Quantum network coding is used to solve the congestion problem in quantum communication, which will promote the transmission efficiency of quantum information and the total throughput of quantum network. We propose a novel controlled quantum network coding without information loss. The effective transmission of quantum states on the butterfly network requires the consent form a thirdparty controller Charlie. Firstly, two pairs of threeparticle nonmaximum entangled states are preshared between senders and controller. By adding auxiliary particles and local operations, the senders can predict whether a certain quantum state can be successfully transmitted within the butterfly network based on the
In 2000, classic network coding was first proposed by Ahlswede et al. [
In 2007, Hayashi [
In addition, Satoh et al. [
When nonmaximum entangled states are used as a quantum channel, the quantum states will be transmitted with a certain probability. If transmission fails, the quantum states will be lost. Therefore, the preservation of quantum states during transmission has become an urgent problem. In 2015, Roa et al. [
Since coupling between the quantum states and the surrounding environment is inevitable in practice [
In the following sections, the paper content is organized as below. Some preliminary definitions and equations involved in our scheme will be given in Section 2. In Section 3, the implementation procedure of our controlled quantum network coding without information loss will be discussed in detail. In addition, the implementation of the quantum circuit implementation, as well as the flow chart and safety analysis for our scheme will be demonstrated in this section as well, which could be of great reference value for future researches. Finally, our conclusions will be stated in Section 4.
In our scheme, we will use a threeparticle nonmaximum entangled state as quantum channel.
where
Some singleparticle gate operations and twoparticle local operations [
The influences from the singleparticle gates on the quantum states are:
The twoparticle local operations are:
This operation is called a controlled NOT gate, in which particle
In our scheme, a controlled unitary operation is applied to ensure that quantum states are not lost.
Here,
In our scheme, controlled quantum teleportation [
Alice, Bob and the controller Charlie share a threeparticle nonmaximum entangled state
where
After that, Charlie integrates the
Therefore, in order to receive the unknown quantum state transmitted by Alice, measurement results from Alice and Charlie are both needed for Bob to realize satisfactory quantum state recovery.
In this section, we propose a controlled quantum network coding scheme without information loss. Our scheme will be discussed based on the measurement results from auxiliary particles. In addition, the flow chart and safety analysis of our scheme will also be given.
In our scheme, a thirdparty controller Charlie is added. As is shown in
In our scheme, the threeparticle nonmaximum entangled states, which are prepared by Charlie, are preshared between the senders and Charlie on the butterfly network. Two pairs of threeparticle nonmaximum entangled states, namely
Firstly, in local operations, the combined state of the unknown state
Sender
Sender
After obtaining
After that,
Secondly, in the stage of encoding,
Subsequently,
When the measurement result is
When the measurement result is
Particularly,
Measurement result  Classic bit 
Unitary operation 

0  
1 
Measurement result  Classic bit 


00  
01  
10  
11 
With the help of
Thirdly, in the transmission stage,
Finally, in the decoding stage,
Next, we present an implementation of our scheme on a quantum circuit. As is shown in
Specifically, only senders are controlled by the controller Charlie, which is a feature of this scheme. The receiver only needs to perform unitary operations on the quantum states it received according to
In our scheme, the measurement results of the auxiliary particle
However, the quantum states will not be lost.
When the measurement given by one party on its own auxiliary particle is
Here, Charlie reprepares one threeparticle nonmaximum entangled state, and distributes the particles to the senders for retransmission of the quantum state. The party who failed in the beginning joins a new round of our scheme until Charlie receives the measurements of its auxiliary particles given by both senders are
In order to demonstrate our scheme more clearly, we hereby give a flow chart in
Quantum network coding is used to solve the congestion problem in the transmission of quantum information, as well as to improve the transmission efficiency, increase network throughput and promote network security. In our scheme, if the sender wants to send a quantum state to the receiver, it needs the consent from a third party Charlie for effective transmission. Therefore, with our scheme, an eavesdropper Eve shall not obtain the original quantum information. In addition, we have not considered inevitable information destruction.
In our scheme, it is Charlie’s responsibility to prepare the threeparticle nonmaximum entangled states and distribute the auxiliary particles to the senders. Information security in this procedure is guaranteed by the BB84 protocol [
To summarize the analysis above, as long as the coding table, which is not seen during the transmission, is not leaked, our scheme is secure. Therefore, our scheme ensures sufficient information security against external eavesdroppers.
In the paper, we propose a controlled quantum network coding without information loss by the employment of threeparticle nonmaximum entangled states on the butterfly network. In our scheme, a third party Charlie is necessary as the controller for perfect and cross transmission of quantum states.
Compared with previous schemes, our scheme is advantageous in several aspects. First of all, compared with the scheme in [
As for the future prospects, we hope that our scheme can be applied in practice. Moreover, this scheme could be extended to a quantum kpair butterfly network to achieve perfect, cross and controlled transmission of k quantum states with further researches. We also hope that our work can contribute to the development of quantum communication [