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Acta Physica Sinica

New formula for calculating the fidelity of teleportation and its applications

Fang Jia, Cunjin Liu, Yin-Quan Hu, Fan Hong‐Yi

Published Jan 1, 2016• 7 citations
Publisher

Abstract

Quantum teleportation plays an important role in quantum information science. In order to obtain the effect of quantum teleportation of a quantum state by using an entangled resource, the fidelity of teleporting the quantum state should be calculated. Braunstein and Kimble[Phys. Rev. Lett. 80 869 (1998)] derived a formula of calculating the fidelity of quantum teleportation for Gaussian entangled resource and any input state to be teleported. Then, the point is how to calculate the quantum teleportation fidelity for any entangled resource. In this paper, werealize this purpose by using the entangled state representation. First, we derive the Weyl expansion of any density operator by using the completeness relation between coherent state and P-representation. Then using the orthogonal property of entangled state representation and the traditional Kimble-Braunstein scheme of quantum teleportation, we further derive the mean density operator of the output state, which means that we establish the relation between the output density operator and the characteristic functions of the input state to be teleported and the entangled resources. The characteristic function of the output state is also derived which is in the concise form relating these two characteristic functions above. Then we further obtain a new formula for calculating the quantum teleportation fidelity for the coherent state input and any two-mode entangled resource. It is shown that the fidelity of teleportation can be easily calculated when the Q-function of the normally ordering form of entangled resource is known. This is a convenient way of obtaining the fidelity of teleportation. As its applications, some Gaussian and non-Gaussian entangled states are examined to teleport the coherent state, whose results are correct.

DOI
10.7498/aps.65.220302

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Quantum teleportation plays an important role in quantum information science. In order to obtain the effect of quantum teleportation of a quantum state by using an entangled resource, the fidelity of teleporting the quantum state should be calculated. Braunstein and Kimble[Phys. Rev. Lett. 80 869 (1998)] derived a formula of calculating the fidelity of quantum teleportation for Gaussian entangled resource and any input state to be teleported. Then, the point is how to calculate the quantum teleportation fidelity for any entangled resource. In this paper, werealize this purpose by using the entangled state representation. First, we derive the Weyl expansion of any density operator by using the completeness relation between coherent state and P-representation. Then using the orthogonal property of entangled state representation and the traditional Kimble-Braunstein scheme of quantum teleportation, we further derive the mean density operator of the output state, which means that we establish the relation between the output density operator and the characteristic functions of the input state to be teleported and the entangled resources. The characteristic function of the output state is also derived which is in the concise form relating these two characteristic functions above. Then we further obtain a new formula for calculating the quantum teleportation fidelity for the coherent state input and any two-mode entangled resource. It is shown that the fidelity of teleportation can be easily calculated when the Q-function of the normally ordering form of entangled resource is known. This is a convenient way of obtaining the fidelity of teleportation. As its applications, some Gaussian and non-Gaussian entangled states are examined to teleport the coherent state, whose results are correct.
In contrast to discrete-variable teleportation, a quantum state is imperfectly transferred from a sender to a remote receiver in a continuous-variable setting. We recall the ingenious scheme proposed by Braunstein and Kimble for teleporting a one-mode state of the quantum radiation field. By analyzing this protocol, we have previously proven the factorization of the characteristic function of the output state. This indicates that teleportation is a noisy process that alters, to some extent, the input state. Teleportation with a two-mode Gaussian EPR state can be described in terms of the superposition of a distorting field with the input one. Here we analyze the one-mode Gaussian distorting-field state. Some of its most important properties are determined by the statistics of a positive EPR operator in the two-mode Gaussian resource state. We finally examine the fidelity of teleportation of a coherent state when using an arbitrary resource state.
We study the Braunstein-Kimble protocol for the continuous variable teleportation of a coherent state. We determine lower and upper bounds for the optimal fidelity of teleportation, maximized over all local Gaussian operations for a given entanglement of the two-mode Gaussian state shared by the sender (Alice) and the receiver (Bob). We also determine the optimal local transformations at Alice and Bob sites and the corresponding maximum fidelity when one restricts to local trace-preserving Gaussian completely positive maps.
We introduce a kind of non-Gaussian entangled state, which can be obtained by operating a non-local coherent photon-subtraction operation on a two-mode squeezed vacuum. It is found that its normalization factor is only related to the Legendre polynomials, which is a compact expression. Its statistical properties are discussed by the negative region Wigner function with the analytical expression. As an application, the quantum teleportation for coherent states is considered by using the non-Gaussian state as an entangled channel. It is found that the teleportation fidelity can be enhanced by this non-Gaussian operation.
Non-Gaussian operations have been shown to enhance the fidelity of continuous-variable quantum teleportation. However, a disadvantage of these non-Gaussian operations is that they are probabilistic in nature. In this article we study the trade-off between teleportation fidelity and success probability for optimal performance of the teleportation protocol. To this end, we first derive a unified expression for the Wigner characteristic function describing three non-Gaussian states: photon-subtracted, photon-added, and photon-catalyzed two-mode squeezed vacuum states. We then utilize it to obtain the fidelity of teleportation for input coherent and squeezed vacuum states using the aforementioned non-Gaussian resource states. We optimize the product of the relative enhancement in fidelity and the probability of state preparation by tuning the transmissivity of the beam splitters involved in implementing non-Gaussian operations. This leads to a scenario that can be effectively implemented in a laboratory to enhance fidelity. It turns out that among all the non-Gaussian resource states considered, the symmetric one-photon-subtracted two-mode squeezed vacuum state is the most advantageous. We provide the associated optimal squeezing and beam splitter transmissivity values for the non-Gaussian resource states considered, which will be of significant interest to the experimental community. We also consider the effect of imperfect photon detectors on teleportation fidelity. Further, we expect the derived Wigner characteristic function to be useful in state characterization and other quantum information processing protocols.
The entanglement properties of non-Gaussian states are investigated, which are obtained by performing the photon addition, photon subtraction, photon-addition-then-subtraction, and photon-subtraction-then-addition operations on the two-mode squeezed vacuum state. We show that the partial von Neumann entropy of all the resulting states is greater than that of the original squeezed state, but only the photon-subtracted states and the photon-added-then-subtracted states have the stronger Einstein-Podolsky-Rosen correlation than the original squeezed state. Quantum teleportation of Braunstein and Kimble protocol is studied for coherent states, squeezed states, and mixed Gaussian states with the non-Gaussian entangled resources. For all the states to be teleported, the fidelity with the photon-subtracted and the photon-added-then-subtracted entangled resources is higher than that with the two-mode squeezed vacuum resource. Based on Bures fidelity, we find that quantum teleportation for mixed and classical single-mode Gaussian states is more faithful than for single-mode Gaussian states with high purity and nonclassicality.