Relativistically covariant state-dependent cloning of photons
aa r X i v : . [ qu a n t - ph ] F e b Relativistically covariant state-dependent cloning of photons
K. Br´adler
1, 2 and R. J´auregui Instituto de F´ısica, Apdo. Postal 20-364, M´exico 01000, M´exico School of Computer Science, McGill University, Montreal, Quebec, Canada (Dated: October 27, 2018)The influence of the relativistic covariance requirement on the optimality of the symmetric state-dependent 1 → / PACS numbers: 03.67.-a, 03.30.+p
I. INTRODUCTION
In recent years the influence of special and general relativity on quantum information processing has begun tobe investigated [1]. Staying just within the realm of special relativity, one of the natural questions is how Lorentztransformations affect the properties of both massive and massless particle states. The basic approach is throughWigner’s little group machinery [2] as the Poincar´e group is non-compact and thus without any finite-dimensionalunitary representations [3, 4]. In this context, the entanglement properties of bipartite states [5] composed of massiveas well as massless particles have been extensively analyzed. The fundamental impact on entanglement when theowners of both entangled subsystems are Lorentz transformed [6] has been recognized since, for physically plausiblestates, entanglement depends on the properties of the frame where it is measured [7, 8].In this paper, we go back to single-qubit transformations and discuss quantum cloning from the relativistic pointof view. We study how the requirement of Lorentz covariance affects the optimality of a cloning protocol. Quantumcloning has come a long way since the discovery of the no-cloning theorem [9] and one can find an extensive variety ofcloners in two recent review articles [10]. Lorentz covariance means that the particular cloning map must be equallyeffective irrespective of how any input state is rotated or boosted. More precisely, choosing the fidelity betweenan input and output state as a figure of merit to measure the quality of the clones, we demand that its value bemaximal and independent on the input qubit. The additional requirement of maximality provides an optimal cloner.As a striking example of how the relativistic covariance constraint modifies the optimality of the fidelity results,we investigate the state dependent 1 → II. WIGNER PHASE AND PHOTONIC WAVE PACKETS
As usual, a standard momentum light-like 4-vector k µ with k > k µ k µ = 0 is chosen. We can transform thisvector into an arbitrary light-like 4-vector p ν = L ( p ) νµ k µ by a standard Lorentz transformation. The most generallittle group element (stabilizer subgroup which leaves k ν invariant) is W (Λ , p ) = L − p Λ L p [3] and consists of rotationsand/or translations in a plane, a group which is isomorphic to the Euclidean group ( ISO (2)).The corresponding Hilbert space is spanned by vectors with two indices since, together with the angular momentumˆ J , the translation operator of the Poincar´e group ˆ P yields the complete set of commuting operators. In any givenreference frame, a rotation around the direction defined by the standard vector induces a phase on the correspondingstate in Hilbert space: e − iγ ˆ J | k ; σ i = e − iσγ | k ; σ i , (1)where for simplicity we took k µ = ω (1 , , ,
1) and σ is the component of the angular momentum in the directionof k µ (helicity). As is well known [3], massless particles have only integer or semi-integer σ values and for photons σ = ± | k ; σ i by applying a standard Lorentz transformation. Thus, if p µ is obtained from k µ by a rotation with longitudinal angle θ and azimuthal angle ϕ , then | p ; σ i = e − iϕ ˆ J e − iθ ˆ J e iϕ ˆ J | k ; σ i (2)Transforming this state vector by an arbitrary Lorentz transformation Λ we arrive at the unitary representation, whichturns out to be one-dimensional D ( W (Λ , p )) = exp( iσϑ W (Λ , p )). Here ϑ W (Λ , p ) is an angle of the rotation dependenton the Lorentz transformation Λ and the initial 4-vector p (the explicit form of ϑ W can be found in [15, 16]). Then,we have U (Λ) | p, σ i = exp( iσϑ W ) | Λ p, σ i . (3)We suppose that a wave packet is prepared in a state | Ψ f i = Z X σ = ± dµ ( p ) f σ ( p ) a † p,σ | vac i , (4)where dµ ( p ) is a Lorentz-invariant measure and f σ ( p ) is a normalized weight function R P σ dµ ( p ) | f σ ( p ) | = 1 thatdescribes the superposition of modes with different frequencies but a common direction of propagation p . Thisselection is made to avoid problems coming from diffraction effects occurring for a general wave packet so that onecannot simply define a polarization matrix [29].Let us examine the action of an arbitrary Lorentz transformation on | Ψ f i . We know [16] that the phase angledoes not depend on the magnitude of p but just on its direction. So making the transformation Λ | Ψ f i the phaseexp( iσϑ W (Λ , p )) is common for the whole wave packet. Considering the choice of our wave packet and also thediscussion in [17], after the Lorentz transformation and tracing over the momenta degree of freedom, we get | ΛΨ f i = Z dµ ( p ) X σ = ± e iσϑ W f σ (Λ p ) | Λ p, σ i Tr p → ̺ = (cid:18) | α | αβ ∗ e iϑ W α ∗ βe − iϑ W | β | (cid:19) , (5)where | α | = R dµ ( p ) | f (Λ p ) | , | β | = R dµ ( p ) | f − (Λ p ) | and αβ ∗ = R dµ ( p ) f (Λ p ) f ∗− (Λ p ). The helicity basis is thelogical basis {| i , | i} for our qubits (we thus do not use the Lorentz invariant logical basis composed of two physicalphotons proposed in [18] - the task is to clone an unknown single-photon state). III. RELATIVISTICALLY COVARIANT CLONING
For the rest of the article, we assume the following spacetime arrangement. In her reference frame, Alice preparesa state which travels in the p -direction. Although this direction is well-defined by the outgoing state, for a subject inanother inertial reference frame who receives the state (Bob) it is not sufficient information. The reason is that thereis the whole group of transformations (rotations around the p -direction) which leaves the given light-like vector intact.This is exactly the ’rotational’ part of Wigner’s little group responsible for inducing the Wigner phase ϑ W in (3) andboth angles (rotation and Wigner phase) coincide [16]. We consider Bob’s rotation to be completely unknown anduniformly distributed.Let us proceed to analyze how the state-dependent cloning setup investigated by Bruß et al. [11] is affected ifrelativistic covariance is incorporated. First, let us remember the original problem and later we formulate howrelativistic covariance enters the game. From now on σ X , σ Y and σ Z denote the Pauli X, Y and Z matrices.The original problem solved in [11] is, in some sense, an opposite extreme compared to the universal cloner [26],where all possible pure states are distributed according to the Haar measure. Here, for a fixed ξ , one of just two realstates {| ψ i = cos( ξ/ | i + sin( ξ/ | i , σ X | ψ i} , ξ ∈ (0 , π/
2) is prepared at random. The authors solved the problemassuming several reasonable invariance constraints. The output states were symmetric with respect to the bit flip σ X and were also permutationally invariant. For later comparison with our results, the fidelity function obtained inRef. [11] is shown in Fig. 1. The angles of the qubit parametrization are re-scaled to conform the parametrizationused here.Now we evaluate (and later generalize) the same setup when the relativistic covariance condition is imposed. Eq. (5)describes the effect of Lorentz transformation on the photonic states that are considered here. On the Bloch sphere,this transformation becomes P ϑ W = exp ( iϑ W / − σ Z )). Since our requirement is that the actual angle of therotation is unknown and uniformly distributed so are the states on the Bloch sphere. Hence, in addition to thesymmetries described in the previous paragraph, we require the invariance of the output with respect to the operator P ϑ W . The invariance reflects the ignorance of the rotation angle that induces the phase angle ϑ W in Eq. (5).Let us pause here and describe the physical situation. We suppose that Alice’s covariant operation is σ X (plussome additional operations which we won’t mention again) which is combined with another covariant operation P ϑ W induced by the Wigner phase ϑ W (that is, Alice sends one of two possible states which could be transformed by Bob’srotation) so we need to compare the action of P ϑ W and P ϑ W σ X . The order is important because the operators do notcommute. It can be easily shown that for single qubits, P ϑ W σ X = σ X P − ϑ W (6)what will prove to be very useful for later calculations.If we want to go beyond the setup studied in [11] and suppose that Alice may prepare a general pure qubit inthe form | ψ i gen = cos( ξ/ | i + e iφ sin( ξ/ | i , ξ ∈ (0 , π/ , φ ∈ (0 , π ) we find that P ϑ W and P ϑ W σ X (our covariantoperations) have a curious behavior since when they are applied to | ψ i gen these transformations appear in generalas two asymmetric oriented arcs on opposite hemispheres (parametrized by ϑ W ). To get a symmetric relativistictransformation we have to assume a different covariant operation, namely Ad( P ϑ W ) Γ Ad( σ X ) where Ad( U )[ ̺ ] = U ̺U − is the conjugation operation so we are in an adjoint representation of a group whose members are U [20]. Γ is the transposition of the density matrix in the standard (logical) basis ̺ Γ → ̺ T (because of this transformationwe traveled into the adjoint representation). The reason for incorporating Γ becomes evident when we compare theaction of Ad( P ϑ W ) and Ad( P ϑ W ) Γ Ad( σ X ) (our new covariant couple) on | ψ i gen . In this case, we will make use ofthe following identity (see proof in Appendix)Ad( P ϑ W ) Γ Ad( σ X ) = Γ Ad( σ X P ϑ W ) . (7)Note that [ Γ, Ad( σ X )] = 0. The motivation for introducing the identity is purely computational (just as for com-mutator (6)) but the physical interpretation is interesting. Since [Ad( P ϑ W ) , Γ Ad( σ X )] = 0 holds the order of thecovariance operations does not matter. Thus, in the next we will investigate both relativistic covariance effects, i.e.when covariance is required with respect to Ad( P ϑ W σ X ) for real states {| ψ i} , and Ad( P ϑ W ) Γ Ad( σ X ) for generalstates {| ψ i gen } . Except where really necessary, we will omit the symbol Ad() for the conjugation operation to avoidthe excessive notation but we have to remember that we keep working in the adjoint representation.Let us rephrase the invariance requirements from the previous paragraph in an appropriate formalism. TheJamio lkowski isomorphism [21] between positive operators and CP maps [22] is a traditional tool for the calcula-tion of optimal and group covariant completely positive (CP) maps. One appreciates the representation even moreby realizing that an implementation of the mentioned transposition operation is particularly easy. Let M be a CPmap, then the corresponding positive operator R M is related by M ( N ) ( ̺ in ) = Tr in h(cid:0) ⊗ Γ ◦ N [ ̺ in ] (cid:1) R ( N ) M i , (8)with N = 1 , Γ ◦ ≡ Γ, Γ ◦ = Γ ◦ Γ =
1. The expression Γ ◦ [ ̺ in ] ≡ ̺ Tin stands for the transposition of the density matrix ̺ in .It is important to stress that the case N = 2 must not be in a contradiction with the definition of the isomorphism( N = 1). Consequently, the net effect is that we require R (2) M to be invariant with respect to the transposition of ̺ in .If M is a cloning CP map then using Eq. (6) for N = 1 and Eq. (7) for N = 2 we can first start by requiringcovariance with respect to P ∓ ϑ W . Then the covariance conditions in both representations (standard and Jamio lkowski,respectively) read M (1) ( ̺ ) = ( P − ϑ W ⊗ P − ϑ W ) † M (cid:16) P − ϑ W ̺P †− ϑ W (cid:17) ( P − ϑ W ⊗ P − ϑ W ) ⇋ [ R (1) M , P − ϑ W ⊗ P − ϑ W ⊗ P ∗− ϑ W ] = 0 (9a) M (2) ( ̺ ) = ( P ϑ W ⊗ P ϑ W ) † M (cid:0) P ∗ ϑ W ̺ T P Tϑ W (cid:1) ( P ϑ W ⊗ P ϑ W ) ⇋ [ R (2) M , P ϑ W ⊗ P ϑ W ⊗ P ϑ W ] = 0 . (9b)Note that P ∗ ϑ W = P † ϑ W = P − ϑ W . Let us explain the use of the Jamio lkowski isomorphism. For N = 1, utilizing Eq. (6)we apply the covariance condition coming from the structure of the phase operator P − ϑ W . We get the basic structureof the Jamio lkowski operator R (1) M and we apply the bit-flip and the output state symmetry covariance conditions.Similarly for N = 2, the strategy is to summon the rhs of Eq. (7) to find how the covariance condition coming from P ϑ W defines the basic structure of R (2) M . Then, in addition to the previously mentioned covariant conditions, werequire the covariance regarding the transposition of an input state. This is the reason why on the lhs of Eq. (9b)there is Γ ◦ [ P ϑ W ̺P † ϑ W ] = P ∗ ϑ W ̺ T P Tϑ W . Finally, we calculate single-copy fidelities of the cloned state for both cases.The operator R (1) M is just a unitary modification of R (2) M as seen from Eqs. (9) so it is sufficient to analyze the structureof the case N = 2 and for N = 1 to subsequently modify the operator by σ Y ⊗ σ Y ⊗ R (2) M . It is a sum of the isomorphismsbetween all equivalent irreducible representations, which in the case of P ϑ W ∈ U (1) are all one-dimensional and aredistinguished by the character values e inϑ W with n ∈ Z . More specifically, P ϑ W ⊗ P ϑ W ⊗ P ϑ W is composed of fourirreducible representations. Two of them are one dimensional (spanned by {| i} , {| i} ) and two are three dimensional {| i , | i , | i} , {| i , | i , | i} where | m i is a decimal record of the 3-qubit basis.Taking into account the above discussed additional symmetries of R (1 , M the number of independent parametersgets limited and we arrive to the following form of R (2) M R (2) M = c ( | ih | + | ih | ) + c ( | ih | + | ih | ) + c ( | ih | + | ih | ) + c ( | ih | + | ih | )+ c ( | ih | + | ih | + | ih | + | ih | )+ c a [( | ih | + | ih | + | ih | + | ih | + h.c. ) + ic b ( | ih | + | ih | + | ih | + | ih | − h.c. )] , (10)where c ij ∈ C (for i = j ) are coefficients of the isomorphisms | i ih i | ↔ | j ih j | and c a = ℜ [ c ] , c b = ℑ [ c ]. Twoadditional conditions come from the trace-preserving constraint Tr out h R ( N ) M i = ⇒ c + c + c + c = 1 (commonfor both N ) and, of course, from the positivity condition R ( N ) M ≥ | ψ i = cos( ξ/ | i + sin( ξ/ | i (for N = 1) and | ψ i gen = cos( ξ/ | i + e iφ sin( ξ/ | i (for N = 2) and thetarget states of the same form F (1) = Tr h ( | ψ ih ψ | ⊗ ⊗ Γ [ | ψ ih ψ | ]) R (1) M i (11a) F (2) = Tr h ( | ψ ih ψ | gen ⊗ ⊗ | ψ ih ψ | gen ) R (2) M i . (11b)Observe that in Eq. (11b) the transposition operator was additionally applied. Case N = 1 (covariance w.r.t. P ϑ W and P ϑ W σ X ) If we apply the covariant operations on an arbitrary real | ψ i then for different values of the Wigner phase ϑ W we generate two symmetric trajectories on the opposite hemispheresof the Bloch sphere. Reformulating the search for the fidelity as a semidefinite program using the SeDuMi solver [24]in the YALMIP environment [25] the number of parameters is reduced and R (1) M can be diagonalized. This leads tothe full analytical derivation of the fidelity function (11a) as a function of the input state | ψ i F (1) = 12 " ξ ξ p ξ + cos ξ ! + sin ξ p ξ + cos ξ . (12)The function is depicted in Fig. 1 and we notice several interesting things. Obviously, the fidelity is lower thanthe original state dependent fidelity. We observe that the minimum moved from ξ B ru ß min = π/ ξ min = arccot p / F (1) min = 5 /
6. This value is ’reserved’ forthe 1 → SU (2) (or,equivalently, to the cloning of all mutually unbiased states of the Bloch sphere [10]). Such a low value for a kind ofphase-covariant cloner we are investigating may be surprising. For ξ = π/ F = + q . This is expected because the bit flip (one of our additional conditions) is unnecessary on the equator(due to the presence of P ϑ W ). Case N = 2 (covariance w.r.t. P ϑ W and P ϑ W Γ ◦ σ X ) Using methods similar to those in the previous paragraphwe arrive with the help of Eq. (11b) and | ψ i gen at the following form of the fidelity function F (2) = max (
14 (cos 2 ξ + 3) , " ξ − ξ p ξ + cos ξ ! + sin ξ p ξ + cos ξ , (13)which is independent on the input state phase φ . This result is no less interesting and the function is again depictedin Fig. 1. The minimum angle is common with the previous case but the corresponding fidelity drops to F (2) min = 2 / F ξ FIG. 1: Illustration of how the local fidelity of a state dependent 1 → F (1) min = 5 / ξ min = arccot p /
2. Furthermore,the transposition transformation corresponding to the finding of an orthogonal complement is considered and for the same ξ min the minimal fidelity (solid line) reaches F (2) min = 2 / This low value can be justified if we realize what kind of operation corresponds to N = 2. We combine two impossibleoperations, quantum cloning and finding the universal-NOT operation, into what is together known as the anti-cloningoperation [27]. This combined requirement is apparently stronger than the universal (i.e. SU (2)) covariance and thereason for the low fidelity values is that the map R (2) M must be of the same form for both ̺ and ̺ T as a result of Eq. (8).Could this result tell us something about, for instance, the security of quantum key distribution (QKD)? Lookingat the most studied protocol BB84 [28] (of course, implemented by the polarization encoding which is preferred for afree-space communication for which the relativistic effects may be very relevant) we see that four qubits equidistantlydistributed on the meridian are used. For N = 1, they form the xz − plane of the Bloch sphere (‘real meridian’)and for N = 2 it is an arbitrary grand circle intersecting the north and south pole (‘complex meridian’). From theviewpoint of an eavesdropper without the knowledge of the Wigner phase ϑ W and decided to clone the quantumstates to get some information, we can now demonstrate that not all quadruples are equally good. If the states { cos( π/ | i ± sin( π/ | i , cos( π/ | i ± sin( π/ | i} are used for the QKD purposes then by inserting ξ = π/ F (1) = (5 + √ / ≃ . F (2) = 3 /
4. On the other hand, using the quadruple {| i , | i , / √ | ± i} we get the fidelity F = 5 / P ϑ W ) we pass the mutually unbiased states of the Bloch sphere. We see that F (2) < F < F (1) corresponds to the factthat for N = 2 the eavesdropper has less information about the input state.Another interesting question is how the relativistic covariance affects the optimality of the universal cloner. Herethe situation is different. In the analysis above we combined two covariant operations ( Γ ◦ σ X and P ϑ W ) which are notgenerally subsets of each other. On the other hand, as we saw, every Wigner rotation is a U (1) covariant rotation andsince U (1) ⊂ SU (2) we may conclude that the optimality of the universal cloner will remain unchanged. Pictorially,it corresponds to the situation where Alice sends a completely unknown photon ( SU (2) covariance) to Bob who, inaddition, does not know how the whole Bloch sphere rigidly rotates (his rotation with respect to Alice). However,this is again a kind of SU (2) rotation. IV. CONCLUSIONS
In conclusion, we investigated the role played by the requirement of relativistic covariance in the problem of theoptimality of one of the most prominent forbidden quantum-mechanical process as quantum cloning. Observing thatthe effect of Wigner’s little group can be translated into the language of so-called phase-covariant processes we studiedhow the effectiveness of the cloning process becomes modified. Particularly, we considered an observer in a differentreference frame with no knowledge of the parameters of the reference frame where the state designated for cloning wasproduced. Here we focused on the class of state-dependent cloners where the effect is especially appreciable. First, asa direct application of the relativistic considerations on the cloning setup studied in [11] where one of two real statesis prepared in one inertial frame and cloned in another inertial frame whose transformation properties regarding thefirst one are completely unknown. Second, we went beyond this setup and supposed that in the first frame two generalpure qubits related by a common action of the Pauli X matrix and the density matrix transposition operator might beprepared. Again, we wanted such a state to be cloned in another inertial frame without knowledge of which state wasactually sent and how it was relativistically transformed. In both cases, we brought analytical expressions for localfidelities of the output states asking the fidelity to be maximal and optimal. One of the intriguing results is that in thesecond case the fidelity drops even below the universal cloner limit. The reason is that we combined the mentionedcloning invariance conditions with another forbidden process - finding the orthogonal complement of an unknownstate. Note that even without the relativistic context we generalized the previous research on the phase-covariantcloning maps and at the same time we studied optimal covariant processes considering covariance operations whichdo not commute.As an example of the consequences for communication security issues we have shown that for an eavesdropperdetermined to get some information by cloning a BB84 quadruple of states, not all possibilities are equally good andsome provide him with more information. Acknowledgments
The authors are grateful to Patrick Hayden for reading the manuscript.
APPENDIX A
In the following we use some of the basic properties of Lie groups [20].Let
A, B, C be invertible linear transformations. We define the conjugation operation c ( B )[ ̺ ] ≡ Ad( B )[ ̺ ] = B̺B − (and similarly c ( C )) satisfying [ A, c ( B )] = 0 , [ A, c ( C )] = 0 , [ c ( B ) , c ( C )] = 0. Then if A = c ( B ) we have A c ( CB ) − c ( BC ) A = 0. Proof.
Noting that
A c ( CB ) − c ( BC ) A = 0 A c ( CB ) A − A c ( BC ) A = 0 multiplied by A from left and right A c ( C ) D − D c ( C ) A = 0 D = A c ( B ) = c ( B ) AA − D c ( C ) A = c ( C ) D. (A1)Similarly, we get c ( B − ) D c ( CB ) = c ( C ) D . Equalling these two expressions we immediately see that Dc ( C ) = A c ( B − ) D c ( CB ) A − holding if A = c ( B ). (cid:3) Now we identify A = Γ, c ( B ) = σ X and c ( C ) = P ϑ W so we have Γ P ϑ W σ X [ ̺ ] = σ X P ϑ W Γ [ ̺ ] and to get Eq. (7) weapply Γ on the equation from the left and right using the fact that Γ ◦ [ ̺ ] = [1] A. Peres and D. R. Terno, Rev. Mod. Phys. 76, 93 (2004).[2] E. P. Wigner, Ann. Math. 40, 149 (1939).[3] S. Weinberg, The quantum theory of fields, vol. 1 (Cambridge University Press, Cambridge, U. K., 1995).[4] W-K. Tung, Group Theory in Physics (World Scientific, Singapore, 1985).[5] M. Czachor, Phys. Rev. A 55, 72 (1997).[6] P. A. Alsing and G. J. Milburn, Quantum Inform. Compu. 2, 487 (2002), H. Terashima and M. Ueda, Int. J. QuantumInf. 1, 93 (2003).[7] R. M. Gingrich and C. Adami, Phys. Rev. Lett. 89, 270402 (2002), C. Soo and C. C. Y. Lin, Int. J. Quantum Inf. 2, 183(2003), D. Ahn, H. J. Lee, S. W. Hwang, and M. S. Kim, arXiv:quant-ph/0304119.[8] R. M. Gingrich, A. J. Bergou, and C. Adami, Phys. Rev. A 68, 042102 (2003).[9] W. K. Wootters and W. H. Zurek, Nature 299, 802 (1982).[10] V. Scarani, S. Iblisdir, N. Gisin, and A. Ac´ın, Rev. Mod. Phys. 77, 1225 (2005), N. J. Cerf and J. Fiur´aˇsek,arXiv:quant-ph/0512172.[11] D. Bruß, D. P. DiVincenzo, A. Ekert, C. A. Fuchs, C. Macchiavello, and J. A. Smolin, Phys. Rev. A 57, 2368 (1998). [12] K. Jarett and T. M. Cover, IEEE Trans. Infor. Theory 27, 152 (1981).[13] C. Adami, G. L. Ver Steeg, arXiv:quant-ph/0601065, C. Adami, G. L. Ver Steeg, arXiv:gr-qc/0407090, P. Hayden and J.Preskill, JHEP09 120 (2007).[14] K. Br´adler, Phys. Rev. A 75, 022311 (2007).[15] N. Mukunda, P. K. Aravind and R. Simon, J. Phys. A: Math. Gen. 36, 2347 (2003).[16] P. Caban and J. Rembieli´nski, Phys. Rev. A 68, 042107 (2003).[17] A. Peres and D. R. Terno, J. Mod. Opt. 50, 1165 (2003).[18] S. D. Bartlett and D. R. Terno, Phys. Rev. A 71, 012302 (2005).[19] A. Aiello and J. P. Woerdman, Phys. Rev. A 70, 023808 (2003), N. H. Lindner and D. R. Terno, J. Mod. Opt. 52, 1177(2005).[20] W. Rossmann, Lie Groups: An Introduction Through Linear Groups. (Oxford University Press, Oxford, 2002).[21] A. Jamio lkowski, Rep. Math. Phys. 3, 275 (1972).[22] G. M. D’Ariano and P. Lo Presti, Phys. Rev. A 64, 042308 (2001).[23] D. Bruß, M. Cinchetti, G. M. D’Ariano, and C. Macchiavello, Phys. Rev. A 62, 012302 (2000).[24] http://sedumi.mcmaster.ca/[25] YALMIP: A Toolbox for Modeling and Optimization in MATLAB. J. L¨ofberg. Proceedings of the CACSD Conference,Taipei, Taiwan, 2004. http://control.ee.ethz.ch/˜joloef/yalmip.php[26] V. Buˇzek and M. Hillery, Phys. Rev. A 54, 1844 (1996), N. Gisin and S. Massar, Phys. Rev. Lett. 79, 2153 (1997).[27] N. Gisin and S. Popescu, Phys. Rev. Lett. 83, 432 (1999).[28] C. H. Bennett and Brassard, Proc. IEEE International Conference on Computers, Systems, and Signal Processing (Ban-galore, India, 1984) p. 175.[29] Considering a general wave packet with a spatial distribution of momenta, a general Lorentz transformation yields anintrinsic entanglement between momenta and polarization degrees of freedom [1]. The wider angular spread the packet hasthe more severe the influence of this entanglement is on the definition of the polarization matrix since such an object is noteven rotationally invariant [19]. The second consequence is that presumably orthogonal helicity states cannot be perfectlydistinguished. If in a given frame the angular and frequency spreads satisfy ∆ ang ≪ ∆ ωω