Lorentz invariant entanglement distribution for the space-based quantum network
Tim Byrnes, Batyr Ilyas, Louis Tessler, Masahiro Takeoka, Segar Jambulingam, Jonathan P. Dowling
aa r X i v : . [ qu a n t - ph ] A p r Lorentz invariant entanglement distribution for the space-based quantum network
Tim Byrnes,
1, 2, 3, 4, 5
Batyr Ilyas, Louis Tessler, MasahiroTakeoka, Segar Jambulingam,
4, 8 and Jonathan P. Dowling State Key Laboratory of Precision Spectroscopy, School of Physical and Material Sciences,East China Normal University, Shanghai 200062, China NYU-ECNU Institute of Physics at NYU Shanghai,3663 Zhongshan Road North, Shanghai 200062, China National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan New York University Shanghai, 1555 Century Ave, Pudong, Shanghai 200122, China Department of Physics, New York University, New York, NY 10003, USA Department of Physics, Nazarbayev University, 53 Kabanbay Batyr Ave., Astana 010000 Kazakhstan National Institute of Information and Communications Technology, Koganei, Tokyo 184-8795, Japan Department of Physics, Ramakrishna Mission Vivekananda College, Mylapore, Chennai 600004, India Hearne Institute for Theoretical Physics, Department of Physics & Astronomy,Louisiana State University, Baton Rouge, Louisiana 70803-4001, USA (Dated: October 9, 2018)In recent years there has been a great deal of focus on a globe-spanning quantum network,including linked satellites for applications ranging from quantum key distribution to distributedsensors and clocks. In many of these schemes, relativistic transformations may have deleteriouseffects on the purity of the distributed entangled pairs. This becomes particularly important for theapplication of distributed clocks. In this paper, we have developed a Lorentz invariant entanglementdistribution protocol that completely removes the effects due to the relative motions of the satellites.
Introduction
One of the main roadblocks to thewidespread utilization of quantum communication suchas quantum cryptography is the difficulty of producinglong-distance entanglement. Photons are a natural wayof generating such entanglement due to their excellentcoherence properties and the fact that they are “flyingqubits”. However optical fiber quantum communicationis limited to distances of approximately ∼
100 km dueto photon loss, which make them practical for only fora limited region and not a global scale. Broadly speak-ing, two approaches have been considered to overcomethis challenge – the use of quantum repeaters to cascadeentanglement generation for longer distances [1, 2], andfree-space schemes [3–5]. While most free-space schemesso far have been ground-to-ground communication, thereis now great activity in towards space-based schemes [6–19]. Quantum communication in space is attractive dueto the negligible effects of the atmosphere, which is theorigin of decoherence effects such as photon loss and de-phasing. The space-based protocol allows for the possi-bility of globe-scale quantum network where the photonscan be transmitted at distances of the order of the diam-eter of the Earth without the need of additional infras-tructure such as quantum repeaters.Arguably the most widespread example of space-basedquantum technology that is in use today is the GlobalPositioning System (GPS), which is based on quadrag-ulation from satellites loaded with atomic clocks trans-mitting their time and position. It is well known thatrelativistic effects due to both special and general rela-tivity must be accounted for an accurate determinationof the position, as time dilation and the gravitationalred shift affect the clock rate due to the orbital motionof the satellites. In addition it is known that relativis- tic effects have an influence upon entanglement [20–22].For instance in Ref. [20], it was shown that entangle-ment may change when viewed from different frames forpolarization-encoded photon pairs, due to the polariza-tion not being a Lorentz invariant (LI) quantity. Sim-ilarly, entanglement encoded in terms of the frequencyof the photon are not immune to relativistic effects asenergy is not a LI quantity. This suggests that investi-gating methods of entanglement distribution that havefavorable properties in relation to relativity should be animportant consideration. One particularly important ap-plication where such effects should be important is clocksynchronization. Several schemes have been discussed toaccurately synchronize atomic clocks on satellites, basedon shared entanglement [23–25]. In view of atomic clockson satellites having a precision of to one part in 10 , andground-based optical atomic clocks reaching one part in10 and beyond, even small effects due to relativity be-come an important consideration. Other potential appli-cations in addition to cryptography and clock synchro-nization are quantum metrology, quantum distributedcomputing, quantum teleportation, quantum simulation,and super-dense coding [26–29]. To date, we are awareof no systematic study has been made to investigate suchrelativistic effects during space-based photonic entangle-ment distribution, and propose favorable ways of over-coming these issues.In this paper, we investigate various strategies forspace-based entanglement distribution using photons.We examine three popular alternatives for entanglementgeneration: (I) a polarization entangled photons; (II) sin-gle photon entangled state; and (III) dual rail entangledphotons. The advantages and disadvantages of each willbe investigated in the context of low Earth orbit (LEO) p qSource satellite Detectorsatellite BDetectorsatellite A vx yz FIG. 1. Entanglement distribution between three satellitesin LEO. The source satellite produces entangled photons asshown in the text. The detector satellites are moving withrespect to the source satellite and each other. The photonsheading to the two satellites may have different momenta p , q ,due to their different directions. We choose Alice’s satelliteto be moving in the z -direction without loss of generality. satellites producing and detecting the photons (see Fig.1). The photonic states (II) and (III) are particularlyinteresting as they are based on Fock states, which areLI quantities [30]. It is therefore natural to choose entan-gled states involving these degrees of freedom to developa truly Lorentz invariant (LI) entanglement distribution.Choosing such manifestly LI states bypasses the need forany correction that would need to be made for states suchas (I). We analyze the prospects for whether making sucha correction would be viable, and examine to what extentthe relativistic effects would be visible. Entangled states
Let us first introduce the three typesof entangled photon states that will be analyzed in thispaper for creating long-distance entanglement using pho-tons. The first is simply a polarization entangled photonpair, produced for example by parametric down conver-sion. The state is written | Ψ ( S )I i = 1 √ | p , h i A | q , h i B − | p , v i A | q , v i B ) , (1)where | p , σ i is a single photon state of four momentum p and polarization σ = h, v , and the S refers to the factthat the photons are in the reference frame of the sourcesatellite. We label the modes for Alice and Bob’s satel-lites with A and B respectively. The second type of en-tangled state is the single photon entangled state, whichcan be produced by a single photon source mounted onthe source satellite entering a 50:50 beamsplitter. Thestate is | Ψ ( S )II i = 1 √ | p , λ i A | i B − | i A | q , λ i B ) . (2)where λ = ± | i is the electro-magnetic vacuum. Finally, the third type of entangledstate is using a dual rail encoding, where Alice and Bobeach posses two distinct modes A , A B , B | Ψ ( S )III i = 1 √ | i A | p , λ i A | i B | q , λ i B − | p , λ i A | i A | q , λ i B | i B ) . (3)Each of these states will have a different behavior undera Lorentz transformation, and our task will be to identifywhich is the best for entanglement generation. Lorentz boost of a single photon
First, let us examinehow single photon states transform. For a photon of he-licity λ and momentum p in the Source frame, the statein Alice’s frame is U (Λ) | p , λ i = e − iλ Θ(Λ , p ) | Λ p , λ i (4)where Θ is the Wigner phase, and Λ is the Lorentz trans-formation to the frame of A . Since we assume that thephoton momentum is in an arbitrary direction, withoutloss of generality we may take the Lorentz transforma-tion to be a pure boost in the z direction Λ = L z ( β ).In this case L z ( β ) is the standard Lorentz transforma-tion matrix with dimensionless velocity β = v/c ( c is thespeed of light). Polarized vectors in the original frameare defined as [20] | p , h i = R ( ˆ p )(0 , cos φ, − sin φ, T | p , v i = R ( ˆ p )(0 , sin φ, cos φ, T | p , λ i = R ( ˆ p )(0 , , iλ, T / √ R ( ˆ p ) = R z ( φ ) R y ( θ ), with R y,z being the standard SO(3) rotation matrices, andˆ p = (sin θ cos φ, sin θ, sin φ, cos θ ) is the normalized 3-momentum. For a pure boost in the z direction, theeffect is to transform the coordinates assin θ → sin θ ′ = sin θ q sin θ + γ (cos θ − β ) φ → φ ′ = φ. (6)To a good approximation, for β ≪ θ ′ ≈ π (cid:18) θπ (cid:19) − π ln 2 β . (7)This effectively broadens or contracts the angular varia-tion around the z -axis. The angular variation is the ori-gin of the variation in entanglement that was observed inworks such as Ref. [20].It is known that polarization is not a LI quantity andhence the state will appear differently in Alice’s frame[20, 31]. To quantify the change we measure the tracedistance of the polarization vector ε = Tr( q ( ρ ( S ) − ρ ( A ) ) ) / ρ ( S ) = Tr p ( | p , σ ih p , σ | ) and ρ ( A ) =Tr p ( | Λ p , σ ih Λ p , σ | ) for this case. Here we trace a b θ θφ φ l og ε l og ε c N α=0α=π/2α=π/4 d log 1 F σ = . σ = . σ = . σ = l ogn FIG. 2. Performance of the entanglement distribution for var-ious protocols. Trace distance ε between the original state andthat observed in a moving frame for (a) a single horizontally(or vertically) polarized photon (b) a polarization entangledphoton pair moving in opposite directions θ = θ A = − θ B . Pa-rameters are β = 10 − . (c) Negativity of (10) under Lorentzboosts with different orientations. Photons are taken to movein opposite directions θ A = − θ B , φ A = φ B and the spread dueto the diffraction is σ = 1. (d) Number of entangled photonstates (10) with σ = 1 required to reach purities as marked.We assume a photon attenuation factor of A = 100, and thenumber of photons required for k purification steps to be 2 k . over the momentum degrees of freedom in order toobtain a 4 × p as observed by the source and Alice’ssatellite. For small velocities β ≪ ε h ≈ β sin θ cos φ, (9)which very accurately summarizes the numerical resultsin Fig. 2(a). For photons traveling along the y or z axisthere is no effect as horizontally polarized photons arealigned along the x -axis. We see that the basic effectof the relativistic correction on the polarization is at thelevel of ε h ∼ O ( β ). We note that the trace distance is themost appropriate quantity (than the fidelity for instancewhich scales as F ∼ − O ( β )), as it is most closelyrelated to distances on the Bloch sphere. For example,in interferometric measurements, the error in the phaseis proportional to the trace distance between the idealand the state with error [23]. Lorentz boost of entangled states
Let us now exam-ine the effect on the entangled states. For the type Ientangled state, in Alice’s frame we have | Ψ ( A )I i = 1 √ | Λ p , h i A | Λ q , h i B − | Λ p , v i A | Λ q , v i B ) . (10) The Wigner phase does not affect the state in this caseas the state is transformed only by a pure Lorentz boost.The sole effect in terms of the trace distance is the ro-tation of the polarization vectors, as given in (6). Thetrace distance between the states in the Source and Al-ice’s frames ρ ( S,A ) = Tr p , q ( | Ψ ( S,A )I ih Ψ ( S,A )I | ) is shown inFig. 2(b). For the case of photons moving in opposite di-rections, the trace distance can be summarized to a verygood approximation by ε I ≈ β sin θ. (11)We again see that the relativistic correction again occursat the level of ∼ O ( β ).For satellites in LEO typically β ≈ − , hence thisis significant effect in comparison with the precision ofatomic clocks. For example, in the clock synchronizationscheme of Ref. [23], if Alice and Bob measure in differentbases, this appears as an offset in the time between theirclocks [32]. One may argue that such systematic errorssuch as (9) or (11) can always be accounted for, and henceremoved. This is indeed true for GPS satellites whererelativistic effects such as time dilation are compensatedout. In this way the errors could potentially be reducedto a level below (9) or (11). Then the real error estimateis then determined by how well the relativistic correctionscan be corrected out, which for the case (9) is related tothe error on the velocity estimate δβ . This gives an errorof ε ∼ O ( δβ ) for (11). Since the precise velocities ofthe satellites are typically not known to extremely highprecision, the relativistic errors can be significant, evenif they are accounted for. For example, if the velocity ofthe satellite is known with relative error of ∼ − [33],thus amounts to an error ε ∼ − , which is still largein comparison to the precision of atomic clocks.In this regard, the type II and III entangled states area better choice. Fock states, including the vacuum, areknown to be invariant states under Lorentz transforms,and remain orthogonal in all reference frames. For thesingle photon entangled states, transforming to the ref-erence frame of satellite A , we find | Ψ ( A )II i = 1 √ e − iλ Θ(Λ , p ) | − Λ p , λ i A | i B − e − iλ Θ(Λ , q ) | i A | Λ q , λ i B ) . (12) | Ψ ( A )III i = e − iλ (Θ(Λ , p )+Θ(Λ , q )) ( | i A | p , λ i A | i B | q , λ i B − | p , λ i A | i A | q , λ i B | i B ) , (13)Similarly to type I, the photons are Lorentz transformedand there are separate Wigner phase terms due to thephoton traveling with different momenta. Helicity is aLorentz invariant quantity. The Wigner phase, whichdepends on the Lorentz transformation and the momen-tum does not show up in the measure we calculate. Inboth these cases, the entanglement is present in the pho-ton number, rather than polarization. We thus define thedensity matrices for these states according to ρ = Tr p , q ,λ ( | Ψ ih Ψ | ) . (14)The trace distance between ρ ( S ) and ρ ( A ) is always zero,hence it is a manifestly LI state. Diffraction effects
Up to this point, we have made oneidealization in that the effects of photon diffraction werenot included. In a more realistic situation, the photonswill have a spread due to diffraction and will have a su-perposition of different momenta p , q . All three types ofstates that were considered (10), (12), (13) in fact havethe same entanglement as a maximally entangled Bellstate in all frames. As discussed in Ref. [20], relativis-tic effects can affect the amount of entanglement as itchanges the diffractive spread of the photons. This typeof error is of relevance to our case as it is not a system-atic error that is correctable through local operations onAlice and Bob’s satellites.We estimate the magnitude of these corrections for thethree types of photonic entangled states. To take intoaccount of diffraction, we integrate with a momentumdistribution [20] | ˜Ψ i = Z ˜ d p ˜ d q f A ( p ) f B ( q ) | Ψ( p , q ) i (15)where the | Ψ( p , q ) i are the states (1), (2), (3) in thesource satellite’s frame. Here ˜ d p ≡ d p | p | is a Lorentz-invariant momentum integration measure and the f ( p )is a normalized diffraction function. For a specific modelof the photon spread, we follow the same form as thatgiven in Ref. [20] where only angular spread of photonswere considered, and the magnitude of the momentum isset to a constant: f ( p ) = 1 √ M e − θ σ δ ( | p | − p ) . (16)This gives a Gaussian spread for a photon traveling inprimarily the z -direction. σ is a parameter controllingthe angular spread of the beam and M is a suitable nor-malization factor. To have photons traveling in directionsother than the z -direction, we make rotation of the co-ordinates around the y -axis by changing variables in theintegrand θ → θ ′′ = cos − (cos α cos θ + sin α sin θ cos φ ) φ → φ ′′ = tan − (cid:18) sin θ sin φ cos α sin θ cos φ − sin α cos θ (cid:19) (17)which gives photons traveling in primarily the direction( θ, φ ) = ( α, z -direction to the states, whichamounts to making the transformation (6).Figure 2(c) shows the entanglement as a function ofthe satellite velocity for type I photons traveling in op-posite directions and various boost angles. In contrastto previous works [20], for boosts aligned to the photonpropagation ( α = 0), we find that the entanglement al-ways degrades regardless of direction. This is due to thedifferent geometry that we consider that is relevant forour case. For photons traveling in opposite directions, the Gaussian distribution tightens for one of the photonsbut broadens for the other photon according to (7), whichalways results in a degradation of the entanglement. Forboosts that are perpendicular to the photon propagation( α = π/ z -axis, re-sulting in an effective tightening of the distribution.We now estimate the order to which the relativisticcorrections affect the entanglement. To gauge this wecalculate the effect of the boost on the purity of the states P = Tr ρ . The purity is directly related to the entangle-ment in this case as for the case with no diffraction, theentanglement is invariant under all boosts. The degrada-tion in the entanglement observed in Fig. 2(c) arises froman effective decoherence entering the system due to trac-ing out the momentum degrees of freedom. Performingan expansion for β ≪ P ≈ − σ (1 + | β | ) . (18)As expected for no diffraction σ = 0, there are no rela-tivistic corrections. The relativistic corrections to lowestorder act to accentuate the diffraction effects which arealready present. In terms of physical parameters, thediffraction angle can be estimated as σ ≈ λ/d , where λ is the photon wavelength and d is the diameter of thetransmitter. For infrared photons, this gives σ ∼ − .We see that in this case the relativistic corrections arequite small as it is a secondary correction.Diffraction effects can be remedied using entanglementpurification methods. We demonstrate that it is possibleto achieve high purities by adapting the purification pro-cedure devised in Ref. [34] to our relativistic entangledphotons containing three components (a photon has spin-1). The procedure is similar to original protocol exceptthat due to the additional components one obtains a mul-tivariable recurrence relation instead of a single variablerecurrence relation [35]. In Fig. 2(d) we show the resultsof the entanglement purification on the state (14) using(15). We calculate the number of photons required as thenumber of photons required for a purification of a partic-ular target fidelity, multiplied by the photon attenuationfactor (the ratio of the number of photons sent to re-ceived), divided by the success probability of the purifica-tion. The photon attenuation is A = L λ /d S d A , whichfor parameters L = 13000 km, λ = 800 nm, d S = d A = 1m gives A ≈
100 photons being sent for each one received[36]. For the various diffractive spreads σ considered, wefind that an improvement in the fidelity is achievable aslong as the original diffractive spread is lower than σ . σ the purification fails and the fidelity de-creases. As typically the spread is σ ≪ A which is the same as theabove. Various methods exist to perform measurementsthat are in a superposition basis of the vacuum and a sin-gle photon [37–39]. For the dual rail type III states, thereis however the issue that the diffraction cone for the tworails will start to overlap unless they are separated by asufficiently large distance, which is impractical for satel-lite based sources and detectors. In this case, time-binentangled modes are a better alternative, with a Fransoninterferometer performing the interference between thetwo modes [40, 41]. Relativistic effects will time dilatethe time bins (in the same way a spatial separated dualrail will undergo Lorentz contraction) but these are ona much longer timescale, hence should not impact theperformance the detection scheme. Conclusions
In summary, we have analyzed severalphoton-based entanglement distribution protocols for thespace-based quantum network. We find that standardpolarization-based photon entanglement (type I) can ex-perience significant errors for satellites that are in LEO.While in principle these are correctable if the velocitiesof the satellites are known to high precision, this can stillintroduce errors at the δβ , which is the error on the es-timate of the satellite velocity.We note that other typesof encodings, such as in energy or time, would also un-dergo Lorentz transformations. Combined with the factthat diffraction effects degrade the entanglement for typeI states, our results point to the fact that single photonentangled states (type II) and dual rail photon entangle- ment (type III) are a superior choice in terms of robust-ness to relativistic transformations.One of the primary applications of space-based en-tanglement is clock synchronization, which is currentlyperformed using classical signals, which requires preciseknowledge of the position of the satellites. Entanglement-based methods can potentially eliminate this require-ment, but as have shown in this paper, to properlytake advantage of this manifestly Lorentz invariant statesshould be used. Encoding in the Fock state basis, usingeither a single photon or dual rail encoding overcomesthis issue. In addition, the entanglement can be used forseveral important tasks such as quantum cryptographywhich can be used without further components such as aquantum memory. For applications that require a quan-tum memory to further manipulate the entanglement, itis likely necessary to have in addition a LI entanglementtransfer and storage, if one requires a high fidelity pro-tocol.T. B. would like to acknowledge support from theShanghai Research Challenge Fund, New York Uni-versity Global Seed Grants for Collaborative Research,NYU-ECNU Institute of Physics at NYU Shanghai, Na-tional Natural Science Foundation of China (Grant No.61571301), the Thousand Talents Program for Distin-guished Young Scholars (Grant No. D1210036A), andthe NSFC Research Fund for International Young Sci-entists (Grant No. 11650110425). J. P. D. would liketo acknowledge support from the US Air Force Office ofScientific Research, the Army Research Office, the Na-tional Science Foundation, and the Northrop-GrummanCorporation. [1] H.-J. Briegel, W. D¨ur, J. I. Cirac, and P. Zoller,Phys. Rev. Lett. , 5932 (1998).[2] N. Sangouard, C. Simon, H. de Riedmatten, andN. Gisin, Rev. Mod. Phys. , 33 (2011).[3] R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach,H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner,T. Jennewein, J. Perdigues, P. Trojek, et al. , Naturephysics , 481 (2007).[4] X.-S. Ma, T. Herbst, T. Scheidl, D. Wang,S. Kropatschek, W. Naylor, B. Wittmann, A. Mech,J. Kofler, E. Anisimova, et al. , Nature , 269 (2012).[5] J. Yin, J.-G. Ren, H. Lu, Y. Cao, H.-L. Yong, Y.-P. Wu,C. Liu, S.-K. Liao, F. Zhou, Y. Jiang, et al. , Nature ,185 (2012).[6] J. Rarity, P. Tapster, P. Gorman, and P. Knight, NewJournal of Physics , 82 (2002).[7] R. Kaltenbaek, M. Aspelmeyer, T. Jennewein,C. Brukner, A. Zeilinger, M. Pfennigbauer, andW. R. Leeb, in Optical Science and Technology, SPIE’s48th Annual Meeting (International Society for Opticsand Photonics, 2004) pp. 252–268.[8] J. M. P. Armengol, B. Furch, C. J. de Matos, O. Min-ster, L. Cacciapuoti, M. Pfennigbauer, M. Aspelmeyer,T. Jennewein, R. Ursin, T. Schmitt-Manderbach, et al. , Acta Astronautica , 165 (2008).[9] P. Villoresi, T. Jennewein, F. Tamburini, M. Aspelmeyer,C. Bonato, R. Ursin, C. Pernechele, V. Luceri, G. Bianco,A. Zeilinger, et al. , New Journal of Physics , 033038(2008).[10] H. Xin, Science , 904 (2011).[11] D. Rideout, T. Jennewein, G. Amelino-Camelia,T. F. Demarie, B. L. Higgins, A. Kempf, A. Kent,R. Laflamme, X. Ma, R. B. Mann, et al. , Classical andQuantum Gravity , 224011 (2012).[12] J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, et al. ,Nature Photonics , 387 (2013).[13] J. Yin, Y. Cao, S.-B. Liu, G.-S. Pan, J.-H. Wang,T. Yang, Z.-P. Zhang, F.-M. Yang, Y.-A. Chen, C.-Z.Peng, et al. , Optics express , 20032 (2013).[14] T. Jennewein, J. Bourgoin, B. Higgins, C. Holloway,E. Meyer-Scott, C. Erven, B. Heim, Z. Yan, H. H¨ubel,G. Weihs, et al. , in SPIE OPTO (International Societyfor Optics and Photonics, 2014) pp. 89970A–89970A.[15] G. Vallone, D. Bacco, D. Dequal, S. Gaiarin, V. Luceri,G. Bianco, and P. Villoresi, Physical review letters ,040502 (2015).[16] Z. Tang, R. Chandrasekara, Y. C. Tan, C. Cheng, L. Sha,
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