content_image/372834.jpg content_image/372835.jpg content_image/372836.jpg content_image/372837.jpg
derive the Bloch vectors r and s and the correlation tensor g$_{ij}$ as follows: 

<img src='content_image/372834.jpg'>

where ρ$_{A}$ = tr$_{B}$ ρ and ρ$_{B}$ = tr$_{A}$ ρ . It is well known that an appropriate local unitary (LU) transformation of ρ can make g$_{mn}$ to be diagonal (see appendix of Ref.[26]). Since entanglement is invariant under the LU transformation, it is in general sufficient to consider the case of diagonal g$_{mn}$ for the discussion of entanglement. Thus, without loss of generality, one can express ρ as 

<img src='content_image/372835.jpg'>

If ρ = | β$_{i}$ 〉〈 β$_{i}$ | , where 

<img src='content_image/372836.jpg'>

it is easy to show that the corresponding Bloch vectors r and s are vanishing and the corresponding correlation tensor g$_{mn}$ become 

<img src='content_image/372837.jpg'>

If, therefore, ρ is Bell-diagonal state, the Bloch vectors r and s are always null vectors. 

Since we are considering on the diagonal case of the correlation tensor, we will regard, from now on, the tensor as a vector, whose components are equal to the diagonal elements. When r = s = 0, Horodecki has shown in Ref.[25] that the total two-qubit states belong to the tetrahedron T with vertices v$_{1}$ = (1 , − 1 , 1), v$_{2}$ = ( − 1 , 1 , 1), v$_{3}$ = (1 , 1 , − 1) and v$_{4}$ = ( − 1 , − 1 , − 1) in the correlation vector space. Ref.[25] also has shown that the separable states (with r = s = 0) belong to the octahedron L with vertices o ( ± ) 1 = ( ± 1 , 0 , 0), o ( ± ) 2 = (0 , ± 1 , 0) and o ( ± ) 3 = (0 , 0 , ± 1). This is pictorially represented in Fig. 1. (+) ( − ) ( − ) (+) (+) (+) ( − ) ( − ) (+) 

As Fig. 1 shows, the planes ( o 1 , o 2 , o 3 ), ( o 1 , o 2 , o 3 ), ( o 1 , o 2 , o 3 ) and ( o ( − ) 1 , o (+) 2 , o ( − ) 3 ) are parts of the planes ( v$_{1}$, v$_{3}$, v$_{4}$ ), ( v$_{1}$, v$_{2}$, v$_{3}$ ), ( v$_{1}$, v$_{2}$, v$_{4}$ ) and ( v$_{2}$, v$_{3}$, v$_{4}$ ) ,re- spectively $^{3}$. Therefore, all entangled Bell-diagonal mixtures belong to the small four tetra- hedra ( v$_{1}$, o (+) 1 , o ( − ) 2 , o (+) 3 ), ( v$_{2}$, o ( − ) 1 , o (+) 2 , o (+) 3 ), ( v$_{3}$, o (+) 1 , o (+) 2 , o ( − ) 3 ) and ( v$_{4}$, o ( − ) 1 , o ( − ) 2 , o ( − ) 3 ). 

