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FIG. 1: The total Bell-diagonal states belong to the tetrahedron ( v$_{1}$, v$_{2}$, v$_{3}$, v$_{4}$ ) and the set of the separable states belong to the octahedron, whose vertices are o ± 1 , o ± 2 and o ± 3 . As this figure 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 contained in 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$_{4}$ ), respectively. Therefore, all entangled Bell-diagonal mixtures belong to the small four tetrahedra ( 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 ). 

Now, we show how to perform the reverse process of Ref.[22] when ρ is an entangled Bell-diagonal state. This can be achieved by following two theorems. 

Theorem 1. Every Bell state has infinite CSS, which cover fully the nearest surface of the octahedron L . 

Proof. It is sufficient to prove this theorem when ρ = | β$_{1}$ 〉〈 β$_{1}$ | . When ρ = | β$_{i}$ 〉〈 β$_{i}$ | ( i = 2 , 3 , 4), one can prove the theorem similarly. 

Let σ be a following Bell-diagonal state: 

<img src='content_image/363311.jpg'>

with p = ( x, y, z ). Then, it is easy to show that the spectral decomposition of σ is 

<img src='content_image/363312.jpg'>

The nearest surface of L from ρ = | β$_{1}$ 〉〈 β$_{1}$ | is ( o (+) 1 , o ( − ) 2 , o (+) 3 ), whose surface equation is x − y + z = 1. If σ belongs to the surface ( o (+) 1 , o ( − ) 2 , o (+) 3 ), it is easy to show that S ( ρ || σ ) = ln 2, which exactly coincides with the REE of the Bell states[11]. Therefore, σ on the surface ( o (+) 1 , o ( − ) 2 , o (+) 3 ) is the CSS of | β$_{1}$ 〉〈 β$_{1}$ | . 

Now, let us consider the case where σ belongs to other surface. For example, let us assume that σ belongs to the surface ( o (+) 1 , o (+) 2 , o ( − ) 3 ), whose surface equation is x + y − z = 1. Then, 

