Quantum teleportation is a linear operation applied to quantum states, and so teleporting multiple degrees of freedom (DoFs) should be possible in theory¹. 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 right-handed and left-handed 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 hyper-entangled 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 hyper-entangled Bell states.

A crucial step in the teleportation is to perform a two-particle joint measurement of photons 1 and 2, projecting them onto the basis of the 16 orthogonal and complete hyper-entangled Bell states, and discriminating one of them; for example

This process is referred to as hyper-entangled Bell state measurement (h-BSM). After the h-BSM 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 hyper-entangled Bell states (Methods). The h-BSM results can be broadcast as four-bit 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 h-BSM, 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 work¹⁴ has suggested that it is impossible to discriminate the hyper-entangled 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 h-BSM is implemented in a step-by-step 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 post-select 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 two-dimensional 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 OAM-preserving¹⁵, 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 molecular-like 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 hyper-entangled 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 output¹⁶, 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 hyper-entangled 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 non-demolition (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 two-photon 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, two-photon 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 hyper-entangled 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 computing⁵.

Figure 2 shows the experimental set-up for quantum teleportation of the spin–orbit composite state of a single photon. Passing a femtosecond-pulsed laser through three type-I β-barium 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 hyper-entangled state . The third pair (4–5) is prepared in the ancillary OAM-entangled state for the teleportation-based 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 single-qubit rotations, using wave plates for SAM and spiral phase plates (SPPs) or binary phase plates (BPPs) for OAM. The entangled state is generated by counter-propagating a SPP inside a Sagnac interferometer (Methods).

The implementation of the h-BSM and the QND measurement requires Hong–Ou–Mandel-type interference²² 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 down-converted photons (~448 fs), which is stretched by narrowband spectral filtering (~3 nm). A step-by-step verification of the two-photon 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 SAM-encoded photons at the PBS, and visibilities of 0.73 ± 0.03 and 0.69 ± 0.03 for the interferences of two OAM-encoded 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 high-brightness, hyper-entangled photons and designing dual-channel and high-efficiency 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 four-photon coincidence after the h-BSM, 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, two-photon 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 state-estimation fidelity on a single copy of a two-qubit system²⁷. 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 entanglement²⁸, 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 hyper-entangled pair, which would lead to the protocol of hyper-entanglement swapping²⁹. 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 h-BSM (1/32 in the present experiment), can be enhanced by using more ancillary photons, quantum encoding, embedded teleportation tricks, high-efficiency single-photon detectors and active feed-forward, in a spirit similar to the Knill–Laflamme–Milburn scheme⁵. We did not implement the feed-forward in the current experiment, but it could be done using electro-optical 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 multi-DoF teleportation protocol is by no means limited to this system, but can also be applied to other quantum systems such as trapped electrons¹², atoms⁹ 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 Bell-state analysis²³ and alignment-free quantum communication³⁰ possible, for example. Carrying multiple DoFs on photons can enhance the information capacity in quantum communication protocols such as quantum super-dense coding²³. Moreover, combining the entanglement of multiple (M) photons and high (N) dimensions from the multiple DoFs would allow the generation of hyper-entanglement 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.

The protocol for teleporting a spin-orbit composite quantum state

The combined state of photons 1, 2 and 3 can be rewritten in the basis of the 16 orthogonal and complete hyper-entangled 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 hyper-entangled Bell state measurement (h-BSM) on photons 1 and 2 to unambiguously distinguish one from the group of 16 hyper-entangled Bell states, Alice can project photon 3 onto one of the 16 corresponding states. After Alice tells Bob her h-BSM result as four-bit classical information via a classical communication channel, Bob can convert the state of his photon 3 into the original state by applying appropriate two-qubit local unitary transformations .

Two-photon 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 OAM-preserving, because the reflection flips the sign of OAM. Therefore, the output state for each of the input 16 hyper-entangled 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 hyper-entangled 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 OAM-preserving, 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 dual-channel OAM readout devices to measure and simultaneously. Therefore, the efficiency of both QND and BSM on OAM is 1/2. The overall efficiency of h-BSM 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 β-barium borate (BBO) crystals (Fig. 2) to generate three photon pairs through type-I spontaneous parametric down-conversion (SPDC). The first and the third crystals are 2 mm-thick BBOs, whereas the second one consists of two 0.6 mm-thick contiguous type-I BBO crystals with optic axes aligned in perpendicular planes. All of the down-converted photons have central wavelengths of 788 nm. The first photon pair (1–t) is initially prepared in the zero-order OAM mode with a coincidence count rate of In SPDC, the zero-order 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 first-order mode), owing to OAM conservation in the type-I SPDC. It has a count rate of and a state fidelity of 0.95. The third photon pair (4–5) is created in the first-order OAM entangled state with a two-photon 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 zero-order OAM. It is then sent into a Sagnac interferometer that consists of a PBS and an SPP, where the counter-propagating and SAMs with zero-order 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.

Two-photon interference on the PBS and beam splitters

For a test of two-photon 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 two-photon interferences on beam splitters 1 and 2 are for the teleportation-based 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 two-photon interference as a function of temporal delay is shown in Extended Data Fig. 1b, c.

Dual-channel and efficient OAM measurement

One of the most frequently used OAM measurement devices in the previous experiments is off-axis 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 six-photon coincidence count rate that scales as p⁶, more challenging than the scaling (p²) in the previous two-photon 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 ‘dual-channel’ 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 inside³² (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 h-BSM. In this way, two of the four OAM Bell states can be discriminated using a beam splitter.

Whereas the first type is like a two-channel readout device (like a PBS for SAM), the second type is like a one-channel 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 zero-order OAM mode so that the photon can be coupled into a single-mode fibre, while all other higher-order OAM modes will be rejected. Here we use an SPP³³ and a BPP³³ for efficient OAM readout.

The SPP³³ 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 first-order subspace with the topological charge l being 1. The conversion between the OAM zero mode and the first-order modes ( and ) is realized using a 16-phase-level SPP with an efficiency of ~97%.

For the coherent transformation between the OAM zero mode and the superposition states and , a 2-phase-level BPP^{33, 34} is used with an efficiency of ~80%. These high-efficiency 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 down-conversion; partial distinguishability of the independent photons that interfere at the PBS (~5%) and the beam splitters (~5%); state measurement error due to zero-order OAM leakage (~2%); fidelity imperfection of entangled photon pairs 2–3 (~5%) and 4–5 (~9%); and fidelity imperfection of the to-be-teleported single-photon 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 two-photon interference at beam splitters 1 and 2 (for the QND and OAM Bell-state measurements, respectively) degrade the teleportation fidelity for all states by ~13%. Third, the imperfect state measurements mainly due to the zero-order OAM leakage cause a degradation of ~2%.

Some error sources can have different effects on different teleported states. The imperfect two-photon 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 two-photon 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 non-polarizing 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 photons⁴⁰ do not suffer the problem of double pair emission. We also plan to develop bright OAM-entangled photons with higher fidelity, and a more precise 32-phase-level SPP for the next experiment of hyper-entanglement 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 hyper-entangled 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 h-BSM, identifying 1 out of the 64 states. The required resources include photon pairs entangled in the Z DoF, pairs hyper-entangled in the Y–Z DoFs, pairs hyper-entangled 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 non-polarizing beam splitter and single-photon 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, molecular-like 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 hyper-entangled 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 bit-flip 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 teleportation-based 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 bit-flip 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 hyper-entangled state, , from the 64 hyper-entangled 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 near-future experimental abilities, given the recent advances in high-efficiency photon collection and detection. It is obvious that the above protocol can be extended to more DoFs as displayed in Extended Data Fig. 2c.

Feed-forward scheme for spin–orbit composite states

To realize a deterministic teleportation, feed-forward Pauli operations on the teleported particle based on the intrinsically random Bell-state measurement results are essential. In our present experiment, no feed-forward has been applied. Here we briefly describe how this can be done for the spin–orbit composite state. For the SAM qubits, active feed-forward has been demonstrated before using fast electro-optical modulators^{39, 41} (EOMs). To take advantage of this technology, which has been demonstrated to have high-speed 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 feed-forward operation on the spin–orbit composite states is shown in Extended Data Fig. 3a. First, the SAM feed-forward 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 feed-forward 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 ‘Dual-channel 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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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).