Table Of ContentRemote State Preparation for Quantum Fields
Ran Ber
School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences,
Tel Aviv University, Tel-Aviv 69978, Israel.
Erez Zohar
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Max-Planck-Institut fu¨r Quantenoptik,
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Hans-Kopfermann-Straße 1, 85748 Garching, Germany.
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(Dated: March24,2016)
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Abstract
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Remote state preparation is generation of a desired state by a remote observer. In spite of causality, it
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h is well known, according to the Reeh-Schlieder theorem, that it is possible for relativistic quantum field
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- theories, and a “physical” process achieving this task, involving superoscillatory functions, has recently
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a been introduced. In this work we deal with non-relativistic fields, and show that remote state preparation
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q is also possible for them, hence obtaining a Reeh-Schlieder-like result for general fields. Interestingly, in
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3 the nonrelativistic case, the process may rely on completely different resources than the ones used in the
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I. INTRODUCTION
Remote State Preparation (RSP) [1–3] is a generation of a desired state by a remote observer,
whomerelyactslocallyonhisowndistantsystem. Suchaprocessisconsideredsuccessful,ifthe
remoteobserverisabletomakesurethatforparticularchoicesofameasurementanditsoutcome
the remote system is in the desired state. Although the success probability might, in general, be
small for a single run of the process, it is required that for events with a successful measurement
result,theremotestatewillbearbitrarilyclosetothedesiredone(i.e.,withfidelityarbitrarilyclose
to1).
A causal theory is a theory with a maximal velocity c, due to which information cannot be
transportedinstantaneouslyacrossadistanceR,butratheratsomefinitetimeT ≥ R/c. Causalityis
animportantpropertyofrelativisticsystems,andinparticularofrelativisticquantumfieldtheories
(which are usually referred to only as ‘quantum field theories’ - QFTs). As the standard model of
high energy physics is a quantum field theory, the importance of causality in our universe and the
physical theories describing its behavior could not be overestimated; However, when one wishes
to relate relativistic theories, and in particular QFTs, with RSP, he could, naively, argue that the
conceptofRSPisimpossibleinacausaltheory.
Surprisingly or not, according to an important theorem established long ago by Reeh and
Schlieder [4–6] in the context of algebraic quantum field theory (AQFT), RSP is possible in rela-
tivisticquantumfields. Accordingtothetheorem,theactionofafieldoperator,orapolynomialof
it,onthevacuumisdenseintheHilbertspace-i.e.,allowingtogeneratestateswhicharearbitrar-
ily close to any desired state in the Hilbert space, regardless of its location in configuration space.
Recently[7],aphysicalprocessresultingwithRSPofanydesiredstatewaspresented,suggesting
a“physical”waytoderivethetheorem(ratherthanAQFT):aprescriptionfortheremoteprepara-
tion of a desired state was given, rather than a general proof of its existence. This prescription is
based on superoscillatory functions, suggesting a deep relation between the task of RSP in QFTs
and the mathematical phenomenon of superoscillations [8, 9]. As the fields are relativistic, it is
clear that the resource which allows for such a task is entanglement [10, 11]. For other quantum
fields, which, in general, do not necessarily posses an entangled vacuum state, it remains unclear
whethersuchataskcouldbeaccomplished.
One might ask then what is the case for nonrelativistic fields. On one hand, these may be
noncausal,andhenceoneshallnotbesurprisedofnoncausaleffectssuchasremotestateprepara-
2
tion. On the other hand, how can one be sure that this still allows to remotely prepare any state?
In this work we show that this task is possible for a very large class of quantum field theories
which satisfy some reasonable physical assumptions. It turns out that RSP is possible, even for
fields with a non-entangled vacuum and a bounded group velocity. Thus, a Reeh-Schlieder-like
result is obtainable for general quantum fields. However, for fields with a non-entangled vacuum
state, the mechanism which produces the remote states relies on an infinite front velocity as a re-
source, rather than entanglement. This has interesting physical consequences. On one hand, the
entanglement-based mechanism, allows generating field states even with a presence of a “wall”
between the operating region and the target region, as nothing actually propagates in the process.
This is not possible in case the mechanism is based on infinite front velocity. On the other hand,
thelattermechanismislesssensitivetonoisescausedbypreviousattemptstoperformtheprocess.
This paper is organized as follows: First, we shall formulate the physical problem we address
anddescribetheprocessofstatepreparationinsectionII.InsectionIIIwearguewhensuperoscil-
lations are required and discuss their implications on the success probability. Finally, in section
IV,wemaketheconnectiontotheReeh-Schliedertheoremandarguethattheprocessesdescribed
herebyisaReeh-Schlieder-likeresultinthesensethatitallowsgenerationofarbitraryfieldstates,
eventhoughitmayutilisecompletelydifferentphysicalresources.
II. STATEMENTOFTHEPROBLEM
Consideraquantumfieldtheoryind+1dimensions,whoseactionisinvariantundertranslations
and rotations (reflections for d = 1), but not necessarily under boosts. The dispersion relation of
suchatheory,duetotherotationalinvariance,satisfies
ω = ω(k), (1)
with k = |k|. Due to the translational invariance, the eigenfunctions take the form f (x,t) =
k
h(ω)ei(k·x−ωt) andthusthefieldoperatormaybeexpandedas
(cid:90)
ddk (cid:16) (cid:17)
φ(x,t) = h(ω ) a ei(k·x−ωkt) +a†e−i(k·x−ωkt) , (2)
(2π)d k k k
(cid:104) (cid:105)
usingannihilationandcreationoperators,satisfyingthecanonicalcommutationrelation a ,a† =
k q
(2π)dδ(d)(k−q). We also assume that h(ω ) is a real function - otherwise, its phase may always
k
beabsorbedinthecreationandannihilationoperators.
3
The field operator (2) is the solution of the Heisenberg equation of motion derived using the
field hamiltonian H . For our purposes, we couple the field to a detector - a two level system,
0
locatedinx = x ,describedbytheHamiltonian H = Ωσ /2((cid:126) = 1). Theinteractionbetweenthe
0 d z
fieldandthedetectorisgivenby
H = λ(σ (cid:15)(t)+σ (cid:15)∗(t))φ(x ), (3)
int + − 0
and is only switched on for a finite period of time - (cid:15)(t) (cid:44) 0 only for t ∈ [−t ,0]. We assume that
0
int ≤ −t thefieldisinitsvacuumstate,andthedetector-initsgroundstate,|↓(cid:105). Assumingthatλ
0
isverysmallandthat(cid:15)(t)isO(1),onemaycalculatethestateint = 0,aftertheinteraction,using
firstorderinteractionpicture. There
H = λ(cid:15)(t)σ eiΩtφ(x ,t)+h.c., (4)
int + 0
andthusthedetector-fieldstateaftertheinteractionisgivenby
(cid:90)
0
|Φ,d(cid:105) = |0,↓(cid:105)−iλ dt(cid:15)(t)eiΩtφ(x ,t)|0,↑(cid:105). (5)
0
−t0
Post-selectingthespinintheexcitedstate|↑(cid:105),oneobtainsthe(normalized)fieldstate
(cid:90)
ddx
|Φ(cid:105) = N−1/2 (cid:15)˜(ω +Ω)h(ω )e−ik·x|k(cid:105), (6)
1 (2π)d k k
(cid:82) (cid:82)
where(cid:15)˜(ω +Ω) ≡ 0 dt(cid:15)(t)ei(ωk+Ω)t,N ≡ ddk h2(ω )|(cid:15)˜(ω +Ω)|2 and|k(cid:105) ≡ a†|0(cid:105).
k −t0 1 (2π)d k k k
Everysingleexcitationofthefieldmaybewrittenas
(cid:90)
|Ψ(cid:105) = N−1/2 ddxF(x)φ(x)|0(cid:105), (7)
2
(cid:82)
whereN ≡ ddxddy ddk h2(ω )F∗(y)eik·(y−x)F(x)[12].
2 (2π)d k
We wish to choose a window function (cid:15)(t), such that the state |Ψ(cid:105) will be created as a result of
theinteraction. Thus,wedemandthat|(cid:104)Ψ|Φ(cid:105)| = 1. Thisissatisfiedifwechoose
(cid:90)
(cid:15)˜ (ω +Ω) = ddxe−ik·xF(x), (8)
des k
where‘des’standsfor‘desired’;i.e.,theFouriertransformofthedesiredwindowfunctionisequal
totheoneof F(x).
The spherical symmetry restricts us to the creation of states which are symmetric around the
detector. Thus, F(x) = F(r), where r ≡ |x−x |; WLOG, we assume next that x = 0. In
0 0
4
(cid:82) (cid:82) (cid:82)
this case, the integration reduces to (cid:15)˜ (ω +Ω) = drrd−1F(r) dθsind−2θe−ikrcosθ dd−2Ω.
des k
(cid:82)
After performing the angular integrals dd−2Ω = (d−1)π(d−1)/2 (for derivation see [13]) and
Γ((d+1)/2)
(cid:82) √
dθsind−2θe−ikrcosθ = π(kr/2)(2−d)/2Γ((d−1)/2)J (kr) (where J (x) is a Bessel func-
(d−2)/2 ν
tion),oneobtains
(cid:90) (cid:32) (cid:33)d/2
2πr
(cid:15)˜ (ω +Ω) = drkF(r) J (kr). (9)
des k d−2
k 2
Thisequationcanbesatisfiedifandonlyifω = ω(k)isbijective. Asr increases,theoscillations
k
of the Bessel function (in k space) become faster, hence (cid:15)˜ (ω +Ω) obtains larger and larger
des k
Fouriercomponents. Atsomepoint,sayr ≥ r ,thefunction(cid:15)˜ (ω +Ω)beginstohavesignificant
0 des k
Fourier components which oscillate (in frequency space) faster than t (below we show that r is
0 0
relatedtotheminimalgroupvelocityofthetheory). Since(cid:15)(t)isnonvanishingonlywithin[−t ,0],
0
the standard frequency-time relations of Fourier transforms suggest that the above relation could
notbesatisfiedforr ≥ r .
0
Itwasshownin[7]that,forrelativisticfields,thisproblemcanbecircumventedusingasuper-
oscillatory (cid:15)˜(ω +Ω). In the following section we introduce this solution briefly. Then, we show
k
thatwhilethesamesolutionholdsfornon-relativisticfields,someconsequencesofitaredifferent.
Thisallowsustoanswerthequestion“whenaresuperoscillationsrequired?” inageneralmanner.
III. SUPEROSCILLATIONS
Superoscillatory functions are functions that oscillate faster than their fastest Fourier compo-
nent [8, 9]. This is due to a destructive interference, and thus they are always accompanied by
exponentially larger amplitudes somewhere outside the so-called superoscillatory region. In our
implementation,wewouldhavetoplacetheexponentiallylargeramplitudesinanon-physicaldo-
main of (cid:15)˜. Another difficulty regarding superoscillations is that these functions can superoscillate
in an arbitrarily large (but not infinite) domain. Therefore, there must also be a physical non-
superoscillatory domain. In our implementation, we shall choose a superoscillatory function that
will not be exponentially amplified in this non-superoscillatory domain. Then, we would have to
findawaytoeliminatethecontributionofthefunctioninthisdomain.
Weshallnowproceedbyfindingfunctionswhichmeetthesedemands. Considerthefollowing
function[7,14,15]:
(cid:90)
(cid:15)˜[h](cid:0)ω(cid:48)(cid:1) = D√ 2πdαeiω(cid:48)t0(cos2α−1)eδi2 cos(α−iA), (10)
2δ 2π
0
5
whereD,δandAaresomeconstants,andω(cid:48) ≡ ω+Ω−ω forsomeω . Sincet = t (cosα−1)/2,
0 0 0
the function (cid:15)[h](t) has support only in [−t ,0]. Nevertheless, we shall now prove that (cid:15)˜[h](ω(cid:48))
0
(cid:16) (cid:17)−1
oscillates arbitrarily fast in ω(cid:48) ∈ [0,ω ] for ω (cid:28) δ2t cosh[A] . Performing the integration
c c 0
explicitlyweobtain
√ (cid:114)
(cid:15)˜[h](cid:0)ω(cid:48)(cid:1) = D√2δπe−21iω(cid:48)t0J0δ12 1+δ2ω(cid:48)t0cosh[A]+14δ4ω(cid:48)2t02. (11)
For ω(cid:48) > 0. Using the asymptotic form of the Bessel function [16] for δ (cid:28) 1, and then taking
δ2 (cid:28) (ω t cosh[A])−1 weobtain
c 0
(cid:32) (cid:33)
(cid:15)˜[h](cid:0)ω(cid:48)(cid:1) (cid:27) De−12iω(cid:48)t0cos δ12 + 21ω(cid:48)t0cosh[A]− π4 (12)
in this domain. Redefining (cid:15)˜[h](ω(cid:48)) to be the summation of two such functions, one having δ−2 =
2πm+π/4,andtheother D → ±iDandδ−2 = 2πm−π/4,wherem (cid:29) 1,weget
(cid:15)˜[h](cid:0)ω(cid:48)(cid:1) = De12iω(cid:48)t0(±cosh[A]−1). (13)
This function oscillates in ω space at “frequency” t(cid:48) = 1t (±cosh[A]−1). By increasing A we
2 0
can set these oscillations to be arbitrarily fast. The superoscillatory domain is finite, therefore the
conditiondescribedinEq. (9)cannotbeexactlysatisfied. However,onecangetarbitrarilycloseto
satisfyingthisconditionbyincreasingthesuperoscillatorydomain. Thisisachievedbydecreasing
δ.
Superoscillations come at the price of an exponential growth outside the superoscillatory do-
main. In our case the growth occurs at ω(cid:48) < 0. For fields whose energy is bounded from below,
we choose ω such that ω ≥ ω , therefore ω(cid:48) < 0 corresponds to ω + Ω < ω , which is in the
0 0 0
non-physical domain. Beyond the superoscillatory domain, the function gradually obtains regular
(slower) oscillations, and in the limit ω(cid:48) (cid:29) ω , it behaves like ω(cid:48)−1/2sin(ω(cid:48)t ). In order to elimi-
c 0
nate this contribution, one can convolute (cid:15)[h](t) with some function h(t) which is differentiable n
times(n (cid:29) 1)andhasasmalltemporalsupport.
Finally, we use a combination of such superoscillatory functions, each with a different t(cid:48), in
ordertogeneratethewindowfunction
(cid:90)
T
(cid:15)˜(cid:0)ω(cid:48)(cid:1) = dt(cid:48)(cid:15)˜[h](cid:0)ω(cid:48);t(cid:48)(cid:1)(cid:15) (cid:0)t(cid:48)(cid:1). (14)
des
−T
In the limits T → ∞ and δ → 0 we get (cid:15)˜(ω(cid:48)) → (cid:15)˜ (ω(cid:48)) in the segment ω(cid:48) ∈ [0,ω ]. (This is
des c
whiletheactualwindowfunction,(cid:15)(t),andthedesiredwindowfunction,(cid:15) (t),areverydifferent:
des
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(cid:15)(t) has temporal support only in [−t ,0], while (cid:15) (t) might have an arbitrarily large temporal
0 des
support.) Therefore,wecangenerateremotesphericalsymmetricalone–particlefieldstatesaround
thespin,uptoanarbitrarilysmallinfidelity. Thegeneralizationtoarbitraryfieldstatesisachieved
using the same method introduced for relativistic field states [7]. Here we only note that the
generalization to one–particle states which are not spherical symmetrical involves an array of
spins(locatedatdifferentpositionsinanarbitrarilysmallregion)ratherthanasinglespin,andthe
generalizationtomany–particlestatesinvolvesasetofsucharrays.
Itisshownin[7]thattheexponentiallysmallamplitudeof(cid:15)˜(ω(cid:48))inthesuperoscillatorydomain
resultsinasuccessprobabilityoftheform
P ∼ e−ωct0T2. (15)
Thefinitenessofthesuperoscillatorydomainisresponsibleforaninfidelity
(cid:90)
∞ 1 (cid:12) (cid:12)
η ∼ (cid:12)(cid:12)F˜ (k)(cid:12)(cid:12)2ddk. (16)
ω
ωc c
SinceF˜ (k)isnormalizable,inthelimitω → ∞oneobtainsη → 0. Invertingthelatterfunctional
c
relation to ω = ω (η) ≡ 1/g(η), and expressing T as a functional of the desired state |Ψ(cid:105), we get
c c
therelation
P ∼ e−Tg2(η[|)Ψt0(cid:105)]. (17)
When F˜ (k) decays with a power law, g(η) behaves according to a power law as well, and when
F˜ (k) decays exponentially g(η) ∼ 1/ln(1/η). For a given dispersion relation, T [|Ψ(cid:105)] can be
(cid:82)
calculated by plugging (cid:15)˜ (ω +Ω) = T dtei(ωk+Ω)t(cid:15) (t) in the l.h.s of Eq. (9) and expressing
des k −T des
ther.h.sasaFouriertransformofatemporalfunction[17].
When are superoscillations required? Mathematically, the answer can be deduced from Eq.
(9). However, in order to understand the meaning of that, let us first consider a very simple,
non-superoscillatory, window function, (cid:15)(t) = δ(t+t ) - a short and impulsive interaction, for a
0
quantum field theory in 1 + 1 dimensions. Assuming the detector is in the origin, the generated
stateofthefield,afterthedetector’spost-selection,willbe
(cid:90)
∞ (cid:16) (cid:17)
|Φ(cid:105) ∝ dkh(ω ) a† e−iωkt0 +a†e−iωkt0 |0(cid:105). (18)
k −k k
0
Both terms represent wave packets propagating out of the detector: one is left-moving and the
otherisright-moving. Assumingthatω ∈ R∀k,andthatthegroupvelocityislessthanthephase
k
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velocity,i.e.,dω/dk < ω/k,eachwavewithk,ωreachesatt distance
0
∂ω
r (ω) = t = v (ω)t , (19)
0 0 g 0
∂k
(theminimalr (ω)isther mentionedaboveinthesimplecaseof1+1dimensionsandaspecific
0 0
desiredstate). Thus,onlywaveswithfrequenciessatisfyingv (ω)t ≥ Lpropagatefastenoughto
g 0
arrive to ±L without superoscillations. In relativistic theories, as well as other theories in which
the group velocity is bounded from above, one can always find L such that all frequencies would
require superoscillations, while in theories in which the group velocity is unbounded from above,
foreveryarbitrary L ,somefrequencieswillnotrequiresuperoscillations.
The above case deals only with waves which are outgoing from the detector. However, one
may also consider the case of ingoing wave packets - for example, if one wishes to generate the
fieldstate
|Ψ(cid:105) = (φ(L)+φ(−L))|0(cid:105)
(cid:90)
∞ (cid:16)(cid:16) (cid:17)(cid:16) (cid:17)(cid:17)
∝ dkh(ω ) a† +a† e−ikL +eikL |0(cid:105), (20)
k −k k
0
whichcontainsbothingoingandoutgoingwavepackets. Theingoingwavesrequirev (ω) < 0for
g
k ≥ 0,whichisimpossible,andthussuperoscillationsarealsorequiredforthiscase.
For general desired states, L is roughly the separation between the operating region and the
farthest place in the target region. Note that the same description holds for higher dimensions; in
these cases (when using a single detector) the resulting state is spherically symmetrical. While
everysinglepointonthespheregenerateswaveswhichpropagateinalldirections,thespherically
symmetricalstateasawholegenerateswaveswhichpropagateonlyinthe±rˆ directionsduetothe
Huygens principle. Thus, superoscillations are required for the generation of wavepackets which
propagatefasterthanthe(slowest)groupvelocity,orinwards,intotheinteractionregion.
IV. RELATIONTOTHEREEH-SCHLIEDERTHEOREM
Itisnowtherighttimetorecallalong-standing,possiblysurprisingresultofAQFT-theReeh-
Schlieder theorem [4–6]. According to this theorem, the set of Hilbert space vectors |ψ(cid:105) ∈ H
generatedfromthevacuum(oranyotherboundedstatewithaboundedenergy[6])ofarelativistic
QFT by operating with polynomials of the field operators in any open region is dense in H. In
other words, by applying certain local operators to the vacuum state in a certain region O , one is
1
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abletogenerate,withnonzerosuccessprobability,astateofthefieldlocalizedatsomeremotere-
gion(s){O } ,arbitrarilyclosetosomedesiredstate. The{O } regionsmayremain,throughout
k k≥2 k k≥2
the process, outside the light-cone of O , and thus this outcome must be the result of pre-existing
1
vacuum correlations (the theorem entails a violation of Bell inequalities as well [18, 19]). Con-
sider, as an example, a relativistic scalar field φ with mass m. There, for |x−x(cid:48)|2 (cid:29) m−2, one
obtainstheequal-timecorrelationfunction
(cid:104)0|φ(x,t)φ(cid:0)x(cid:48),t(cid:1)|0(cid:105) ∼ e−m|x−x(cid:48)|, (21)
-thecorrelationsdonotvanishevenbetweenspacelikeseparatedregions[20].
In this paper, we have shown that RSP is theoretically possible for general quantum fields,
includingnonrelativisticones. Thus,wehavedescribedaReeh-Schlieder-likeprocessforgeneral
fields. Note that while the Reeh-Schlieder theorem does not involve time dependence (as one
would expect in a relativistic context), our process does. In relativistic theories, adding time
dependenceismeaningless,asforanytwospacelikeseparatedpointsinspacetime, xµ = (t,x)and
x(cid:48)µ = (t(cid:48),x(cid:48)), there is a reference frame (connected by a Lorentz transformation), in which t = t(cid:48)
andx (cid:44) x(cid:48). Thus, ifwedefine∆xµ ≡ xµ − x(cid:48)µ,andr2 ≡ −∆x ∆xµ = −(t−t(cid:48))2 +(x−x(cid:48))2 > 0,due
µ
toLorentzinvariance,inthelimitr2 (cid:29) m−2,oneobtains(cid:104)0|φ(x,t)φ(x(cid:48),t(cid:48))|0(cid:105) ∼ e−mr. Intheother
extremecaseoffieldswhichdonotpossessanycorrelationsatall-i.e.,fieldsforwhich
(cid:0) (cid:1) (cid:0) (cid:1)
(cid:104)0|φ(x,t)φ x(cid:48),t |0(cid:105) ∝ δ(d) x−x(cid:48) , (22)
the time dependence is crucial, because correlations are generated in time. For example, the
Schro¨dingerfieldsatisfies
(cid:104)0|φ(x,t)φ(cid:0)x(cid:48),t(cid:1)|0(cid:105) = δ(d)(cid:0)x−x(cid:48)(cid:1), (23)
and thus, without the time dependence, the overlap between the generated state and the desired
stateiszeroandRSPisnotpossible. Therefore,onecandeducethatRSPispossibleonlywhen
(cid:104)0|φ(x,t)φ(cid:0)x(cid:48),t(cid:48)(cid:1)|0(cid:105) (cid:44) 0, (24)
for every x, x’ and t (cid:44) t(cid:48). It means that for every quantum field (in which RSP is possible) either
thereexistcorrelationsint = t(cid:48) and/orsomecomponentsofthequantumfieldpropagateinfinitely
fast, i.e., the front velocity is infinite. This is yet another formulation of a result discovered in
[21, 22] and widely discussed in [23, 24]. In the first scenario, our mechanism uses vacuum
9
correlationsasa‘resource’forRSP,whileinthesecondscenariothe’resource’istheinfinitefront
velocity. Remarkably,inbothcasesthe(gedanken)prescriptionforRSPisthesame.
The different ‘resources’ have different physical implications. Consider for example a case
where one puts a wall between the operating region and the target region. A good model for such
awall,inthecaseofascalarfield,forexample,isatimedependentpotential,
(cid:90)
H = limΓ ddxW(x)θ(t+t )φ2(x), (25)
wall 0
Γ→∞
where θ(t+t ) is the Heaviside step function and W(x) = 1 where the wall is placed and 0
0
elsewhere. This potential adds an “infinite mass” to the field in particular space points where
W(x) = 1,startingfromt = −t ,andthusitactsasawall.
0
IncausaltheorieswiththeequationofmotionOˆφ(xµ) = 0,theGreen’sfunctiondefinedby
OˆG(xµ −yµ) = −iδ(4)(xµ −yµ), (26)
may be chosen to be a causal, retarded Green’s function, G(xµ −yµ) = D (xµ −yµ), which van-
R
ishes when yµ is outside the past light-cone of xµ, and thus the field state during the preparation
processcanbeexpressedasasuperpositionoffieldstatesattimet = −t -allinthepastlight-cone
0
of the generating region. Therefore, if the wall is placed outside the past light-cone of both the
operating region and the target region throughout the process, its effect, which is propagating at
the speed of light at most, will not travel far enough to destroy the vacuum correlations between
thetworegionsandRSPwouldbepossible.
On the other hand, when the process is due to infinite front velocity, the wave front will en-
counter the wall and RSP will not be possible. Mathematically, this is manifested in the fact that
inthiscaseofanon-causalfield,therearenolight-coneand“causal”Green’sfunction. Therefore,
the field state φ(x,−t +(cid:15))|0(cid:105) (where (cid:15) > 0 is arbitrarily small) will involve field states at t = −t
0 0
fromregionswherethewallisplaced.
While the previous example was in favor of the vacuum correlations of relativistic fields, one
can also come up with different examples which favor fields with an infinite front velocity. Con-
sider,forexample,acaseinwhichonefailstoremotelygenerateafieldstateandthentriesagain.
The unsuccessful attempt contaminates the vacuum state of the field. Due to high order terms in
Eq. (6), in relativistic fields, the field state will no longer be of bounded energy (i.e. it will be a
superposition of energy states with ever increasing energies) [25]. This will destroy the delicate
vacuumcorrelationsandsoRSPwillnotbepossible. However,forsomenon-relativisticfields,at
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