# Simultaneous teleportation of composite quantum states

In the science fiction dream of teleportation,^{1} an object is transported by disintegrating in one place and reappearing intact in another distant location. When only classical information is of interest, or when the object can be fully characterized by classical information (which can, in principle, be measured precisely), the object can be perfectly reconstructed (i.e., copied) in a remote location from the measurement results. The properties of microscopic quantum systems (e.g., single electrons, atoms, or molecules), however, are described by quantum wave functions that may be in superposition states. Furthermore, perfect measurement, or cloning, of unknown quantum states is impossible under the laws of quantum mechanics and a quantum teleportation scheme was proposed in 1993 to circumvent this problem.^{2} In this quantum teleportation method—given a classical communication channel, as well as a quantum channel of shared entangled states—arbitrary unknown quantum states can be transferred from a sender to a spatially distant receiver, without actual transmission of the object itself. Quantum teleportation now attracts much attention—not just from the quantum physics community (i.e., as a key element in long-distance quantum communication technology, distributed quantum networks, and quantum computation)—but also from more general audiences because of its connection to science fiction.

Although extensive efforts have been made to experimentally demonstrate teleportation in various physical systems (e.g., for photons,^{3} atoms,^{4} ions,^{5} electrons,^{6} and superconducting circuits^{7}) all of these studies shared a fundamental limitation, i.e., only one degree of freedom (DOF) could be transferred during the teleportation process. The transfer of only one DOF is insufficient for complete teleportation of an object (which can naturally possess several DOFs). For example, even in the simplest case of a single photon, the elementary quanta of electromagnetic radiation has many intrinsic properties. These include its frequency, momentum, polarization, and orbital angular momentum. A hydrogen atom (i.e., the simplest atom) also has several properties (e.g., principle quantum number, as well as the spin and orbital angular momentums of its electron and nucleus), and various couplings between these DOFs, which can cause hybrid entangled quantum states.

We have, therefore, started to develop a new approach for the simultaneous teleportation of composite quantum states of single photons.^{8} Such complete teleportation of an object requires that all information, i.e., in various DOFs, is transferred over a distance. As quantum teleportation is a linear operation that can be applied to quantum states, it is theoretically possible to achieve teleportation of multiple DOFs. Before now, however, it has been difficult to experimentally demonstrate this process because it is a challenge to coherently control multiple quantum bits (qubits) and DOFs. In our method, we encode the quantum states in both polarizations, i.e., spin angular momentum (SAM) and orbital angular momentum (OAM). In addition, we prepare hyper-entangled states (simultaneous entanglement among multiple DOFs) in both these DOFs as the quantum channel for teleportation.

Our linear optical scheme for teleporting the spin–orbit composite state of a single photon is illustrated in Figure 1. We implement a hyper-entangled Bell-state measurement (h-BSM) in a step-by-step manner using a combination of two separate BSMs. Bell-state (i.e., the most simple type of entanglement) measurements normally require coherent interactions between independent qubits. This can become difficult with multiple DOFs (because it is necessary to measure one DOF without disturbing any others). In the first stage of our method, two photons (1 and 2) are sent through a polarizing beam splitter. We then post-select the event where there is only one photon in each output. Such an event is only possible when the two input photons have the same SAM. At both outputs of the polarizing beam splitter (PBS), we add two polarizers that project the two photons on a diagonal basis. The PBS in our setup does not preserve the OAM because the reflection that occurs at the PBS flips the sign of the OAM qubit. Both the SAM and the OAM must, therefore, be taken into account (as a molecular-like coupled state) in our technique. Together, the combination of the PBS and two polarizers allow us to select four states out of the total 16 hyper-entangled states.

We then perform a BSM on the remaining OAM qubit and the two single photons that emerge from the PBS are superposed on a beam splitter. Only the asymmetric Bell state leads to a coincidence detection where there is only one photon in each output. For the other three symmetric Bell states, the two input photons coalesce to a single output mode. In previous work, however, it has been suggested that it is impossible to discriminate hyper-entangled states unambiguously.^{9} Nevertheless, in our technique, we exploit a quantum non-demolition measurement, and we are able to overcome this conventional wisdom. We can thus unambiguously distinguish one hyper-entangled state from 16 different possibilities. In the final step of our process, Pauli operations (for both the SAM and OAM DOFs) are applied to the received h-BSM so that the final photon (3) is converted to the original state of photon 1.

We conducted proof-of-concept experiments to verify our quantum teleportation approach. In these experiments, we prepared five different states for teleportation. These were grouped into three different categories, i.e., product states of the two DOFs on a computational basis, product states of the DOFs on a superposition basis, and a spin–orbit hybrid-entangled state. To evaluate the performance of our teleportation, we measured the teleported state fidelity (defined as the overlap of the ideal teleported state and the measured density matrix). From our measurements, we obtained an average fidelity value of 0.63. This result is well beyond the classical limit, which is defined as the optimal state estimation fidelity on a single copy of a two-qubit system. Our results therefore illustrate the successful realization of quantum teleportation of the spin–orbit composite state of a single photon. Furthermore, for the entangled state, our teleportation fidelity result (0.57) exceeds the threshold of 0.5 and thus proves the presence of entanglement. As such, we have demonstrated that the hybrid entanglement of different DOFs inside a quantum particle can be preserved during teleportation.

We have begun development of a new quantum teleportation technique, in which multiple DOFs of single photons can be simultaneously transferred. In addition, we have experimentally demonstrated the success of our approach. Our experimental results show that the teleported state fidelity exceeds both the classical limit and the entanglement threshold. In principle, our methods can be generalized to even more DOFs (e.g., involving the photon's momentum, time, and frequency). It should also be possible to enhance the efficiency of our teleportation by using more ancillary entangled photons, quantum encoding, embedded teleportation tricks, and high-efficiency single-photon detectors. Furthermore, our multi-DOF teleportation protocol can also be applied to other quantum systems such as trapped electrons, atoms, and ions. We plan to test these applications in the near future. Besides the fundamental interest of our work, the methods we develop can open up new possibilities for quantum technologies.

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