code without entanglement. Suppose W is a unitary Clifford operator that commutes with Z$_{1}$, · · · ,Z$_{r}$ such that after the operation of W , the simplified check matrix of the initial state (13) becomes where A and B are two r × ( n − r ) binary matrices, and C is an r × r binary matrix. The simplified check matrix satisfies the commutation relations (7)-(10) with    O$_{r}$ × n I$_{r}$ × r O$_{r}$ × ( $^{n}$−r ) I$_{r}$ × r A C B    , (17) C T + AB T + C + BA T = O. (18)    O$_{k}$ × n A T I$_{k}$ × k O$_{k}$ × n I$_{k}$ × k B T O$_{k}$ × k    ( U$_{E}$W ) Z$_{i}$ ( U$_{E}$W ) † = U$_{E}$Z$_{i}$U † E = g ′ i In addition, it can be checked that the simplified logical matrix is of the form after Gaussian elimination such that (15) and (16) hold. Since for i = 1 , · · · ,r , U$_{E}$W is also an encoding operator of the quantum stabilizer code C ( S ). However, the symplectic partners of g ′ $_{i}$’s, may differ from U$_{E}$X$_{i}$U † E for i = 1 , · · · , r and the logical operators U$_{E}$ ( WX$_{i}$W † ) U † E U$_{E}$ ( WX$_{i}$W † ) U † $_{E}$, U$_{E}$ ( WZ$_{j}$W † ) U † $_{E}$, 15