Einstein-Podolsky-Rosen-steering swapping between two Gaussian multipartite entangled states

Multipartite Einstein-Podolsky-Rosen (EPR) steering is a useful quantum resource for quantum communication in quantum networks. It has potential applications in secure quantum communication, such as one-sided device-independent quantum key distribution and quantum secret sharing. By distributing optical modes of a multipartite entangled state to space-separated quantum nodes, a local quantum network can be established. Based on the existing multipartite EPR steering in a local quantum network, secure quantum communication protocol can be accomplished. In this manuscript, we present swapping schemes for EPR steering between two space-separated Gaussian multipartite entangled states, which can be used to connect two space-separated quantum networks. Two swapping schemes, including the swapping between a tripartite Greenberger-Horne-Zeilinger (GHZ) entangled state and an EPR entangled state and that between two tripartite GHZ entangled states, are analyzed. Various types of EPR steering are presented after the swapping of two space-separated independent multipartite entanglement states without direct interaction, which can be used to implement quantum communication between two quantum networks. The presented schemes provide technical reference for more complicated quantum networks with EPR steering.

Figure 1

Schematic of two multipartite steering swapping schemes. Networks A and B consist of two multipartite entangled states. A joint measurement is performed at local quantum network B and the measurement results are fed forward to the remaining modes of network B by classical channels. (a) Scheme for Gaussian EPR-steering swapping between a tripartite GHZ state and an EPR entangled state. (b) Scheme for Gaussian EPR-steering swapping between two tripartite GHZ entangled states. HD: homodyne detector; EOMp and EOMx: phase and amplitude electro-optical modulators; $T$: transmission efficiencies of the quantum channels.

Figure 2

Dependence of steering parameter between one and the other two modes on the gain factor and squeezing parameter in steering swapping scheme I. (a), (b) Dependence of steering parameter between one and the other two modes on gain factor with two different squeezing parameters. Red dashed-dotted and dotted lines represent $r=0.57$, blue solid and dashed lines represent $r=1.15$. (c), (d) Dependence of EPR-steering parameter between one and the remaining two modes on squeezing parameter with unit gain (red dashed-dotted and dotted lines) and optimal gain (blue solid and dashed lines), respectively.

Figure 3

Dependence of EPR-steering parameter in steering swapping scheme I on transmission efficiency. (a) There is no EPR steering between any two modes of output modes (black dashed-dotted line). (b) EPR-steering parameter between one and the other two modes of output modes. Red dashed-dotted and dotted lines represent the situation where one mode is located in network A. Blue solid and dashed lines correspond to the situation where one mode is located in network B.

Figure 4

EPR-steering parameter between any two modes for the steering swapping scheme II. The black dashed-dotted line represents the situation that the steering does not exist. The red dashed-dotted and dotted lines represent the steering parameter when the optimal gain factor $g_{\text{opt}}^{B_3^{\prime }\to B_2^{\prime }}$ is chosen.

Figure 5

Dependence of EPR steering on the transmission efficiency between one and the other two modes in swapping scheme II. (a) Dependence of steerabilities between one mode located in network A and the other two modes of the output state. (b) Dependence of steerabilities between one mode located in network B and the other two modes of the output state.

Figure 6

(a) Dependence of EPR-steering parameter on the transmission efficiency between one mode located in network B and the remaining three modes. Red dashed-dotted and dotted lines represent the EPR steering between ${\widehat{B}}_2^{\prime }$ and the remaining three modes. Blue dashed-dotted and dotted lines represent the EPR steering between ${\widehat{B}}_3^{\prime }$ and the remaining three modes. (b) Dependence of EPR steering between the one mode located in network A and the remaining three modes on the transmission efficiency.

Figure 7

Dependence of EPR steering parameter on the transmission efficiency between two modes and the remaining two modes. Red dashed-dotted and dotted lines represent the EPR steering between two modes located in one local network and the remaining two modes. Blue solid and dashed lines represent the EPR steering between two modes located in two local networks and the remaining two modes.