Complementary error records in a Kraus basis

Let $H$ and $K$ be nonzero finite-dimensional complex inner-product spaces. Let $\mathcal{N}:\mathcal{L}(H)\to \mathcal{L}(K)$ be a quantum channel, meaning a completely positive, trace-preserving linear map, with a finite Kraus representation $\mathcal{N}(X)={\sum }_a{E}_{a}X{E}_a^{†}$. Here $\mathcal{L}(H)$ denotes the linear endomorphisms of $H$, and $E_a^{†}$ is the adjoint of $E_a:H\to K$. Choose a nonempty finite label set for the family and an environment space $E$ with orthonormal basis $\{\mathrm{\mid }a⟩\}$ indexed by that set. Define $W:H\to K\otimes E$ by $W\mathrm{\mid }\psi ⟩={\sum }_a{E}_a\mathrm{\mid }\psi ⟩\otimes \mathrm{\mid }a⟩$. The complementary error record associated with this representation is ${\mathcal{N}}^c(X)={\mathrm{Tr}}_K(WX{W}^{†})$, where the partial trace contracts the first tensor factor. In the chosen label basis its $(a,b)$ entry is $\mathrm{Tr}(E_{a}X{E}_b^{†})$.

For an isometric code encoding $V:H_L\to H$ with nonzero finite-dimensional logical space and code subspace $C=\mathrm{im}V$, the record is logical-state independent on the code when ${\mathcal{N}}^c(\rho )$ is the same operator for every density operator $\rho$ on $H$ supported on $C$. A density operator is a positive semidefinite operator of trace one; supported on $C$ means $P\rho P=\rho$ for the orthogonal projector $P=V{V}^{†}$. This condition concerns the entire record, including off-diagonal entries, and is stronger than merely requiring the diagonal entries to be independent of the code state.

Trace preservation and the Kraus formula give $\mathrm{Tr}(X{\sum }_a{E}_a^{†}{E}_a)=\mathrm{Tr}(X)$ for every $X$. Taking matrix units for $X$ shows ${\sum }_a{E}_a^{†}{E}_a=I_H$. The product inner product and orthonormality of the label basis give $⟨W\psi ,W\phi ⟩={\sum }_a⟨E_a\psi ,E_a\phi ⟩=⟨\psi ,\phi ⟩$, hence $W^{†}W=I_H$, so $W$ is an isometry. Expanding $WX{W}^{†}$ and tracing over $E$ gives $\mathcal{N}(X)$; thus $W$ is a dilation of the given channel. Tracing instead over $K$ gives the displayed entries, so the defined record is the complementary channel for this chosen dilation. The choice of labels and basis is part of the representation.

The code-state condition is well-defined because the code projector specifies exactly which density operators are supported on $C$. Equality of full operators entails equality of their diagonal entries, while diagonal entries alone do not specify off-diagonal entries. For example, the two density matrices $\frac{1}{2}\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)$ and $\frac{1}{2}\left(\begin{array}{cc}1& -1\\ -1& 1\end{array}\right)$ have the same diagonal and are different. These matrices occur as complementary records for a single channel on the full two-dimensional code: take $H=K=H_L={\mathbb{C}}^2$, $V=I$, $Z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$, $E_0=I\mathrm{/}\sqrt{2}$ and $E_1=Z\mathrm{/}\sqrt{2}$. The map $\mathcal{N}(X)=(X+ZXZ)\mathrm{/}2$ is completely positive, since each amplification is a sum of conjugations, and is trace-preserving, since $E_0^{†}{E}_0+E_1^{†}{E}_1=I$. The record formula gives ${\mathcal{N}}^c(\rho )=\frac{1}{2}\left(\begin{array}{cc}1& \mathrm{Tr}(\rho Z)\\ \mathrm{Tr}(\rho Z)& 1\end{array}\right)$ for every density operator $\rho$. Its diagonal entries are both $1\mathrm{/}2$ for every code state, whereas the inputs $\mathrm{\mid }0⟩⟨0\mathrm{\mid }$ and $\mathrm{\mid }1⟩⟨1\mathrm{\mid }$ yield the two distinct matrices above. Thus diagonal independence over all code states does not imply full-record independence.