For example, the simplified check matrix corresponding to the set of generators (2) of a stabilizer group of the initial state | ψ 〉 EA is 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    O$_{r}$ × n I$_{r}$ × r O$_{r}$ × ( $^{n}$−r ) I$_{r}$ × r O$_{r}$ × ( $^{n}$−r ) O$_{r}$ × n    . (13) c = 1 2 rank( ˜ H Λ ˜ H $^{T}$) . (14) 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 Since the logical operators satisfy the commutative relations (3)-(6), we have H ′ Λ$_{2}$$_{n}$ L ′ T = O. (15) L ′ Λ$_{2}$$_{n}$ L ′ T =    O$_{k}$ × k I$_{k}$ × k I$_{k}$ × k O$_{k}$ × k    . 12