Code and channels

Choose $D=2g$ orthonormal even eigenvectors ${\beta }_{ia}$, $i=0,1$, $a=1,\dots ,g$, with fixed spectral, phase, and degeneracy conventions. Set

The full code algebra is $V\mathcal{B}(MK)V^{†}$. The message algebra is $V(\mathcal{B}(M)\otimes I_K)V^{†}$.

For a message channel prepare ${\rho }_{QM}\otimes {\sigma }_K$, with independent ${\sigma }_K$. Uniform subsystem recovery instead allows arbitrary ${\rho }_{QMK}$.