content_image/295974.jpg content_image/295975.jpg content_image/295976.jpg content_image/295977.jpg content_image/295978.jpg content_image/295979.jpg content_image/295980.jpg content_image/295981.jpg content_image/295982.jpg
group of | ψ 〉 EA is 

<img src='content_image/295974.jpg'>

Note that the operators on the left and right of the tensor product ⊗ are applied to Alice’s qubits and Bob’s qubits, respectively, and the superscripts A and B will be omitted through- out the rest of this article. The logical operators on | ψ 〉 EA are Z$_{r}$$_{+1}$ ⊗ I , · · · , Z$_{n}$ ⊗ I, and X$_{r}$$_{+1}$ ⊗ I , · · · , X$_{n}$ ⊗ I . Now consider the operators on Alice’s qubits. It can be observed that 

where [ g, h ] = gh − hg and { g, h } = gh + hg , or 

<img src='content_image/295975.jpg'>

<img src='content_image/295976.jpg'>

<img src='content_image/295977.jpg'>

<img src='content_image/295978.jpg'>

<img src='content_image/295979.jpg'>

<img src='content_image/295980.jpg'>

<img src='content_image/295981.jpg'>

<img src='content_image/295982.jpg'>

If a set of ( r + c ) operators satisfy equations (3)-(6) or equations (7)-(10), we say that the two operators in (5) or the two vectors in (9) form a symplectic pair , and they are symplectic partners of each other. Hence Z$_{t}$$_{i}$ and X$_{t}$$_{i}$ form a symplectic pair. 

An encoding operation U$_{E}$ is applied on Alice’s n qubits, while no operation is performed 

