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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 

<img src='content_image/295637.jpg'>

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 

<img src='content_image/295638.jpg'>

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 

<img src='content_image/295639.jpg'>

<img src='content_image/295640.jpg'>

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 

<img src='content_image/295646.jpg'>

<img src='content_image/295645.jpg'>

