11.5 Same evolution, different errors

We can always pick up an orthonormal basis |u_i\rangle in the environment and express the system–environment evolution as \begin{aligned} |\psi\rangle|e\rangle \longmapsto &\sum_{ij} E_i|\psi\rangle|u_j\rangle\langle u_j|e_i\rangle \\&= \sum_{j}\Big( \sum_i \langle u_j|e_i\rangle E_i\Big)|\psi\rangle|u_j\rangle \\&= \sum_j M_j|\psi\rangle|u_j\rangle. \end{aligned} The new “error” operators M_j satisfy \sum_j M_j^\dagger M_j =\mathbf{1} and, in general, they are not unitary. Now, the evolution of the density operator |\psi\rangle\langle\psi| can be written as |\psi\rangle\langle\psi|\longmapsto \sum_j M_j|\psi\rangle\langle\psi| M_j^\dagger. Which particular errors you choose depends of your choice of the basis in the environment. If, instead of |u_j\rangle, you pick up a different basis, say |v_k\rangle, then \begin{aligned} |\psi\rangle|e\rangle \longmapsto &\sum_j M_j|\psi\rangle|u_j\rangle \\&= \sum_j M_j |\psi\rangle\sum_k|v_k\rangle\langle v_k|u_j\rangle \\&= \sum_k \Big(\sum_j \langle v_k|u_j\rangle M_j \Big)|\psi\rangle|v_k\rangle \\&= \sum_k N_k|\psi\rangle|v_k\rangle, \end{aligned} and, consequently, |\psi\rangle\langle\psi|\longmapsto \sum_k N_k|\psi\rangle\langle\psi| N_k^\dagger. The new “errors” satisfy \sum_k N_k^\dagger N_k = \mathbf{1}, and the error operators N_k and M_j are related by the unitary matrix U_{kj}=\langle v_k|u_j\rangle.