Generation of Entanglement Outside of the Light Cone
Superluminal Generation of Entanglement
Generation of Entanglement Outside of the Light Cone 
J.D. Franson 
University of Maryland, Baltimore County, Baltimore, MD 21250 
The Feynman propagator has nonzero values outside of the forward light cone.  
That does not allow messages to be transmitted faster than the speed of light, but it 
is shown here that it does allow entanglement and mutual information to be 
generated at space-like separated points.  These effects can be interpreted as being 
due to the propagation of virtual photons outside of the light cone or as a transfer 
of pre-existing entanglement from the quantum vacuum.  The differences between 
these two interpretations are discussed. 
1.  INTRODUCTION 
 Entanglement is one of the most fundamental and nonclassical aspects of quantum 
mechanics.  It is shown here that entanglement, mutual information, and optical coherence can all 
be generated at two distant locations in less time than it would take for light to travel between 
them.  These counterintuitive effects are possible because the probability amplitude to emit a 
photon at one location and annihilate it at another location is proportional to the Feynman 
propagator FD  [1].  As noted by Feynman himself [2, 3], FD  has nonzero values outside of the 
forward light cone, as illustrated in Fig. 1.  It should be emphasized that this property of the 
Feynman propagator does not allow messages to be transmitted between space-like separated 
points, but it will be shown here that it does allow entanglement to be generated outside of the 
light cone.   
 The fact that the Feynman propagator is not confined to the light cone raises some issues with 
regard to causality that have previously been discussed by Feynman [2, 3], Hegerfeldt [4], and 
others [5-13].  Much of this discussion has been concerned with the probability that an excited 
atom at one location will emit a photon that is absorbed by a second, distant atom after some time 
interval , as illustrated in Fig. 2.  The probability  that the second atom will be excited was 
first considered by Fermi [14], who made several unwarranted assumptions [15] and concluded 
that there were no effects at all outside of the forward light cone.  Other authors [4-13, 15-20] 
considered the problem in more detail, some of whom [4, 20] concluded that there should be an 
increase in  in apparent contradiction with causality.  It was subsequently shown that  
cannot change outside the light cone when all possible effects are included [5-13], but that 
correlations between the states of the two atoms could be produced [5, 7, 11].   
tΔ 2P
2P 2P
 This paper generalizes the earlier results to show that entanglement can be generated outside of 
the light cone as well.  It should be emphasized that the generation of correlations or 
entanglement in this way does not contradict the results of an earlier paper by Milonni et al [9], 
who considered the expectation values associated with the state of a distant atom and showed that 
causality is maintained.  That paper did not consider the generation of correlations or 
entanglement between two distant atoms, which do not violate causality.    
 This paper and the references quoted above are all based on the usual perturbative approach to 
quantum optics or quantum electrodynamics, where these effects can be interpreted as being due 
to the propagation of virtual photons outside of the light cone.  In a separate series of papers, 
Summers and Werner [21-25] used the more abstract techniques of algebraic quantum field 
theory [21-28] to show that Bell’s inequality can be violated in the vacuum state of any quantum 
field theory that satisfies certain assumptions.  In a similar manner, Reznik and his collaborators 
[29-33] have discussed the generation of entanglement outside of the light cone by scalar or 
Dirac fields, which they interpreted as being due to the transfer of pre-existing entanglement in 
the quantum vacuum to the atomic states.  Thus the usual perturbative approach and the algebraic 
quantum field approach lead to very different interpretations of these effects.   
 FIG. 1.  A plot of the Feynman propagator FD  in the xy plane, which is proportional to the probability 
amplitude to emit a photon at one location and annihilate it at another location.  The plot shown here corresponds to 
a 1 ns delay after a photon was emitted at the origin.  The colors of the contour levels were arbitrarily chosen to 
reflect the wave-like nature of a photon.  The value of FD  was truncated on the light cone (the yellow ring), where 
it diverges, but it can be seen that it has a nonzero value arbitrarily far outside the forward light cone as well.  
 Despite the long history of papers on this subject, there is still no consensus of opinion 
regarding the significance or interpretation of effects of this kind.  The goal of this paper is to 
consider these issues in more detail and within the context of entanglement, quantum information 
theory, and the Feynman propagator.  An attempt will be made to address all of the significant 
objections that have been raised in the past, such as the difficulty in localizing relativistic free 
particles, the use of the dipole approximation, and the virtual photon "cloud" associated with 
dressed atomic states.  It is hoped that this will provide some additional insight into what is 
entailed by Fig. 1 as well as the difference between these two interpretations. 
 The remainder of the paper is organized as follows.  In Section 2, perturbation theory and the 
Feynman propagator are used to calculate the probability amplitude to excite the second atom in 
the limit where  is much smaller than the time required for light to travel between the two 
atoms.  Those results are used in Section 3 to show that the two systems become entangled 
outside of the forward light cone, and that the entanglement of the two atoms can be increased 
using post-selection and entanglement concentration.  These effects are compared with quantum 
teleportation and entanglement swapping in Section 4. 
FIG. 2.  Two distant atoms, one initially in its excited state and the other in its ground state.  A photon emitted by 
atom 1 can be absorbed by atom 2 to produce a transition to its excited state.  The probability amplitude for this 
process is determined by the Feynman propagator of Fig. 1 and is nonzero outside the forward light cone. 
 Section 5 considers the impact of these effects on special relativity, including a proof based on 
the commutation relations of the field operators that superluminal messages cannot be 
transmitted; it will be shown that the generation of entanglement and mutual information is not 
limited by that argument.  The generation of optical coherence (as conventionally defined) 
outside the forward light cone is discussed in Section 6, and it is suggested that the definition of 
optical coherence may need to be reconsidered.  The possibility of new types of quantum 
information protocols, including a quantum time capsule, is considered in section 7.  The two 
interpretations mentioned above are contrasted and compared in Section 8, while a summary and 
conclusions are given in Section 9. 
 For simplicity, all of the calculations in the main text will be based on the dipole 
approximation.  The effects of higher-order multipoles are included in Appendix A, where it is 
shown that they are negligible for the situation of interest here.  The calculations in the main text 
consider only bare atomic states, which provide the most straightforward way to derive the 
effects of interest.  Appendix B includes the effects of the virtual photons associated with the use 
of dressed atomic states.  It is found that similar results are obtained in either the bare-state or 
dressed-state basis, which justifies the use of bare atomic states in the main text.  These results do 
not depend on any relativistic properties of the atoms, but a covariant calculation using second-
quantized Dirac operators is discussed in Appendix C nevertheless. 
2.  PERTURBATION THEORY CALCULATION 
 The effects of interest can be understood by considering two distant atoms labeled 1 and 2 and 
located at positions  and , as illustrated in Fig. 2.  At the initial time , atom 1 is 
assumed to be in its first excited state 
1x 2x 0t =
1E  while atom 2 is in its ground state 2G , with the field 
in its vacuum state (no photons present).  The system is then allowed to evolve for a time interval 
, during which there is some probability amplitude that atom 1 may emit a photon and make a 
transition back to its ground state 
1G  while atom 2 absorbs the photon and makes a transition to 
its excited state 2E .  Since the Feynman propagator is nonzero outside the forward light cone, 
the question is whether or not this process could produce a state of the form 
                              1 2 1 2a E G b G Eψ γ φ⊥= + +                                   (1)   
even if 2 1 c t− > Δx x .  Here the state φ⊥  is orthogonal to the other two states and it includes 
the possibility that atom 1 may have emitted a photon that was not absorbed by atom 2, for 
example. 
 Another scenario would be to assume that the excited states 1E  and 2E  correspond to 
metastable states with zero dipole moment, such as the 2S state of hydrogen.  In that case, atom 1 
can be assumed to have been in state 1E  for times  0t <  with negligible interaction with the 
electromagnetic field, and similarly for atom 2 in state 2G .  A dipole moment could then be 
produced by applying an external electric field to both atoms over the time interval , as in the 
Lamb shift experiments.  To the extent that the interaction with the field is negligible outside of 
the time interval , the analysis of this situation is the same as that of the original example of 
Fig. 2.  This example has several advantages when dressed atomic states are considered and it 
will be discussed in more detail in Appendix B.  
 The probability amplitude  to find the system in the state b 1 2G E  with no photons present 
[34] at time  can be calculated in a straightforward way using commutators and the Feynman 
propagator.  From second-order perturbation theory, which is equivalent [35] to the use of Wick’s 
theorem in scattering (S-matrix) calculations, the change in the state of the system at time  is 
given by 
                                        
1 ˆ ˆ( ) ' '' '( ') '( '') .
t dt dt H t H t
Δ = ∫ ∫                                (2) 
The interaction Hamiltonian ˆ '( )H t  is given [36] by 
                                                       3 ˆ ˆˆ '( ) ( , ) ( , )
H t d t
= ⋅∫ r j r A r t                                                    (3) 
where ( , )tj r  is the current operator, is the vector potential operator, and  is the charge 
of the electron.  The minimum coupling Hamiltonian of Eq. (3) has the advantage of being 
manifestly covariant in the Lorentz gauge (see appendix C) and is used in quantum 
electrodynamics.  Similar results for the correlations between the atoms were also obtained by 
Power and Thirunamachandran [11] using the 
ˆ ( , )tA r e−
⋅r E  form of the Hamiltonian.  
 In the limit of small atomic dimensions (dipole approximation), the vector potential can be 
evaluated at the centers of the atoms and the atomic matrix elements of j  reduce to , 
where 
/AiE− d
G E=d x  is the atomic dipole moment, AE  is the energy of the excited atomic states, 
and  is Planck’s constant divided by 2π  [36].  (The contribution from higher-order multipoles 
is negligible, as shown in Appendix A.)  The projection of equation (2) onto the state 1 2 0G E  
then gives 
                   
( '' ') /
1 ˆ ˆ' '' 0 ( , ') ( , '') 0
iE t tA
b dt dt e A t A
− −⎛ ⎞= ⎜ ⎟
∫ ∫ 2 1x x t                           (4) 
where 0  is the vacuum state of the field with no photons and it has been assumed for simplicity 
that the dipole moment is perpendicular to 2 1−x x  and along the x-axis. 
 As usual [37], we can write the vector potential as the sum of its positive and negative 
frequency components 
                                                    ( ) ( )ˆ ˆ ˆ( , ) ( , ) ( , )x x xA t A t A t
+ −= +x x x                                                  (5) 
where ( )ˆ ( , )xA t
− x  creates a photon and ( )ˆ ( , )xA t
+ x  annihilates a photon.  Only the product 
( ) ( )ˆ ˆ( , ') ( , '')x xA t A t
2 1x x  contributes to the matrix element M  in Eq. (4) and   
            ( ) ( )ˆ ˆ ˆ ˆ0 ( , ') ( , '') 0 0 ( , ') ( , '') 0 .x x x xM A t A t A t A t
+ −≡ =2 1 2 1x x x x                      (6) 
Equation (6) can be simplified using the commutator of the field operators to give  
              
( ) ( ) ( ) ( )
( ) ( )
ˆ ˆ ˆ ˆ0 ( , '') ( , ') [ ( , '), ( , '')] 0
ˆ ˆ0 [ ( , '), ( , '')] 0 .
x x x x
M A t A t A t A t
M A t A t
− + + −
1 2 2 1
x x x x
                        (7) 
 The commutator has a simple form in the Lorentz gauge [37], where  
                    ( ) ( )ˆ ˆ[ ( , '), ( , '')] ( , ' '')x x FA t A t ic D t t
+ − = − −2 1 2 1x x x x −                                      (8) 
for '  [where ' 't t> ( ' '') 1t tθ − = ].  Combining Eqs. (4), (7), and (8) gives 
                                
( )2 ' ( '' ') /
' '' ( , ' '')A
A iE t t
b dt dt e D
− −= − − −∫ ∫ 2 1x x t t                                (9) 
 For a massless particle such as a photon, FD  has the explicit value [38, 39] 
                     
2 1 2 2
22 2 2
4 ( )
4 ( ' '')
FD x x i x x i
i c t t i
− = −
− − − −x x
                                 (10) 
Here ε  is an infinitesimally small quantity that determines the value of any integrals and we 
have used the 4-vector notation ( , )x ict= x , which demonstrates the covariant nature of the 
propagator.  We will consider a space-like separation with 2 1r c t= − >> Δx x , which gives 
2 21/ 4FD i rπ= − .  This approximation is valid far outside of the light cone and it is equivalent to 
neglecting terms that are of order  smaller than the remaining terms, as can be seen from 
a power series expansion of Eq. (10). 
2( / )t rΔ
 The two integrals over time in Eq. (9) can then be evaluated to give 
                                                (
2 2 14
b i t e
Δ= − Δ + − )                                                   (11) 
Here  is the fine structure constant and  is the resonant frequency of the 
atomic transition.   
2 /e cα = /A AEω =
 Eq. (11) corresponds to the probability amplitude for atom 1 to emit a photon that is absorbed 
by atom 2.  Although it may seem counterintuitive, there is also a probability amplitude for atom 
2 to emit a photon and make a transition to its excited state while atom 1 absorbs the photon and 
makes a transition to its ground state, since a virtual process of that kind need not conserve 
energy in the intermediate state.  This probability amplitude can be calculated in the same way as 
that leading to Eq. (11) and the total probability amplitude from both processes is given by 
                                        [
02 2 1 cos( ) .2
= − − Δ ]t                                                  (12) 
 Eq. (12) shows that there is a probability amplitude for the two atoms to exchange a photon 
even though they are space-like separated, and this entangles the two systems as discussed in the 
next section.  It can be seen that the effects are only appreciable for atomic separations that are at 
most a few orders of magnitude larger than .  As a result, the predicted correlations are 
experimentally observable in principle but an actual experimental test would be difficult.   
 It should be emphasized once again that these results do not contradict an earlier paper by 
Milonni et al. [9], which only considered the expectation value of the state of atom 2 and not the 
correlations between the two atomic states.  Commutator techniques are used in section 5 to give 
a more general proof that the expectation values associated with atom 2 cannot change outside of 
the light cone, as would be expected from causality. 
3.  ENTANGLEMENT GENERATION AND CONCENTRATION 
 By definition, an entangled state is any quantum state that cannot be written as the product of 
two or more single-particle states.  Eq. (1) does not appear to be factorable, but the situation is 
complicated by the presence of the φ⊥  term and a more careful analysis is required in order to 
demonstrate that the state ψ  actually is entangled. 
 If we could simply ignore the φ⊥  term we would have the entangled state 
1 2 1 2' a E G b G Eψ = + , where the coefficients  and  have been normalized to give unit 
probability. That term cannot be ignored, however, and one way to include its effects is to ask 
whether or not the state 
ψ  of Eq. (1) can be written in the form 
                                                        1 2ψ = Ψ Ψ                                                                  (13) 
where 1Ψ  represents the most general form of the system  consisting of atom 1 and any 
localized field associated with it: 
                 
( ) ( )
( ) ( )
1 1 1 1
1 1 1
ˆ ˆ( ) 0
ˆ ˆ( ) 0
G c n a a
E d n a a
Ψ = ⋅⋅ ⋅
+ ⋅ ⋅ ⋅
                                        (14) 
Here  creates a photon with wave vector  and †ˆia ik { }1( ic n )  and { }1( id n )  are arbitrary complex 
coefficients.  A similar expression exists for the most general form of the system  consisting 
of atom 2 and any field associated with it.  Taking the projection onto the vacuum state gives 
                              
1 1 1 1
2 2 2 2
0 0 0 (0) (0)
(0) (0) ' .
c G d E
c G d E
= ⎡ + ⎤⎣ ⎦
×⎡ + ⎤ =⎣ ⎦
                                            (15) 
But 'ψ  is entangled and cannot be written in this form [40], which implies that Eq. (13) cannot 
hold.  Thus the two systems  and  must be entangled as well, even without the post-
selection process described below.  This result corresponds to the use of bare atomic states, but a 
more general result is derived in Appendix B for dressed atomic states. 
1S 2S
 Most applications of quantum information would use only the atoms independent of the state of 
the field, and an average (trace) over the field states would then have to be taken.  Although ψ  
itself is a pure state, the trace would produce a mixed state with a density matrix ρ̂  that would no 
longer be entangled.  It can be shown [11], however, that the atoms are still correlated if a trace is 
taken over the field states. 
 Entangled states of the atoms alone could be produced, at least in principle, by using an array 
of single-photon detectors to determine whether or not any photons were present in the final state.  
It will be assumed that suitable detectors are located throughout the region surrounding the two 
atoms and that they are all turned on shortly after time tΔ , with a response time that is 
comparable to .  If no photons are detected, the system would be projected into the entangled 
state 
'ψ  at a time , which is outside the forward light cone.   ' ~tΔ Δt
 The use of an array of detectors would allow the post-selection process illustrated in Fig. 3a, 
where the event is rejected if a photon is found after time tΔ , as indicated by the red X in the 
figure.  In principle, this post-selection process could be performed using a large number of pairs 
of atoms to generate a smaller number  of pairs of atoms in entangled states.  In order to 
actually use this entanglement in a quantum information protocol, it would be necessary to wait 
for a much longer time 
2 1 /TΔ ≥ −x x c  in order to distribute the results from the detectors to all 
of the relevant locations.  Nevertheless, the entangled states were created at time  as recorded 
by the detectors. 
FIG. 3.  Generation of maximally entangled atoms using post-selection.  (a)  If a photon is detected after time , 
the event is rejected.  (b)  If atom 2 is found in state F, the event is also rejected as described in more detail in Fig. 
4.  (c)  If no photon is found and atom 2 is not in state F, the two atoms are known to have been maximally 
entangled outside of the light cone.  An entangled state of the atoms and their associated fields will be generated 
outside the light cone without any need for post-selection. 
 Most quantum information protocols would also require that the entanglement of the two atoms 
be nearly perfect (maximal entanglement).  This could be achieved (in principle) using 
entanglement concentration [41], such as the protocol illustrated in Fig. 4.  Here the ground state 
2G  of atom 2 is coupled with a laser pulse to a third atomic level 2F .  By adjusting the 
intensity of the laser pulse, it is possible to transfer any desired amount of probability amplitude 
from 2G  to 2F  and to convert the post-selected state 'ψ  into the state ''ψ  given by 
                                   1 2 1 2 1 2'' 'b E G b G E E Fψ = + + γ                                (16) 
where 'γ  is a complex constant of no interest.  A measurement is then performed to determine 
whether or not atom 2 is in state 2F .  If it is, then the event is discarded as illustrated by the red 
X in Fig. 3b.  If not, then the system will be projected into the maximally entangled state 
( )1 2 1 2 / 2E G G E+ .  A protocol of this kind can be used to convert  pairs of partially 
entangled atoms into  pairs of maximally entangled Bell states.   
'N N<
 It should be noted that the laser pulse and measurements can be completed outside the light 
cone in a small time interval after time tΔ .  Similar techniques have been proposed [42] for use 
in Zeno quantum logic gates [43], and Reznik et al. [29-33] have considered the use of 
entanglement distillation with scalar fields.   
laser
FIG. 4.  Concentration of entanglement in which  partially-entangled pairs of atoms are converted into a smaller 
number  of maximally entangled atoms.  A laser beam is used to partially excite atom 2 from its ground state to 
state F.  A pair of atoms is subsequently rejected if a measurement shows that atom 2 is in state F.  This process 
creates a balanced superposition of states 
1 2E G  and 1 2G E . 
 To summarize this section, it has been shown that an entangled state of the atoms and their 
associated fields will be generated outside the light cone without any need for post-selection.  A 
maximally entangled state of the two atoms alone can be prepared using post-selection and 
entanglement concentration, where the entangled states are known to have been produced within 
time 2 1'tΔ < −x x / c .  The atoms are still correlated but not entangled if an average over the 
field states is taken instead. 
4.  COMPARISON WITH QUANTUM TELEPORTATION AND ENTANGLEMENT 
SWAPPING 
 The post-selection processes described above could be used to prepare  pairs of maximally 
entangled atoms.  Entanglement between two distant systems can also be prepared using quantum 
teleportation [44] or entanglement swapping [45], and it may be useful to discuss the 
fundamental differences between these techniques. 
 In quantum teleportation, two experimenters A (Alice) and B (Bob) would like to transport an 
unknown quantum state from Alice to Bob.  It is assumed that they already share an entangled 
pair of particles, such as a pair of photons that were generated in an entangled state and then 
propagated through optical fibers to Alice and Bob.  Alice performs a Bell state measurement 
between her unknown state and her member of the entangled pair, which produces two bits of 
classical information.  Once the classical information is transmitted to Bob, it can be used to 
regenerate the unknown quantum state by applying a suitable transformation to his member of the 
entangled pair of particles.   
 Although quantum teleportation is a remarkable and useful process, it is apparent that the 
information needed to recreate Alice’s unknown state is transmitted in the form of classical 
information from Alice to Bob at ordinary velocities.  This is very different from the situation 
described above, where the entanglement was created in a time interval 'tΔ  outside of the 
forward light cone and the classical information is used only to determine which events were 
successful; no corrections are applied to the atoms based on that information.   
 In entanglement swapping [45], a pair of entangled particles is created locally at Alice and 
Bob.  One member of each pair is then transmitted at ordinary velocities to a central location, 
where a Bell state measurement is made.  If we post-select on the results of the measurement, the 
two distant particles will be left in an entangled state.  But in this case the entangled state is not 
generated until sufficient time has elapsed for two of the particles to have traveled to the same 
location.  In addition, there is no physical interaction of the distant particles via virtual photons as 
there is in the situation of interest here.  
5.  SPECIAL RELATIVITY AND CAUSALITY 
 The generation of entanglement outside of the forward light cone may seem counterintuitive, 
but it does not allow messages to be transmitted faster than the speed of light.  These effects are 
thus consistent with special relativity to that extent, although they raise some questions regarding 
the postulates of relativity theory. 
 Commutator techniques can be used to provide a simple proof that there can be no net change 
in the probability  of finding atom 2 in its excited state.  In analogy with the usual perturbation 
theory expression for the state vector in Eq. (2), the expectation value of any observable quantity 
corresponding to an operator  at time t is given in the interaction picture [46] by  
                  
0 0 0
1ˆ ˆ ˆ ˆ( ) ( ) ' , '( ')
1 ˆ ˆ ˆ' '' , '( ') , '( '') ...
t Q t Q dt Q H t
dt dt Q H t H t
0ψ ψ ψ ψ ψ ψ
⎡ ⎤= + ⎣ ⎦
⎡ ⎤⎡ ⎤+ +⎣ ⎦⎣ ⎦
                          (17) 
Here the interaction Hamiltonian ˆ '( )H t  involves the free-field (Heisenberg picture) operators 
 and , which satisfy the usual commutation relations. The probability of finding 
atom 2 in its excited state is given by the expectation value of the projection operator 
ˆ ( , )tA x 2ˆ ( , )tA x
2 2 2p̂ E E= .  Since this operator and the corresponding field operator commute with the 
Hamiltonian of atom 1 and the field at that location, it follows immediately that  cannot 
depend on the state of atom 1 outside the light cone.  On the other hand, the correlated probability 
described by the operator 
12 2 1 1 2p̂ E G G E=  does not commute with atom 1 and need not be 
zero outside of the forward light cone. 
 The fact that there is no net change in  can be understood from the fact that there are other 
processes that must be included as well.  For example, energy conservation does not strictly 
apply for a finite time interval 
tΔ , where the uncertainty relation / 2E tΔ Δ ≥  holds.  As a 
result, atom 2 could emit a photon and make a transition to its excited state even if atom 1 were 
not present.  The probability to find atom 2 in its excited state with a photon present is reduced if 
atom 1 can absorb the photon, and this tends to cancel the increase in the probability  2PΔ
calculated above.  A complete calculation of  must include all fourth-order terms and gives no 
dependence on the initial state of atom 1 [5-13].  Thus the exchange of a photon can produce 
correlations and entanglement between the two atoms, but those correlations cannot be controlled 
by an experimenter to send superluminal messages between the two locations. 
 The postulates of special relativity are that (i) the laws of nature are the same in all inertial 
reference frames and (ii) the speed of light is a constant independent of its source.  In order to 
investigate the impact on postulate (i), consider another reference frame R’ that is moving at a 
velocity  with respect to the original reference frame R in which the two atoms are allowed to 
interact with the field over the same time interval 
tΔ .  (Here we assume metastable excited states 
with the dipole interaction controlled by an external electric field.)  Since the events of interest 
are space-like separated, it is possible to choose  in such a way that atom 1 is allowed to 
interact over a time interval  that does not overlap with the time interval 
1tΔ 2tΔ  over which atom 
2 is allowed to interact.  Moreover, the interaction at atom 1 can occur before or after the 
interaction at atom 2 depending on the choice of .  It can be shown that the same entanglement 
is produced in any inertial reference frame, in agreement with postulate (i).  This result is ensured 
by the fact that perturbation theory can be put into an invariant form [47].   
 Although the same entanglement is produced in any reference frame, the photon can only be 
emitted by atom 1 and travel to atom 2 in some reference frames, whereas it can only travel from 
atom 2 to atom 1 in other reference frames.  As a result, any causal interpretation of this process 
would depend on the choice of reference frame.  In that sense, there is some similarity to the 
collapse or reduction of the wave function in experiments based on Bell’s inequality [48]. 
 The second postulate assumes that the speed of light is a constant in all reference frames 
regardless of its source.  Special relativity was proposed in the context of classical 
electromagnetism where the speed of light has a well-defined value, and quantum 
electrodynamics did not exist at the time.  The effects described above correspond to the 
generation of entanglement outside of the forward light cone by the exchange of virtual photons, 
which raises some possible issues regarding the definition of the speed of light.  For example, do 
individual photons travel faster than the classical speed of light during the time that they are 
being exchanged?  Are the photons really being exchanged if we are not allowed to detect them 
in the process?  And what is the nature of the photons if their direction of travel depends on the 
choice of reference frame?   
 Similar questions arise in the interpretation of Feynman diagrams, which consist of space-time 
points (vertices) where two or more particles interact.  The particles propagate between the 
vertices as described by the Feynman propagator, which connects points that are space-like 
separated with a small but nonzero probability amplitude.  One might ask once again whether or 
not the particles travel faster than the speed of light over such trajectories.  As Feynman put it [3], 
“possible trajectories are not limited to regions within the light cone” but “in reality, not much of 
the t − x  space outside the light cone is accessible”.   
 It seems apparent that these effects are due to the properties of the Feynman propagator and 
that the atoms can be treated nonrelativistically.  Nevertheless, a covariant calculation using 
second-quantized Dirac operators is outlined in Appendix C.  
6.  OPTICAL COHERENCE 
 In the conventional theory of single-photon detection developed by R.J. Glauber [49], the 
probability  of detecting a single photon at position  and time  is given by dP x t
                                             ( ) ( )ˆ ˆ( , ) ( , )dP E t E tη
− += x x                                                    (18) 
Here η  is a constant related to the detector efficiency and time window.  This expression can be 
evaluated in the same way as  using the Feynman propagator, with the result that  would 
be nonzero outside of the forward light cone, in apparent violation of causality.  However, Eq. 
(18) is based on an approximation [8, 9, 50] that neglects the possibility that the detector may 
emit a photon as well as absorb one (the rotating wave approximation).  As described above, that 
process will cancel the more intuitive one and the actual detection probability is unchanged 
outside of the forward light cone, as can be shown using Eq. (17) or other methods. 
2PΔ dP
 The concept of higher-order optical coherence plays an important role in quantum optics.  If we 
assume that atom 1 was initially in its ground state and then excited with a short laser pulse at 
time , we can ask whether or not there is any coherence between the remaining laser pulse 
and the field in the vicinity of atom 2.  In analogy with Eq. (18), the second-order optical 
coherence at two locations is defined [49] as  
                 
( ) ( ) ( ) ( )
1 2 2 1(2)
1 1 2 2 ( ) ( ) ( ) ( )
1 1 2
ˆ ˆ ˆ ˆ( , ) ( , ) ( , ) ( , )
( , ; , )
ˆ ˆ ˆ ˆ( , ) ( , ) ( , ) ( , )
E t E t E t E t
g t t
E t E t E t E t
− − + +
− + − +
1 2 2 1
1 1 2 2
x x x x
x x x x 2
               (19) 
If we accept this definition, then it can be shown that second-order coherence would be generated 
between the laser pulse used to excite atom 1 and the field at a space-like separated point.  Once 
again, this is a non-physical result due to the use of the rotating-wave approximation, and it 
suggests that the definition of optical coherence may need to be reconsidered [9,50].  
7.  QUANTUM INFORMATION PROTOCOLS 
 The nonclassical nature of quantum information allows a number of potentially useful practical 
applications, such as quantum computing and quantum key distribution.  The ability to generate 
entanglement outside of the forward light cone would, in principle, allow the possibility of 
additional protocols that would not be possible otherwise.  As a practical matter, the rate at which 
the entanglement can be generated would be too small to be of any real use, but it is interesting to 
consider protocols of this kind nonetheless. 
 One example of a quantum protocol of this kind is the quantum time capsule.  In a 
conventional time capsule, information and artifacts describing the current environment are 
sealed into a capsule that is to be left unopened for some length of time, say 100 years.  
Classically, there is no way to ensure that the capsule might not be opened sooner than intended.  
This would be a potential problem, for example, if a well-known politician sealed his or her notes 
and memoirs in a time capsule with the understanding that the material would remain unavailable 
until long after his or her death.   
 A quantum-mechanical version of a time capsule that cannot be prematurely opened could (in 
principle) be implemented if entanglement were generated between two distant locations, say 100 
light years apart, as illustrated in Fig. 5.  Information could be encoded in the qubits at one 
location by taking the XOR (exclusive OR operation) between the classical information and the 
entangled qubits.  The classical information could then be destroyed, and it could not be recreated 
from the qubits at the first location since they have random values [51].  The information could 
only be retrieved by bringing together the entangled pairs of atoms from the two distant locations 
and comparing the qubits (a second XOR operation), which could only be done after 100 years in 
this example.  Generating the entanglement at a distance eliminates the possibility of cheating 
that would occur if the entangled pairs were generated at one location and then allegedly 
separated by a large distance.  Although protocols of this kind are not practical, they do illustrate 
the fact that the entanglement generated in this way can have unique properties. 
FIG. 5.  Implementation of a quantum time capsule whose contents cannot be accessed for a specified period of 
time.  (a)  Two distant quantum memories contain a large number of atoms that are pair-wise entangled.  Classical 
information is stored in one of the quantum memories by taking the XOR of the classical bits with the qubits in the 
memory and destroying the classical information.  (b)  The information can only be retrieved when the two 
quantum memories are brought together at less than or equal to the speed of light and a second XOR is performed. 
 It may also be useful to note that these effects allow  mutual information as conventionally 
defined to be established outside of the light cone.  Suppose that  pairs of maximally 
entangled qubits are generated at two space-like separated locations as described above.  If the 
values of the qubits are simply measured, then the classical mutual information is defined by 
                                 2
( , )
( ; ) ( , ) log
( ) ( )y Y x X
p x y
I X Y p x y
p x p y∈ ∈
= ∑∑                                       (20) 
where ( , )p x y  is the joint probability distribution of the variables  and  and X Y ( )p x  and 
( )p y  are their marginal probability distributions.  Since the resulting measurements are totally 
correlated, the process yields  bits of mutual information that were generated outside of the 
forward light cone.  
8.  INTERPRETATION OF RESULTS 
 Effects similar to those described here were independently predicted based on two different 
approaches.  Starting from the work of Fermi [14] and Hegerfeldt [4], the usual perturbative 
approach to quantum optics or quantum field theory led to the realization that correlations could 
be generated between two distant atoms.  Here that approach was generalized to show that 
entanglement and mutual information could be generated outside the light cone as well.  This 
approach involves the exchange of virtual photons and the Feynman propagator (or equivalent 
calculations).  Similar effects were also predicted using algebraic quantum field theory, which 
leads to a different interpretation that does not involve virtual photons [21-33]. 
   Electromagnetic interactions are generally viewed as being produced by the exchange of real 
or virtual photons, which is consistent with the usual perturbative treatment of quantum 
electrodynamics.  The photons are assumed to carry momentum and energy and that is 
responsible for the force between two charges, for example.  In that case, the most natural 
interpretation of the results obtained here would be to assume that the entanglement is generated 
by the propagation of virtual photons outside the forward light, as suggested by Fig. 1 and 
substantiated by the calculations of Section 2. This interpretation is complicated by the fact that 
the photons cannot be directly observed without destroying the effects of interest, while their 
direction of travel depends on the choice of reference frame, as discussed in Section 5.    
 Summers and Werner independently used the more abstract techniques of algebraic quantum 
field theory to show that Bell’s inequality is violated by the field operators in the vacuum state of 
a quantum field [21-25].  Later work in algebraic field theory showed that the vacuum state of the 
field is entangled provided that the theory satisfies certain assumptions [26-33].  This has 
prompted an interpretation by Reznik and his collaborators [29-33] in which effects similar to 
those of interest here (but for a scalar field)  were assumed to be due to a transfer of pre-existing 
entanglement from the quantum vacuum to the atoms, with no requirement for any transfer of 
information outside of the light cone.   
 The assertion that the quantum vacuum is entangled may seem surprising within the context of 
quantum optics.  Whether a system is entangled or not depends on the choice of basis vectors.  In 
the usual plane-wave number-state basis used in quantum optics, the quantum vacuum 
corresponds to the product of a large number of ground states of independent harmonic 
oscillators, which is not an entangled state.  But the plane waves do not correspond to spatially 
separated systems and they do not form a suitable basis for a discussion of entanglement.  In 
algebraic quantum field theory, the states of interest are localized to two or more separated 
regions and the quantum vacuum corresponds to an entangled state in that basis. 
  Regardless of whether the quantum vacuum is entangled or not, the vacuum fluctuations at 
two different locations are correlated.  For example, it follows from Eq. (8) that  
             ˆ ˆ0 ( , ') ( , '') 0 ( , ' '')x x FA t A t ic D t t= − −2 1 2 1x x x x −                          (22) 
which shows that the field operators at two different locations are correlated, as qualitatively 
indicated in Fig. 6a.  The algebraic quantum field approach suggests that these local fluctuations 
may be responsible for the change in the state of the two atoms without any requirement for a 
transfer of information outside of the light cone. 
position
vacuum state:
virtual photon
position
vacuum state:
FIG. 6.  (a)  Correlations between the electric fields E due to vacuum fluctuations at two distant locations, which 
could produce correlated changes in the states of two atoms.    (b)  The state of an atom cannot change without a 
change in the state of the field, which corresponds to the emission or absorption of virtual photons. 
 But it seems to me that there is something missing here, namely the virtual photons illustrated 
in Fig. 6b [53].  If a test particle is placed in the electromagnetic field in its vacuum state, then 
conservation of momentum does not allow a change in the momentum of the test particle unless 
the state of the field changes as well.  Such a change in the field corresponds to the emission or 
absorption of a virtual photon.  The form of the field operators also shows that there can be no 
change in the state of the atoms unless there is a change in the state of the field corresponding to 
the emission or absorption of a virtual photon.  This suggests that the quantum vacuum alone is 
not sufficient to produce the effects of interest here, and that the emission and absorption of 
virtual photons is required.   
 The correlations between the field operators can be further understood if we insert a complete 
set of basis states in the left-hand side of Eq. (22).  Since the vector potential only couples the 
vacuum to single-photon states, this can be written as  
            ˆ ˆ ˆ ˆ0 ( , ') ( , '') 0 0 ( , ') ( , '') 0x x x x
A t A t A t k k A t=∑2 1 2 1x x x x               (23) 
where k  is a single-photon state with wave vector .  In the presence of two test charges, 
each of the matrix elements on the right-hand side of the equation correspond to the 
emission or absorption of a photon, whose propagation is described by the Feynman 
propagator as before.   Eq. (23) suggests that the correlations between the vacuum 
fluctuations at two locations are maintained by the exchange of virtual photons, which is 
consistent with the fact that the states of the two atoms cannot change in a correlated way 
without such an exchange.   
 In contrast, the algebraic quantum field theory approach considers the correlations 
between the field operators without considering their origin.  As a result, the propagation of 
virtual photons does not enter into those discussions.  Unless we are willing to abandon the 
role of photons in electromagnetic interactions, the discussion above shows that the effects 
of interest here are due to the propagation of virtual photons outside of the forward light 
cone.  
9.  SUMMARY AND CONCLUSIONS 
 It has been shown that entanglement, optical coherence, and mutual information can all be 
generated between two space-like separated points.  An entangled state of the atoms and the field 
can be generated outside the light cone without any need for post-selection, while the latter can 
be used to identify pairs of atoms that were maximally entangled after an arbitrarily short amount 
of time.   
 The analysis in the main text was based on the use of bare atomic states, but similar results are 
obtained in the dressed-state basis in Appendix B, where it is shown that the entanglement 
generated during the time interval tΔ  is independent of the “cloud” of virtual photons that may 
have existed before that time.  This justifies the use of bare states in the main text. 
 It has been suggested that these results do not demonstrate any “real” entanglement because 
useful results can only be obtained using post-selection.  But Eq. (15) clearly shows that the two 
systems (including the field) become entangled without any post-selection or entanglement 
concentration; a generalization of this result to dressed atomic states is given in Appendix B.  
Even in the case where post-selection is used, no corrections need be applied to the atoms and 
they must have been entangled during a time interval 2 1'tΔ < −x x / c , as recorded by the photon 
detectors.   
   These counterintuitive effects are due to the fact that the Feynman propagator has nonzero 
values outside of the forward light cone.  Hegerfeldt [52] has noted that this is an unavoidable 
feature of any quantum field theory where the energies of the particles are bounded from below.  
If electromagnetic interactions are viewed as being produced by the exchange of virtual photons 
that carry energy and momentum, then the results obtained here can be interpreted as being due 
to the propagation of virtual photons outside of the light cone.  Whether such effects should be 
considered to be superluminal is a semantic issue, since the photons cannot be directly observed 
and only the resulting correlations between the atoms can be measured.  
 An alternative interpretation has been suggested based on algebraic quantum field theory, 
where the quantum vacuum is considered to be an entangled state [21-33]. In that case, the effects 
of interest here could be interpreted instead as being due to the transfer of entanglement from the 
quantum vacuum to the atoms without the need for an exchange of particles or information 
outside of the forward light cone.  Although that is perhaps a valid interpretation, it does not 
explain how the correlations in the vacuum fluctuations are produced or maintained.  My 
personal preference would be to retain the usual role of virtual photons in quantum 
electrodynamics.  
 Regardless of the interpretation, a number of questions remain [53]:  Are there any feasible 
experiments to test effects of this kind outside of the light cone?  If an experiment were to be 
performed, would these effects be observed?  And are there any alternative theories that do 
not have this property?   Further research of this kind appears to be a natural extension of the 
earlier work on Bell’s inequality. 
ACKNOWLEDGEMENTS 
 I am grateful to B.C. Jacobs, T.B. Pittman, and M.H. Rubin for their comments on the 
manuscript. 
APPENDIX A:  DIPOLE APPROXIMATION 
 Eq. (4) in the text is based on the dipole approximation, which is valid when the characteristic 
dimensions of the atom are much smaller than the wavelength of the photons.  Since the 
wavelength does not explicitly appear in the commutator approach used here, it may be useful to 
verify that the dipole approximation is valid for the situation of interest.  It will be shown that the 
contribution from the higher-order multipoles is negligible for r c tΔ  and the only significant 
contribution is from the dipole term.  The basic approach will be to evaluate the commutator of 
the field operators before the atomic matrix elements are calculated. 
 Combining Eqs. (2) and (3) in the text gives  
               
1 ˆ ˆ0 ( ) ' '' 0 ' ( ', ') ( ', ')
ˆ ˆ'' ( '', '') ( '', '') 0 .
t dt dt d t
d t t E G
t⎡ ⎤Δ = ⋅⎣ ⎦
⎡ ⎤× ⋅⎣ ⎦
∫ ∫ ∫
r j r A r
r j r A r
                    (A1) 
From Eq. (7) in the text, the matrix element M  of the field operators is given by 
       
22 2 2
ˆ ˆ0 ( ', ') ( '', '') 0 ( ' '', ' '')
4 ' '' ( ' '')
x x FM A t A t ic D t t
c t tπ
≡ = −
− − −
r r r r
                                  (A2) 
We can expand this in a Taylor series about the point p  given by 2' =r x , 1'' =r x , , and 
' 0t =
'' 0t =
                                       
( ) ( )2 22 2
' ' ...
4 ' '
c f f
M x x y
r x y
⎡ ⎤∂ ∂⎡ ⎤
= + − + −⎢ ⎥⎢ ⎥∂ ∂⎣ ⎦ ⎣ ⎦
∂ ∂⎡ ⎤ ⎡ ⎤
+ +⎢ ⎥ ⎢ ⎥∂ ∂⎣ ⎦ ⎣ ⎦
                        (A3) 
Here we have defined 2 1r = −x x  as before and the function f  is defined as  
                                              2 2
' '' ( ' '')
c t t
− − −r r 2
                                                             (A4) 
The partial derivatives are given by 
                                 
2 12 42 2 2
22 2 2
2 ' '' 2
' ' '' ( ' '')
2 ' ''
' ' '' ( ' '')
x rc t t
c t tf
t c t t
−∂⎡ ⎤ ⎢ ⎥= − = − −⎢ ⎥ ⎢ ⎥∂⎣ ⎦ − − −⎢ ⎥⎣ ⎦
−∂⎡ ⎤ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥∂⎣ ⎦ − − −⎢ ⎥⎣ ⎦
                              (A5) 
for example, with similar expressions for the other terms.  For simplicity, we can take  
along the z axis, so that only the partial derivatives with respect to  and  are nonzero. 
2 1−x x
'z ''z
 Inserting this into Eq. (A1) and taking the projection onto 1 2G E  gives the probability 
amplitude : b
                                           ( )
2 2 2
( ) 4
2ˆ' ( ', ') 1 ' ...
2ˆ'' ( '', '') 1 '' ... .
b dt dt
i c r
E d j t z z G
G d j t z z E
× − − +⎜⎢ ⎥
⎝ ⎠⎣ ⎦
⎡ ⎤⎛ ⎞
× + −⎜ ⎟⎢ ⎥
⎝ ⎠⎣ ⎦
                              (A6) 
It can be seen that the second term in the expansion is smaller than the first by a factor of , 
where  is the characteristic dimensions of the atom.  Thus in the limit of  we get  
/Ad r
Ad /Ad r 1
                                      
2 22 2 2
1 1 ˆ' '' ' ( ', ')
( ) 4
ˆ'' ( '', '') .
b dt dt E d j
i c r
G d j t E
⎡ ⎤= ⎣
⎡ ⎤× ⎣ ⎦
∫ ∫ ∫
t G⎦                           
(A7) 
The identity /x A xj iE d= −  does not depend on the dipole approximation and can be derived 
from the fact that  [36].  Eq. (A7) is thus equivalent to Eqs. (9) and 
(10) in the text. 
ˆˆ/ / , /xdx dt p m x H i⎡ ⎤= = ⎣ ⎦
 These results show that the contributions from the higher-order multipoles are negligible when 
the separation between the atoms is much larger than their dimensions.  This situation is very 
different from the usual case in which electromagnetic energy is emitted by atom 1 and then 
travels at the speed of light to atom 2, where it can be absorbed; there the ratio of the dipole to 
quadrupole contributions is independent of the distance, for example.  In that case the largest 
contribution is on the light cone where FD  diverges and the Taylor’s series expansion is not 
valid.  Here we are considering the opposite limit, where the multipole contributions can be 
completely neglected. 
APPENDIX B:  DRESSED ATOMIC STATES AND VIRTUAL PHOTONS 
 The analysis in the text assumed an initial state in which atom 1 was in its bare excited state 
1E  and atom 2 was in its bare ground state 2G  with no photons present initially.  It is valid to 
consider the time evolution of an initial state of this kind, at least as a Gedanken experiment, and 
most of the previous discussion has been based on this example.  It is shown here that similar 
results are obtained in the more realistic case in which the initial state corresponds to dressed 
atomic states that have some probability amplitude to include one or more virtual photons.  The 
entanglement generated during the time interval tΔ  will be shown to be independent of any 
virtual photons that existed before that time to lowest order in perturbation theory. 
 The presence of divergent diagrams, such as the self-corrections to the mass and charge of the 
electron, require that the theory be renormalized.  The calculations can then be performed using 
the physical mass and charge, as is implicitly done in most quantum optics calculations.  It will 
be assumed that the theory has been renormalized in this way, so that the remaining interaction is 
small ( 1α << ) and perturbation theory can be used. 
 After the renormalization, the remaining interaction of the atoms with the field will produce 
energy eigenstates with some probability amplitude for the presence of a virtual photon, which 
we will refer to as dressed states.  The virtual photon “cloud” associated with a dressed atomic 
state is illustrated in a very schematic way by the black arrows in Fig. 7a.  We will refer to an 
eigenstate of the Hamiltonian that does not include the interaction with the field (but with the 
theory renormalized to use the physical mass and charge of the electron) as a bare atomic state.  
 Including dressed atomic states in the analysis raises a number of questions:  Does the presence 
of the virtual photons in the initial dressed states affect the matrix elements and the transition 
amplitudes?  Is the entanglement between dressed or bare states, and how would that affect the 
results of any measurements?   And would the virtual photons in the final dressed states have any 
effect on the outcome of the detectors used in the post-selection process?  
 In order to investigate these issues, we will make two basic assumptions:  (i)  The interaction 
between the field and the atoms is weak ( 1α << ) after renormalization, so that we can consider 
only the lowest-order effects.   (ii) The interaction Hamiltonian that couples the atomic states of 
interest ( E  and G ) can be effectively turned off before time 0t =  and after time , or at 
least reduced to a negligible  value outside the time interval 
t = Δt
tΔ .   
 FIG.7.  Comparison of the dressed state basis with the bare state basis.  (a)  After renormalization, the dressed 
atomic states will include virtual photons (represented by black arrows) produced by atomic transitions other than 
the one of interest for times .   (b)  During the time interval 0t < tΔ , the interaction between the ground state and 
first excited state is turned on by an external field, which produces additional virtual photons represented by red 
arrows.  (c)  The effects of the additional photons are independent of the virtual photons that existed before  to 
lowest order, so that the situation is equivalent to using the bare initial state with no virtual photons.  (d)  During the 
time interval , the analysis in the bare state basis includes only the virtual photons generated by the atomic 
transition of interest.  To lowest order, this gives the same results as the dressed state analysis of (a) and (b). 
 As mentioned in the text, an example of a situation in which the interaction Hamiltonian can be 
modulated is the case in which the excited dressed states 1 'E  and 2 'E  correspond to 
metastable states with zero dipole moment.  Atom 1 can then be assumed to have been in dressed 
state 1 'E  for times  and similarly for atom 2 in dressed state 0t < 2 'G .  The application of an 
external electric field  can be used to produce a dipole moment for both atoms over the time 
interval , as in the Lamb shift experiments.  We will work in the interaction picture where the 
unperturbed Hamiltonian 
ˆ ( )H t  is chosen to include the effects of  and the coupling to the 
field that exists for , while the interaction Hamiltonian 
0t < ˆ '( )H t  will include only the coupling 
term that is turned on during time tΔ .   
 Modulating the interaction in this way provides a clear definition of the initial conditions that 
apply to the problem.  Moreover, the modulation could be applied using electromagnetic pulses 
traveling in free space, so that there is no change in the boundary conditions associated with the 
modulation.  This avoids the ambiguities that have been associated with some of the earlier 
discussions.  It should also be noted that the virtual photons that exist prior to  all 0t =
correspond to atomic transitions other than the G E↔  transition of interest, which has zero 
matrix elements for . 0t <
 We begin with a generalization of Eqs. (13-15) in the text, which showed that 1 2ψ ≠ Ψ Ψ  
in the bare state basis.  Here the dressed atomic states may contain correlations between the two 
atoms even before , and the bare atomic states used in the main text need to be replaced 
with more general eigenstates, such as  
                                           1 2 1 20 ', ',0E G E G→ ' .                                         (B1) 
Here 1 2', ',0 'E G  corresponds to the perturbed eigenstate produced by the interaction with the 
field in 0Ĥ , which will no longer factor into two single-atom states as before.  The  0 '  in Eq. 
(B1) reflects the fact that the field will no longer be in the bare vacuum state and it may contain 
virtual photons. Similar notation will be used for the other dressed states of interest. 
  The dressed atomic states that exist outside of the time interval tΔ  are eigenstates of 0Ĥ .  To 
lowest order in perturbation theory, they can be written in the form 
          
1 2 1 2 1 2
1 2 1 2
1 2 1 2
', ',0' 0 ( , )
( , ) ( , , , ) ,
( , ) 0 0
E G E G c i F G
d i E F e i j F F
f i j F F g G G
h E E
k k k p k p
                   (B2) 
for example, with similar expressions for the other dressed states.  Here ε  is a constant of the 
same order as ˆ 'H , β  is a normalization constant on the order of unity, c  through  represent 
arbitrary complex coefficients, and 
1iF  and 2iF  represent atomic states other than the ground 
state or first excited state.  The form of this equation follows from the fact that 0Ĥ  does not 
directly couple the ground and first excited states. 
 The second and third terms in Eq. (B2) correspond to the emission of a single virtual photon 
and an atomic transition to  1iF  or 2iF . The fourth term corresponds to the emission of two 
virtual photons and two atomic transitions, while the fifth term corresponds to the emission of a 
virtual photon by one atom followed by its absorption by the other atom.  The final two terms 
correspond to the emission and absorption of a virtual photon by the same atom, which can 
produce a transition from 1E  to 1G , for example, via a virtual state involving 1iF .  (Similar 
transitions to 1 2 0iF G  and 1 2 0iE F  are of no interest and have not been included.)  It 
should be noted that the term 1 2 0G E  can only be produced via two virtual transitions of that 
kind and is therefore fourth order and negligible. 
 We will consider the subspace S of Hilbert space corresponding to the projection ÊGP  onto the 
ground and first excited atomic states: 
                  1 2 2 1 1 2 2 1
1 2 2 1 1 2 2 1
EGP G G G G G E E G
E G G E E E E E
                            (B3) 
The projection of the initial dressed state onto S gives 
                                 
1 2 1 2 1 2
ˆ ', ',0 ' 0 0
EGP E G E G g G G
h E E
                       (B4) 
This state can be prepared by local operations on the two atoms and it can be written as the 
product of two single-atom states to order 2ε .  This shows that the initial state is not entangled in 
the subspace S before the interaction at time 0t =  to lowest order. 
 We will now show that the system is entangled in subspace S after the interaction over time 
interval .  The Hamiltonian tΔ 0Ĥ  cannot produce any change ψΔ  in the state of the system 
and ψΔ  must involve at least one factor of ˆ 'H .  To second order in ε , we can therefore drop 
the 2ε  terms in Eq. (B2) and only the first three terms in the initial state can contribute to ψΔ .  
Since ˆ 'H  does not produce any transitions involving the states 1iF , the projection onto the 
subspace S also has no contribution from the second and third terms of Eq. (B2), so that 
                             
1ˆ ˆ ˆ' '' '( ') '( '') 0 .
EGP dt dt H t H t Ei
Δ = ∫ ∫ G                          (B5) 
where we have taken 1β → .  This is identical to Eq. (2) in the main text, and the same 
techniques used there can be used to show that 
                                                    1 2ÊGP ψ ≠ Ψ Ψ                                          (B6) 
for  . t t> Δ
 These results show that the system was not entangled in subspace S before the interaction (to 
second order) while it is entangled in subspace S after the interaction.  It is also apparent that the 
entanglement that is generated during time interval tΔ  is independent of any entanglement or 
virtual photons that may have existed prior to the interaction to lowest order. 
 The situation here is analogous to entanglement in quantum optics, where a photon can be 
independently entangled in polarization or in energy-time variables, for example.  The 
entanglement in polarization is routinely measured experimentally while simply ignoring any 
energy-time entanglement.  Here the entanglement generated during the interaction over time 
interval  is orthogonal to any entanglement that previously existed, and the latter can be 
ignored in the same way. 
 We will now consider the effects of using dressed atomic states for post-selection and 
entanglement concentration, as in Fig. 3.  The analysis can be repeated using the same 
perturbation theory techniques that were used in the main text or by using Eq. (17), where the 
matrix elements will now be taken in the dressed-state basis.  A transition from the dressed state 
1 2', ',0 'E G  to 1 2', ',0 'G E  will require two factors of ˆ 'H  as described above, so that a typical 
term  in the integrand is T
                                    1 2 1 2 1 2 1 2ˆ ˆ' '0 ' ' ' ' ' ' ' ' '0 'T G E H G G G G H E G= k k                                 (B7) 
for example.  Since the factors of ˆ 'H  already introduce a factor of 2ε , to second order in 
perturbation theory this reduces to  
                                                  
( ) ( )
( ) ( )
2 1 1 2
2 1 1 2
ˆ ' 0
T E G H G G
G G H E G .
⎡ ⎤= ⎣ ⎦
⎡ ⎤×⎣ ⎦
                                        (B8) 
These matrix elements are the same as in the bare-state basis, and the perturbation calculation 
will therefore give the same results as were obtained in the main text to lowest order.  Measuring 
the joint state of the two systems is a nonlocal operation [54, 55] that will require a longer time 
interval  as before, but that does not alter the basic conclusion that entanglement can be 
generated outside of the forward light cone. 
 We can model the detectors as additional two-level atoms initially prepared in their dressed 
ground states, where a detection event will correspond to finding the atom in its excited dressed 
state.  We do not require a fast response from the detectors, so that they can be turned on slowly 
after time  and the system allowed to evolve over a relatively long time period during the 
post-selection process (but still shorter than ).  As a result, energy conservation will apply.  
For times , the system is in an eigenstate of the Hamiltonian where no transitions can occur 
and the detectors could not have registered a count due to any virtual photons in the initial 
dressed atomic states.  The same situation will hold after time 
tΔ , where atoms 1 and 2 are either 
in their ground state or a metastable state and cannot supply any energy to excite the detector 
atoms.  As a result, the detector atoms cannot respond to any virtual photons associated with the 
dressed atomic eigenstates and they can only register photons that were emitted by atom 1 or 
atom 2 during the time interval tΔ .  Thus the post-selection process can be performed as 
described in the main text. 
 Since we are working in the dressed state basis, Eq. (B8) combined with the post-selection and 
entanglement concentration described in the main text will produce a final state of the form 
                                      1 2 1 2
1 2 1 2
'' ', ',0' ', ',0' / 2
0 0 / 2
E G G E
E G G E
ψ = ⎡ + ⎤⎣
= ⎡ + ⎤⎣ ⎦ .
⎦                                (B9) 
The second line of the equation holds to second order in the interaction from Eq. (B2), since the 
dominant terms are already second order.  Thus the dressed-state analysis gives the same results 
as the bare-state analysis in the text to lowest order. 
 A comparison of the dressed-state and bare-state analyses is summarized in Fig. 7. In the 
dressed-state basis, there will be virtual photons associated with other atomic transitions for 
, as illustrated by the black arrows in Fig. 7a.  Turning on the coupling between the ground 
and first excited states over time interval 
tΔ  will produce additional virtual photons, as 
illustrated by the red arrows in Fig. 7b.  The effects of these additional virtual photons are 
independent of the virtual photons that were present in the initial state to lowest order, so that the 
net result is the same as in the bare-state basis illustrated in Figs. 7c and 7d.  This justifies the use 
of the bare-state basis in the main text and in the discussion of Appendix C. 
APPENDIX C:  COVARIANT CALCULATION 
 It seems apparent that these results are due to the nature of the Feynman propagator for the 
photons and that the atoms can just as well be treated nonrelativistically.  It has been suggested, 
however, that the calculations should be performed in a covariant way nevertheless to ensure that 
the results are not an artifact of the nonrelativistic treatment of the atoms.  It will be shown in this 
appendix that the same results are obtained using the second-quantized Dirac theory for the 
electrons, aside from a small relativistic correction to the atomic matrix elements.   In the 
Lorentz gauge, the interaction Hamiltonian can be written in the form 
                                                         3 ˆˆ ˆ'H d j Aμμ= −∫ r                                                                 (C1) 
Here  ĵμ  is the current 4-vector in the second-quantized Dirac theory with components 
                                                                                                                  (C2) 
ˆ ˆ ˆ( ) ( ) ( )
ˆ ˆ ˆ( ) ( ) ( )
ρ ψ ψ
r r r
j r r α r
where  is the field operator for the electron-positron field and  corresponds to the Dirac 
matrices.  The use of perturbation theory will give rise to integrals 
ˆ ( )ψ r α
I  of the form 
                                            3 ˆˆ ˆ'( ') ' '
I H t dt dt d j Aμμ
−∞ −∞
= = −∫ ∫ ∫ r                                                   (C3) 
for example.  If the interaction goes to zero after the time of interest, which it does in S-matrix 
theory as well as in the metastable state example discussed above, then the integral over time can 
be extended to infinity to give 
                                                            4 ˆˆI d x j Aμμ
= − ∫                                                               (C4) 
Eq. (C4) is an invariant under Lorentz transformations, and it can be shown that the results of 
perturbation theory are the same in any reference frame and that perturbation theory is equivalent 
to the use of Wick’s theorem in scattering (S-matrix) calculations [35]. 
 In the Lorentz gauge, the interaction Hamiltonian includes both the vector and scalar potential 
operators: 
                                                      3 ˆ ˆˆ ˆˆ' (H d )ρ= − ⋅ + Φ∫ r j A                                                         (C5) 
Here  is the scalar potential operator and Φ̂ ρ̂  is the charge density operator.  The contribution 
from the scalar potential term can be evaluated using the commutator techniques of Appendix A.  
That contribution vanishes in the limit of r c tΔ , since the only nonzero matrix elements of ρ̂  
depend on the partial derivatives of Eq. (A5).  Thus the only significant contribution is from the 
vector potential term that was used in Eq. (3) of the main text.    
 Although the nonrelativistic Schrodinger equation was tacitly assumed in the main text, the 
actual form of the current operator was never used.  Instead, the dipole approximation was 
derived from the identity 
                                                                                                       (C6) 0ˆˆ/ / , /xdx dt p m x H i⎡ ⎤= = ⎣ ⎦
This relationship must hold at least approximately for the Dirac theory, since the Schrodinger 
equation corresponds to its nonrelativistic limit.  As a result, the matrix elements of  in the 
Dirac theory are related at least approximately to the dipole moment by , just as in the 
nonrelativistic Schrodinger equation.  The dipole moment is to be evaluated between the 
relativistic eigenstates of the atom [56], which will give a small correction to their value.   
/A iiE d−
 Since the matrix elements of the interaction Hamiltonian are the same in the Dirac theory as in 
the nonrelativistic Schrodinger equation, aside from a small relativistic correction to the dipole 
moments, it follows from perturbation theory or Eq. (17) that the Dirac theory will give all the 
same results that were previously described in the main text.  This discussion also shows that the 
same results would be observed in any reference frame, as can be explicitly demonstrated. 
 Hegerfeldt [52] showed that, in relativistic quantum field theory, a free particle that is 
completely localized inside a finite volume at time 0t =  will subsequently have some probability 
to be found arbitrarily far away after a short time interval.  It has been argued [13, 54, 55] that 
this difficulty in localizing free particles invalidates the usual assumption that the initial state is 
localized.  But Hegerfeldt’s theorem applies to free particles, whereas the electrons in the atomic 
states of interest here are bound.  In addition, they are not strictly localized in the initial state, 
since the relativistic atomic state corresponds to exponentially decaying probability amplitudes 
[56], and Hegerfeldt’s theorem does not apply.  To within an exponentially small error, the 
atomic eigenstates can be modulated by an external electric field of finite extent and the effects of 
that modulation  outside of the forward light cone can be determined as discussed in the text.   
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