Secure Continuous Variable Quantum Cloning Based on EPR Steering

Quantum communication is based on the three principles of uncertainty, measurement collapse, and no-cloning in quantum mechanics. Compared with traditional classical communication methods, quantum communication features security and high efficiency and has great application significance and prospect in information security. In recent years, domestic and international scientists have conducted a lot of research on theories and experiments and made outstanding achievements in long-distance transmission and practical network of quantum communication. Quantum teleportation and quantum cloning have caught extensive attention as important protocols in quantum communication. With the help of quantum entanglement and classical communication, the transmission of any unknown quantum state from one location to another can be realized. As important resources of quantum information, quantum entanglement and EPR steering are widely adopted in various quantum communication tasks. The natural asymmetry of EPR steering makes quantum steering a helpful resource in various quantum information processes. In the tasks of single-side device-independent quantum-key distribution, secure quantum teleportation, and subchannel discrimination, quantum steering can improve key acquisition rate, and enhance the protocol efficiency and security. In 2000, Cerf N J et al. proposed quantum cloning of Gaussian states with continuous variables and gave the fidelity boundary of quantum cloning as 2/3. In 2001, the Grangier P group presented the quantum and classical fidelity boundary of coherent state continuous variable quantum cloning under Heisenberg representation. For coherent state input, quantum teleportation is achieved when the fidelity exceeds the classical limit of 1/2, which is the best value that can be obtained without entanglement. However, it is necessary to have certain requirements for entangled beams to realize quantum teleportation with a fidelity greater than 2/3. In 2004, the Furusawa group applied three single-mode OPOs to obtain a continuous variable quantum teleportation network with an optimal fidelity of 0.64, and then they utilized four OPOs to achieve quantum teleportation with a fidelity of 0.7. In 2012, Pan J W group experimentally realized long-distance quantum teleportation. In 2018, Wei J H et al. put forward a quantum teleportation scheme using non-maximum entangled states for measurement. In 2018, Wang K et al. studied teleportation by partially entangled GHZ states. The analysis based on quantum cloning shows that for coherent state inputs, secure teleportation is guaranteed if the teleportation fidelity is greater than 2/3. To sum up, the research on remote transmission security is still a long-term important topic.

Methods

Based on the basic idea of quantum teleportation, we employ the method of combining quantum channel and classical channel to design a $1 \to 2$ quantum cloning scheme with continuous variables by partially disembodied transport. The relationship between the fidelity of a partially disembodied transport cloning scheme and EPR entanglement source is studied theoretically. Firstly, the fidelity of two output modes in $1 \to 2$ cloning scheme, the entanglement and steering of EPR shared entanglement source are analyzed. Secondly, the relationship between the fidelity of the output mode $C l o n e 1$ and the steering characteristics under the optimal gain is studied. Thirdly, the fidelity of the output mode $C l o n e 2$ varies with the reflectance and squeezing parameters under the optimal gain of the output mode $C l o n e 1$ .

Results and Discussions

First, we analyze the variation of the steering between entanglement sources ${\hat{b}}_{1}$ and ${\hat{b}}_{2}$ and optimal gain with $η_{1}$ and $η_{2}$ . Only if $η_{1} > 0.5$ there is a steering of ${\hat{b}}_{2}$ by ${\hat{b}}_{1}$ , and if $η_{2} > 0.5$ there is a steering of ${\hat{b}}_{1}$ by ${\hat{b}}_{2}$ . The results are as follows: when $η_{1} > 0.5$ and $η_{2} > 0.5$ , there is a two-way steering between ${\hat{b}}_{1}$ and ${\hat{b}}_{2}$ , and the entanglement amount between the sources increases with the improving transmission efficiency $η_{1}$ and $η_{2}$ . The range of optimal gain $g_{o p t} = m a x \{g_{b_{2} | b_{1}}, g_{b_{1} | b_{2}}\}$ is $\sqrt[]{2} \leq g_{o p t} < 5$ , and the optimal gain corresponds to the optimal gain of output mode $C l o n e 1$ , which is not optimal for output mode $C l o n e 2$ . Second, the fidelity of output modes $C l o n e 1$ and $C l o n e 2$ varies with $η_{1}$ and $η_{2}$ under different reflectance when the optimal gain $g_{o p t}$ is taken. The fidelity $F_{1} > \frac{2}{3}$ should be in the two-way steering region, but the fidelity of the two-way steering region may not always meet $F_{1} > \frac{2}{3}$ . Meanwhile, the fidelity of output mode $C l o n e 1$ decreases with the increasing reflectivity, and that of output mode $C l o n e 2$ reduces with the rising reflectance. Third, the fidelity of output modes $C l o n e 1$ and $C l o n e 2$ varies with $η_{1}$ and $η_{2}$ under different squeezing parameters when the optimal gain $g_{o p t}$ is taken. The fidelity of output mode $C l o n e 1$ in the two-way steering region is greater than 2/3, and the fidelity beyond the no-cloning threshold can also be achieved by two-way steering under smaller squeezing parameters. The fidelity of the output mode $C l o n e 2$ decreases with the increase in squeezing parameters.

Conclusions

In summary, we theoretically investigate the relationship between the fidelity of cloning and EPR steering based on the partially disembodied transport continuous variable $1 \to 2$ quantum cloning scheme. Meanwhile, we explore the fidelity variation with the reflectance of the beam-splitter and squeezing parameters at a given gain. The results show that for the output mode $C l o n e 1$ , when the optimal gain is obtained, the two-way steering of the entanglement source should be shared when the fidelity exceeds the no-cloning threshold, but not all two-way steering resources can make the cloning fidelity greater than 2/3. The fidelity of output mode $C l o n e 1$ decreases with the rising reflectance and decreasing squeezing parameters, and the two-way steering can also achieve fidelity beyond the no-cloning threshold under smaller squeezing parameters. Additionally, the fidelity of the output mode $C l o n e 2$ reduces with the increasing reflectance and squeezing parameters. Therefore, high cloning fidelity does not require significant squeezing and high reflectivity. Therefore, we can employ the combination of quantum channel and classical channel to improve the cloning fidelity. The two-way quantum steering state is the necessary resource for secure quantum cloning of the coherent states. The research results provide certain references for the security of quantum communication networks.