Nature  Letter
Quantum teleportation of multiple degrees of freedom of a single photon
 XiLin Wang,^{1, 2}^{, }
 XinDong Cai,^{1, 2}^{, }
 ZuEn Su,^{1, 2}^{, }
 MingCheng Chen,^{1, 2}^{, }
 Dian Wu,^{1, 2}^{, }
 Li Li,^{1, 2}^{, }
 NaiLe Liu,^{1, 2}^{, }
 ChaoYang Lu^{1, 2}^{, }
 & JianWei Pan^{1, 2}^{, }
 Journal name:
 Nature
 Volume:
 518,
 Pages:
 516–519
 Date published:
 DOI:
 doi:10.1038/nature14246
 Received
 Accepted
 Published online
Quantum teleportation^{1} provides a ‘disembodied’ way to transfer quantum states from one object to another at a distant location, assisted by previously shared entangled states and a classical communication channel. As well as being of fundamental interest, teleportation has been recognized as an important element in longdistance quantum communication^{2}, distributed quantum networks^{3} and measurementbased quantum computation^{4, 5}. There have been numerous demonstrations of teleportation in different physical systems such as photons^{6, 7, 8}, atoms^{9}, ions^{10, 11}, electrons^{12} and superconducting circuits^{13}. All the previous experiments were limited to the teleportation of one degree of freedom only. However, a single quantum particle can naturally possess various degrees of freedom—internal and external—and with coherent coupling among them. A fundamental open challenge is to teleport multiple degrees of freedom simultaneously, which is necessary to describe a quantum particle fully and, therefore, to teleport it intact. Here we demonstrate quantum teleportation of the composite quantum states of a single photon encoded in both spin and orbital angular momentum. We use photon pairs entangled in both degrees of freedom (that is, hyperentangled) as the quantum channel for teleportation, and develop a method to project and discriminate hyperentangled Bell states by exploiting probabilistic quantum nondemolition measurement, which can be extended to more degrees of freedom. We verify the teleportation for both spin–orbit product states and hybrid entangled states, and achieve a teleportation fidelity ranging from 0.57 to 0.68, above the classical limit. Our work is a step towards the teleportation of more complex quantum systems, and demonstrates an increase in our technical control of scalable quantum technologies.
Subject terms:
Main
Quantum teleportation is a linear operation applied to quantum states, and so teleporting multiple degrees of freedom (DoFs) should be possible in theory^{1}. Suppose Alice wishes to teleport to Bob the composite quantum state of a single photon (photon 1; Fig. 1a), encoded in both the spin angular momentum (SAM) and the orbital angular momentum (OAM) as follows:
Here and denote horizontal and vertical polarizations of the SAM; and refer to righthanded and lefthanded OAMs of and , respectively; and α, β, γ and δ are complex numbers satisfying . For Alice to do so, she and Bob first need to share a hyperentangled photon pair (photons 2 and 3), which is simultaneously entangled in both SAM and OAM:
Here we use and to denote the four Bell states encoded in SAM, and and to denote the four Bell states in OAM. Their tensor products result in a group of 16 hyperentangled Bell states.
A crucial step in the teleportation is to perform a twoparticle joint measurement of photons 1 and 2, projecting them onto the basis of the 16 orthogonal and complete hyperentangled Bell states, and discriminating one of them; for example
This process is referred to as hyperentangled Bell state measurement (hBSM). After the hBSM that projects photons 1 and 2 onto the state , photon 3 will be projected onto the initial state of photon 1:
With equal probabilities of 1/16, photons 1 and 2 can also be projected onto one of the other 15 hyperentangled Bell states (Methods). The hBSM results can be broadcast as fourbit classical information, which will allow Bob to apply appropriate Pauli operations to perfectly reconstruct the initial composite state of photon 1.
Experimental realization of the above teleportation protocol poses significant challenges to the coherent control of multiple particles and multiple DoFs simultaneously. The most difficult task is to implement the hBSM, because it would normally require coherently controlled gates between independent quantum bits (qubits) of different DoFs. Moreover, with multiple DoFs, it is necessary to measure one DoF without disturbing any other. With linear operations only, previous theoretical work^{14} has suggested that it is impossible to discriminate the hyperentangled states unambiguously. This challenge has been overcome in our work.
Figure 1 illustrates our linear optical scheme for teleporting the spin–orbit composite state. The hBSM is implemented in a stepbystep manner, as a combination of two separate BSMs. First, photons 1 and 2 are sent through a polarizing beam splitter (PBS), an optical device that transmits horizontal polarizations ( ) and reflects vertical polarization ( ). After the PBS, we postselect the event that there is one and only one photon in each output. Such an event can occur only if the two input photons have the same SAM (both are transmitted ( ) or reflected ( )), which projects the SAM part of the wavefunction into the twodimensional subspace spanned by . At both outputs of the PBS, we add two polarizers, projecting the two photons into the diagonal basis . It should be noted that the PBS is not OAMpreserving^{15}, because the reflection at the PBS flips the sign of the OAM qubit; that is, , , and . Thus, both the SAM and the OAM must be taken into account as in a molecularlike coupled state. A detailed mathematical treatment is presented in Methods, showing that the PBS and two polarizers select the four following states out of the total 16 hyperentangled Bell states: , , and .
Second, having measured and filtered out the SAM, we perform BSM on the remaining OAM qubit. The two single photons that emerge from the PBS are superposed on a beam splitter (Fig. 1a). Only the asymmetric Bell state will lead to a coincidence detection where there is one and only one photon in each output^{16}, whereas for the three other symmetric Bell states, the two input photons will coalesce to a single output mode. Note again that reflection at the beam splitter inverts the OAM sign. Therefore, the state can be distinguished by a coincidence detection in separate outputs, and can be discriminated by measuring two orthogonal OAMs in either output. In total, these two steps would allow an unambiguous discrimination of the two hyperentangled Bell states and .
However, these interferometric processes cannot be simply cascaded. They depend on the assumption that two input photons are from different paths. When directly connected, the 50% chance of both photons coalescing into a single output spatial mode after the first ‘interferometer’ would remain undetected (no ‘coincidence’ event occurs), and would further induce erroneous detection of a coincidence event after the second interferometer, thus failing both BSMs. This is a ubiquitous and important problem in linear optical quantum information processing^{5, 17}.
Our remedy is to exploit quantum nondemolition (QND) measurement, whereby a single photon is observed without destroying it and keeping its quantum information intact. Interestingly, quantum teleportation itself can be used for probabilistic QND detection^{7, 18}. As shown in Fig. 1b, another pair of photons entangled in OAM is used as ancillary. The procedure is a standard teleportation. If there is an incoming photon, a twophoton coincidence detection behind a beam splitter can occur with 50% efficiency, triggering a successful BSM, which heralds the presence of the incoming photon and teleports the full quantum state of the incoming photon to a freely propagating photon. If there is no incoming photon, twophoton coincidence behind the beam splitter cannot occur, in which case we will know and will ignore the outgoing photon. Thus, using the QND measurement, the BSM interferometers can be concatenated. We note that using a QND in one of the arm of the beam splitter is sufficient owing to the conserved total number of eventually registered photons. Our protocol can identify two hyperentangled Bell states with an overall efficiency of 1/32 (Methods). The QND method can also be used to boost the efficiency of photonic quantum logic gates in the Knill–Laflamme–Milburn scheme for scalable optical quantum computing^{5}.
Figure 2 shows the experimental setup for quantum teleportation of the spin–orbit composite state of a single photon. Passing a femtosecondpulsed laser through three typeI βbarium borate crystals generates three photon pairs^{19, 20, 21}, engineered in different forms (Methods). The first photon pair (1−t) is used to prepare a heralded single photon (1) to be teleported, triggered by the detection of its sister photon (t). The second pair (2–3) is created in the hyperentangled state . The third pair (4–5) is prepared in the ancillary OAMentangled state for the teleportationbased QND measurement.
We prepare five different initial states to be teleported: , , , and . These states can be grouped into three categories: and are product states of the two DoFs in the computational basis; and are products states of the two DoFs in the superposition basis; and is a spin–orbit hybrid entangled state. The four product states are prepared by independent singlequbit rotations, using wave plates for SAM and spiral phase plates (SPPs) or binary phase plates (BPPs) for OAM. The entangled state is generated by counterpropagating a SPP inside a Sagnac interferometer (Methods).
The implementation of the hBSM and the QND measurement requires Hong–Ou–Mandeltype interference^{22} between indistinguishable single photons with good time, spatial and spectral overlap. The two photons are synchronized to arrive at the PBS and beam splitters within 10 fs of each other, a delay that is much smaller than the coherence time of the downconverted photons (~448 fs), which is stretched by narrowband spectral filtering (~3 nm). A stepbystep verification of the twophoton interference as a function of temporal delay is presented in Methods and Extended Data Fig. 1. We observe a visibility of 0.75 ± 0.03 for the interference of the two SAMencoded photons at the PBS, and visibilities of 0.73 ± 0.03 and 0.69 ± 0.03 for the interferences of two OAMencoded photons at beams splitter 1 and, respectively, beam splitter 2.
The final verification of the teleportation results relies on the coincidence detection counts of the six photons encoded in both DoFs, which would suffer from a low rate. It should be noted that all the previous experiments^{15, 20, 21, 23, 24, 25, 26} with OAM states have never gone beyond two single photons. We overcome this technical challenge by preparing highbrightness, hyperentangled photons and designing dualchannel and highefficiency OAM measurement devices (Methods).
To evaluate the performance of the teleportation operation, we measure the fidelity of the teleported state, , which is defined as the overlap of the ideal teleported state ( ) and the measured density matrix ( ). Conditioned on the detection of the trigger photon t and the fourphoton coincidence after the hBSM, we register the photon counts of teleported photon 3 and analyse its composite state. The fidelity measurements for the product states are straightforward because we can measure the SAM and OAM qubits separately. We measure the final state in the and bases for the teleportation of and , in the and bases for the teleportation of , and in the and bases for the teleportation of . The data in Fig. 3 yield teleportation fidelities of 0.68 ± 0.04, 0.66 ± 0.04, 0.62 ± 0.04 and 0.63 ± 0.04 for , , and, respectively, .
The fidelity of the entangled state , where the SAM and OAM qubits are not separable, can be decomposed as , where σ_{x}, σ_{y} and σ_{z} are the Pauli operators. The expectation values of the joint observables , and can be obtained by local measurements in the corresponding basis for the two DoFs, , and , respectively. Our experimental results on the three different bases are presented in Fig. 4a–c, from which we determine a teleportation fidelity of 0.57 ± 0.02 for the entangled state.
We note that all reported data are without background subtraction. The main sources of error include double pair emission, imperfection in the initial states, entanglement of photons 2–3 and 4–5, twophoton interference and OAM measurement. We note that the teleportation fidelities of the states in the three categories are affected differently by errors from different sources (Methods). Despite the experimental noise, the measured fidelities (summarized in Fig. 4d) of the five teleported states are all well above 0.40—the classical limit, defined as the optimal stateestimation fidelity on a single copy of a twoqubit system^{27}. These results prove the successful realization of quantum teleportation of the spin–orbit composite state of a single photon. Furthermore, for the entangled state, , we emphasize that the teleportation fidelity exceeds the threshold of 0.5 required to prove the presence of entanglement^{28}, which demonstrates that the hybrid entanglement of different DoFs inside a quantum particle can survive teleportation.
We have reported the quantum teleportation of multiple properties of a single quantum particle, demonstrating the ability to coherently control and simultaneously teleport a single object with multiple DoFs that forms in a hybrid entangled state. It is interesting to note that these DoFs can also be in a fully undefined state, such as being part of a hyperentangled pair, which would lead to the protocol of hyperentanglement swapping^{29}. Our methods can be generalized to more DoFs (see Methods for a universal scheme). The efficiency of teleportation, which is limited mainly by the efficiency of hBSM (1/32 in the present experiment), can be enhanced by using more ancillary photons, quantum encoding, embedded teleportation tricks, highefficiency singlephoton detectors and active feedforward, in a spirit similar to the Knill–Laflamme–Milburn scheme^{5}. We did not implement the feedforward in the current experiment, but it could be done using electrooptical modulators for both the SAM and the OAM qubits (see Methods for a detailed protocol). Although the present work is based on linear optics and single photons, the multiDoF teleportation protocol is by no means limited to this system, but can also be applied to other quantum systems such as trapped electrons^{12}, atoms^{9} and ions^{10, 11}.
As well as being of fundamental interest, the methods developed in this work on the manipulation of quantum states of multiple DoFs will open up new possibilities in quantum technologies. Controlling multiple DoFs makes complete SAM Bellstate analysis^{23} and alignmentfree quantum communication^{30} possible, for example. Carrying multiple DoFs on photons can enhance the information capacity in quantum communication protocols such as quantum superdense coding^{23}. Moreover, combining the entanglement of multiple (M) photons and high (N) dimensions from the multiple DoFs would allow the generation of hyperentanglement with an expanded Hilbert space that grows in size as N^{M}, which would provide a versatile platform in the near future for demonstrations of complex quantum communication and quantum computing protocols, as well as extreme violations of Bell inequalities.
Methods
The protocol for teleporting a spinorbit composite quantum state
The combined state of photons 1, 2 and 3 can be rewritten in the basis of the 16 orthogonal and complete hyperentangled Bell states as follows:
where the triples ( , , ) and ( , , ) are the Pauli operators for the SAM and, respectively, the OAM qubits. It indicates that, regardless of the unknown state , the 16 measurement outcomes are equally likely, each with a probability of 1/16. By carrying out the hyperentangled Bell state measurement (hBSM) on photons 1 and 2 to unambiguously distinguish one from the group of 16 hyperentangled Bell states, Alice can project photon 3 onto one of the 16 corresponding states. After Alice tells Bob her hBSM result as fourbit classical information via a classical communication channel, Bob can convert the state of his photon 3 into the original state by applying appropriate twoqubit local unitary transformations .
Twophoton interference of spin–orbit composite state on a PBS
The input and output of photons 1 and 2 encoded in SAM and OAM on a PBS as shown in Fig. 1 can be summarized as follows: , , , . Note that the PBS is not OAMpreserving, because the reflection flips the sign of OAM. Therefore, the output state for each of the input 16 hyperentangled Bell states can be listed below:
where , , and . Then both of the output modes are passing through two polarizers at . Only the output with the term will result in there being one photon in each output mode with an efficiency of 1/8. The other terms will be rejected by the conditional detection. According to equations (1)–(16), 4 of the 16 output modes have the term and the 4 corresponding input hyperentangled Bell states are , , and , which will be sent to the next stage of BSM on the OAM qubits.
BSM and teleportation of OAM qubits
Having measured and filtered out the SAM qubit, next we perform BSM on the OAM qubit. Dealing with a single degree of freedom is more straightforward, and can be implemented using a beam splitter. Like the PBS, the beam splitter also is not OAMpreserving, because the reflection at the beam splitter will flip the sign of OAM. Therefore, the transformation rules at a beam splitter for the four OAM Bell states are
We can see that only the Bell state will result in there being one and only one photon in each output, whereas for the three other Bell states the two input photons will coalesce into a single output mode. Among these three Bell states, the state can be further distinguished by measuring two single photons in either output with the OAM orthogonal basis . Thus, this would allow us to unambiguously discriminate two from the four Bell states. Experimentally, we design dualchannel OAM readout devices to measure and simultaneously. Therefore, the efficiency of both QND and BSM on OAM is 1/2. The overall efficiency of hBSM combining all three steps is .
Generating three photon pairs
Ultrafast laser pulses with an average power of 800 mW, central wavelength of 394 nm, pulse duration of 120 fs and repetition rate of 76 MHz successively pass through three βbarium borate (BBO) crystals (Fig. 2) to generate three photon pairs through typeI spontaneous parametric downconversion (SPDC). The first and the third crystals are 2 mmthick BBOs, whereas the second one consists of two 0.6 mmthick contiguous typeI BBO crystals with optic axes aligned in perpendicular planes. All of the downconverted photons have central wavelengths of 788 nm. The first photon pair (1–t) is initially prepared in the zeroorder OAM mode with a coincidence count rate of In SPDC, the zeroorder OAM mode has the highest weight among all OAM modes and thus has the highest brightness. The second photon pair (2–3) is simultaneously entangled in both the SAM and the OAM (in the firstorder mode), owing to OAM conservation in the typeI SPDC. It has a count rate of and a state fidelity of 0.95. The third photon pair (4–5) is created in the firstorder OAM entangled state with a twophoton count rate of and a fidelity of 0.91. We estimate the mean numbers of photon pairs generated per pulse as ~0.1, ~0.01 and ~0.05 for the first, second and third pairs, respectively.
Preparing spin–orbit entanglement
As illustrated in the yellow panel of Fig. 2, we use a Sagnac interferometer to prepare the spin–orbit hybrid entangled state to be teleported, . Photon 1 is initially prepared in the SAM state with zeroorder OAM. It is then sent into a Sagnac interferometer that consists of a PBS and an SPP, where the counterpropagating and SAMs with zeroorder OAM pass through the SPP in opposite directions and are converted into and , respectively. Considering an extra π phase shift for the 1^{s} SAM from double reflections in the PBS, the final output state is . We note that the whole conversion process is deterministic.
Twophoton interference on the PBS and beam splitters
For a test of twophoton interference visibility at the PBS, input photons 1 and 2 are first intentionally prepared in the states (orthogonal SAMs; see open squares in Extended Data Fig. 1a) and (parallel SAMs; see solid circles in Extended Data Fig. 1a), respectively. We measure each single photon from the two outputs of the PBS in the basis (1′ and 2′ denote the output spatial modes). At zero delay, where the photons are optimally overlapped in time, the orthogonal SAM input yields an output state of conditioned on a coincidence detection, which can be decomposed in the diagonal basis as , thus showing an enhancement. For the parallel SAM input, the output state is , which shows a reduction. The increase in the delay gradually destroys the indistinguishability of the two photons, such that their quantum state becomes a classical mixture. Thus, at large delays the counts appear flat. Interferometers of this type are sensitive only to length changes of the order of the coherence length of the detected photons and stay stable for weeks.
The twophoton interferences on beam splitters 1 and 2 are for the teleportationbased QND measurement of OAM qubits and the BSM of OAM qubits, respectively. As a test, the two input photons are prepared in the same SAM but orthogonal OAM states. It is interesting to note that, in stark contrast to the conventional Hong–Ou–Mandel interference, only having the two input OAM states orthogonal can lead to an interference dip, because the reflection at the beam splitter flips the sign of OAM. The twophoton interference as a function of temporal delay is shown in Extended Data Fig. 1b, c.
Dualchannel and efficient OAM measurement
One of the most frequently used OAM measurement devices in the previous experiments is offaxis hologram gratings^{21, 23, 24, 31}, which typically have a practical efficiency (p) of about 30% (ref. 31). This low efficiency would cause an extremely low sixphoton coincidence count rate that scales as p^{6}, more challenging than the scaling (p^{2}) in the previous twophoton OAM experiments. To overcome this challenge, we use two different types of device for efficient OAM readout.
The first type is what we refer to as ‘dualchannel’ OAM measurement devices, used after the two beam splitters. The strategy is to transfer the OAM information to the photon’s SAM, and measure it using a PBS with two output channels. The method is as follows. After the beam splitter, each photon passes through a HWP and is prepared in the state . They are then sent into Sagnac interferometers with a Dove prism inside^{32} (Fig. 2), which is placed at a π/8 angle with respect to the interferometer plane. The photon is rotated by an angle of π/4 for the SAM component when passing through the Dove prism forwards and by an angle of –π/4 for the SAM component when passing through the Dove prism backwards. OAM states and with respective phases and , where is the azimuthal angle in the polar coordinate system, which pass through a Dove prism rotated by π/8, will be rotated by an angle of π/4 and their phases will change to and , respectively. Therefore, two opposite phases, and , will be added to the two orthogonal OAM modes. Finally, the output photons are transformed into the and polarizations using a QWP. The overall transformations can be summarized as a CNOT gate between the OAM and SAM:
Effectively, the OAM qubit is deterministically and redundantly encoded by the SAM qubit, that is, the SAM becomes identical to the OAM state. Thus, by measuring the SAM using a PBS with two output channels, we can recover information about the OAM with a high efficiency of ~97%. In our experiment, four such Sagnac interferometers are used in the hBSM. In this way, two of the four OAM Bell states can be discriminated using a beam splitter.
Whereas the first type is like a twochannel readout device (like a PBS for SAM), the second type is like a onechannel readout device (like a polarizer for SAM), and is used in the final stage of state verification after teleportation. The strategy for the projective OAM measurement is to transform it into the zeroorder OAM mode so that the photon can be coupled into a singlemode fibre, while all other higherorder OAM modes will be rejected. Here we use an SPP^{33} and a BPP^{33} for efficient OAM readout.
The SPP^{33} is designed with a spiral shape to create a vortex phase of , where l is an integer and is referred to as topological charge. The SPP can attach vortex phases of and to a photon when it passes through the SPP forwards and, respectively, backwards. When vortex phases of and are attached, the corresponding OAM values will increase by and, respectively, decrease by . Therefore, a SPP can be used as an OAM mode converter. In our experiment, the OAM qubits are encoded in the OAM firstorder subspace with the topological charge l being 1. The conversion between the OAM zero mode and the firstorder modes ( and ) is realized using a 16phaselevel SPP with an efficiency of ~97%.
For the coherent transformation between the OAM zero mode and the superposition states and , a 2phaselevel BPP^{33, 34} is used with an efficiency of ~80%. These highefficiency OAM measurement devices boost the sixfold coincidence count rate in our present experiment by more than two orders of magnitude, compared with the previous use of hologram gratings.
Error budget
The sources of error in our experiment include double pair emission in spontaneous parametric downconversion; partial distinguishability of the independent photons that interfere at the PBS (~5%) and the beam splitters (~5%); state measurement error due to zeroorder OAM leakage (~2%); fidelity imperfection of entangled photon pairs 2–3 (~5%) and 4–5 (~9%); and fidelity imperfection of the tobeteleported singlephoton hybrid entangled state (~8%).
Some error sources affect all the teleported states. First, the double pair emission contributed a ~15% background to the overall sixfold coincidence counts. If this were subtracted, the average teleportation fidelity would be improved to ~0.74. Second, the imperfectly entangled photon pair 4–5 and the imperfect twophoton interference at beam splitters 1 and 2 (for the QND and OAM Bellstate measurements, respectively) degrade the teleportation fidelity for all states by ~13%. Third, the imperfect state measurements mainly due to the zeroorder OAM leakage cause a degradation of ~2%.
Some error sources can have different effects on different teleported states. The imperfect twophoton interference at the PBS degrades the teleportation fidelities for the states , and by ~5%. However, for the states and where the photon is horizontally or vertically polarized, the actual teleportation does not require twophoton interference at the PBS, and is therefore immune to the imperfection of the interference. This explains why the teleportation fidelities for the states and are the highest. This is inconsistent with the previous results: for the experiments using a PBS^{35, 36, 37}, the teleportation fidelities in the horizontal–vertical basis are higher than those in the ± 45° linear and circular polarization bases; whereas for the experiments^{38, 39} using a nonpolarizing beam splitter, the teleportation fidelities in all polarization bases are largely unbiased.
We note that the teleportation fidelity for the hybrid entangled state, , is the lowest, which is affected by the imperfection (~8%) in the state preparation of the initial state as well as by imperfections in and , that is, essentially all error and noise present in the experiment. It can be expected that all entangled states are subject to the same decoherence mechanism as the state as demonstrated, and should undergo similar reductions in fidelity.
These sources of noise can in principle be eliminated in future by various methods. For instance, deterministic entangled photons^{40} do not suffer the problem of double pair emission. We also plan to develop bright OAMentangled photons with higher fidelity, and a more precise 32phaselevel SPP for the next experiment of hyperentanglement swapping.
A universal scheme for teleporting N DoFs
We illustrate in Extended Data Fig. 2 a universal scheme for teleporting N DoFs of a single photon. For simplicity, we discuss an example for three DoFs (Extended Data Fig. 2b), labelled X, Y and Z. There are in total 64 hyperentangled Bell states:
These the products of the Bell states of each DoF, defined as
where i = X, Y, Z. The aim is to perform an hBSM, identifying 1 out of the 64 states. The required resources include photon pairs entangled in the Z DoF, pairs hyperentangled in the Y–Z DoFs, pairs hyperentangled in the X–Y–Z DoFs, filters for the three DoFs that can project the state to or (with a functionality similar to the polarizers for SAM), qubit flip ( ) operations for the DoFs, a 50:50 nonpolarizing beam splitter and singlephoton detectors, all of which are commercially available or have been experimentally demonstrated previously.
It has been known that if two single photons are superposed at a beam splitter, only asymmetric quantum states can result in one and only one photon exiting from each output of the splitter. In the simplest case of one DoF, this is the asymmetric state. As now we have three DoFs, we have to consider the combined, molecularlike quantum states. There are in total 28 possible combinations that are asymmetric states, as follows:
After the photons have passed through the first beam splitter, we can filter out and retain these 28 from the 64 hyperentangled Bell states conditioned on seeing one and only one photon in each output of the splitter. Next we apply two filters in the two output of the splitter to project the X DoF into . One filter is set to pass the state and the other is set to pass the state. This results in the following 16 states being filtered from the 28:
We perform a bitflip operation on the X DoF on one of the arms of the interferometer, erasing the information in the X DoF. We then pass them into the second beam splitter, filtering out and retaining the six asymmetric combinations
We emphasize that, before sending the photons into the second beam splitter, for the reason discussed in the main text we use teleportationbased QND measurement to ensure the two photons can be fed into the subsequent cascaded interferometers. Here the QND should preserve the quantum information in the Y and Z DoFs. Thus, quantum teleportation of two DoFs of a single photon is required (Extended Data Fig. 2a), which is exactly what we demonstrated in the experiment presented in the main text.
After that, we again pass the photons through two filters on the Y DoF, one set in the state and the other set in the state, which leaves four asymmetric combinations:
Similarly, we perform a bitflip operation on the Y DoF on one of the arms, erasing the information in the Y DoF. We then do a QND measurement on the Z DoF, and pass the two photons into the third beam splitter, finally filtering out the only remaining asymmetric state:
By detecting one and only one photon in the output of the third splitter, we can thus discriminate this particular hyperentangled state, , from the 64 hyperentangled Bell states on three DoFs.
To experimentally demonstrate the teleportation of three DoFs, the scheme would need in total ten photons (or five entangled photon pairs from SPDC), which is within the reach of nearfuture experimental abilities, given the recent advances in highefficiency photon collection and detection. It is obvious that the above protocol can be extended to more DoFs as displayed in Extended Data Fig. 2c.
Feedforward scheme for spin–orbit composite states
To realize a deterministic teleportation, feedforward Pauli operations on the teleported particle based on the intrinsically random Bellstate measurement results are essential. In our present experiment, no feedforward has been applied. Here we briefly describe how this can be done for the spin–orbit composite state. For the SAM qubits, active feedforward has been demonstrated before using fast electrooptical modulators^{39, 41} (EOMs). To take advantage of this technology, which has been demonstrated to have highspeed operation and high gate fidelity, we use a coherent quantum SWAP gate between the OAM and SAM qubits. The SWAP gate is defined as
The operation sequence for the feedforward operation on the spin–orbit composite states is shown in Extended Data Fig. 3a. First, the SAM feedforward is done with an EOM. Second, the SAM and OAM qubits undergo a SWAP operation. Third, an EOM is used to operate on the ‘new’ SAM that is converted from the OAM. We note that the EOM does not affect the OAM, and so the previous operation on the SAM is unaffected. Lastly, a final SWAP gate converts the SAM back to OAM, which completes the feedforward for spin–orbit composite states.
The SWAP gate is composed of three CNOT gates (Extended Data Fig. 3b). In the first and third CNOT gates (blue shading), the SAM qubit is the control qubit that acts on the OAM as target qubit, realized by sending a photon through, and recombined at a PBS. On the reflection arm only, a Dove prism is inserted to induce an OAM bit flip. In the second CNOT (yellow shading), the control and target qubits are respectively the SAM and OAM qubits, as explained in the Methods section ‘Dualchannel and efficient OAM measurement’. The extra Dove prism and HWP in front of the PBS are used to compensate for the phase shift inside the Sagnac interferometer.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China, the Chinese Academy of Sciences and the National Fundamental Research Program (grant no. 2011CB921300).
Author information
Affiliations

Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
 XiLin Wang,
 XinDong Cai,
 ZuEn Su,
 MingCheng Chen,
 Dian Wu,
 Li Li,
 NaiLe Liu,
 ChaoYang Lu &
 JianWei Pan

CAS Centre for Excellence and Synergetic Innovation Centre in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
 XiLin Wang,
 XinDong Cai,
 ZuEn Su,
 MingCheng Chen,
 Dian Wu,
 Li Li,
 NaiLe Liu,
 ChaoYang Lu &
 JianWei Pan
Contributions
C.Y.L. and J.W.P. had the idea for and designed the research; X.L.W., X.D.C., Z.E.S., M.C.C., D.W., L.L., N.L.L. and C.Y.L. performed the experiment; X.L.W., M.C.C., C.Y.L. and J.W.P. analysed the data; C.Y.L. and J.W.P. wrote the paper with input from all authors; and N.L.L., C.Y.L. and J.W.P. supervised the whole project.
Competing financial interests
The authors declare no competing financial interests.
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XiLin Wang
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MingCheng Chen
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Dian Wu
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Li Li
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NaiLe Liu
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Extended data figures and tables
Extended Data Figures
 Extended Data Figure 1: Hong–Ou–Mandel interference of multiple independent photons encoded with SAM or OAM. (263 KB)
a, Interference at the PBS where input photons 1 and 2 are intentionally prepared in the states (orthogonal SAMs; open squares) and (parallel SAMs; solid circles). The y axis shows the raw fourfold (the trigger photon t and photons 1, 2 and 3) coincidence counts. The extracted visibility is 0.75 ± 0.03, calculated from V(0) = (C_{+} − C_{//})/(C_{+} + C_{//}), where C_{//} and C_{+} are the coincidence counts without any background subtraction at zero delay for parallel and, respectively, orthogonal SAMs. The red and blue lines are Gaussian fits to the raw data. b, Twophoton interference on beam splitter 1, where photons 1 and 4 are prepared in orthogonal OAM states. The black line is a Gaussian fit to the raw data of fourfold (the trigger photon and photons 1, 4 and 5) coincidence counts. The visibility is 0.73 ± 0.03, calculated from V(0) = 1 – C_{0}/C_{∞}, where C_{0} and C_{∞} is the fitted counts at zero and, respectively, infinite delays. c, Twophoton interference at beam splitter 2, where input photons 1 and 5 are prepared in the orthogonal OAM states. The black line is a Gaussian fit to the data points. The interference visibility is 0.69 ± 0.03 calculated in the same way as in b. Error bars, 1 s.d., calculated from Poissonian counting statistics of the raw detection events.
 Extended Data Figure 2: A universal scheme for teleporting N DoFs of a single photons. (75 KB)
a, A scheme for teleporting two DoFs of a single photon using three beam splitters, which is slightly different from the one presented in the main text using a PBS and two beam splitters. Through the first beam splitter, six asymmetric states, and , can result in one photon in each output, which is ensured by teleportationbased QND on the Y DoF. After passing the two photons through the two filters that project them into the and states for the X DoF, four states, and , survive. Through the second beam splitter, only the asymmetric state of the Y DoF can result in one photon in each output. Finally we can discriminate the state from the 16 hyperentangled Bell states. b, Teleportation of three DoFs of a single photons (Methods). Note that to ensure that there is one and only one photon in the output of the first beam splitter, we can use the teleportationbased QND on two DoFs in a (dashed circle). c, Generalized teleportation of N DoFs of a single photons. The hBSM on N DoFs can be implemented as follows: (1) the beam splitter postselects the asymmetric hyperentangled Bell states in N DoFs which contain an odd number of asymmetric Bell states in one DoF, (2) two filters and one bitflip operation erase the information on the measured DoF and further postselect asymmetric states, and (3) teleportationbased QND.
 Extended Data Figure 3: Active feedforward for spin–orbit composite states. (70 KB)
a, The active feedforward scheme. This composite active feedforward could be completed in a stepbystep manner. First, we use an EOM to implement the active feedforward for SAM qubits. It is important to note that EOM does not affect OAM. Second, we use a coherent quantum SWAP gate between the OAM and SAM qubits. The original OAM is converted into a ‘new’ SAM, whose active feedforward operation is done by a second EOM. Then the OAM and SAM qubits undergo a second SWAP operation and are converted to the original DoFs. b, The quantum circuit for a SWAP gate between the OAM and SAM qubits. The SWAP gate is composed of three CNOT gates: in the first and third CNOT gates, the SAM and OAM qubits act as the control and target qubits, respectively, whereas in the second CNOT gate this is reversed.
Additional data

Extended Data Figure 1: Hong–Ou–Mandel interference of multiple independent photons encoded with SAM or OAM.Hover over figure to zoom

Extended Data Figure 2: A universal scheme for teleporting N DoFs of a single photons.Hover over figure to zoom

Extended Data Figure 3: Active feedforward for spin–orbit composite states.Hover over figure to zoom