1. INTRODUCTIONThe helicity density matrix of a generic momentum-helicitywave packet under the action of the Lorentz group is ill de-fined. The underlying reason is the explicit helicity depen-dence on the momentum eigenstates when transformed fromone reference frame into another. Consequently, tracing overthe momentum is an act of violence against the rules of quan-tum mechanics because one intends to sum helicity densitymatrices from different Hilbert spaces. It makes no senseto talk about transformation properties of the helicity densitymatrices, and there is no group representation that the helicitymatrix can transform according to. The situation is furtherworsened by attempts to measure such a state to recover theencoded information. This problem was first systematicallystudied in [1], but some explicit comments were already madein [2], and probably even before. A partial remedy was foundin [1], too, where the authors first realized the importance of aspecific helicity measurement and then introduced the con-cept of a three-dimensional helicity density matrix to deal withthe problems mentioned above. Regarding the helicitymeasurement, it is a simple projective measurement ontothe plane perpendicular to the direction of propagation ofthe wave packet, and, due to its simplicity, it will be usedin this work as well. We will refer to this kind of measurementas the Peres–Terno (measurement) scheme. We will, how-ever, abandon the concept of a three-dimensional helicity den-sity matrix. The additional longitudinal degree of freedomintroduced in [1] for the sake of having at the least definitetransformation properties under ordinary rotations will beof no use here. Moreover, the introduction of a longitudinaldegree of freedom sounds like there is something forbiddengoing on, but it is not the case. This expression, however, sim-ply refers to the third degree of freedom of a specific positiveoperator-valued measure (POVM). One could, in principle,have POVMs with arbitrarily many outcomes to be able to re-construct the wave packet to an arbitrary precision. Thewhole problem of the helicity density matrix has been furtherclarified and generalized to other (nonrelativistic) situationsin [3] and [4]. Problems of density matrix construction were
considered in other scenarios, too [5]. For the purpose of thisarticle, we will stick to the relativistic context.
There exists a class of wave packets in which the definitionof the helicity density matrices causes no troubles. These arelinear polarization wave packets, in which case, loosely speak-ing, all the momentum three-vectors point in the same direc-tion (the direction of propagation). This class of wave packetshas very simple transformation properties and allows for in-troduction of the invariant helicity density matrix [6–8]. Thatis, tracing over the momentum is allowed, and, by invariance,we mean that the coefficients of the resulting density matrixare independent of the reference frame. States transformingcovariantly under a general action of the Lorentz group area subject of interest. The classification of two-photon stateshas been provided in Ref. [9]. Let us emphasize that contraryto our paper, only states with sharp momenta were consideredso the physical picture considerably differs.
The aim of this paper is twofold. In the first part, weconstruct localized wave packets where the information isencoded in the helicity degrees of freedom (Section 2). Theeffect of an arbitrary Lorentz transformation is studied, andthe conditions for invariant transformation of these wavepackets are found. The helicity density matrix derived fromthese states is well defined and invariant under the generalaction of the Lorentz group. Finally, the effect of thePeres–Terno helicity measurement on the receiver’s side isinvestigated, concluding the perfect discriminability of twoinitially orthogonal states in contrast to the original solution.Appendix A contains some further technical details.
If we ask for a specific communication scenario, one mightimagine the following situation. Suppose there are two ormore satellites far from a massive body (that is, in approxi-mately Minkowski space–time). As they pass by each otherand communicate by sending wave packets, theWigner phase,whose definition we recall in Appendix B, does not stay con-stant. Instead of tracking down the senders’ velocities and thespatial angle from which the packets come to adjust the de-tectors, the receiving satellite does not need to do anything.The encoding we will present here takes care of the changing
Kamil Brádler Vol. 28, No. 4 / April 2011 / J. Opt. Soc. Am. B 727
Wigner phase, and the helicity density matrix stays invariant.It might actually be impossible to tune just one detector toreceive transmissions frommore than one satellite at the sametime. Note that a completely different task would be to con-sider communication between a satellite and a ground-basedbase. We do not cover this topic here, but similar ideaspresented in this article might be considered to includeweak-gravitational fields as well. The starting point for sucha calculation would be Ref. [7].
The second part is solely aimed at nonexperts on the pro-blem of the photonic wave packets’ construction and theirrelativistic transformations. It consists of Appendices B andC, where we first briefly review Wigner’s phenomenal contri-bution to the understanding of the role that is played by thePoincaré group in quantum field theory in Minkowski space–time [10]. Appendix C contains a discussion of a rathercomplicated issue of measurement for generic photonic wavepackets, and the solution of this problem found in [1].
2. RELATIVISTICALLY INVARIANTPHOTONIC WAVE PACKETSA. Photonic Wave PacketsWe adopt the following notation throughout the article. Thecomponents of a general momentum four-vector (brieflyfour-momentum) p is written as pμ. We recall that indices fromthe Greek alphabet are reserved for labeling all four com-ponents of the four-momentum. The four-momenta pμ livingin the four-momentum space can be of null (pμpμ ¼ 0), time-like (pμpμ > 0) or spacelike (pμpμ < 0) character (omitting thezero vector). For the purpose of this article, we are interestedin the first class. Hence, we set c ¼ 1 and normalize energysuch that p0 ¼ 1, so we get pip
i ¼ 1 (i ∈ f1; 2; 3g ¼fx; y; zg) for all null four-momenta from the future light cone.For further details about the Poincaré group and how it acts,the reader can take a look at Appendix B. Unless explicitlymentioned, the manifold we will consider in this paper is thefour-momentum space and not the usual Minkowski space–time (one temporal and three spatial coordinates). The differ-ence is again sketched in Appendix B. For a certainly betterexposition, one might also consult extensive literature on thesubject of the representations of the Poincaré group, forexample, in [11–14].
A generic single-particle pure photonic wave packet can bewritten in the form
dμðpÞ ¼ 1=ð2ð2πÞ3Þd3p is the Lorentz invariant measure andf ðpÞ is an envelope (scalar) function, therefore staying invar-iant under the Lorentz transformation [12]. The helicity eigen-states span a two-dimensional complex Hilbert space, andthese degrees of freedom are perfectly suited for transmissionof quantum or classical information in free space. We keep thezeroth component of all four-momentum vectors equal to one(or equivalently that pipi ¼ 1 holds), so we restrict ourselvesto monochromatic wave packets. We did not make our con-siderations less general by restricting to monochromatic wavepackets. The measurement introduced later will simply be
color-blind, that is, it will act trivially on the color subspaceCp for every direction p.
Furthermore, we naturally assume that α, β are identical forall momentum directions, so we may drop the subscript. Thepreparation of the wave packet is assumed to be under ourcontrol. Moreover, it would be a strange way of encodingquantum information into the wave packet if the quantumstate differed for different momentum four-vectors alreadyat the preparation stage. However, the momentum directions(the envelope function f ðpÞ) continue to be distributed arbi-trarily. We thus rewrite Eq. (1) as
jψi ¼ αZ
dμðpÞf ðpÞjp;þi þ βZ
dμðpÞf ðpÞjp;−i: ð2Þ
Let us see what happens if we first just rotate the wavepacket:
UðRðϑ;ϕÞÞjψi ¼ αZ
dμðqÞf ðqÞ exp½iθp;q�jq;þi
þ βZ
dμðqÞf ðqÞ exp½−iθp;q�jq;−i; ð3Þ
where Rðϑ;ϕÞ determines the spatial rotation of the coordi-nate system (or of the wave packet if we adopt the active pointof view) in the direction given by ϑ and ϕ, and from the scalarcharacter of the envelope function, it is understood thatf ðqÞ≡ f ðpÞ. The transformation in Eq. (3) is indeed nothingelse than a Uð1Þ transformation and the parameter is theWigner angle from Eq. (B8). Also, from Eq. (B8) follows thatthe Wigner angle is explicitly dependent on p and so every ketunder the above integral acquires a different phase. Aspointed out by the authors of [1], when we trace over the non-interesting degrees of freedom (the momentum), the resultingtwo-dimensional helicity density matrix does not transformaccording to a representation of the SUð2Þ group. It doesnot actually have transformation properties at all. We canget an insight into why tracing over the momentum is ill de-fined. Take just two different directions p1, p2 in Eq. (2). Therotation of the wave packet given by Rðϑ;ϕÞ results in an un-equal phase change, and, thus, the coefficients originally equalto each other will generally become different. So by a suitablerotation, two vectors with the same components can be madeorthogonal and vice versa. Note, however, that even beforethe rotation the operation of tracing over the momentum isnot valid despite the condition of bringing up a reasonabledensity matrix.
B. Assembling of Invariant Photonic Wave PacketsWe will take a slightly different path toward a reasonablewave packet density matrix. While we stick to the helicitymeasurement studied by Peres and Terno [1], we changethe building block from which a wave packet is constructed.The choice of the Peres–Terno measurement scheme is wellmotivated. This helicity measurement is probably the simplestone possible. We recall the whole issue for readers unfamiliarwith the problem in Appendix C. The fixed projection mea-surement breaks the rotational symmetry. This is the reasonwhy we cannot hope for a construction of a covariant densitymatrix.
728 J. Opt. Soc. Am. B / Vol. 28, No. 4 / April 2011 Kamil Brádler
Nevertheless, we will be able to construct an invariant den-sity matrix. That is a matrix whose components will stay con-stant independently on the receivers’ reference frame. How dowe achieve that? Contrary to [1] we use helicity-momentumentangled two-mode states as the construction blocks. Welet the helicity-momentum entangled two-mode states propa-gate in the z direction in a fixed coordinate system. We firstfind the conditions for the entangled states under which theydo not acquire any phase in a different (spatially rotated) re-ference frame. Having these conditions imposed, we investi-gate the effect of the helicity measurement on the entangledstates and also see the effect of a boost in the z direction.Based on the results, we shall construct wave packets thatcan perfectly transfer quantum or classical information, evenin the relativistic regime.
C. Effect of Spatial RotationsFor the purpose of the helicity wave packet construction in-troduced later, we will now show that for the Bell statesEq. (7) with specifically correlated momenta, the effect of aspatial rotation is harmless.
Following Appendix B, let us introduce the helicity eigen-states of the standard momentum k:
jþik ¼ 100
!; j−ik ¼
010
!: ð4Þ
The axial component can be disregarded as a result of the ap-plication of the Coulomb gauge, which explicitly sets the com-ponent to zero. The helicity eigenstates can be transformedinto the linear polarization eigenstates corresponding to thestandard momentum k by means of the following unitaryoperation:
S ¼ 1ffiffiffi2
p 1 −i 01 i 00 0
ffiffiffi2
p!: ð5Þ
We will need this operation to transform the rotation operatorfrom the linear polarization basis Rðϑ;ϕÞ [1] to the helicitybasis ~Rðϑ;ϕÞ. Therefore, to determine the helicity vectorfor an arbitrary direction parametrized by two angles ϑ, ϕ, wetransform the helicity eigenstates as j�ip ¼ ~Rðϑ;ϕÞj�ik×SRðϑ;ϕÞS−1j�ik. Then
~Rðϑ;ϕÞ ¼ 12
ðcosϑþ 1Þ expð−iϕÞ ðcosϑ − 1Þ expð−iϕÞ ffiffiffi2
psinϑ expð−iϕÞ
ðcosϑ − 1Þ expðiϕÞ ðcosϑþ 1Þ expðiϕÞ ffiffiffi2
psinϑ expðiϕÞ
−ffiffiffi2
psinϑ −
ffiffiffi2
psinϑ 2 cosϕ
!; ð6Þ
and so throughout the article, we work strictly in the helicitybasis. Note that we get the helicity eigenstates in the linearpolarization basis by the action of S−1j�ik. Before proceeding,let us mention that there exist many conventions for thematrix S. One can, for example, often find
S0 ¼ 1ffiffiffi2
p −1 1 0−i −i 00 0
ffiffiffi2
p!:
The rotation matrix then looks different as well as other ob-jects we will meet later, but the physical consequences areequivalent. By working with Eq. (5), we employed the conven-tion of [11]. We write down the usual four Bell states:
jΦðp1; p2Þi� ¼ 1ffiffiffi2
p ðjp1;þijp2;þi � jp1;−ijp2;−iÞ; ð7aÞ
jΨðp1; p2Þi� ¼ 1ffiffiffi2
p ðjp1;þijp2;þi � jp1;−ijp2;−iÞ; ð7bÞ
where jpi;�i ¼ a†ðpi;�Þjvaci using the notation ofAppendix B (from now on, the Latin subscripts label the cor-responding four-vectors and not their components). There isno need to repeat that the states above are not wave packetsyet. Using Eqs. (5) and (6), we rewrite states (7a) and (7b).The helicity triads then read
jþip ≡ jp;þi ¼ 12
ðcosϑþ 1Þ expð−iϕÞðcosϑ − 1Þ expðiϕÞ
−ffiffiffi2
psinϑ
!; ð8Þ
j−ip ≡ jp;−i ¼ 12
ðcosϑ − 1Þ expð−iϕÞðcosϑþ 1Þ expðiϕÞ
−ffiffiffi2
psinϑ
!: ð9Þ
An arbitrary rotation can be decomposed into rotationsaround the z axis followed by another rotation around they axis. Following the prescription sketched in Remark 1 {orlooking into [15] or eventually [6], where the Wigner anglewas calculated in the representation of the cover of theLorentz group—the group SLð2;ℂÞ}, we find that the rotationRzðλÞ around the z axis through angle λ induces the Wignerphase angle θz;p ≡ θp ¼ 0 for all p as long as p ≠ k. Exactlythe measure zero possibility p ¼ k is excluded from the defi-nition of jΦðp1; p2Þi because that would imply p1 ¼ p2 ≡ k
and jΦðp1; p2Þi to be converted into jϕi ∝ ð½a†ðk;þÞ�2 þ½a†ðk;−Þ�2Þjvaci due to the indistinguishability of suchphotons. Hence the Bell states transform as
UðRzðλÞÞ:jΦðp1; p2Þi� → jΦðRzp1; Rzp2Þi�; ð10Þ
UðRzðλÞÞ:jΨðp1; p2Þi� → jΨðRzp1; Rzp2Þi�: ð11Þ
The rotation RyðϖÞ around the y axis results in a morecomplicated formula for the Wigner angle:
Kamil Brádler Vol. 28, No. 4 / April 2011 / J. Opt. Soc. Am. B 729
θp ¼ arctan
�sinϖ sinϕ
sinϖ cosϑ cosϕþ cosϖ sinϑ
�: ð12Þ
To tackle the momentum dependence, we will assume the mo-menta p1, p2 to be correlated such that the net effect of therotation will be the overall phase equal to zero for all p1. Thatimplies for Eq. (7a) to find such p2 that the correspondingWigner phases are conjugated (because the helicity is equalfor each product). Having pi ¼ ð1; sinϑi cosϕi; sinϑi sinϕi;
cosϑiÞ, this condition implies that we have to findsolutions of the equation
sinϖ sinϕ1
sinϖ cosϑ1 cosϕ1 þ cosϖ sinϑ1
¼ −sinϖ sinðϕ1 þ xÞ
sinϖ cosðϑ1 þ yÞ cosðϕ1 þ xÞ þ cosϖ sinðϑ1 þ yÞ ;
ð13Þwhere ϕ1 þ x ¼ ϕ2, ϑ1 þ y ¼ ϑ2. This equation deserves anexplanation. To find a solution means to find x, y independenton ϖ. In other words, we want to locate all fixed points. Notethat these points are not completely fixed because they varywith ϕ1, ϑ1.
Similarly, for the second couple [Eq. (7b)], we have to findp1, p2 such that the Wigner phases are equal (the helicity isopposite for each product):
To have solutions independent on ϖ and satisfying the aboveequation, the only possibility is when both sides are equal tozero. That implies
cosðϑ1 þ yÞ cosðϕ1 þ xÞsinðϕ1 þ xÞ ¼ −
cosϑ1 cosϕ1
sinϕ1; ð16Þ
sinðϑ1 þ yÞsinðϕ1 þ xÞ ¼ −
sinϑ1
sinϕ1: ð17Þ
It is easy to enumerate all possibilities for which, for example,the first equation is satisfied and then to check if it also holdsfor the second one. We end up with four different solutions(modulo 2π):
Having the solutions of Eq. (13), we immediately get the solu-tions for Eq. (14). The later equation is obtained from theformer one by a mere shift x↦x, y↦yþ π. Hence
In Figs. 1 and 2 we see the position of four fixed points forsome ϑ1, ϕ1. If we had in Fig. 2 the same parameters ϑ1,ϕ1 as in Fig. 1, the picture would indeed be only shifted alongthe y. We can check that all solutions satisfy the correspond-ing Eqs. (13) and (14).
The first solution Eq. (18a) is the most interesting one be-cause p2 dwells in the same hemisphere as p1. Commenting onthe rest of the solutions, Eq. (18b) is equivalent to Eq. (18a)and the remaining two solutions are a bit awkward because p2points in the opposite direction. We might, however, say that
UðRyðϖÞÞ : jΦðp1; p2Þi� → jΦðRyp1; Ryp2Þi�: ð20Þ
From the physical point of view, there is a noteworthy asym-metry in the solutions for Eq. (14). The first two fixed pointsagain correspond to a wave packet with the second momen-tum pointing in the opposite direction, and the last two solu-tions are just an identity operation (p2 ¼ p1). So there are twosolutions where
UðRyðϖÞÞ:jΨðp1; p2Þi� → jΨðRyp1; Ryp2Þi� ð21Þholds, but, for practical purposes, the opposite momentumdirection is an obvious obstacle.
Fig. 1. (Color online) Four fixed points corresponding to Eqs. (18a)–(18d) are indicated by points a, b, c, and d. Curves corresponding to adifferent rotation angle ϖ about the y axis will always meet in thesepoints. The fixed points satisfy Eq. (13). One can notice several gapson every curve. These are singularities of Eq. (13).
730 J. Opt. Soc. Am. B / Vol. 28, No. 4 / April 2011 Kamil Brádler
The conclusion is that as long as we keep the momenta inthe Bell states correlated as described above, it is not neces-sary to introduce three-dimensional helicity density matricesbecause there is no phase change due to the spatial rotation atall [see Equations (10), (11), (20), and (21)].
The way of getting rid of the Wigner phase for spatial rota-tions is reminiscent of the method presented in [16]. Theauthors constructed wave packets by entangling two spatiallydistinguishable wave packets carrying the same momentumand opposite helicities. Every spatial rotation induces an op-posite Wigner phase for each wave packet, so the resultingphase is zero. Note that this construction is not workingfor arbitrary wave packets. It works only for wave packetswhere the rotation induces the same Wigner phase for allmomenta from which the wave packet is “assembled.” Aswe already stressed, however, the Bell states we work withso far are not wave packets, so the construction presentedhere is qualitatively different.
D. Effect of Peres–Terno MeasurementThe projection present in the Peres–Terno measurement pro-cess Eq. (C2) acts by “cutting off” the z component of Eqs. (8)and (9). Henceforth, after the proper normalization, we get
Πk:jp;þi → ⌊p;þi ¼ 1N
� ðcosϑþ 1Þ expð−iϕÞðcosϑ − 1Þ expðiϕÞ
�; ð22Þ
Πk:jp;−i → ⌊p;−i ¼ 1N
� ðcosϑ − 1Þ expð−iϕÞðcosϑþ 1Þ expðiϕÞ
�; ð23Þ
where N ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2þ 2 cos2 ϑ
p. Note that hp;þjp;−i ¼ 0 for all p,
but hp;þ⌋⌊ p;−i ≠ 0 as expected. The “floored” kets indicatethe action of the projection Πk on a momentum-helicity state.
Interesting things start to happen when we ask how the pro-jection Πk acts on the Bell states. First of all, these states aretwo-mode states, so the projection is actually Πð1Þ
k ⊗ Πð2Þk ,
where the superscripts indicate which mode is beingmeasured. The action of the projection can be found in Appen-dix A. In general, these states are subnormalized and non-orthogonal and, thus, resemble a general situation studiedin [1]. But, intriguingly, when plugging in some of the solutionsfor the momentum correlations [namely, Eqs. (18a), (18c),(18d), (19a), and (19b)] found from the study of fixed pointsfor arbitrary rotations about the y axis, we find that (i) theresulting floored Bell states are orthogonal for all ϑ1, ϕ1
and (ii) one of the states is always (up to a normalization) in-variant. This has some interesting consequences, particularlyfor the solution of Eq. (18a). Recall that this is the only non-trivial solution where the resulting state “points” in the samedirection as the propagation direction. From now on, we willstick just to this solution because it will be later relevant forour discussion of localized wave packets (the analysis for therest of the nontrivial solutions gives similar results, but theiruse as localized wave packets is obviously none).
When the previous claim written in detail for ϕ2 ¼ −ϕ1,ϑ2 ¼ ϑ1 we get from Eqs. (A2) and (A3)
Πð1Þk ⊗ Πð2Þ
k :jΦðp1; p2Þiþ → ⌊Φðp1; p2Þiþ
¼
0BBBBB@
1
expð−2iϕ1Þ cos2ϑ1−1cos2ϑ1þ1
expð2iϕ1Þ cos2ϑ1−1cos2ϑ1þ1
1
1CCCCCA
→
normalizationa1⌊Φþi þ a2⌊Ψþi þ ia3⌊Ψ−i; ð24Þ
Πð1Þk ⊗ Πð2Þ
k :jΦðp1; p2Þi− → ⌊Φðp1; p2Þi−
¼ 1
1þ cos2ϑ1
0BBBBB@
2 cosϑ1
0
0
−2 cosϑ1
1CCCCCA →
normalization⌊Φ−i; ð25Þ
where ai ∈ ℂ and Σia2i ¼ 1. We denoted ⌊Φþi ¼
1=ffiffiffi2
p ð⌊01i⌊02i þ ⌊11i⌊12iÞ and similarly for the rest of thestates. Note that both output states are orthogonal for allϑ1 and ϕ1. Also, the resulting states are subnormalized eventhough states (22) and (23) are normalized. This is again theeffect of cutting off (alias projection) the “longitudinal” part ofEqs. (8) and (9). The normalization factor for Eq. (24) and (25)is
respectively. Recall that the measurement indeed acts as atrace-decreasing noncompletely positive map [17]. We alsowant to draw attention to the similarity between the behaviorof the investigated Bell states under the irrep of the product
Fig. 2. (Color online) Here we demonstrate how the position of fixedpoints changes for a different momentum direction. The fixed pointssatisfy Eq. (14), so in contrast to Fig. 1, there is a global (trivial) fixedpoint in the center that is constant for all ϑ1, ϕ1. One can notice sev-eral gaps on every curve. These are singularities of Eq. (14).
Kamil Brádler Vol. 28, No. 4 / April 2011 / J. Opt. Soc. Am. B 731
SUð2Þ × SUð2Þ and the projection Πð1Þk ⊗ Πð2Þ
k . In the firstcase, the singlet is preserved and the triplet states transformamong themselves. This is a starting point to considerationsabout reference frames and decoherence-free subspaces [18].In our case, one of the triplets is invariant; meanwhile, thesecond triplet mingles with the singlet and the remainingtriplet state.
An important comment also related to decoherence-free subspaces has to made. As we have seen, the Bell statesremain orthogonal after the projection:
hΦðp1; p2ÞþjΦðp1; p2Þ−i ¼ 0
→
Πð1Þk⊗Πð2Þ
k hΦðp1; p2Þþ ⌋⌊Φðp1; p2Þ−i ¼ 0:
However, the action of the projection is not SUð2Þ × SUð2Þcovariant. In other words, there is a preferred basis spannedby states from Eq. (7). One faces a slightly analogous situationencountered in the study of decoherence-free subspaces [19].However, the presented results still hold for any orthogonalbasis jpi;þi; jpi;−i forming the actual entangled states inEq. (7). The only difference is given by the fact that our under-lying Hilbert space is infinite dimensional.
E. Boost Along the z Axis Followed by MeasurementThe found result has several serious consequences. First, if weprepare two orthogonal pure qubits having the basis formedby the Bell pair jΦðp1; p2Þi�, they will be perfectly distinguish-able by the measurement. This was not the case for the ori-ginal Peres–Terno scheme. But more important, if we boostalong the z axis and then perform the same projective mea-surement, we see that state (24) remains invariant in the sensethat the boost only modifies ϑ1ð¼ ϑ2Þ and keeps ϕ1ð¼ −ϕ2Þintact. Of course, by the change of ϑ1, ϑ2 the four-momentap1, p2 change, too, as Eq. (B1) dictates. They symmetricallyclose in or open up depending on the observer’s velocitydirection. In a real experiment, this would be registered asthe color change (this is the Doppler effect that we formallysuppress by the renormalization of the new four-vectors).
Hence the effect of a boost in the z direction BzðηÞ followedby the projective measurement Πð1Þ
k ⊗ Πð2Þk is
Πð1Þk ⊗ Πð2Þ
k :jΦðBzp1; Bzp2Þiþ → ⌊ΦðBzp1; Bzp2Þiþ; ð26Þ
Πð1Þk ⊗ Πð2Þ
k :jΦðBzp1; Bzp2Þi− → ⌊ΦðBzp1; Bzp2Þi−: ð27Þ
Let us conclude this section with two comments. (i) As inthe previous case, all output states are orthogonal for allboosts and every p1. We have to stress, however, that theabove transformations are not unitary. We recall that the nor-malization of Eqs. (22)–(25) is explicitly ϑ dependent. It wouldbe a clear sign of inconsistency if we had a finite-dimensionalunitary representation for boost transformations—the boostgenerators are the noncompact part of the Lorentz algebra.Similar inconveniences were encountered in [1], but in ourcase it does not cause serious troubles. (ii) For a state propa-gating in the z direction, the most general Lorentz transforma-tion can be expressed as RzðλÞRyðϖÞBzðηÞ. Hence, we haveshown that the Bell states propagating in the z directionand with the appropriately correlated momenta are unaf-
fected by an arbitrary rotation (and, therefore, a generalboost) if measured according to the Peres–Terno scheme.
F. Relativistically Invariant Photonic Wave PacketsThe real touchstone for quality will be the behavior of wavepackets composed from the previously studied entangledstates under the Lorentz transformation. As indicated, we willdemonstrate the construction and properties of invariantwave packets with jΦðp1; p2Þi� serving as the logical basisfor construction of a (pure) wave packet qubit. We will nothave anything specific to say about the actual feasibility ofproducing of such states in a laboratory. Also, the dispersioneffects of other than relativistic origins are not consid-ered here.
Let us follow the prescription and notation for the single-mode case Eq. (1):
Similarly to the situation of ordinary wave packets we assumeα and β to be momentum independent [see the discussionbelow Eq. (2)].
Taking into account Eqs. (10) and (20) and how the boostacts, we easily find
j℧12i ¼ UðRzðλÞRyðϖÞBzðηÞÞjΩ12i
¼ αZ
dμðq1Þdμðq2Þf ðq1; q2ÞjΦðq1; q2Þiþ
þ βZ
dμðq1Þdμðq2Þf ðq1; q2ÞjΦðq1; q2Þi−; ð29Þ
where qi ¼ RzðλÞRyðϖÞBzðηÞpi. As discussed earlier in Eq. (3)the measure and the envelope function are Lorentz invariant[f ðq1; q2Þ≡ f ðp1; p2Þ]. Hence, we call the transformed wavepacket invariant because the state amplitudes remain unal-tered by an arbitrary Lorentz transformation. In other words,otherwise disastrous decoherence between helicity and mo-mentum degrees of freedom is kept under control by a carefulchoice of the logical basis.
As a final step, the measurement described by Πð1Þk ⊗ Πð2Þ
k
is realized, due to Eqs. (24) and (25), by the projections
Γ1 ¼ ⌊Φ−ihΦ−⌋; Γ2 ¼ 1 − Γ1: ð30Þ
To be more precise, we have seen that the map related to themeasurement operator Πð1Þ
k ⊗ Πð2Þk is trace decreasing. Con-
sequently, this measurement is a conditional (or postselected)measurement. The overall probability of measurement is low-er than one and depends on the details of the wave packet,namely on the choice of its envelope function. When the mea-surement event occurs, the observer can always perfectly dis-tinguish both orthogonal basis states because the twosubspaces from Eq. (30) are orthogonal and independenton the observer’s reference frame. Note that we tacitly butnaturally assume the existence of a global coordinate system.
732 J. Opt. Soc. Am. B / Vol. 28, No. 4 / April 2011 Kamil Brádler
3. CONCLUSIONSThere exists a class of photonic stateswe call the realisticwavepackets. It is a weighted and normalized superposition of com-mon eigenvectors of the momentum and helicity operators.The reason we call them realistic is twofold. From the mathe-matical point of view, if the corresponding envelope function issquare integrable (but otherwise almost arbitrary) function,such an object lives in a Hilbert space. From the physical pointof view, this object can, in principle, be prepared, transmitted,andmanipulated in a localizedmanner and finally detected in alaboratory. The problem appears when we consider wavepackets in a relativistic regime. The general Lorentz transfor-mation related to the change of a reference frame acts indivi-dually on every momentum and induces momentum-helicityentanglement. The situation is even worse when we try to re-cover the information by an act of measurement. The simplest(and probably the only feasible) way is the Peres–Ternoscheme of the helicity projection onto the plane perpendicularto the direction of propagation.
Surprisingly, in this paper we have shown that under theseunfavorable conditions, we are still able to prepare localizedwave packets for which the information can be, at least inprinciple, transmitted and recovered in every reference framewith perfect fidelity. There are two key ingredients. First, thelogical qubit basis are two-mode maximally entangled helicitystates. Second, the momentum degrees of freedom of thesestates are correlated in a precise manner. This results in can-celing the explicit momentum dependence for an arbitrary ro-tation and boost. Intriguingly, the same condition implies thatduring the measurement, according to the above scheme, theorthogonal logical basis states forming a qubit are projectedinto two orthogonal subspaces, enabling us to perfectly distin-guish them.
It has been known from the previous analysis, that the mapgoverning the evolution and measurement of generic wavepackets in the relativistic setting is trace decreasing and non-completely positive. It is a consequence of the noncovarianthelicity measurement scheme, and this leads to all aforemen-tioned effects and troubles. Here we have encountered pre-cisely the same behavior, but contrary to the previous work,these effects are now harmless. Consequently, using the inves-tigated encoding, the problem of noncovariant transformationof the von Neumann entropy of helicity density matricesdisappears.
Finally, let us stress that the choice of the Peres–Ternomeasurement (arguably spoiling all nice transformation prop-erties of the momentum-helicity eigenstates) is directly moti-vated by its relative feasibility in a laboratory. Should moresophisticated helicity measurements be available, we mightobtain more elegant results.
APPENDIX A—COMPUTATIONAL DETAILSOF PROJECTION ACTIONThe projection corresponding to the Peres–Terno measure-ment scheme acts as
Πð1Þk ⊗ Πð2Þ
k :jpi; σiijpj; σji → ⌊pi; σii⌊pj; σji; ðA1Þwhere the “floored” quantities indicate the projection action.Denoting
APPENDIX B—REVIEW OF THE UNITARYREPRESENTATION OF THE POINCARÉGROUPThe purpose of Appendices B and C is to recall the problem ofpolarization encoding for realistic photonic wave packets.Readers not familiar with this problem, full of intricate details,might find this short review useful.
The Poincaré algebra has two Casimir operators C1 ¼ PχPχ
and C2 ¼ W χWχ , where W χ ¼ −1=2ϵχραωPρJαω is the Pauli-
Lubanski vector, fJ23; J31; J12g≡ fJ1; J2; J3g are the total an-gular momentum generators, and fJ01; J02; J03g≡fK1; K2; K3g are the boost generators. We recall that ϵχραωis the Levi-Civita tensor and ϵ0123 ¼ −1. The correspondingPoincaré group induces translations xμ → Λμ
νxν þ aμ (inthe coordinate representation), where ΛðζÞ ∈ SOð3; 1Þ is aLorentz transformation ΛðζÞ ¼ expð−i=2ζσςJσςÞ and TðaÞ ¼expð−iaμPμÞ. It is well known that the Poincaré group is asemidirect product of the proper orthochronous Lorentzgroup and a group of translations SOð3; 1Þ⋉R4.
The group acts on set K , which is a linear vector spaceequipped with the Minkowski metric with the signaturef−þþþg. Hence, set K is a manifold rich in structure. Notethat in the case of Minkowski space–time, the action of thePoincaré group is uninteresting because it acts transitively.It is not so in the four-momentum space. The Lorentz group
Kamil Brádler Vol. 28, No. 4 / April 2011 / J. Opt. Soc. Am. B 733
acts in a similar manner for both spaces, but the action of thetranslation operator differs. One can see it easily (in an admit-tedly handwaving way) realizing that the Fourier transform ofa function f : xi↦xi − ai for some constant a is a simple phasetransformation in the three-momentum space.
The analysis of the four-momentum space K shows thatthere are six orbits. The whole classification from the physicalpoint of view can be found, for instance, in [11,12]. We areinterested in the orbit, which physically corresponds to nullvectors with a positive zero component of the momentumfour-vector. They correspond (after quantization) to thescarce but important class of free massless particles contain-ing photons and possibly gravitons. By the very definition oforbit, no Lorentz transformation can change the character of anull vector. In other words, photons always travel at the speedof light, no matter the reference frame a potential observerresides in. We are free to generate the “null orbit” from anynull momentum four-vector, but it is customary and advanta-geous to choose the simplest possible one—the standard four-momentum kμ ¼ ð1; 0; 0; 1Þ. Note that other choices arecompletely equivalent, as discussed in detail in [13]. Fromthe physical perspective, the standard direction correspondsto (eventually quantized) plane electromagnetic waves(photons) traveling at the normalized speed of light alongthe z axis. Nevertheless, the word “physical” should be usedcarefully. It is not, in principle, possible to generate such afield (classical or quantum). The reason is that the sharp mo-mentum value implies that the classical wave is “infinitely”spread in position, and one would need an infinite amountof energy to prepare such a field. In the quantum case, thecorresponding state does not even occupy a Hilbert spaceof square integrable functions and is completely delocalizedin Minkowski space–time. Classical or quantum fields createdin a laboratory are, in reality, wave packets with a finitespread in momentum and position.
Using another property of the algebra ½Pμ; Pν� ¼ 0 andC1 ¼ C2 ¼ 0, which holds in the massless case only, we de-note a “single-particle” state jp; σi as an eigenstate of Pμ,where pμ are eigenvalues of Pμ. Other possible degrees of free-dom related to Wμ rather than to C2 (note that ½Wμ; Pν� ¼ 0)are gathered in σ:
Pμjp; σi ¼ pμjp; σi: ðB1Þ
To be able to use quantummechanics in the relativistic con-text, as we habitually do in ordinary quantum mechanics, weneed to introduce a Hilbert space and a unitary representationin it, which respects the composition law for the Poincarégroup:
The appearance of the phase is due to the projective nature ofquantum states (rays instead of vectors). For simply con-nected groups, it is possible to get rid of the phase factor(ϒ ¼ 0). This is not the case of the Lorentz (or Poincaré)group, and it will have some impact at a later stage.
The Poincaré group is noncompact, and, therefore, there isno faithful finite-dimensional unitary representation. Thismight have been a huge obstacle for studying the evolutionof relativistic quantum states, but Wigner realized how to cir-
cumvent this problem. He found that the unitary action of theLorentz group is governed by the irreps of its stabilizer sub-groups [12]. Loosely speaking, the little group “induces” its ac-tion on the Lorentz group. The method is called the “method ofinduced representations,” and was later studied and general-ized by [20] and others. The Wigner method prescribes thatthe unitary action of a Lorentz transformation Λν
μpμ ¼ p0ν
reads
UðΛÞjp; σi ¼Xσ0Dσ;σ0 ðSkÞjp0; σ0i; ðB3Þ
where Dσ;σ0 ðSkÞ is the irrep of the stabilizer group Sk, keepingthe standard null direction k intact. The stabilizer group canbe written
Sk ¼ L−1ðΛpÞΛLðpÞ; ðB4Þ
where LðpÞ:k → p takes the standard four-vector to any othernull vector (i.e., to a vector on the same orbit). It turns outthat the stabilizer group for the standard null orbit is theEuclidean group ISOð2Þ [also known as Eð2Þ] transforminga two-dimensional (Euclidean) plane into itself. It is a threeparametric group generated by translations in two directionsand a rotation around the axis perpendicular to the plane. Thecomponents of the Pauli-Lubanski vector are the generators ofthe corresponding Lie algebra. The group is noncompact, so itmight seem that we did not improve our situation but we ac-tually did. To make a long story short [11], it appears that thenoncompact part of the stabilizer group is not physically re-levant to the evolution of the photon under the Lorentz group,so the only SOð2Þ subgroup of ISOð2Þ remains. The corre-sponding algebra generator is W0 ¼ W3 ≡ J3, and we find
J3jk; σi ¼ σjk; σi: ðB5Þ
We can see from Eq. (B5) that
UðRðϑ;ϕÞÞJ3U−1ðRðϑ;ϕÞÞjp; σi ¼ σjp; σi; ðB6Þ
so we define helicity as the projection of the angular momen-tum along the direction of motion, and σ is clearly a relativisticinvariant.
The SOð2Þ group element is, as usual, recovered by expo-nentiation RzðθÞ ¼ expð−iθJ3Þ, so for the purpose of Eq. (B3),we get
Dσ;σ0 ðRzðθÞÞ ¼ exp½−iσθ�δσσ0 ; ðB7Þ
and, so, for an arbitrary Lorentz group Λ:p → q
UðΛÞjp; σi ¼ exp½−iσθp;q�jq; σi: ðB8Þ
Remark 1. We will call θp;q the Wigner angle, and it can beexplicitly calculated for a given Lorentz transformationΛ:p →
q from Eq. (B4) by setting Rzðθp;qÞ ¼ Sk. Notice that the phasefactor exp½−iσθp;q� ∈ Uð1Þ is indeed a unitary representa-tion of Rzðθp;qÞ ∈ SOð2Þ. The two groups are known to beisomorphic.
Remark 2. The parameter σ can take just integer or half-integer values as the result of the inability to set the phase inEq. (B2) to zero [11]. Moreover, the momentum projection in
734 J. Opt. Soc. Am. B / Vol. 28, No. 4 / April 2011 Kamil Brádler
two opposite directions p and −p are related by the parityoperator, so for photons as massless vector bosons, we getσ ¼ �1. We see that it is necessary to enlarge the Poincarégroup by including the parity transformation [2]. Clearly, thereis no helicity equal to zero because there is no rest frame for aphoton.
Remark 3. A state jp; σi coincides with the action of crea-tion operators a†ðp; σÞjvaci to the vacuum state from the usualquantization procedure of a free massless vector field yieldinga single-particle state of a momentum p and helicity σ ¼ �1(alias circular polarization). It is straightforward to generatemultiparticle states by a repeated application of the creationoperators for different modes satisfying ½aðp0; σ0Þ; a†ðp; σÞ� ¼δðp0 − pÞδσ0σ and thus to create the familiar Fock space as adirect sum of the completely symmetric Hilbert space of nphotons F ¼ ⊕∞
n¼0Hsymn .
Also notice that one of the consequences of the Wigner pro-cedure is a clear indication that there are just two “spin” de-grees of freedom, and they are always perpendicular to thedirection of motion and to each other. We know this fact with-out mentioning the Coulomb gauge whatsoever. The Coulombgauge can achieve the same goal, but one has to pay the priceof not having a manifestly covariant gauge condition. That is,the gauge condition must be imposed for every referenceframe separately. On the other hand, the Coulomb gauge isimportant and useful in quantum field theory from a broaderpoint of view.
APPENDIX C—MEASUREMENT TROUBLESAND WAVE PACKET HELICITY DENSITYMATRICESThe authors of [1] proposed a very simple measurement thatcorresponds to what actually might happen in a laboratorywhen measuring the helicity (polarization) degrees of free-dom. Assume that the direction of propagation of the wavepacket is the standard direction k. It is customary to placea polarization analyzer perpendicular to the direction of mo-tion. Our helicity eigenstates jp;�i can be represented as twoorthogonal complex four-vectors ϵμp;�. For a free field, theCoulomb gauge piϵip;� ¼ 0 implies the axial component ofthe helicity vector to be zero ϵ0p;� ¼ 0 (this holds for all p be-cause just rotations are considered now). Quantum electro-dynamics indicates what happens next. We summon up thewell-known relation for the helicity eigenvectors [11]:
Πijp ¼
Xσ¼�
ϵip;σ�ϵjp;σ ¼ δij − pipj
pkpk; ðC1Þ
where the overbar indicates complex conjugation. Followingthe notation of Eqs. (8) and (9) and Appendix B, the helicitymeasurement for p ¼ k corresponds to the trivial projection:
Πk ¼Xσ¼�
jσihσjk ¼ 1 0 00 1 00 0 0
!;
jþik ¼ 100
!jþik ¼
010
!;
ðC2Þ
where special symbols have been assigned for the helicity ei-genstates because we work with them in the main body. The
problem is that the projection is of rank two. When a singlephoton arrives in a direction p ≠ k, it has a longitudinal com-ponent and, hence, it gets cut during the course of measure-ment. Then, to have a consistent definition of the helicitydensity matrix in a rotated reference frame, one has to recon-struct the longitudinal part because that is the place wheresome parts of the wave function “got lost.” This basicallymeans to measure the helicity in the planes perpendicularto the x and y axes in our fixed coordinate system to be ableto build a three-dimensional helicity density matrix. Finally,we get a two-dimensional effective density matrix ϱeff by cut-ting a 2 × 2 block from the whole three-dimensional matrix.The resulting state is positive semidefinite and subnormalized.It is the consequence of the lemma following the definition.
Definition 1. For a matrix of dimension n, the leading prin-cipal submatrices are all the upper-left submatrices of dimen-sion k ≤ n.
Lemma 2. Let ϱ be a density matrix written in terms of(not necessarily orthogonal) pure states jφii, such thatϱ ¼Pn
i¼1 pijφiihφij. Then every leading principal submatrixof ϱ is positive semidefinite.
Proof. Taking the k-dimensional leading principal subma-trix of jφii means to replace the last n − k entries of jφii byzeros. We will label the resulting vectors as ⌊φii. The corre-sponding matrices ⌊φiihφi⌋ are still positive semidefinite, andthe positivity stays preserved by taking their convex com-bination ~ϱ ¼Pk
i¼1 pi⌊φiihφi⌋. The resulting density matrixis subnormalized.
Corollary. The projection Πk does exactly the job of cut-ting out the principal two-dimensional submatrix from the3 × 3 helicity matrix (recall that the axial part is set to zerofor all p by the gauge condition). Therefore,
ϱeff ¼1N
ZdμðpÞjf ðpÞj2
×Πk½ðαpjþip þ βpj−ipÞð�αphþjp þ �βph−jp�Πk; ðC3Þ
where N is a normalization constant.Equation (C3) says nothing other than if we measure the
helicity perpendicular to the propagation direction [the z axisin this case, and so Πk is of the simple form Eq. (C2)], a sta-tistical mixture of all x, y helicity components for all p is gen-erated. This is exactly the result from [1]. Of course, we maydecide to measure along a general axis g ≠ k. In this case,Eq. (C3) is still valid if we take Πk → Πg and render thehelicity vectors in the new basis.
Remark 4. One can adopt the approach of [3] where theabove Peres–Terno measurement scheme was reformulatedin terms of measurement of the Stokes parameters of the in-coming wave packet. We will not go into detail, but we justpoint out the Lie algebraic aspect of this approach. In a similarmanner as above, the authors basically constructed three dif-ferent effective helicity matrices, which correspond to theabove-mentioned measurement of the helicity in three ortho-gonal spatial directions. From these they constructed a three-dimensional helicity density matrix, which, in general, can bewritten in terms of the suð3Þ Lie algebra generators λsuð3Þi [21]:
ϱ ¼ 13þX8i¼1
ssuð3Þi λsuð3Þi ; ðC4Þ
Kamil Brádler Vol. 28, No. 4 / April 2011 / J. Opt. Soc. Am. B 735
where ssuð3Þi are the actual measured Stokes parameters. We
now realize that the suð3Þ Lie algebra is composed of threemutually dependent suð2Þ Lie algebras, and they preciselyform the three effective helicity density matrices calculatedin [22] [and one of them is Eq. (C3)].
Remark 5.We should stress that in the above process, oneindeed gets a longitudinal part of the wave function but thereis no need to call it a nonphysical situation [1]. We simply get ahelicity component in the plane not parallel to the plane wherethe helicity is measured. In principle, this situation cannot beavoided unless one works with single-particle states or verynarrow photonic wave packets.
Also, the three-dimensional helicity density matrix concepthas only little to do with the bosonic nature of the photon. It isjust a coincidence that we measure along three perpendiculardirections corresponding to our usual spatial dimensions. Ifwe would have had a more sophisticated measurement de-vice, we could have measured the helicity in each and everyplane perpendicular to every p direction and map it to someother (nonphotonic) multilevel quantum system. We wouldavoid projecting it down to our usual three-dimensional spacewhere helicity measurement devices usually operate. Pushingit to the limit, this way we would have been able to recover thewhole density matrix because hp;�jp0;�i ¼ δðp − p0Þ holds inthe infinite-dimensional helicity-momentum Hilbert space.
So far we explained just how a wave packet transforms un-der an ordinary rotation. The remedy was to introduce a three-dimensional helicity density matrix or to eventually create anarrow wave packet where the “longitudinal” component isnegligible [1]. The effect of a boost on a wave packet is moredisastrous. Assume again that the wave packet [Eq. (1)] is pro-pagating along the z axis with the standard momentum k andan observer is also moving along the z axis with a constantvelocity v ¼ vz ∈ ð−1; 1Þ. Having η ¼ atanh vz the boostBz:p → q, where BzðηÞ ¼ expð−iηK3Þ does not induce anyphase change [15], so the boost transforms a general sin-gle-particle state as UðBzðηÞÞ:jp; σi → jBzp; σi. The effect ofthe boost on a four-momentum vector is twofold. (i) The mag-nitude of the three-vector and the zeroth component are nolonger equal to one. This is the Doppler effect that we sup-press by renormalization, as discussed below Eq. (1). (ii)The gauge condition is violated because it is not a Lorentz cov-ariant condition and must be imposed for every q as is usuallydone in quantum electrodynamics [11]. Thus Bz:ϵp;σ → ϵq;σ isfollowed by
Coulomb gauge :ϵq;σ → ϵq;σ − gq;σq; ðC5Þ
where gq;σ ¼ ϵ0q;σ=q0 [q0 ¼ 1 following from (i)]. The gaugecondition [Eq. (C5)] assures that the new helicity four-vectorsstay orthogonal in a plane orthogonal to q. This is a final blowto our effort to have nice transformation properties of thehelicity density matrix between two different frames ofreference.
Remark 6. Things get more technically complicated if weallow for a boost in a general direction. Then, a Wigner phaseappears that can be easily seen if we realize that such a boostcan be decomposed in a boost in the z direction followed by arotation in the required direction.
ACKNOWLEDGMENTSIt is a pleasure to thank Rocío Jáuregui and Prakash Panan-gaden for their comments on the draft. The work was sup-ported by a grant from the Office of Naval Research (ONR)(N000140811249).
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