group of | ψ 〉 EA is 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     Z A i ⊗ I $^{B}$, if i / ∈ T ; Z A i ⊗ Z B j , if i = t$_{j}$ ∈ T ; for i = 1 , · · · , r , X A t$_{j}$ ⊗ X B j , for j = 1 , · · · , c . (2) [ Z$_{i}$, Z$_{j}$ ] = 0 , for 0 ≤ i, j ≤ r, (3) [ X$_{t}$$_{i}$, X$_{t}$$_{j}$ ] = 0 , for 0 ≤ i, j ≤ c, (4) { Z$_{t}$$_{i}$, X$_{t}$$_{i}$ } = 0 , for 0 ≤ i ≤ c, (5) [ Z$_{i}$, X$_{t}$$_{j}$ ] = 0 , for i = t$_{j}$, (6) ϕ ( Z$_{i}$ ) ⊙ ϕ ( Z$_{j}$ ) = 0 , for 0 ≤ i, j ≤ r, (7) ϕ ( X$_{t}$$_{i}$ ) ⊙ ϕ ( X$_{t}$$_{j}$ ) = 0 , for 0 ≤ i, j ≤ c, (8) ϕ ( Z$_{t}$$_{i}$ ) ⊙ ϕ ( X$_{t}$$_{i}$ ) = 1 , for 0 ≤ i ≤ c, (9) ϕ ( Z$_{i}$ ) ⊙ ϕ ( X$_{t}$$_{j}$ ) = 0 , for i = t$_{j}$. (10) where [ g, h ] = gh − hg and { g, h } = gh + hg , or 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 9