If there exists a standard [[ n + c, k, d ]] quantum code, then we may not be achieving the maximum boost to our error correcting power from the c ebits of shared entanglement. We expect added entanglement in general to increase the error-correcting ability of a quantum error-correcting code such that the EAQEC code is not equivalent to any standard quantum stabilizer code, and indeed this turns out to be possible by our encoding optimization pro- cedure. (Note that this is not always possible—the smallest examples of the [[3 , 1 , 3; 2]] and [[4 , 1 , 3; 1]] codes are both equivalent to the standard [[5 , 1 , 3]] QECC, and this is the best that can be done.) We now consider how added entanglement affects standard quantum stabilizer codes. Suppose { g ′ $_{1}$, g ′ $_{2}$, · · · , g ′ r } forms a set of independent generators of a stabilizer group S for an [[ n, k, d ]] quantum stabilizer code C ( S ), and U$_{E}$ is the encoding unitary operator obtained by Wilde’s encoding circuit algorithm [12], such that U$_{E}$Z$_{1}$U † E = g ′ $_{1}$, · · · , U$_{E}$Z$_{r}$U † E = g ′ $_{r}$. The logical operators are ¯ Z ′ 1 = U$_{E}$Z$_{r}$$_{+1}$U † $_{E}$, · · · , ¯ Z ′ k = U$_{E}$Z$_{n}$U † $_{E}$, ¯ X ′ 1 = U$_{E}$X$_{r}$$_{+1}$U † $_{E}$, · · · , ¯ X ′ k = U$_{E}$X$_{n}$U † $_{E}$. Suppose entanglement between the encoder and decoder is available, and a set T of c ancilla qubits are replaced by ebits. An EAQEC code is obtained. This introduces c simplified generators h ′$_{j}$ = U$_{E}$X$_{t}$$_{j}$U † $_{E}$’s, t$_{j}$ ∈ T , j = 1 , · · · , c , to the generating set of the stabilizer group S . As we will examine in detail below, the encoding unitary operator is not uniquely defined. The EAQEC code defined by S ′ = 〈 g ′ $_{1}$, · · · , , g ′ $_{r}$, h ′ $_{1}$, · · · , , h $^{′$_{c}$}$〉 may gain higher error-correcting ability by modifying the encoding operator. We first discuss the case c = r , where the generators h ′$_{i}$ ’s are symplectic partners of g ′ $_{i}$’s, respectively. We will treat the case c < r later, by optimizing the choice of c linearly independent generators from the group 〈 h ′ $_{1}$, · · · , , h $^{′$_{r}$}$〉 . Since the symplectic partners of g ′ $_{1}$, · · · , g ′ r are not unique, we now explain how to select these partners such that the minimum distance of the EAQEC codes is higher than the 3.1 Selecting Symplectic Partners and Logical Operators 14