Quantum Scissors

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  Teleporting photonic qudits usingmultimode quantum scissors Sandeep K. Goyal 1 & Thomas Konrad 1,2 1 School of Chemistry and Physics, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa,  2 NationalInstitute of Theoretical Physics (NITheP), University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa. Teleportation playsanimportant rolein thecommunication of quantuminformation between thenodesof a quantum network and is viewed as an essential ingredient for long-distance Quantum Cryptography. Wedescribeamethodtoteleportthequantuminformationcarriedbyaphotoninasuperpositionofanumber d  oflightmodes(a‘‘qudit’’)bythehelpof  d  additionalphotonsbasedontranscription.Aquditencodedintoa single excitation of   d   light modes (in our case Laguerre-Gauss modes which carry orbital angularmomentum) is transcribed to  d   single-rail photonic qubits, which are spatially separated. Each single-railqubit consists of a superposition of vacuum and a single photon in each one of the modes. After successfulteleportationofeachofthe d  single-railqubitsbymeansof‘‘quantumscissors’’theyareconvertedbackintoa qudit carried by a single photon which completes the teleportation scheme. Q uantumteleportation, thecarrier-less transmission of quantum informationby transferring astate fromonequantumsystemtoaremoteonewasdescribedbyBennett etal. 1 andsoonafterplayedanimportantrole in photonic quantum computing  2–5 as well as secure communication by means of quantum key distribution (e.g. 6,7 ). The fragile nature of quantum systems and nearly omnipresent dissipative environmentsmakeitchallengingtorealize quantumteleportation experimentally. Bouwmeester  et al. 8 were thefirsttoachievequantumteleportationfollowedbymanyothersfordiscrete-levelquantumsystems 9–15 aswellaswithcontinuous variables 16–19 . In the case of discrete-level quantum systems so far only the state of the most simple quantumsystems, i.e., two-level systems, and therewith the smallest unit of quantum information (a ‘‘qubit’’) could beteleported. A new teleportation scheme, proposed recently  20 , is capable to transmit the quantum informationcarried by an elementary excitation of a superposition of an arbitrary number  d   of co-propagating light modes (aphotonic ‘‘qudit’’). The teleportation of photonic qudits increases the quantum information sent per carrierphoton.Currently,thelowtransmissionratesareoneofthebottlenecksofquantumcommunicationascomparedto its classical counterpart. However, the scheme proposed in 20 requires to prepare  d   additional photons in ahighlyentangledstate.Herewepresentanalternativeschemebasedonthetranscriptionofthequditencodedinasingle photon to  d   qubits carried by light modes which propagate along different optical paths. Each qubitcontains the quantum information about the excitation of a particular of the  d   srcinal light modes and isteleported individually by means of an additional photon using quantum scissors.Quantum scissors 15,21–24 is a device to teleport only the vacuum and the single-photon component of a single-mode state (a so-called single-rail qubit), while it truncates (‘‘cuts off’’) higher photon-number components. Inparticular, if an input light mode  c   (cp. Fig. (1)) is prepared in a superposition of states with different photonnumbers, quantum scissors projects its vacuum and single photon component to an ouput mode  b : x j i c  ~  a 0  0 j i c  z a 1  1 j i c  z a 2  2 j i c  z ...   ? x 0 j i b ~ 12  a 0  0 j i b z a 1  1 j i b   ,  ð 1 Þ where j n æ c  ( b ) representsaso-called Fockstate with n 5 0,1,2…photonsinlightmode c  ( b )andthecoefficients a n are the corresponding probability amplitudes. This process occurs upon conditioning on a single-photon detec-tionwiththeprobability given bythesquareofthenorm ofthefinalstate  j x 9 æ b , i.e,( j a 0 j 2 1 j a 1 j 2 )/4. However, thisprobability can be doubled by conditioning on one of two possible single-photon detections (cp. Methods). Theworking principle of quantum scissors is explained in the caption of Fig. (1).Ontheotherhand,iftheinputstateinmode c  consistsalreadyofasingle-railqubit,i.e. j x æ c  5 ( a 0  j 0 æ c  1 a 1  j 1 æ c  ),it is transferred according to transformation (1) without truncation, and hence teleported, to mode  b . A gen-eralization of quantum scissors which cutsoff allstate components with a number of  d  or morephotons andthusteleports multi-photon states of the form  x j i c  ~ X d  { 1 n ~ 0  a n  n j i c   (‘‘single-rail qudits’’) can be achieved using  OPEN SUBJECT AREAS: QUANTUM PHYSICSOPTICS AND PHOTONICS Received5 August 2013 Accepted19 November 2013Published19 December 2013 Correspondence andrequests for materialsshouldbeaddressedtoT.K. (konradt@ukzn.ac.za) SCIENTIFIC  REPORTS  | 3 : 3548 | DOI: 10.1038/srep03548  1  multiports and  d   2  1 additional input photons 25,26 . However, theencoding ofan arbitrarysuperposition  j x æ c  of multiple photon-num-ber Fock states is in practice difficult and requires non-linear opticalmedia leading to small efficiencies 27–29 .Moreover, one can teleport  n  single-rail qubits simultaneously,provided theyarestored inlight modeswhichpropagateondifferentpaths, by applying   n  quantum scissor setups in parallel, one for eachsingle-rail qubit. Obviously, the simultaneous teleportation works if the single-rail qubits in the individual modes are not correlated. Butnote,thatalsothe stateof  n entangled qubits canbeteleported inthisway. Results This feature of quantum scissors can be exploited to teleport a quditencoded into a single photon which is shared by   d   spatial modes of paraxial light. For this purpose the  d  -level state of the photon is transcribed   into  d   single-rail qubits carried by light modes propagat-ingalongdifferentpathswiththehelpofamodesorter.Suchadevicehas the task to transfer orthogonal light modes within a single lightbeam to different optical paths, similar to a polarizing beam splitter,which conveys light with horizontal and vertical polarization toorthogonal paths. For example, consider a single-photon state  j x æ given by an elementary excitation of a superposition of   d   paraxialLaguerre-Gauss modes  LG l  ,  p 5 0  corresponding to different values  l   h oforbitalangularmomentum(OAM) 30 whichco-propagatealonganoptical path  o , i.e, x j i o ~ X d  { 1 l  ~ 0 l   1 l  j i o  with X l l  j j 2 ~ 1 ,  ð 2 Þ where  j 1 l  æ  denotes the state of a single photon with OAM  l   h . Wespatially separate the OAM modes by diverting them into differentoptical paths  c  l   depending on their OAM value  l   h  with the help of anOAM mode sorter 31,32 . For  d  5 3 this transformation reads: x j i o ? x j i c  0  c  1  c  2 ~  0  1 c  0  0 c  1  0 c  2 j i z  1  0 c  0  1 c  1  0 c  2 j i z  2  0 c  0  0 c  1  1 c  2 j i : X 2 l  ~ 0 l  c  { l   0 j i , ð 3 Þ where 1 c  0  0 c  1  0 c  2 j i  represents a single photon with OAM quantumnumber l  5 0inpath c  0 andnophotoninallotherpaths(accordingly for the remaining terms). The single photon states are conveniently expressed by the creation operators  c  { l   acting on the global vacuumstate  j 0 æ , cp. the right-hand side of (3). This transformation transcribes  the state of a qudit into  d   entangled single-rail qubits.After the transcription the  i th qubit contains the quantum informa-tionaboutwhetherthecorrespondingOAMmode l  5 i ofthephoto-nic qudit was occupied 1 i j i o ? 1 c  i j i    or not 0 i j i o ? 0 c  i j i   .Now each single-rail qubit can be teleported individually (cp.Fig. (2)) using quantum scissors. This is accomplished as follows:each of the  d   spatial modes are inserted into  d   independent quantumscissorssetups(seeFig.(3)for d  5 2).Therearethus d  inputmodes c  i and  d   output modes  b i  with  i 5 0, 1 …  d  2 1 to carry the quantuminformation. In addition, the quantum scissors require a total of   d  singlephotonsenteredseparatelyinmodes a i .Uponconditioningonthe detection of a single photon in each of the quantum scissordevices (success probability 1/2 d  with ideal detectors, for non-unitdetection efficiencies see Methods) the state  j x æ  carried by the inputmodes  c  i  is transferred to the output modes  b i  (cp. Methods): x j i c  0 ... c  d  { 1 ? x j i b 0 ... b d  { 1 :  ð 4 Þ Since the mode  b i  in the  i th quantum scissors device srcinates fromthe reflection of mode  a i  both are identical except for their propaga-tion direction, and they should carry the same OAM value as inputmode  c  i , the state of which is supposed to be transferred to  b i . Hence,the photon entering in mode  a i  should be prepared with OAM value i  h .However, as shown under Methods, this is not necessary if themodes  b i  are transformed into the appropriate OAM mode after thestate transfer. In fact, preparing the additional photons in a differentsystem of basis modes enables a transcription of the quantuminformationstoredinaspecificbasis(hereOAMmodes)intheinputmodes of the quantum scissors devices to another basis (for exampleHermiteGaussianmodes 33 )initsoutput modes. Bysuch atranscrip-tion any unitary gate acting on the Hilbert space of the qudit can berealized, however only with limited success probability which isdetermined by the quantum scissors involved.After successful teleportation by the quantum scissors we canconverttheentangled d  single-railqubitsbacktotheoriginal d  -modeOAM state (2) with the help of a mixer which is a sorter run inreverse. This completes the teleportation of a photonic qudit (cp.Fig. (2)). In order to realize an additional unitary qudit-gate (seeabove) together with the teleportation, the mixer has to map thenew basis modes in the outputs of the  d   quantum scissors into asinglebeam,i.e.,itmustbeareversesorterfortheseparticularmodes,which exists for example for Hermite Gaussian modes 33 . Discussion In this article we have presented a scheme to teleport a photonicqudit carried by OAM modes. The scheme requires linear opticaldevices, OAM mode sorters as well as single-photon sources andphoton-number resolving detectors. The essential step is to tran-scribethestate ofthequditto d  single-rail qubitsbymeansofamode a a abbcc DD 12 BS BS 1 2 Figure 1  |  Schematic diagram for quantum scissors.  In the quantumscissors setup there are two 50 5 50 beam splitters (BS 1  and BS 2 ) and twonumber-resolving photon detectors ( D  1  and  D  2 ). An optical state  | x æ  isinsertedinmode(c)alongwithasinglephotoninmode(a)andvacuumin(b). Beam splitter BS 1  entangles mode (a) and (b) by distributing theincoming photon. A detection of a single photon in  D  1  or  D  2  leaves mode(b) in a superposition of vacuum (in case the detection annihilated thephoton in mode (a)) and a single photon state (in case the detected lightsrcinated not from mode (a) but (c)). The superposition state in (b) iscaused by beam splitter BS 2 , which deletes the path information about thesrcin of the detected light.    S  o  r   t  e  r    P   S   +   M   i  x  e  r QS l = d−1QS l = 1QS l = 0 Figure 2  |  Qudit teleportation setup using multiple quantum scissors. The d  -dimensionalstateofasinglephoton,whichisinasuperpositionof  d  OAM modes, is mapped by a mode sorter to  d   single-rail qubits. This isfollowed by the teleportation of the individual single-rail qubits using  d  quantum scissors. Eventually, a  p  phase shift ( PS  ) is applied to the single-rail qubits where necessary and their composite state is transcribed back into a photonic qubit by a mixer.  www.nature.com/ scientificreports SCIENTIFIC  REPORTS  | 3 : 3548 | DOI: 10.1038/srep03548  2  sorter and to teleport the qubits individually by quantum scissors. Inasfarassuchsorterdevicescanbedesignedforotherlightmodes,forexample Hermite-Gaussian modes 33 , the proposed teleportationschemeisuniversalandcanbeimplementedwithanysystemofbasismodes. Using quantum scissors a single-rail qubit can be teleportedwith success probability 1/2, therefore, the success probability toteleport  d   single-rail qubits and thus the encoded qudit amounts to1/2 d  .In principle, the probability to teleport a single-rail qubit can beincreased to  N  /( N   1  1) by employing   N   additional entangledphotons and a balanced multiport with  N  1 1 inputs and outputsas described by Knill et al. 4 instead of one additional photon and abalanced beam splitter in each of the quantum scissors setups. Thisleads to a success probability for the qudit teleportation of ( N  /( N  1 1)) d  but requires  d   highly entangled  N  -photon states, which can beprepared probabilistically offline 4,34 .On the other hand, the encoding of one qudit into  d   two-levelsystems (such as single-rail qubits) represents an inefficient use of storage capacity. The amount of quantum information present in asingle qudit actually corresponds to log  2  d   qubit units of quantuminformation and could thus be stored efficiently in log  2  d   single-railqubits.Givenaschemewhichisabletotranscribetheinitialphotonicqudit into log  2  d   single-rail qubits, a subsequent teleportation couldbe achieved by means of log  2  d   quantum scissors with a successprobability of 1  2 log  2 d  ~ 1 = d  . This would mean an exponentialdecrease of the resources needed to teleport a qudit.Alternatively, using the transcription based on a OAM mode sor-ter, as described above, the present scheme allows, instead of a singlequdit, to teleport  d   single-rail qubits encoded in co-propagating OAM modes, with the same resources as before. This correspondsto an exponential increase of quantum information sent per use of the teleportation protocol. However, the preparation and manipula-tion of single-rail qubits seems problematic compared to qudits car-ried by single-photon states of OAM modes, which can be prepared,transformed and measured with standard techniques 35 . For single-rail qubits, general deterministic single- and two-qubit gates are notavailable 36 . Moreover, the vacuum component makes state tomo-graphy of single-rail qubits difficult.The present scheme has certain advantages as well as disadvan-tages over a recently proposed alternative teleportation method 20 .Unlike the latter, it  does not   require highly sensitive multipartiteentangled states to perform the quantum teleportation. On the otherhand,thealternativemethodyieldsagreatersuccessprobabilityof1/ d  2 for qudit teleportation, and requires less additional photons.Remarkably, it yields for the joint teleportation of the state of many photonsthesamemaximalteleportationrateasquantumscissorsforsingle rail qubits, namely one qubit per additional photon. However,by improving the transcription efficiency one could overcome thesedrawbacks of the present scheme. Methods In the following we show that  d   quantum scissors enable a transfer of the state x j i c  0 ... c  d  { 1 obtained after the transformation (3) to output modes  b 0  …  b d  2 1  (4). Thestate  x j i c  0 ... c  d  { 1 along with  d   photons at ports  a i  constitute the initial state entering the d   quantum scissors (cp. Fig. (3) for  d  5 2)1 a 0  ... 1 a d  { 1 j i 6 x j i c  0 ... c  d  { 1 ~ P d  { 1 i ~ 0 a { i X d  { 1 l  ~ 0 l  c  { l   0 j i ,  ð 5 Þ where we have introduced the creation operators  a { i  to denote single photons in themodes  a i . Each quantum scissors device contains two 50 5 50 beam splitters, cp. Fig. 1.The first beam splitter BS 1  of the  i th device distributes the incoming photon in mode a i  equally over both modes,  a i  and  b i , represented by the transformation rule in termsof the corresponding creation operators  a { i ? 1  ffiffiffi 2 p   a { i z b { i   . Also the action of thesecondbeamsplitterBS 2 ineachquantumscissorsdeviceisconvenientlydescribedby similar rules: a { i ? 1  ffiffiffi 2 p   a { i z c  { i   ,  ð 6 Þ c  { i ? 1  ffiffiffi 2 p   a { i { c  { i   ,  ð 7 Þ Consecutive application of these transformations for beam splitters BS 1  and BS 2  forall quantum scissors  i  to the initial state (5) yields the total state change:1 a 0  ... 1 a d  { 1 j i 6 x j i c  0 ... c  d  { 1 ? W j i ~ P d  { 1 i ~ 0 1  ffiffiffi 2 p   1  ffiffiffi 2 p   a { i z c  { i   z b { i  X d  { 1 l  ~ 0 l  1  ffiffiffi 2 p   a { l  { c  { l      0 j i : ð 8 Þ Thesecondandfinalsteptocompletethestatetransferbymeansofquantumscissorsprovides a photon-number measurement in modes  a i  and  c  i  conditioned on thedetectionofasinglephotonin a i andvacuumin c  i ,cp.Fig.1.Sincethereareatotalof  d  1 1 photons in the system, a detection of one photon in each of the  d   modes  a i  andzero photons in the modes  c  i  results in a single photon in one of the modes  b i according to photon-number conservation. The measurement projects onto thosecomponents of state  j W æ  in Eq. (8) which allow for such a detection event: W j i ? 1 a { 2     a { d  b { 1 a { 1 z  2 a { 1     a { d  b { 2 a { 2     z  d  a { 1     a { d  { 1 b { d  a { d     0 j i ~ a { 1 a { 2     a { d  X l l  b { l   0 j i ~ 1 a 0  ... 1 a d  { 1 j i 6 x j i b 0 ... b d  { 1 , ð 9 Þ Therefore the state of light in the output modes  b 0  …  b d  2 1  of the quantum scissorsreads: x j i b 0 ... b d  { 1 ~ X l l  b { l   0 j i ð 10 Þ which is the state initially carried by the input modes  c  l  , cf. (5).Pleasenote,thatateleportationoftheinitialstatecarriedbytheinputmodes c  i ontodifferent output modes  ~ b i  can also be achieved: x j i c  0 ... c  d  { 1 ? x j i ~ b 0 ...~ b d  { 1 ~ X l l  ~ b { l   0 j i ð 11 Þ For this purpose, photons of modes  ~ a i  corresponding to the targeted modes  ~ b i  areinserted into the ports  a i , together with the initial state  j x æ  in modes  c  i ;1 ~ a 0  ... 1 ~ a d  { 1 j i 6 x j i c  0 ... c  d  { 1 ~ P d  { 1 i ~ 0 ~ a { i X d  { 1 l  ~ 0 l  c  { l   0 j i :  ð 12 Þ The consecutive actions of the beam splitters  BS 1  and  BS 2  in the quantum scissordevices, given respectively by   ~ a { i ? 1  ffiffiffi 2 p   ~ a { i z ~ b { i    and  ~ a { i ? 1  ffiffiffi 2 p   ~ a { i z ~ c  { i    together device 1device 2 c 2 c 1 c 2 c 1 a 2 a 2 a 1 a 11 b 1 bb 2 b 2    S  o  r   t  e  r   M   i  x  e  r t t 3 Figure 3  |  Schematic diagram of teleportation of a photonic qubit.  In thissetup,the modesorter  transferstheinputqubitencodedintwobasismodesof the input light beam on the left to different paths  c  1  and  c  2 . Twoquantum scissors devices teleport the states of light of the spatially separated modes  c  1  and  c  2  individually to  b  1  and  b  2 , respectively. The latterareretransferredbyamixerintoaphotonicqubitcarriedbyasingleoutputbeam on the right, which is a sorter run in reverse.  www.nature.com/ scientificreports SCIENTIFIC  REPORTS  | 3 : 3548 | DOI: 10.1038/srep03548  3  with (7), transform state (12) into Y j i ~ P d  { 1 i ~ 0 1  ffiffiffi 2 p   1  ffiffiffi 2 p   ~ a { i z ~ c  { i   z ~ b { i  X d  { 1 l  ~ 0 l  1  ffiffiffi 2 p   a { l  { c  { l      0 j i :  ð 13 Þ The only components of state  j Y æ  that can contribute to a coincidence detection of asinglephotonbythedetectors D 1 (cp.Fig.1)ineachofthe d  quantumscissorsdevicesare given by  1 ~ a { 2     ~ a { d  ~ b { 1 a { 1 z  2 ~ a { 1     ~ a { d  ~ b { 2 a { 2     z  d  ~ a { 1     ~ a { d  { 1 ~ b { d  a { d     0 j i ð 14 Þ If we further assume that each detectors  D 1  absorbs a single photon in the detectionprocess without distinguishing between both kinds of photons 24 ,  a i  and  ~ a i , then theremaining state is given as claimed by  x j i ~ b 0 ...~ b d  { 1 ~ X l l  ~ b { l   0 j i :  ð 15 Þ Thedetection eventindicating successful teleportation occurs forideal detectors withprobability 1/2 2 d  which is obtained from a normalization factor in projection (9).However, the success probability can be increased by considering other detectionevents.For example, if onephoton isdetected in mode c   j instead of mode a  j ,as wellasone photon in each of the remaining modes  a i , the state of   b  collapses into:  x j i b ~ X l   l  b { l   0 j i ,  ð 16 Þ where   l   is  l   for  l  ?  j  and 2  l   for  l  5  j . The minus sign can be compensated by applyinga p -phaseshifttomode b  j whichcausesthestatechange   x j i b ? x j i b .Hence,itdoes not matter whether the detectors in modes  a  j  or  c   j  register a single photon countas long as there is only one count in each quantum scissors setup. Thus there are 2 d  detection events corresponding to successful teleportation, which increases theprobability of success to 2 d  /2 2 d  5 1/2 d  .The success probability of the teleportation scheme depends on the efficiencies of the detectors used with the quantum scissors. For detectors which count a singlephoton with probability (i.e., efficiency)  g  ,  1 the success probability of the schemereduces to ( g /2) d  . Moreover, a restricted detection efficiency can induce the falseidentification of a two-photon detection event as a single-photon count with prob-ability2 g (1 2 g ).Suchamistakenidentificationinoneofthequantumscissorset-upstogether with single photon counts in the remaining ones leads to vacuum in theoutputmodes,whilethedetectorsseeminglyannounceasuccessfulteleportation.Theprobability for such false announcement equals  g (1 2 g ) and can be calculated fromthe probability to obtain a two-photon detection event with ideal detectors whichamounts to 1/2, independent of the number of quantum scissor setups (cp. Eq. (8)),multiplied with the probability for a false identificaton due to the non-unit detectionefficiency. Therefore, the teleportation fidelity defined as the overlap between theinput and the outputstate of the teleportationscheme decreases from  f  5 1with idealdetectors to  f  5 1 2 g (1 2 g ) for detectors with efficiency   g .1. Bennett, C. H.  et al  . 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T.K. acknowledges the partialsupport from National Research Foundation of South Africa (Grant specific uniquereference number (UID) 86325).  Authorcontributions S.K.G. and T.K. contributed equally to the content of this article.  Additionalinformation Competing financial interests:  The authors declare no competing financial interests. How to cite this article:  Goyal, S.K. & Konrad, T. Teleporting photonic qudits using multimode quantum scissors.  Sci. Rep.  3 , 3548; DOI:10.1038/srep03548 (2013). This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported license. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-nd/3.0  www.nature.com/ scientificreports SCIENTIFIC  REPORTS  | 3 : 3548 | DOI: 10.1038/srep03548  4
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