Detecting ${\boldsymbol~A}{\boldsymbol~B}{\boldsymbol~\rightarrow}{\boldsymbol~C}$ steering in tripartite quantum systems

Abstract

In this paper, the detection problem of $AB\\rightarrow~C$ steering is discussed. Firstly, we give some relationships between the $AB\\rightarrow~C$ steering of $\\rho^{ABC}$ and $A\\rightarrow~C$ steering of the reduced state $\\rho^{AC}$ as well as $B\\rightarrow~C$ steering of the reduced state $\\rho^{BC}$ based ontheir definitions. Secondly, by deducing an inequality (called an steering inequality) for $AB\\rightarrow~C$ unsteerable states, a necessary condition for a quantum state to be $AB\\rightarrow~C$ unsteerable is obtained. Furthermore, the steering radius $S(\\rho)$ of a state $\\rho$ is defined and it is proved that when $\\rho$ is $AB\\rightarrow~C$ unsteerable, $S(\\rho)\\le1$ and that when $S(\\rho)>1$, $t\\rho+(1-t)\\eta$ is $AB\\rightarrow~C$ steerable for each $t_0<t\\le1$ and for each $AB\\rightarrow~C$ unsteerable state $\\eta$ where $t_0=\\frac{2}{S(\\rho)+1}$.Finally, $AB\\rightarrow~C$ steerability of the “partial convex combinations of the GHZ state, generalized GHZ state and the W-state with a $AB\\rightarrow~C$ unsteerable state are detected, respectively by using the obtained steering inequality.