Figure 1

(Color online) The initial system-bath state is the generically non-vanishing quantum discord (VQD) state ${\rho }_{SB}\left(t_0\right)$. The encoded system $S=D+E$ consists of data qubits $D$ and encoding ancillas $E$. We also include the recovery ancillas $R$, which are assumed to be completely isolated until they are brought into contact with $S$ and $B$ at a later time. Thus the full initial state is ${\rho }_{SB}\left(t_0\right)\otimes {\rho }_R\left(t_0\right)$. The overall evolution is governed by the unitary $U_{SRB}$, which acts on the system $S$, the bath $B$, and eventually the recovery ancillas $R$, and is denoted by the large gray box. The state of the data qubits is ${\rho }_{|\psi ⟩}={\text{Tr}}_{E,B}\left[{\rho }_{SB}\left(t_0\right)\right]$, a state which is as close as possible (by isolating the system) to the desired pure data state $|\psi ⟩$. The state of each of the encoding ancillas is ${\rho }_{|0⟩}={\text{Tr}}_{E^{\prime },B}\left[{\rho }_{SB}\left(t_0\right)\right]$, a state which is as close as possible (again, by isolating the system) to the desired pure encoding ancilla state $|0⟩$. Here ${\text{Tr}}_{E,B}$ denotes a partial trace over all encoding ancillas and the bath, ${\text{Tr}}_{E^{\prime },B}$ denotes a partial trace over all but one of the encoding ancillas and the bath. Ideally, the encoding unitary $U_S$ is then applied to the encoded system. This is of course an idealization since in reality the encoding operation will not be a perfect unitary; instead what is really applied is $U_{SRB}\left(t_1,t_0\right)$, which is supposedly close to the ideal $U_S\otimes I_R\otimes I_B$. Thus, after the encoding the total state is ${\rho }_{SRB}\left(t_1\right)=U_{SRB}\left(t_1,t_0\right)\left[{\rho }_{SB}\left(t_0\right)\otimes {\rho }_R\left(t_0\right)\right]U_{SRB}^{†}\left(t_1,t_0\right)$ and the encoded system state is ${\rho }_S\left(t_1\right)={\text{Tr}}_{R,B}\left[{\rho }_{SRB}\left(t_1\right)\right]$. The system is then passed through the noise channel for the purpose of either computation or communication, i.e., ${\rho }_{SRB}\left(t_2\right)=U_{SRB}\left(t_2,t_1\right){\rho }_{SRB}\left(t_1\right)U_{SRB}^{†}\left(t_2,t_1\right)$, whence ${\rho }_S\left(t_2\right)={\text{Tr}}_{R,B}\left[{\rho }_{SRB}\left(t_2\right)\right]={\Phi }_{\text{H}}\left[{\rho }_S\left(t_1\right)\right]$, where ${\Phi }_{\text{H}}$ is a Hermitian noise map since ${\rho }_{SRB}\left(t_1\right)$ is generically a non-VQD state due to the initial nonclassical correlations between $S$ and $B$. The goal of the error correction procedure is to recover the original encoded system state from ${\rho }_S\left(t_2\right)$, and to this end we introduce recovery ancillas $R$ at $t_2$. Similarly to the encoding ancillas, these recovery ancillas are each in the state ${\rho }_{|0⟩}={\text{Tr}}_{S,R^{\prime },B}\left[{\rho }_{SRB}\left(t_2\right)\right]$, a state which is as close as possible to the desired pure recovery ancilla state $|0⟩$. Next, ideally the recovery unitary $U_{SR}\otimes I_B$ is applied. In reality what is applied is $U_{SRB}\left(T,t_2\right)$, which is supposedly close to the ideal $U_{SR}\otimes I_B$. Then the recovery ancillas are discarded and possibly recycled, leaving the encoded system in the final state ${\rho }_S\left(T\right)={\text{Tr}}_{R,B}\left[{\rho }_{SRB}\left(T\right)\right]={\mathcal{R}}_S\left[{\rho }_S\left(t_2\right)\right]$, which can be measured. Since ${\rho }_{SRB}\left(t_2\right)$ is generically not a VQD state (due to nonclassical correlations between $S$ and $R$, mediated by their mutual interaction with $B$), it is clear that the recovery map ${\mathcal{R}}_S$ is generically a non-CP Hermitian map. We recover the CP recovery map scenario if, for example, ${\rho }_{SRB}\left(t_2\right)={\rho }_{SB}\left(t_2\right)\otimes {\rho }_R\left(t_2\right)$. The assumption that this is not the case is consistent with the working premise of this paper and is equivalent in that regard to the assumption that the initial system-bath state is not of the form ${\rho }_{SB}\left(t_0\right)={\rho }_S\left(t_0\right)\otimes {\rho }_B\left(t_0\right)$.Reuse & Permissions