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on Bob’s c qubits. A set of generators of a stabilizer group S of the encoded state ( U$_{E}$ ⊗ I ) | ψ 〉 EA is { g$_{1}$, · · · , g$_{r}$, h$_{1}$, · · · , h$_{c}$ } , where 

and 

The logical operators on ( U$_{E}$ ⊗ I ) | ψ 〉 EA are 

and 

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The 2 $^{k}$− dimensional subspace of the ( n + c ) − qubit state space fixed by the stabilizer group S with a set of generators { g$_{1}$, · · · , g$_{r}$, h$_{1}$, · · · , h$_{c}$ } is called an entanglement-assisted quantum error-correcting (EAQEC) code with parameters [[ n, k, d ; c ]] for some minimum distance d . With the help of c pairs of maximally-entangled states, the stabilizer group of an [[ n, k, d ; c ]] EAQEC code has c more generators than that of an [[ n, k, d ]] standard quantum stabilizer code. The operators on Bob’s side have no direct effect on the error- correcting ability of the EAQEC code if we assume that the c qubits of Bob suffer no error. These operators serve to make the stabilizer abelian, so that the generators can be measured simultaneously. For convenience, we denote 

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