I'm John and I have a question which I have been thinking about. I'm studying quantum information, especially, quantum error-correcting codes. When I learned some types of quantum codes (e.g. 5 qubits code or quantum Hamming code), I thought the following condition is also a necessary condition for quantum codes, but I have no idea to prove it. Could anyone prove it although it might be false?
Let $\mathbb{P}=\{P_1,...,P_n\}$ be a projection measurement, $\rho$ an $m$ qubits codeword and $U_i$ a unitary operator which represents an error ($i=1,...,m$:position of error). Then, the condition is $$\forall k_1,k_2\in \{1,...,n\}, \forall i\in \{1,...,m\},\mathrm{Tr}(P_{k_1}U_i\rho U_i^\dagger)\neq 0 \land \mathrm{Tr}(P_{k_2}U_i\rho U_i^\dagger)\neq 0\\ \Rightarrow \frac{P_{k_1}U_i\rho U_i^\dagger P_{k_1}}{\mathrm{Tr}(P_{k_1}U_i\rho U_i^\dagger)}=\frac{P_{k_2}U_i\rho U_i^\dagger P_{k_2}}{\mathrm{Tr}(P_{k_2}U_i\rho U_i^\dagger)}$$