content_image/320426.jpg content_image/320427.jpg content_image/320431.jpg content_image/320432.jpg
For example, the simplified check matrix corresponding to the set of generators (2) of a stabilizer group of the initial state | ψ 〉 EA is 

<img src='content_image/320426.jpg'>

Conversely, an ( r + c ) × 2 n binary matrix ˜ H , serving as a simplified check matrix, can define a stabilizer group and hence an EAQEC code. The number of ebits required to construct an EAQEC code [13] is 

<img src='content_image/320427.jpg'>

Like a check matrix of a standard quantum error-correcting code, the simplified check matrix H ′ can be used to determine the minimum distance of nondegenerate EAQEC codes. Note that Wilde’s encoding circuit algorithm [12] can also be applied to a simplified check matrix to find an encoding unitary operator of the EAQEC code, just as for a standard stabilizer code. 

Similarly, we define a simplified logical matrix L ′ corresponding to the logical operators by putting ϕ ( ¯ Z ′ $_{i}$) to be the i − th row vector of L ′ for i = 1 , · · · , k , and ϕ ( ¯ X ′ $_{j}$) to be the ( j + k ) − th row vector of L ′ for j = 1 , · · · , k. Since the logical operators commute with { g ′ $_{1}$, · · · , g ′ $_{r}$, h ′ $_{1}$, · · · , h ′ r } , we have 

<img src='content_image/320431.jpg'>

Since the logical operators satisfy the commutative relations (3)-(6), we have 

<img src='content_image/320432.jpg'>

