λ λ
Fig. 3 puts also in evidence the previously mentioned
time diffraction pattern.
t˜= 1 t˜= 2 0 1 2 n 3 4 5 0 1 2 n 3 4 5
2 2 FIG. 4: Eigenvalues λn versusn at times t˜=1,2,3,4.
q q
andisusuallycalledSlaterdecomposition. Itisobtained
2 2 antisymmetrizing the amplitude for channel t
− −
2 p 2 2 p 2 F(2)(p,s1;q,s2;t)∝ λn(t˜)
− −
t˜= 3 t˜= 4 Xn q
ψ(1)(p,t˜) ψ(2)(q,t˜) ψ(2)(p,t˜) ψ(1)(q,t˜)
n n n n
|↑i |↓i− |↓i |↑i,
× √2
2 2 (29)
q q wherethemodesψn(1)(k,t˜)andψn(2)(k,t˜)aretheSchmidt
modes of the channel t obtained for particles 1 and 2
2 2 respectively, and they correspond to the modes of the
− − channel u for particles 2 and 1 respectively.
Ameasureoftheentanglementofapurebipartitestate
2 2 2 2
− p − p of the form of Eq. (29), equivalent to the entropy of en-
tanglement S, is given by the Slater number [19]
FIG. 3: (Color online) |F(2)(p,↑;q,↓;t)t| versus p, q at t˜= 1
K . (30)
1,2,3,4 . ≡ ∞n=0λ2n
WehaveappliedthemethodforobtainingtheSchmidt K gives the number ofPeffective Slater determinants
decomposition given in Ref. [22] to Eq. (27), considering whichappearinacertainpurebipartitestateintheform
for the orthonormal functions O(1)(p) , O(2)(q) Her- ofEq.(29). The largerthe valueofK, the largerthe en-
{ } { }
mite polynomials with their weights, to take advantage tanglement. For K = 1 (one Slater determinant) there
of the two Gaussian functions. We obtain the Schmidt is no entanglement. This measure is obtained consider-
decompositionfor t˜=1,2,3,4,where the errorwith ma- ing the average probability, which is given by ∞n=0λ2n
trices Cmn 12×12 or smaller is d2m0,n0 ≤ 7·10−3 in all ( ∞n=0λn =1, and thus {λn} can be seen as a pProbabil-
considered cases. We plot in Fig. 4 the coefficients λ ity distribution). The inverse of the average probability
n
P
of the Schmidt decomposition of Eq. (27) as a function isthe Slater number. Its attractivepropertiesarethatit
of n, for times t˜= 1,2,3,4. The number of λ differ- isindependentofthe representationofthewavefunction,
n
ent from zero increases as time is elapsed, and thus the it is gauge invariant, and it reaches its minimum value
entanglement grows. of 1 for the separable state (single Slater determinant).
The complete Schmidtdecomposition,including chan- In Fig. 5, we show the Slater number K as a function of
nelstandu,isgivenintermsofSlaterdeterminants[23], elapsedtime t˜, verifyingthat the entanglementincreases
6
n = 0 n = 1
2
K 1 0.8
1.5
1 0 −0.8
0 1 2 3 4
n = 2 n = 3
t˜
0.8 0.8
FIG.5: Slater numberK as afunction oftheelapsed timet˜.
as the system evolves. It can be appreciated in this fig-
0.4 0.8
ure the monotonic growth of entanglement, due to the − 4 2 0 2 4 − 4 2 0 2 4
− − p − − p
fact that we have considered an incident electron with
well defined momentum. In realistic physical situations
with wave packets, this growth would stop, due to the
momentum spread of the initial electrons. The general FIG. 6: (Color online) Schmidt modes ψn(1)(p,t˜) at times
t˜= 1,2,3,4 for n = 0,1,2,3. The sharper modes for each
trendisthatthehighertheprecisionintheincidentelec-
n correspond to thelater times.
tron momentum, the larger the resulting asymptotic en-
tanglement. The factthatthe entanglementinmomenta
between the two fermions increaseswith time is a conse-
δ(p+q) and thus the amplitude gets sharper.
quenceoftheinteractionbetweenthem. Weremarkthat
Now we consider the Schmidt modes in configuration
the entanglement cannot grow unless the two particles
space. To obtain them, we just Fourier transform the
“feel” each other. The correlations in momenta are not
modes of Eq. (31) with respect to the momenta p , p
specific of QED: the effect of any interaction producing 1 2
momentum exchange while conserving total momentum
wilTlhteraSncshlamteiditntmoomdoesmienntmumomceonrtraelsaptiaocnes.for the ampli- ψ˜m(α)(xα,t˜)= √12π ∞ dpαei(pαxα−2pm2αt)ψm(α)(k(pα),t˜),
tude of Eq. (27) are given by Z−∞ (32)
whereα=1,2. Thedependenceofponp andq onp is
1 2
ψ(α)(k,t˜) e k2/2 n0 (√π2nn!) 1/2A(α)(t˜)H (k) given through Eq. (22). The factor e−i2pm2αt in Eq. (32) is
m ≃ − − mn n the one which commutes the states between the interac-
n=0
X tion picture (considered in Eq. (31) and in the previous
α = 1,2, (31)
calculationsin Secs. II and III) andthe Schr¨odingerpic-
where n is the corresponding cut-off and the values of ture.
0
the coefficientsA(α)(t˜)areobtainedthroughthe method The Hermite polynomials obey the following expres-
mn
sion [27]
given in Ref. [22]. The modes in momenta space de-
pendontime becausethey arenotstationarystates: the
QED dynamics between the two fermions and the inde- ∞ dxe (x y)2H (αx)=√π(1 α2)n/2H αy .
− − n n
terminacyontheenergyatearlystagesoftheinteraction − √1 α2
give this dependence. By construction, the coefficients Z−∞ (cid:18) − (33(cid:19))
A(α)(t˜) do not depend on p, q. With the help of Eq. (33) and by linearity of the Fourier
mn
transforms, we are able to obtain analytic expressions
WeplotinFig.6theSchmidtmodesψ(1)(p,t˜)attimes
n for the Schmidt modes in configuration space (to a cer-
t˜=1,2,3,4 for n=0,1,2,3 (we are plotting specifically
tainaccuracy,whichdependsonthecut-offsconsidered).
therealpartofeachmodeonly,whichapproximateswell
This is possible because the dispersion relation of the
the whole mode, because Eq. (27) is almost real for the
massive fermions in the considered non-relativistic limit
cases considered). The sharper modes for each n cor- is E = p2α, and thus the integral of Eq. (32) can be
respond to the later times. Each Schmidt mode is well α 2m
obtained analytically using Eq. (33).
approximated at early times by the corresponding Her-
The corresponding Schmidt modes in configuration
mite orthonormal function, and afterwards it sharpens
space are then given by
anddeviates fromthatfunction: it getscorrectionsfrom
higher order polynomials. The fact that the modes get
n0
thinner with time is related to the behavior of Eq. (27) ψ˜(α)(x˜ ,t˜) A(α)(t˜)O˜(α)(x˜ ,t˜), α=1,2, (34)
at large times. In particular the sinc function goes to m α ≃ mn n α
n=0
X
7
where the orthonormal functions in configuration space proximitytothestationaryvalueofthephasedointerfere
are constructivelyandareappreciable. Whatwedisplayhere
is the weak convergence to zero for the functions Υ (t)
O˜(α)(x˜ ,t˜)=in(√π2nn!) 1/2e−inarctan(σ˜t˜)+iσ˜−1(x˜α−t˜/2) and Υu(t). t
n α −
1+iσ˜t˜
(x˜α−t˜)2 t˜ x˜αp
e−2(1+iσ˜t˜)Hn − . (35)
× " 1+(σ˜t˜)2#
p
In Eqs. (34) and (35), we are using dimensionless vari-
ables,x˜ = σxα,α=1,2,σ˜ = σ ,andthe dimensionless ˜t)]
time deαfined√b2efore, t˜= p0aσt.p0aThe modes in Eqs. (34) .,12
2m =
and (35) are normalized in the variables x˜ . The or-
α q
thonormal functions of Eq. (35) propagate in space at ,1
a speed p0a and they spread in their evolution. Addi- =
√2m p
(
tionally,the modes ofEq.(34)havealsothe time depen- g
dence ofA(α)(t˜). The Slaterdecompositioninconfigura- [ℑ
mn
tion space, obtained Fourier transforming the modes of
Eq. (31) is
F˜(2)(x˜1,x˜2,t˜) ∝ sλn2(t˜)[ψ˜n(1)(x˜1,t˜)|↑iψ˜n(2)(x˜2,t˜)|↓i ℜ[g(p=1,q =1.2,t˜)]
n
X
− ψ˜n(2)(x˜1,t˜)|↓iψ˜n(1)(x˜2,t˜)|↑i]. (36) fFoIrGp.=7:1R, eqa=l a1n.2d,iamnadgt˜in∈ar(y1,p1a.0rt0s1)of(atrhbeitarmarpyliutundites)g.(p,q,t˜)
Thecoefficientsλ (t˜)areunaffectedbytheFouriertrans-
n
formation, and thus the degree of entanglement in con-
In this section, we investigated the generation of en-
figuration space is the same as in momenta space.
tanglement in momenta between two identical spin-1/2
We consider now the initial spin configuration
particles which interact via QED. We showed how the
correlations grow as the energy conservation is increas-
s0s0 = , (37)
| a bi |↑↑i ingly fulfilled with time. The previous calculation had,
however, the approximation of considering a projectile
where,the onlypossiblefinalstateinthe non-relativistic
particle with perfectly well defined momentum, some-
limit is
thing not achievable in practice. This is a first step to-
wards a real experiment, where both fermions will have
s s = . (38)
1 2
| i |↑↑i a dispersion in momenta and thus infinite entanglement
In this case, the sinc term goes to zero, because the mo- will never be reached, due to the additional integrals of
mentum part of this term is antisymmetric in p2, q2 and the Dirac delta δ(∆E) over the momentum spread.
the sinc function goes to δ(p+q), which has support (as
a distribution) on q = p. We point out that the sinc
contributiontothisamp−litudeisnegligiblebecauseofthe IV. CONCLUSIONS
particular setup chosen. In other experiment configura-
tions the amplitude in Eq. (19) associated to the spin Weanalyzedthedynamicalgenerationofentanglement
states of Eqs. (37) and (38) would have appreciable sinc betweentwoelectronsdue totheirmutualinteractionby
term and thus increasing momenta entanglement with meansoftheSlaternumber. Intheasymptoticlimit,and
time. On the other hand, in this case the contribution for electrons whose initial momenta are sharply defined,
fromΥ (t)inEq.(10)andΥ (t)isevensmallerthanthe entanglementdivergenciesmay appear. We observethat
t u
sinc term, and converges weakly to zero. considering finite-time intervals of interaction, and/or a
We plot in Fig. 7 the real and imaginary parts of the certainspreading in the particles momentum, the entan-
termassociatedtoΥ (t)andΥ (t)inEq.(23),whichwe glementremainsfinite. Wehavestudiedforthefirsttime
t u
denote by g(p,q,˜t), for spin states of Eqs. (37) and (38) the dynamicalgenerationofmomentum entanglementof
as a function of time t˜ (1,1.001) and having p = 1, two spin-1/2 particles at lowest order QED, observing
∈
q = 1.2. We want to show with it the strong oscilla- how the correlations increase as the particles exchange
tory character of the amplitude with time, and how all virtual photons. We obtain the Schmidt decomposition
the contributions interfere destructively with each other of the scattering amplitude, given in terms of Slater de-
giving a zero final value. This is similar to the station- terminants, and the Slater number, which clearly shows
ary phase procedure, in which only the contributions in the growth of entanglement with time. We also obtain
8
analytic approximations of the Schmidt modes, both in where p denotes the corresponding continuous variable.
momentum and configuration spaces. For pure bipartite states a relevant measure of entan-
glementis the entropy of entanglement,S. Givenastate
Ψ , it is defined as the von Neumann entropy of the
ACKNOWLEDGEMENTS |rediuced density matrix with respect to S or S ,
1 2
J.L. thanks I. Bialynicki-Birula for useful discussions d−1
S = λ log λ , (A5)
andprivatecommunications. L.L.thanksI.Ciracforhos- − n 2 n
pitalityatMax-PlanckInstituteforQuantumOpticsand nX=0
acknowledgesfinancialsupportfromFPUgrantAP2003-
where the λ ’s are given in Eq. (A3). In general, S 0,
0014. The workof J.L.and L.L.was partially supported n ≥
S = 0 for a product state, and the more entangled a
by the Spanish Ministerio de Educaci´on y Ciencia under
state is, the larger S. For a maximally entangled state,
project BMF 2002-00834. E.S. acknowledges financial
S =log d, and if d= , then S diverges.
support from EU Project RESQ. 2 ∞
Aninterestingworkwherethe Schmidt decomposition
for continuous variables is analyzed discretizing the cor-
responding integral equations can be found in Ref. [29].
APPENDIX A: ENTANGLEMENT IN FINITE
Another method for obtaining the continuous variables
AND INFINITE DIMENSIONAL HILBERT
SPACES Schmidt decomposition, based in decomposing the bi-
partite wave function in complete sets of orthonormal
functions, is developed in Ref. [22].
We consider a composite system described by a
S
Hilbert space , which may be either finite or infinite
H
dimensional. This space is constructed as the tensor
APPENDIX B: ENTANGLEMENT TRANSFER
product of the Hilbert spaces associated to each of the
BETWEEN MOMENTUM AND SPIN
subsystems, , of . For simplicity, and in view of our
α
S S
present purposes, we restrict ourselves to pure states of
1. Dynamical transfer and distillation
a bipartite system . Thus, α=1,2,and = .
1 2
S H H ⊗H
Definition: product state. A vector state Ψ of the
| i
system is a product state if it can be written as In Sec.III we computed the entanglement in momenta
S
for a pair of identical spin-1/2 particles which interact
Ψ = Ψ(1) Ψ(2) , (A1) through exchange of a virtual photon. The sharper the
| i | i| i
initial momentum distribution of the incident fermion,
where Ψ(1) and Ψ(2) .
| i∈H1 | i∈H2 andthelongertheinteractiontime,thelargertheentan-
Definition: entangled state. A vector state Ψ of the
| i glementinmomenta. Heisenberg’sprinciple,ontheother
system is entangled if it is not a product state. A
S hand, establishes a limit to the precision with which the
relevant example, where dim( ) = dim( ) = 2, is
H1 H2 momentum may be defined and hence to the achievable
called the singlet state,
degree of entanglement.
1 It is possible, in principle, to transform the entangle-
Ψ = (Ψ(1) Ψ(2) Ψ(1) Ψ(2) ). (A2)
| −i √2 | 1 i| 2 i−| 2 i| 1 i ment in momenta into entanglement in spins. This is
easily seen in terms of the majorization criterion [6, 30],
Averyusefultoolforanalyzingtheentanglementofpure which is of practical interest because the experimental-
statesofbipartitesystemsisgivenbytheSchmidtdecom- ist usually manipulates spins. Here, we will analyze this
position [28, 29]. It basically consists in expressing the entanglement transfer.
pure bipartite state asa sum ofbiorthonormalproducts, Majorization is an area of mathematics which pre-
with positive coefficients √λ , as follows datesquantummechanics. QuotingNielsenandChuang,
n
“Majorization is an ordering on d-dimensional real vec-
d−1 tors intended to capture the notion that one vector is
Ψ = λ Ψ(1) Ψ(2) , (A3)
| i n| n i| n i more or less disordered than another”. We consider
nX=0p a pair of d-dimensional vectors, x = (x1,...,xd) and
where Ψ(1) , Ψ(2) , are orthonormal bases associ- y = (y1,...,yd). We say x is majorized by y, written
ated to{| n1 ia}nd{| 2n, ir}espectively. In Eq. (A3), d = x≺y, if kj=1x↓j ≤ kj=1yj↓ for k =1,...,d, with equal-
min{dim(HH1),dimH(H2)}, and it may be infinite, as for ity insteaPd of inequaPlity for k = d. We denote by z↓ the
in continuous variable systems, describing momentum, componentsofz indecreasingorder(z1↓ ≥z2↓ ≥...≥zd↓).
energy, position, frequency, or the like. In those cases, The interest of this work in the majorization concept
the states Ψ(α) would be (square integrable) L2 wave comes froma theorem whichstates that a bipartite pure
n
functions, | i state ψ maybetransformedtoanotherpurestate φ by
| i | i
LocalOperationsandClassicalCommunication(LOCC)
pΨ(α) = ψ(α)(p), α=1,2, (A4) if and only if λ λ , where λ , λ are the vectors
h | n i n ψ ≺ φ ψ φ
9
of (square) coefficients of the Schmidt decomposition of transformed frame as
the states ψ , φ , respectively. LOCC adds to those
| i | i
qofuacnlatsusmicaolpceormatmiounnsiceaffteicotnedbeotnwlyeelnocsapllaytiathlleypsoespsairbailtietdy gλσ(p,q) Λ Uλ(Λλ′−1p)Uσ(Λσ′−1q)gλ′σ′(Λ−1p,Λ−1q),
−→ λ′σ′
partsofthe system. Accordingto thiscriterion,itwould X
(B3)
be possible in principle to obtain a singlet spin state φ
| i where
beginning with a momentum entangled state ψ when-
| i
eveTrhλeψp≺ossλiφb.ility of obtaining a singlet spin state from Uλ(pλ)′ ≡Dλ(1λ/′2)(R(Λ,p)) (B4)
a momentum-entangledstate can be extended to a more
is the spin 1/2 representation of the Wigner rotation
efficient situation: the possibility ofdistillation ofentan-
R(Λ,p). The Wigner rotations of Eq. (B4) can be seen
glement. This ideaconsistsonobtainingmultiple singlet
as conditional logical operators, which rotate the spin a
statesbeginning withseveralcopiesofagivenpure state
certain angle depending on the value of the momentum.
ψ . The distillable entanglement of ψ consists in the
| i | i Thus, a Lorentz transformation will modify in general
ratio n/m, where m is the number of copies of ψ we
| i the entanglement between momentum and spin of each
have initially, and n the number of singlet states we are
individual electron. We distinguish the following three
able to obtain via LOCC acting on these copies. It can
cases.
be shown [6] that for pure states the distillable entan-
glement equals the entropy of entanglement, S. Thus, in i) Product state in all variables. In this case,
the continuous case (infinite-dimensional Hilbert space),
the distillable entanglement is not bounded from above, g (p,q)=g (p)g (q)λ σ , (B5)
λσ 1 2
| i| i
becauseneitherisS.Accordingtothis,thelargertheen-
tanglement in momenta the more singlet states could be andthe entanglement atthe rest referenceframe is zero.
obtained with LOCC. Underaboost,theWignerrotationsofEq.(B4)entangle
To illustrate the possibility of entanglement trans- the momentum of each fermion with its spin, and thus
fer with a specific example, we consider a momentum- the entanglement momentum-spin grows [10].
entangled state for two distinguishable fermions
ii) Entangled state spin-spin and/or momentum-
momentum. We consider now a state
1
ψ = [ψ(1)(p)ψ(2)(q)+ψ(1)(p)ψ(2)(q)] . (B1)
| i √2 1 1 2 2 |↑↑i
g (p,q)=f(p,q)φ (B6)
λσ
| i
This state has associated a vector λ↓ψ =
with φ an arbitrary state of the spins, and f(p,q)
(1/2,1/2,0,0,...). On the other hand, the singlet
| i
an arbitrary state of the momenta. In this case, a
state
Lorentz boost will entangle in general each spin with its
φ =ψ(1)(p)ψ(2)(q) 1 ( ) (B2) corresponding momentum, and a careful analysis shows
| i 1 1 √2 |↑↓i−|↓↑i that the spin-spin entanglement never grows [12]. Of
course,by applying the reversedboost the entanglement
has associated a vector λ↓φ =(1/2,1/2,0,0...). momentum-spin would be transferred back to the spin-
These vectors obey λ λ , and thus the state en- spinone,andthusthelatterwouldgrow. Thisparticular
ψ φ
≺
tangled in momenta may be transformed into the state case shows that, for the state we considered in Sec. III,
entangled in spins via LOCC. givenbyEqs.(24),(25)and(27),theentanglementcould
not be transferred from momenta into spins via Lorentz
transformations. Thus, the dynamical approach would
2. Kinematical transfer and Lorentz boosts be here more suitable.
iii) Entangled state momentum-spin. According to
Another approachto the study ofentanglementtrans- theprevioustheorem,the momentum-spinentanglement
fer between momentum and spin degrees of freedom is may be lowered, transferring part of the correlations to
the kinematical one. In fact, the Lorentz transforma- the spins, or increased, taking some part of the correla-
tions may entangle the spin and momentum degrees of tionsfromthem. Toourknowledge,thereisnotasimilar
freedom. Tobe moreexplicit,andfollowingthe notation result for the momentum, that is, whether the momen-
of Ref. [12], we consider a certain bipartite pure wave tum entanglement can be preserved under boosts, or it
functiong (p,q)fortwospin-1/2fermions,whereλand suffers decoherence similarly to the spins, and part of it
λσ
σ denoterespectivelythe spindegreesoffreedomofeach istransferredtothe momentum-spinpart. Thisisavery
of the two fermions, and p and q the corresponding mo- interesting question, which we will treat more deeply in
menta. This would appear to an observer in a Lorentz future works.
10
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