Definition: Information targets

Let $H$ be the input space.

1. A named effect $0\le F\le I_H$ defines the binary readout channel

   $${\mathcal{M}}_F(\rho )=\mathrm{Tr}(F\rho )|1⟩⟨1|+\mathrm{Tr}((I_H-F)\rho )|0⟩⟨0|.$$

2. Full-code recovery has target ${\mathrm{id}}_H$.

3. If $H=C_L\otimes S$, subsystem recovery has target

   $${\mathcal{T}}_L={\mathrm{Tr}}_S:\mathcal{B}(C_L\otimes S)⟶\mathcal{B}(C_L).$$

4. If

   $$H=\underset{\alpha }{⨁}(C_{\alpha }\otimes S_{\alpha }),\mathcal{M}=\underset{\alpha }{⨁}(\mathcal{B}(C_{\alpha })\otimes I_{S_{\alpha }}),$$

   set $K={⨁}_{\alpha }{C}_{\alpha }$. The algebra-restriction target is

   $${\mathcal{R}}_{\mathcal{M}}(\rho )=\underset{\alpha }{⨁}{\mathrm{Tr}}_{S_{\alpha }}(P_{\alpha }\rho P_{\alpha }).$$