Local Hidden State Models For Bell Diagonal States And Beyond

In 1935,the steering phenomenon was first proposed by Einstein et al.It describes a two-qubit entangled state in which any measurement of one can remotely manipulate the other's state.If the change by one party's measurement cannot be simulated by the LHS(Local Hidden State)model,then the two quantum states are said to be steerable.Studying how the LHS model can simulate a particular entangled state is important for understanding the EPR(Einstein-Podolsky-Rosen)steering.However,even for the simplest two quantum states,the construction of the local hidden state model is very difficult.In 1989,Werner showed that not all entangled states are nonlocal.He introduced the family state now known as the Werner state and gave a LHV(local hidden variable)model to simulate the measurement of some entangled states in the family.This result is the theoretical source of all subsequent local hidden state models.This paper focuses on the construction of the local hidden state model of Bell diagonal state.In the paper,the LHS model construction problem of two-qubit Bell diagonal state is transformed into the LHS model of Werner state,and the optimal model given by Jevtic et al.using EPR(Einstein-Podolsky-Rosen)steering ellipsoid is obtained.This conversion also allows us to derive sufficient conditions for the unsteering of any two qubit states.When Alice's Bloch vector a faces the largest component of the correlation matrix,our criterion is superior to the Bowles criterion.When the hidden variable is taken as a finite dimension,Bowles et al.use 12 discrete values(corresponding to 12 vertices of the icosahedron),and give the local hidden state model of the Werner state when ?=?l/6.On this basis,we use 12 random variables distributed non-uniformly on Bloch sphere to construct a local hidden model of Bell diagonal state when t=2?l/?i=112|T0vi|.As the orientation of the icosahedron changes,the value of the parameter t also changes.When the correlation matrix on the EPR steering boundary is axisymmetric,after numerical calculation,we obtain three special directions of the optimal positive icosahedron(the value t takes the maximum value),they are:vertex,the midpoint of the face and the center of the face.It can be seen that as the anisotropy of the correlation matrix increases,the maximum value t of the Bell diagonal state also increases,while the entanglement and the shared randomness decrease.When the classical random variables are finite and discrete,Bowles et al.have implemented the construction of the Werner state LHS model.On this basis,we use the non-uniform distribution of 12 vertices to correspond to the T-state LHS modelIn order to construct a more economical model,this paper also pays attention to the case of the classical random variable dimension D=4,which corresponds to the four vertices of the tetrahedron,which is equivalent to the use of classical bits to simulate the correlation in the quantum state.Using the transform method when studying EPR steering,this paper obtains the separable Bell diagonal state LHS model from the separable Werner state LHS model.The model is optimal in the sense of reaching a separable boundary.