Quantum teleportation, which is the transfer of an unknown quantum state from one station to another over a certain distance with the help of nonlocal entanglement shared by a sender and a receiver, has been widely used as a fundamental element in quantum communication and quantum computation. Optical fibers are crucial information channels, but teleportation of continuous variable optical modes through fibers has not been realized so far. Here, we experimentally demonstrate deterministic quantum teleportation of an optical coherent state through fiber channels. Two sub-modes of an Einstein-Podolsky-Rosen entangled state are distributed to a sender and a receiver through a 3.0-km fiber, which acts as a quantum resource. The deterministic teleportation of optical modes over a fiber channel of 6.0 km is realized. A fidelity of 0.62 ± 0.03 is achieved for the retrieved quantum state, which breaks through the classical limit of 1 / 2 . Our work provides a feasible scheme to implement deterministic quantum teleportation in communication networks.

Here, we report the first experimental realization of CV quantum teleportation through optical fiber channels. Two sub-modes of a CV entangled state of light are distributed to the sender (Alice) and the receiver (Bob) through a 3.0-km-long optical fiber which means that the total fiber length between Alice and Bob is 6.0 km. The fidelity of the retrieved coherent state is about 0.62 ± 0.03, which is higher than the classical limit of 1 / 2 . Furthermore, a fidelity of 0.69 ± 0.03 breaking through the no-cloning limit of 2 / 3 ( 28 , 29 ) has also been achieved when the transmission distance is 2.0 km.

Quantum teleportation is a reliable protocol for transferring quantum states under the help of entanglement. At first, two subsystems of a prepared entangled state are distributed to a sender and a distant receiver. Then, an input quantum state is jointly measured together with one-half of the entangled state hold by the sender. Successively, the measurement results are transmitted to the receiver through classical channels. Last, the other half of the entangled state hold by the receiver is transformed via a basic operation with the received measurement results to retrieve the teleported state. During the teleporting process, the quantum state is not transferred directly; only its quantum and classical information are sent to the receiver by means of quantum entanglement and classical channels, respectively. The input quantum state is destroyed by the joint measurement at the sending station and retrieved at the receiving station; thus, the purification of the input state will not be influenced by the loss and extra noise in the transmission channels ( 1 ). Since Bennett et al. proposed quantum teleportation in 1993, various researches on theoretical analysis and experimental implementation have been successively completed ( 1 – 6 ). Quantum teleportation serves as the cornerstone for building quantum information networks, and it also greatly contributes to completing quantum computation and quantum communication ( 7 – 9 ). A variety of quantum information protocols, such as entanglement swapping, quantum repeaters, quantum teleportation networks, quantum gate teleportation, and quantum computation, have already been realized by applying quantum teleportation ( 10 – 17 ). Because of the relatively simple generation system and negligible decoherence from noise environment, single-photon qubits have become important physical carriers to realize quantum teleportation over long distances ( 18 – 23 ). In 2003, Gisin and colleagues accomplished quantum teleportation of qubits with a 2.0-km standard telecommunication fiber, in which the transmission distance of a quantum state on the order of kilometers was first reached in the discrete variable region ( 18 ). Then, quantum teleportation over 100 km was implemented by Pan’s and Zeilinger’s groups separately ( 19 , 20 ). Very recently, by means of a low–Earth orbit satellite, ground-to-satellite quantum teleportation with a single photon over 1400 km was achieved, which provided a feasible protocol to realize quantum communication at a global scale ( 23 ). Although great progress has been made for demonstrating quantum teleportation of photonic qubits, a probabilistic generation method forms an obstacle to develop instantaneous transfer of quantum states without post-selection. Thus, it is necessary to explore near-deterministic quantum teleportation protocols in quantum communication and teleportation-based quantum computation ( 9 ). Continuous variable (CV) quantum teleportation of optical modes, which is based on entangled states of light, can realize unconditional and deterministic transfer of arbitrary unknown quantum states ( 3 , 4 ). Transfer and retrieval for both coherent and nonclassical states, such as squeezed state, entangled state, photonic quantum bits, and Schrödinger’s cat state, have been experimentally realized with CV quantum teleportation method in free space ( 10 , 24 – 27 ). Nevertheless, all these quantum teleportation experiments in the CV region are implemented in laboratories, and the transmission distance is very short. For practical applications of CV quantum teleportation, a key challenge is to extend possible transfer distance.

RESULTS

Schematic of CV fiber-channel quantum teleportation The schematic for CV quantum teleportation through fiber channels is shown in Fig. 1, which includes a resource station for providing an Einstein-Podolsky-Rosen (EPR) entangled state of light, a sending station (Alice), and a receiving station (Bob). These stations are connected by optical fibers that are used as quantum channels. In quantum optics theory, a coherent state is defined as the eigenstate of an annihilation operator, and its dynamics most closely resembles that of a classical harmonic oscillator. The expectation values of amplitude and phase quadrature operators of a coherent state are equal to their classical values. The coherent state is a minimum uncertainty state, and its uncertainty is split equally between two quadrature components. It is the closest quantum approximation of an optical field generated by a laser; thus, the coherent state is usually selected as the input state in CV quantum optical experiments (4). In the experiment, EPR entanglement is obtained by coupling two single-mode squeezed states of light, which are generated from a pair of DOPAs operating below their oscillation threshold, at a 50/50 beam splitter (50/50 BS). The wavelength of EPR entangled optical modes is chosen at 1.34 μm (30), which is close to the transmission window of the commercial fiber at 1.3 μm. The amplitude quadrature ( ) and the phase quadrature ( ) of two sub-modes of an EPR entangled state (EPR1 and EPR2) are expressed by and , respectively, where â and â+ are annihilation and creation operators of the electromagnetic field, respectively. There are strong quantum correlations between quadrature components of two EPR sub-modes; i.e., the correlation variances of both quadrature amplitude sum and quadrature phase difference are lower than the corresponding quantum noise limits (QNLs), where r (0 ≤ r < ∞) is the correlation factor (30). Meanwhile, the anti-squeezing quantum noise levels of amplitude difference and phase sum are much higher than the corresponding QNLs (8). During quantum teleportation, two sub-modes of the EPR entangled state are first sent to Alice and Bob through an optical fiber. Then, the sub-mode received by Alice (â EPR1 ) and the unknown quantum state (input state, ) are combined on a 50/50 BS. The amplitude quadrature and the phase quadrature of two output fields of the 50/50 BS are measured by two sets of balanced homodyne detectors (D x and D p ) with local oscillators (LOs) (local x and local p), respectively. These joint measurements of the input state and the sub-mode EPR1 provide an analogy of Bell-state measurement in the CV region (5, 31, 32). If a perfect EPR entangled state (r→∞) is used, then Alice will not be able to obtain any information about the input state. The results measured by Alice (i x , i p ) are transmitted to Bob via two classical channels. Bob modulates his own sub-mode of the EPR entangled state (â EPR2 ) with the received measurement results, which is realized by means of an amplitude electro-optic modulator (AM) and a phase electro-optic modulator (PM). In this way, the input quantum state destroyed by the joint measurements at Alice is recovered by Bob under the help of nonlocal quantum entanglement (3). Last, Victor performs the verification measurements of teleportation results with a homodyne detector (D V ). Fig. 1 Experimental scheme of fiber-channel CV quantum teleportation. Two single-mode squeezed states generated by a pair of degenerate optical parametric amplifiers (DOPAs) are coupled to produce an EPR entangled state. The two sub-modes of the EPR entangled state are sent to Alice and Bob through two optical fiber channels, respectively. Then, Alice implements a joint measurement on the unknown input state and the sub-mode EPR1 and sends the measured results to Bob through classical channels. Bob implements a translation for EPR2 by coupling a coherent beam, which is modulated by two joint-measured classical signals, respectively, via an AM and a PM. Last, Victor accomplishes the verification for quantum teleportation. 98/2 BS, beam splitter with reflectivity of 98%; HR, mirror with a reflectivity larger than 99.9%; fiber coupler, used to couple optical modes into the fiber; BHD, balance homodyne detector.

Fidelity of recovered quantum state In quantum teleportation experiments, the output state of Bob is sent to Victor to verify whether quantum teleportation has been successfully implemented. Fidelity F is usually used to quantify the performance of quantum teleportation (29) (1)which represents the overlap between the input state |ψ in 〉 and the output state characterized by the density matrix ρ out . If detectors with perfect unitary efficiencies are used in the experiment, then the fidelity of quantum teleportation for a coherent input state is expressed by (31) (2)where (3) σ Q is the variance of the teleported state in representation of the Q function, which depends on fluctuation variances of amplitude and phase quadratures ( and ). β in and β out are amplitudes of the input state at Alice and the output state at Bob, respectively. g is the gain factor of the classical channels, which usually has an equivalent value for amplitude (g x ) and phase (g p ) quadratures. In the fiber-channel quantum teleportation system, the influence of transmission efficiency and extra noise inside fibers cannot be neglected because of their observable effect on quantum features of the entanglement. Thus, the coupling efficiency of the fiber coupler (η C ) and the transmission efficiency in fiber (η F ) have to be involved in the calculation of fidelity. The extra noises resulting from fiber channels will reduce entanglement and thus decrease the distance of quantum teleportation. In general, the potential sources of noise in fibers include guided acoustic wave Brillouin scattering (GAWBS), Rayleigh scattering, Raman scattering, and so on. Because of its scattering level and impact frequency range, the extra noise generated by the GAWBS forms a notable thermal noise in fiber channels, the effect of which on quantum entanglement distribution and quantum communication is notable and thus has to be considered (33, 34). In our experiment, a sub-mode of the EPR entangled state and the LO beam are simultaneously transferred in an optical fiber of length l with polarization multiplexing to conveniently lock their relative phase. The depolarized GAWBS scatters some horizontal polarization photons of the LO beam into the signal beam with vertical polarization, which constitutes a thermal-noise source: , where ξ is the scattering efficiency per kilometer of fiber and is the average photon number of the corresponding LO beam (34). The imperfect detection efficiencies ( ) and finite EPR entanglement have to be considered as well. Hence, the variances ( and ) measured by the verifier Victor are expressed by (4)where (5) is the reflectivity of the coupling mirror (M B ) in Bob’s station. g x (g p ) is the gain factor of the classical channel for the amplitude (phase) quadrature component of the input state. In our experiment, the teleportation gains of amplitude quadrature and phase quadrature always take the same values (g x = g p = g) because the two quadrature components are symmetric. The transmission efficiency of EPR sub-modes in the optical fiber consists of the coupling efficiency η C = 0.9 of the fiber coupler and the transmission efficiency inside the fiber. and are quantum efficiencies of the photoelectric detectors in Alice’s and Victor’s stations, respectively. With the increasing power of the LO beam, the induced GAWBS extra noise is enhanced, and thus, the quantum entanglement between the sender (Alice) and the receiver (Bob) is decreased. Obviously, the transmission distance is notably influenced by the GAWBS extra noise. The dependences of fidelities of quantum teleportation on the communication distance between Alice and Bob for different powers of LO beams are shown in Fig. 2, where the actual physical parameters of our experimental system are applied in the calculation. The blue, red, yellow, and green curves express the calculated dependences of fidelities on the communication distance between Alice and Bob when the powers of LO beams are 0.25, 0.50, 1.00, and 2.50 mW, respectively. If the fidelity of quantum teleportation decreases below the classical limit of 1/ 2 , then the quantum teleportation is unsuccessful. The squares mark the experimental results, which are in reasonable agreement with the theoretical values. Fig. 2 Fidelity of quantum teleportation versus the communication distance between Alice and Bob for different powers of the LO beam. The blue, red, yellow, and green curves express the calculated dependences of fidelities on the communication distance between Alice and Bob when the power of the LO beam is 0.25, 0.50, 1.00, and 2.50 mW, respectively. With the increasing power of the LO beam, the induced GAWBS extra noise gradually reduces the quantum entanglement between the sender (Alice) and the receiver (Bob). If the fidelity of quantum teleportation decreases below the classical limit of 1/ 2 , then the process is unsuccessful. It can be seen that the fidelity drops quickly when the power of the LO beam is increased because the LO beam with higher power induces more GAWBS noise in the fiber channels. The squares mark the experimental results, which are in reasonable agreement with the theoretical values. Error bars represent the SE and are obtained from the statistics of the fidelity.