changes, and so does the set N ( S ′ ) − S ′ I . In addition, the span of a subset of { h ′ $_{1}$, · · · , h ′ r } can change after the operation of V , though the span of the full set remains unchanged. This means that if we add less than the maximum amount of entanglement to a code, we must optimize over such unitary row operations. Since the group S ′ I and the set N ( S ′ ) − S ′ I remain the same under the operation of type 1 unitary row operators that operate on the h ′ j for j / ∈ T , it suffices to assume that the operation V consists of type 1 unitary row operators that operate only on the h ′ j for j ∈ T . Let M$_{V}$ be a c × r matrix such that the i − th row of M$_{V}$ is the t$_{i}$ − th row of M$_{Z}$ for i = 1 , · · · , c . It is obvious that some M$_{V}$ ’s have the same effect on the row space of H ′ . For example, if c = 2 , { g ′ $_{1}$g ′ $_{2}$, g ′ $_{2}$, · · · , g ′ $_{r}$, h ′ $_{1}$, h ′ $_{1}$h ′ 2 } and { g ′ $_{1}$, g ′ $_{2}$, · · · , g ′ $_{r}$, h ′ $_{1}$, h ′ 2 } are two different sets of generators but they generate the same space and hence their corresponding EAQEC codes have the same minimum distance. Therefore, a distinct unitary row operation V is assumed to be be represented by a matrix M$_{V}$ in reduced row echelon form. Theorem 3. The operation of V is equivalent to applying a series of type 1 unitary row operators on the h ′ j for j ∈ T . In addition, there are distinct unitary row operations. Proof. The total number of distinct unitary row operations N ( r, c ) is determined as follows. If we begin with matrices of the form 23    $^{1}$ 0 · · · 0 □ · · · □ 0 1 · · · 0 □ · · · □ . . . . . . . . . . . 0 0 · · · 1 □ · · · □    $^{,}$ N ( r, c ) ≜ $^{r}$−c ∑ l$_{c}$ =0 l$_{c}$ ∑ l$_{c}$$_{−}$$_{1}$ =0 l$_{c}$$_{−}$$_{1}$ ∑ l$_{c}$$_{−}$$_{2}$ · · · l$_{2}$ ∑ l$_{1}$ =0 2 c ( $^{r}$−c $^{)}$− ∑ c i =1 l$_{i}$