The Knill–Laflamme criterion for a protected subsystem

For necessity, let $\mathcal{R}:\mathcal{B}(H_O)\to \mathcal{B}(H_G\otimes H_L)$ be an exact decoded recovery. By [F3], choose a finite Kraus family $(S_r)$ for $\mathcal{R}$. The corrected map $\mathrm{\Phi }=\mathcal{R}\circ \mathcal{N}\circ {\mathrm{Ad}}_J$ has Kraus operators $K_{ra}=S_r{E}_{a}J$. It is completely positive by [F3], and ${\sum }_{r,a}{K}_{ra}^{\ast }{K}_{ra}=J^{\ast }{\sum }_a{E}_a^{\ast }\left({\sum }_r{S}_r^{\ast }{S}_r\right)E_{a}J=J^{\ast }\left({\sum }_a{E}_a^{\ast }{E}_a\right)J=J^{\ast }J=I_{H_G\otimes H_L}$ by [F2] and [F3]. Thus $\mathrm{\Phi }$ is a channel. By exactness [F1], it preserves the $H_L$ marginal; [F4] gives $S_r{E}_{a}J=A_{ra}\otimes I_{H_L}$ for some $A_{ra}:H_G\to H_G$.

For sufficiency, assume $J^{\ast }{E}_a^{\ast }{E}_{b}J=M_{ab}\otimes I_{H_L}$. Let $d_G=\dim H_G>0$ and choose an orthonormal basis $(g_k{)}_{k<d_G}$ by [F7]. Define ${\iota }_k:H_L\to H_G\otimes H_L$ by ${\iota }_{k}x=g_k\otimes x$, and put $F_{ak}=E_{a}J{\iota }_k$ and ${\tilde{F}}_{ak}=F_{ak}\mathrm{/}\sqrt{d_G}$. By trace preservation [F3] and $J^{\ast }J=I$ from [F2], ${\sum }_{a,k}{F}_{ak}^{\ast }{F}_{ak}={\sum }_k{\iota }_k^{\ast }{J}^{\ast }\left({\sum }_a{E}_a^{\ast }{E}_a\right)J{\iota }_k=d_G{I}_{H_L}\mathrm{.}$ Thus $({\tilde{F}}_{ak})$ is a finite Kraus representation of a channel $\tilde{\mathcal{N}}:\mathcal{B}(H_L)\to \mathcal{B}(H_O)$.

Trace preservation of $\mathcal{R}$ and $\mathcal{N}$ gives ${\sum }_r{S}_r^{\ast }{S}_r=I_{H_O}$ and ${\sum }_a{E}_a^{\ast }{E}_a=I_{H_P}$ by [F3]. Hence $J^{\ast }{E}_a^{\ast }{E}_{b}J=J^{\ast }{E}_a^{\ast }\left({\sum }_r{S}_r^{\ast }{S}_r\right)E_{b}J={\sum }_r(S_r{E}_{a}J{)}^{\ast }(S_r{E}_{b}J)=\left({\sum }_r{A}_{ra}^{\ast }{A}_{rb}\right)\otimes I_{H_L}\mathrm{.}$ Set $M_{ab}={\sum }_r{A}_{ra}^{\ast }{A}_{rb}$. This proves the operator-valued condition.

For all $a,b,k,\mathrm{\ell }$, the assumed condition gives ${\tilde{F}}_{ak}^{\ast }{\tilde{F}}_{b\mathrm{\ell }}=\frac{⟨M_{ab}{g}_{\mathrm{\ell }},g_k⟩}{d_G}{I}_{H_L}\mathrm{.}$ Apply [F5] to $\tilde{\mathcal{N}}$ with the identity encoding of $H_L$. It supplies a channel $\mathcal{S}:\mathcal{B}(H_O)\to \mathcal{B}(H_L)$ and a finite Kraus family $(S_r)$ for which $\mathcal{S}({\tilde{F}}_{ak}X{\tilde{F}}_{b\mathrm{\ell }}^{\ast })=\frac{⟨M_{ba}{g}_k,g_{\mathrm{\ell }}⟩}{d_G}X$ for every $X\in \mathcal{B}(H_L)$.

Fix an orthonormal gauge basis $(g_k)$ using [F7]. The matrix entry of $M_{ab}$ in row $k$, column $\mathrm{\ell }$ is $⟨M_{ab}{g}_{\mathrm{\ell }},g_k⟩$ by [F8]. Therefore $P_{kk}{E}_a^{\ast }{E}_b{P}_{\mathrm{\ell }\mathrm{\ell }}=J(Q_{kk}(M_{ab}\otimes I_{H_L})Q_{\mathrm{\ell }\mathrm{\ell }})J^{\ast }=⟨M_{ab}{g}_{\mathrm{\ell }},g_k⟩P_{k\mathrm{\ell }}\mathrm{.}$ This proves the matrix-unit statement in the necessary direction.

Trace preservation and $J^{\ast }J=I$ imply ${\sum }_a{M}_{aa}\otimes I_{H_L}=J^{\ast }\left({\sum }_a{E}_a^{\ast }{E}_a\right)J=I_{H_G\otimes H_L},$ so ${\sum }_a{M}_{aa}=I_{H_G}$: applying the equality to each $g\otimes {\mathrm{\ell }}_0$ for fixed unit ${\mathrm{\ell }}_0$ gives $({\sum }_a{M}_{aa}g-g)\otimes {\mathrm{\ell }}_0=0$, and tensoring with nonzero ${\mathrm{\ell }}_0$ is injective. Define the decoded recovery $\mathcal{T}(X)=P_{g_0}\otimes \mathcal{S}(X)$ for a fixed unit $g_0\in H_G$. For a gauge matrix unit $\mathrm{\mid }{g}_k⟩⟨g_{\mathrm{\ell }}\mathrm{\mid }\otimes X$, the encoded channel output is ${\sum }_a{F}_{ak}X{F}_{a\mathrm{\ell }}^{\ast }$; applying $\mathcal{S}$ and scaling the cross-term formula in step 2.2 from ${\tilde{F}}_{ak}$ to $F_{ak}=\sqrt{d_G}{\tilde{F}}_{ak}$ gives logical output ${\sum }_a\mathcal{S}(F_{ak}X{F}_{a\mathrm{\ell }}^{\ast })={\sum }_a⟨M_{aa}{g}_k,g_{\mathrm{\ell }}⟩X=⟨g_k,g_{\mathrm{\ell }}⟩X={\delta }_{k\mathrm{\ell }}X\mathrm{.}$ Here the scaling is $\mathcal{S}(F_{ak}X{F}_{b\mathrm{\ell }}^{\ast })=d_G\mathcal{S}({\tilde{F}}_{ak}X{\tilde{F}}_{b\mathrm{\ell }}^{\ast })$, so the factor $d_G$ cancels the denominator in step 2.2. The input logical marginal of that matrix unit is also ${\delta }_{k\mathrm{\ell }}X$. By linearity and the product matrix-unit basis in [F12], the decoded map $\mathcal{T}$ with output gauge reset preserves the logical marginal on every operator; [F9] extends the equality to every finite reference.

Since $P=J{J}^{\ast }$, the operator-valued condition gives $P{E}_a^{\ast }{E}_{b}P=J(M_{ab}\otimes I_{H_L})J^{\ast }$. Conversely, multiplying this compressed equality by $J^{\ast }$ and $J$ and using $J^{\ast }J=I$ recovers the operator-valued condition.

The Kraus operators of $\mathcal{T}$ are $T_{r}y=g_0\otimes S_{r}y$. Since $g_0$ is unit, ${\sum }_r{T}_r^{\ast }{T}_r={\sum }_r{S}_r^{\ast }{S}_r=I_{H_O}$; hence $\mathcal{T}$ is a channel by [F3]. Step 3.2 proves it preserves the logical marginal on every operator, and [F9] extends that identity to every finite reference. The physical reset-and-encode map is a channel by [F4], so the protected subsystem is exactly correctable.

For a fixed orthonormal gauge basis, if the operator-valued condition holds, then $Q_{kk}(J^{\ast }{E}_a^{\ast }{E}_{b}J)Q_{\mathrm{\ell }\mathrm{\ell }}=⟨M_{ab}{g}_{\mathrm{\ell }},g_k⟩Q_{k\mathrm{\ell }}$, which gives the embedded matrix-unit condition after multiplying by $J$ and $J^{\ast }$. Conversely, sum the embedded matrix-unit identities over $k,\mathrm{\ell }$. Since ${\sum }_k{P}_{kk}=P$ and $J^{\ast }{P}_{k\mathrm{\ell }}J=Q_{k\mathrm{\ell }}$, the resulting operator is $J^{\ast }{E}_a^{\ast }{E}_{b}J=({\sum }_{k,\mathrm{\ell }}{\lambda }_{abk\mathrm{\ell }}\mathrm{\mid }{g}_k⟩⟨g_{\mathrm{\ell }}\mathrm{\mid })\otimes I_{H_L}$. This establishes equivalence for the fixed basis; because the operator-valued condition is basis-independent, it holds for every orthonormal gauge basis.

If $\dim H_G=1$, every $M_{ab}$ is a scalar, so the criterion becomes $P{E}_a^{\ast }{E}_{b}P=C_{ab}P$. Under the unitary identification $H_G\otimes H_L\cong H_L$, the first-factor trace leaves the logical operator unchanged, so the subsystem recovery definition [F11] is ordinary exact recovery. The necessity and sufficiency directions are [F6].