Abstract
Teleportation plays an important role in the communication of quantum information between the nodes of a quantum network and is viewed as an essential ingredient for longdistance Quantum Cryptography. We describe a method to teleport the quantum information carried by a photon in a superposition of a number d of light modes (a “qudit”) by the help of d additional photons based on transcription. A qudit encoded into a single excitation of d light modes (in our case LaguerreGauss modes which carry orbital angular momentum) is transcribed to d singlerail photonic qubits, which are spatially separated. Each singlerail qubit consists of a superposition of vacuum and a single photon in each one of the modes. After successful teleportation of each of the d singlerail qubits by means of “quantum scissors” they are converted back into a qudit carried by a single photon which completes the teleportation scheme.
Introduction
Quantum teleportation, the carrierless transmission of quantum information by transferring a state from one quantum system to a remote one was described by Bennett et al.^{1} and soon after played an important role in photonic quantum computing^{2,3,4,5} 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 environments make it challenging to realize quantum teleportation experimentally. Bouwmeester et al.^{8} were the first to achieve quantum teleportation followed by many others for discretelevel quantum systems^{9,10,11,12,13,14,15} as well as with continuous variables^{16,17,18,19}. In the case of discretelevel quantum systems so far only the state of the most simple quantum systems, i.e., twolevel systems, and therewith the smallest unit of quantum information (a “qubit”) could be teleported. A new teleportation scheme, proposed recently^{20}, is capable to transmit the quantum information carried by an elementary excitation of a superposition of an arbitrary number d of copropagating light modes (a photonic “qudit”). The teleportation of photonic qudits increases the quantum information sent per carrier photon. Currently, the low transmission rates are one of the bottlenecks of quantum communication as compared to its classical counterpart. However, the scheme proposed in^{20} requires to prepare d additional photons in a highly entangled state. Here we present an alternative scheme based on the transcription of the qudit encoded in a single photon to d qubits carried by light modes which propagate along different optical paths. Each qubit contains the quantum information about the excitation of a particular of the d original light modes and is teleported individually by means of an additional photon using quantum scissors.
Quantum scissors^{15,21,22,23,24} is a device to teleport only the vacuum and the singlephoton component of a singlemode state (a socalled singlerail qubit), while it truncates (“cuts off”) higher photonnumber components. In particular, if an input light mode c (cp. Fig. (1)) is prepared in a superposition of states with different photon numbers, quantum scissors projects its vacuum and single photon component to an ouput mode b: where n〉_{c}_{(b)} represents a socalled Fock state with n = 0, 1, 2 … photons in light mode c (b) and the coefficients α_{n} are the corresponding probability amplitudes. This process occurs upon conditioning on a singlephoton detection with the probability given by the square of the norm of the final state χ′〉_{b}, i.e, (α_{0}^{2} + α_{1}^{2})/4. However, this probability can be doubled by conditioning on one of two possible singlephoton detections (cp. Methods). The working principle of quantum scissors is explained in the caption of Fig. (1).
On the other hand, if the input state in mode c consists already of a singlerail qubit, i.e. χ〉_{c} = (α_{0} 0〉_{c} + α_{1} 1〉_{c}), it is transferred according to transformation (1) without truncation, and hence teleported, to mode b. A generalization of quantum scissors which cuts off all state components with a number of d or more photons and thus teleports multiphoton states of the form (“singlerail qudits”) can be achieved using multiports and d − 1 additional input photons^{25,26}. However, the encoding of an arbitrary superposition χ〉_{c} of multiple photonnumber Fock states is in practice difficult and requires nonlinear optical media leading to small efficiencies^{27,28,29}.
Moreover, one can teleport n singlerail qubits simultaneously, provided they are stored in light modes which propagate on different paths, by applying n quantum scissor setups in parallel, one for each singlerail qubit. Obviously, the simultaneous teleportation works if the singlerail qubits in the individual modes are not correlated. But note, that also the state of n entangled qubits can be teleported in this way.
Results
This feature of quantum scissors can be exploited to teleport a qudit encoded into a single photon which is shared by d spatial modes of paraxial light. For this purpose the dlevel state of the photon is transcribed into d singlerail qubits carried by light modes propagating along different paths with the help of a mode sorter. Such a device has the task to transfer orthogonal light modes within a single light beam to different optical paths, similar to a polarizing beam splitter, which conveys light with horizontal and vertical polarization to orthogonal paths. For example, consider a singlephoton state χ〉 given by an elementary excitation of a superposition of d paraxial LaguerreGauss modes LG_{l}_{,p = 0} corresponding to different values of orbital angular momentum (OAM)^{30} which copropagate along an optical path o, i.e, where 1_{l}〉 denotes the state of a single photon with OAM . We spatially separate the OAM modes by diverting them into different optical paths c_{l} depending on their OAM value with the help of an OAM mode sorter^{31,32}. For d = 3 this transformation reads: where represents a single photon with OAM quantum number l = 0 in path c_{0} and no photon in all other paths (accordingly for the remaining terms). The single photon states are conveniently expressed by the creation operators acting on the global vacuum state 0〉, cp. the righthand side of (3). This transformation transcribes the state of a qudit into d entangled singlerail qubits. After the transcription the ith qubit contains the quantum information about whether the corresponding OAM mode l = i of the photonic qudit was occupied or not .
Now each singlerail 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 quantum scissors setups (see Fig. (3) for d = 2). There are thus d input modes c_{i} and d output modes b_{i} with i = 0, 1 … d − 1 to carry the quantum information. In addition, the quantum scissors require a total of d single photons entered separately in modes a_{i}. Upon conditioning on the detection of a single photon in each of the quantum scissor devices (success probability 1/2^{d} with ideal detectors, for nonunit detection efficiencies see Methods) the state χ〉 carried by the input modes c_{i} is transferred to the output modes b_{i} (cp. Methods): Since the mode b_{i} in the ith quantum scissors device originates from the reflection of mode a_{i} both are identical except for their propagation direction, and they should carry the same OAM value as input mode 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 .
However, as shown under Methods, this is not necessary if the modes b_{i} are transformed into the appropriate OAM mode after the state transfer. In fact, preparing the additional photons in a different system of basis modes enables a transcription of the quantum information stored in a specific basis (here OAM modes) in the input modes of the quantum scissors devices to another basis (for example Hermite Gaussian modes^{33}) in its output modes. By such a transcription any unitary gate acting on the Hilbert space of the qudit can be realized, however only with limited success probability which is determined by the quantum scissors involved.
After successful teleportation by the quantum scissors we can convert the entangled d singlerail qubits back to the original dmode OAM state (2) with the help of a mixer which is a sorter run in reverse. This completes the teleportation of a photonic qudit (cp. Fig. (2)). In order to realize an additional unitary quditgate (see above) together with the teleportation, the mixer has to map the new basis modes in the outputs of the d quantum scissors into a single beam, i.e., it must be a reverse sorter for these particular modes, which exists for example for Hermite Gaussian modes^{33}.
Discussion
In this article we have presented a scheme to teleport a photonic qudit carried by OAM modes. The scheme requires linear optical devices, OAM mode sorters as well as singlephoton sources and photonnumber resolving detectors. The essential step is to transcribe the state of the qudit to d singlerail qubits by means of a mode sorter and to teleport the qubits individually by quantum scissors. In as far as such sorter devices can be designed for other light modes, for example HermiteGaussian modes^{33}, the proposed teleportation scheme is universal and can be implemented with any system of basis modes. Using quantum scissors a singlerail qubit can be teleported with success probability 1/2, therefore, the success probability to teleport d singlerail qubits and thus the encoded qudit amounts to 1/2^{d}.
In principle, the probability to teleport a singlerail qubit can be increased to N/(N + 1) by employing N additional entangled photons and a balanced multiport with N + 1 inputs and outputs as described by Knill et al.^{4} instead of one additional photon and a balanced beam splitter in each of the quantum scissors setups. This leads to a success probability for the qudit teleportation of (N/(N + 1))^{d} but requires d highly entangled Nphoton states, which can be prepared probabilistically offline^{4,34}.
On the other hand, the encoding of one qudit into d twolevel systems (such as singlerail qubits) represents an inefficient use of storage capacity. The amount of quantum information present in a single qudit actually corresponds to log_{2} d qubit units of quantum information and could thus be stored efficiently in log_{2} d singlerail qubits. Given a scheme which is able to transcribe the initial photonic qudit into log_{2} d singlerail qubits, a subsequent teleportation could be achieved by means of log_{2} d quantum scissors with a success probability of . This would mean an exponential decrease of the resources needed to teleport a qudit.
Alternatively, using the transcription based on a OAM mode sorter, as described above, the present scheme allows, instead of a single qudit, to teleport d singlerail qubits encoded in copropagating OAM modes, with the same resources as before. This corresponds to an exponential increase of quantum information sent per use of the teleportation protocol. However, the preparation and manipulation of singlerail qubits seems problematic compared to qudits carried by singlephoton states of OAM modes, which can be prepared, transformed and measured with standard techniques^{35}. For singlerail qubits, general deterministic single and twoqubit gates are not available^{36}. Moreover, the vacuum component makes state tomography of singlerail qubits difficult.
The present scheme has certain advantages as well as disadvantages over a recently proposed alternative teleportation method^{20}. Unlike the latter, it does not require highly sensitive multipartite entangled states to perform the quantum teleportation. On the other hand, the alternative method yields a greater success probability of 1/d^{2} for qudit teleportation, and requires less additional photons. Remarkably, it yields for the joint teleportation of the state of many photons the same maximal teleportation rate as quantum scissors for single rail qubits, namely one qubit per additional photon. However, by improving the transcription efficiency one could overcome these drawbacks of the present scheme.
Methods
In the following we show that d quantum scissors enable a transfer of the state obtained after the transformation (3) to output modes b_{0} … b_{d}_{ − 1} (4). The state along with d photons at ports a_{i} constitute the initial state entering the d quantum scissors (cp. Fig. (3) for d = 2) where we have introduced the creation operators to denote single photons in the modes a_{i}. Each quantum scissors device contains two 50:50 beam splitters, cp. Fig. 1. The first beam splitter BS_{1} of the ith device distributes the incoming photon in mode a_{i} equally over both modes, a_{i} and b_{i}, represented by the transformation rule in terms of the corresponding creation operators . Also the action of the second beam splitter BS_{2} in each quantum scissors device is conveniently described by similar rules: Consecutive application of these transformations for beam splitters BS_{1} and BS_{2} for all quantum scissors i to the initial state (5) yields the total state change: The second and final step to complete the state transfer by means of quantum scissors provides a photonnumber measurement in modes a_{i} and c_{i} conditioned on the detection of a single photon in a_{i} and vacuum in c_{i}, cp. Fig. 1. Since there are a total of d + 1 photons in the system, a detection of one photon in each of the d modes a_{i} and zero photons in the modes c_{i} results in a single photon in one of the modes b_{i} according to photonnumber conservation. The measurement projects onto those components of state Φ〉 in Eq. (8) which allow for such a detection event: Therefore the state of light in the output modes b_{0} … b_{d}_{ − 1} of the quantum scissors reads: which is the state initially carried by the input modes c_{l}, cf. (5).
Please note, that a teleportation of the initial state carried by the input modes c_{i} onto different output modes can also be achieved: For this purpose, photons of modes corresponding to the targeted modes are inserted into the ports a_{i}, together with the initial state χ〉 in modes c_{i}; The consecutive actions of the beam splitters BS_{1} and BS_{2} in the quantum scissor devices, given respectively by and together with (7), transform state (12) into The only components of state Ψ〉 that can contribute to a coincidence detection of a single photon by the detectors D_{1} (cp. Fig. 1) in each of the d quantum scissors devices are given by If we further assume that each detectors D_{1} absorbs a single photon in the detection process without distinguishing between both kinds of photons^{24}, a_{i} and , then the remaining state is given as claimed by The detection event indicating successful teleportation occurs for ideal detectors with probability 1/2^{2d} which is obtained from a normalization factor in projection (9). However, the success probability can be increased by considering other detection events. For example, if one photon is detected in mode c_{j} instead of mode a_{j}, as well as one photon in each of the remaining modes a_{i}, the state of b collapses into: where is for l ≠ j and for l = j. The minus sign can be compensated by applying a πphase shift to mode b_{j} which causes the state change . Hence, it does not matter whether the detectors in modes a_{j} or c_{j} register a single photon count as 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 the probability of success to 2^{d}/2^{2d} = 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 single photon with probability (i.e., efficiency) η < 1 the success probability of the scheme reduces to (η/2)^{d}. Moreover, a restricted detection efficiency can induce the false identification of a twophoton detection event as a singlephoton count with probability 2η(1 − η). Such a mistaken identification in one of the quantum scissor setups together with single photon counts in the remaining ones leads to vacuum in the output modes, while the detectors seemingly announce a successful teleportation. The probability for such false announcement equals η(1 − η) and can be calculated from the probability to obtain a twophoton detection event with ideal detectors which amounts 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 nonunit detection efficiency. Therefore, the teleportation fidelity defined as the overlap between the input and the output state of the teleportation scheme decreases from f = 1 with ideal detectors to f = 1 − η(1 − η) for detectors with efficiency η.
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Acknowledgements
We thank A. Forbes and F.S. Roux for useful discussions. T.K. acknowledges the partial support from National Research Foundation of South Africa (Grant specific unique reference number (UID) 86325).
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School of Chemistry and Physics, University of KwaZuluNatal, Private Bag X54001, Durban 4000, South Africa
 Sandeep K. Goyal
 & Thomas Konrad
National Institute of Theoretical Physics (NITheP), University of KwaZuluNatal, Private Bag X54001, Durban 4000, South Africa
 Thomas Konrad
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S.K.G. and T.K. contributed equally to the content of this article.
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Correspondence to Thomas Konrad.
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