Generalized isometric tensor based quantum key distribution protocols of squeezed multiphoton entangled states

Isometric tensor offers a novel and powerful tool that can compress an entangled state into its tensor network state (TNS). The resulting quantum compression provides a new opportunity for enhancing quantum key distribution (QKD) protocols. The main idea explored in this work is to use the quantum compression to improve the efficiency of QKD. In a nut-shell, a collection of any multi-photon entangled states that carry encoded classical bits is compressed into a single-photon state before the corresponding photon is sent to the receiver that measures the qubit and decompresses it. In this paper, we first show how to obtain the generalized isometric tensors for compressing any entangled states and their inverse isometric tensors for decompression. In our proposed QKD protocol, the input state consists of any multi-photon entangled states, which are first compressed into a single-photon state &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ \left| 0 \right\rangle $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20230589_M7.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20230589_M7.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; or &lt;inline-formula&gt;&lt;tex-math id="M8"&gt;\begin{document}$ \left| 1 \right\rangle $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20230589_M8.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20230589_M8.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; or Bell states by the sender Alice. A sequence of single-photon states &lt;inline-formula&gt;&lt;tex-math id="M9"&gt;\begin{document}$ \left| 0 \right\rangle $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20230589_M9.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20230589_M9.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M10"&gt;\begin{document}$ \left| 1 \right\rangle $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20230589_M10.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20230589_M10.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; and one photon from the Bell state mixed with decoy qubits is sent to the receiver Bob via a quantum channel. Bob obtains the final sifted compressed states &lt;inline-formula&gt;&lt;tex-math id="M11"&gt;\begin{document}$ \left| 0 \right\rangle $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20230589_M11.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20230589_M11.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M12"&gt;\begin{document}$ \left| 1 \right\rangle $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20230589_M12.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20230589_M12.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; and conjugate transpose of the isometric tensors. Using our protocols, Bob can decompress the received states &lt;inline-formula&gt;&lt;tex-math id="M13"&gt;\begin{document}$ \left| 0 \right\rangle $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20230589_M13.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20230589_M13.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M14"&gt;\begin{document}$ \left| 1 \right\rangle $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20230589_M14.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20230589_M14.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; into original entangled states. Since quantum processors that are used to send quantum information between nodes are relatively primitive and low in power and the preparation of many-photon entanglement is relatively difficult at present, finding suitable protocols for the compression of transmitted quantum data brings important practical benefits. More generally, the quantum information theory primarily investigates quantum data manipulation under locality constraints, so our protocols connect naturally to these investigations. Our protocols increase the encoding capacity of QKD protocols. Not only our proposed processes of compression and decompression are very simple, but also entanglement compression using isometric tensors can be implemented by using quantum circuits and current technology. Because many ideas for designing of quantum information processing equipment envision that a network composed of relatively small quantum processors sending quantum information between nodes, it is greatly significant to find appropriate protocols for compressing the transmitted quantum data .