† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant No. 61172071), the Scientific Research Program Funded by Shaanxi Provincial Education Department, China (Grant No. 16JK1711), the International Scientific Cooperation Program of Shaanxi Province, China (Grant No. 2015KW-013), and the Natural Science Foundation Research Project of Shaanxi Province, China (Grant No. 2016JQ6033).
Recently, bidirectional quantum teleportation has attracted a great deal of research attention. However, existing bidirectional teleportation schemes are normally discussed on the basis of perfect quantum environments. In this paper, we first put forward a bidirectional teleportation scheme to transport three-qubit Greenberger–Horne–Zeilinger (GHZ) states based on controled-not (CNOT) operation and single-qubit measurement. Then, we generalize it to the teleportation of multi-qubit GHZ states. Further, we discuss the influence of quantum noise on our scheme by the example of an amplitude damping channel, then we obtain the fidelity of the teleportation. Finally, we utilize the weak measurement and the corresponding reversing measurement to protect the quantum entanglement, which shows an effective enhancement of the teleportation fidelity.
Quantum teleportation (QT) plays an important role in quantum computing and quantum communication, which utilizes pre-shared quantum entanglement and classical communication to transmit an unknown quantum state from one location to another. Since the first QT protocol was introduced by Bennnet et al. in 1993,[1] it has become a research hotspot. Many modified QT schemes have been proposed, such as controlled teleportation,[2–5] probabilistic teleportation,[6–9] and bidirectional quantum teleportation (BQT).[10–13]
Having the capability of transmitting quantum information between Alice and Bob in two directions simultaneously, BQT has drawn more and more attention in recent years. Huelga et al. discussed the possibility of using BQT to implement the nonlocal quantum gates.[10,11] In 2013, Zha et al. proposed a controlled BQT scheme by using a five-qubit cluster state to transmit arbitrary single-qubit states.[12] Other similar BQT protocols have been presented, by using different types of entanglement states as the teleportation channels, such as five-qubit composite GHZ–Bell state,[13] six-qubit cluster state,[14] and genuine five-qubit entangled state.[15] After that, asymmetric controlled BQT schemes were devised to transmit a single-qubit state and a two-qubit entanglement state.[16–18] Binayak and Arpan proposed a BQT protocol to transmit arbitrary two-qubit entanglement states via a ten-qubit entangled state.[19] Hassanpour and Houshmand presented a BQT scheme to transmit pure EPR states by using a composite GHZ state.[20]
However, current research work relating to the above BQT seldom thinks about the problem of quantum noise. In a practical situation, a quantum state will couple with the environment inevitably, which breaks the coherence of the quantum entanglement state and gives rise to quantum noise. This does occur during the entanglement distribution in BQT. So, it is very important to analyze the influence of the quantum noise in the process of BQT, thus to determine whether a BQT is successful in a certain circumstance.
On the other hand, to overcome the influence of decoherence on quantum entanglement, several methods have been proposed, including quantum entanglement purification,[21–23] quantum error coding,[24–26] and decoherence-free subspace.[27–29] In recent years, weak measurement and its reversing measurement were widely studied to protect quantum state and quantum entanglement. Korotkov and Jordan demonstrated that the weak measurement of a solid-state qubit could be recovered with a reversal procedure.[30] The similar recovery methods of a superconducting phase qubit and an arbitrary cavity field with finite photon number were discussed and demonstrated in Refs. [31] and [32]. Liao et al. proposed an alternative way based on a Hadamard gate and CNOT gate to recover an arbitrary pure two-qubit entanglement state that has undergone a weak measurement.[33] Korotkov and Keane,[34] Lee et al.,[35] and Kim et al.[36] demonstrated weak measurement and the reversing measurement can protect a single-qubit state and two-qubit entanglement state from decohering the amplitude damping (AD). In Refs. [37] and [38], it was demonstrated that the weak measurement method can increase the fidelities of a three-qubit GHZ state and a W-like state in an AD channel.
In this paper, we first propose a BQT scheme to transmit three-qubit GHZ states based on the work of Ref. [20], then we generalize this scheme to teleport multi-qubit GHZ states. As an example, we use the AD channel model to analyze the influence of quantum noise on this BQT scheme and give the fidelity of the result of the teleportation. Finally, we use the weak measurement and the reversing measurement on part of the qubits of the teleportation channel state before and after the entanglement distribution. The comparison between the results shows the weak measurement method is helpful for enhancing the fidelity of the bidirectional teleportation in an AD channel.
In this section, we first discuss the bidirectional teleportation scheme to transport three-qubit GHZ states in a noiseless quantum channel. Suppose that Alice and Bob have the following three-qubit GHZ states to be teleported, respectively, which are described as
The scheme in subSection
In Eq. (
In this section, we will discuss how the noise in a quantum channel influences the procedure of bidirectional teleportation. As an example, we assume that the channel model is an AD channel. It can model many practical physical processes, such as spontaneous emission and super-conduction with zero-temperature energy relaxation. In this model, the evolutions of a single-qubit system state and the environment can be described as the following quantum transformation:[39]
We still consider the bidirectional teleportations of three-qubit GHZ states. Assume the noise mainly exists in the process of entanglement distribution in the quantum channel. To reduce the influence of quantum noise as much as possible, Bob should prepare the four-qubit GHZ state
After the bidirectional teleportation described in Subsection
According to Ref. [35], the weak measurement and the corresponding reversing measurement on a single-qubit system can be described as the following transformations:
During the entanglement distribution, only qubit
After the weak measurement, qubit
After the processes above, Alice receives qubit
With a similar derivation, the four-qubit GHZ state
Using the shared channel state in Eq. (
Performing a partial trace over the environment in Eq. (
The fidelity is an important metric to display the distance between quantum states. The fidelity between states
If state
In an AD channel, if Alice and Bob teleport the three-qubit GHZ states
In Fig.
According to Section
The comparison between
The fidelities
From Eqs. (
According to subSection
The success probability
In this work, we present a BQT scheme to transmit three-qubit and multi-qubit GHZ states. This scheme mainly utilizes CNOT operation and single-qubit measurement, thus it has a higher practicability. Unlike previous research that usually discuss BQT in an ideal quantum channel, we intended to study the influence of quantum noise on BQT. As an example, we analyze our BQT scheme in an amplitude damping channel and obtain the fidelity of the teleportation, which decreases with the amplitude damping factor increasing. In order to suppress this influence, we use the weak measurement and the reversing measurement before and after the quantum entanglement distribution, demonstrating an effective enhancement of the teleportation fidelity. On the other hand, we find that a higher fidelity needs a stronger weak measurement strength, which leads to a smaller success probability. This is a contradiction we must consider, and the solution is expected to be found in future work.
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