All-Optical Tripartite Entanglement Swapping (Invited)

Objective In quantum information science, quantum entanglement enables quantum information processing and communication beyond classical limits through its unique nonlocal property. Consequently, quantum entanglement has been widely applied in quantum communications, including quantum teleportation, entanglement swapping, and quantum state sharing. Among these, entanglement swapping serves as the core technology for quantum repeaters and a critical component in building quantum networks, allowing two initially independent particles without direct interaction to establish entanglement. Since its proposal in 1993, entanglement swapping has been extensively studied in both discrete variable and continuous variable (CV) systems. In 2004, based on the entangled state generated from a non-degenerate optical parametric amplifier (OPA), unconditional quantum entanglement swapping was first achieved in the CV system through Bell-state measurement. In 2016, further entanglement swapping between two independent multipartite entangled states was successfully accomplished via Bell-state measurement. However, the Bell-state measurement process requires optic-electro and electro-optic conversions, which inherently limit the bandwidth of quantum entanglement swapping. To avoid this limitation, in 2022, a low-noise OPA based on a four-wave mixing process (FWM) was experimentally demonstrated to achieve measurement-free all-optical entanglement swapping between two optical fields. This approach replaces optic-electro and electro-optic conversions with a low-noise OPA, providing a methodology for all-optical entanglement swapping among independent multipartite entangled states. Here, we theoretically propose an all-optical tripartite entanglement swapping scheme based on this framework. Our results not only offer an all-optical approach to multipartite entanglement swapping but also provide theoretical foundations for constructing measurement-free all-optical broadband quantum networks. Methods In present study, a low-noise OPA based on the FWM is used to replace the Bell-state measurement procedure in conventional entanglement swapping schemes, thereby realizing all-optical tripartite entanglement swapping. This approach avoids the bandwidth limitations introduced by optic-electro and electro-optic conversions in traditional entanglement swapping scheme. The tripartite entangled state is generated by combining two amplitude-squeezed states and one phase-squeezed state in orthogonal quadratures. Theoretical analysis demonstrates that all-optical tripartite entanglement swapping can be achieved through two distinct schemes: one utilizing a pair of EPR (Einstein-Podolsky-Rosen) entangled states as the quantum channel, and the other one employing two pairs of EPR entangled states. In our scheme, the positivity under partial transposition (PPT) criterion, which serves as a necessary and sufficient condition for bipartite and tripartite entanglement in the CV regime, is used to verify whether entanglement exists in the system. The PPT criterion checks for entanglement via the smallest symplectic eigenvalue of its transposed covariance matrix. For tripartite entangled states, it is generally common to partition all subsystems into two parts to analyze the correlations between each subsystem. A tripartite system has three types of 1× 2 partitions. When the smallest symplectic eigenvalues υ₁, υ₂, and υ₃of these three partitions are all less than 1, it confirms the presence of tripartite entanglement in the system. Results and Discussions For the entanglement swapping scheme using a pair of EPR entangled states as the quantum channel (Fig. 4), tripartite entanglement only exists among Alice, Bob, and Claire when G > 1. At this point, the smallest symplectic eigenvalues υ₁, υ₂, and υ₃are less than 1 and decrease with the increase of G, indicating that all-optical tripartite entanglement swapping can be accomplished when G > 1 and its performance exhibits positive correlation with G. While this scheme successfully establishes tripartite entanglement among Alice, Bob, and Claire, it’s crucial to note that Alice and Bob were already correlated prior to the entanglement swapping. Therefore, this approach cannot achieve our fundamental objective: creating tripartite entanglement among three initially independent systems without direct interaction. To accomplish this goal, we implemented an alternative scheme using two pairs of EPR entangled states as quantum channels (Fig. 5). In this configuration, tripartite entanglement among Alice, Bob, and Claire emerges under the condition G > 1.0145. As G increases, the smallest symplectic eigenvalues υ₁, υ₂, and υ₃gradually decrease. It means that all-optical tripartite entanglement swapping of the second scheme can be successfully accomplished only under the condition G > 1.0145, and its performance maintains a positive correlation with G. Conclusions We propose two feasible all-optical tripartite entanglement swapping schemes by replacing the function of Bell-state measurement with a low-noise OPA. The one using a pair of EPR entangled states as the quantum channel enables three systems that do not initially share tripartite entanglement to establish entanglement. The other, using two pairs of EPR entangled states as the quantum channel, establishes tripartite entanglement among three initially independent systems without direct interaction. By PPT criterion, we analyze the tripartite entanglement characteristics among Alice, Bob, and Claire. Theoretical investigations into the influence of gain on all-optical quantum entanglement swapping further improve the existed theoretical framework. These results provide critical theoretical support for the development of all-optical quantum networks and multi-node all-optical quantum communication.