Complementary channels and coherent information

Let $A$ and $B$ be nonzero finite-dimensional complex inner-product spaces and let $\mathcal{N}:\mathcal{L}(A)\to \mathcal{L}(B)$ be a quantum channel as in Finite-dimensional quantum channel. Choose a finite-dimensional environment $E$ and a linear isometry $V:A\to B\otimes E$ such that

$\mathcal{N}(X)={\mathrm{Tr}}_E(VX{V}^{\ast })$

for every $X\in \mathcal{L}(A)$; such a dilation exists by Every finite-dimensional channel has a finite isometric dilation. The channel

${\mathcal{N}}^c(X):={\mathrm{Tr}}_B(VX{V}^{\ast })\in \mathcal{L}(E)$

is the complementary channel associated with this chosen dilation. The partial trace ${\mathrm{Tr}}_B$ contracts the first factor of the ordered product $B\otimes E$ (Partial trace over the first tensor factor).

It is a quantum channel: reorder $B\otimes E$ to $E\otimes B$ by the finite wire-permutation unitary (Ordered tensor registers, their inner product, and wire permutations, Finite wire permutations are unitary basis permutations); the reordered $V$ is still an isometry, so tracing its second factor $B$ gives a channel by An isometry followed by partial trace is a channel.

For a density operator $\rho$ on $A$, choose a finite purification $\mathrm{\mid }\psi ⟩\in A\otimes R$ with ${\mathrm{Tr}}_R\mathrm{\mid }\psi ⟩⟨\psi \mathrm{\mid }=\rho$ (Ensemble and purification), and define ${\omega }_{BR}:=(\mathcal{N}\otimes {\mathrm{id}}_R)(\mathrm{\mid }\psi ⟩⟨\psi \mathrm{\mid })$. Its coherent information from $A$ to $B$ is

$I_c(\rho ,\mathcal{N}):=S(\mathcal{N}(\rho ))-S({\omega }_{BR}),$

where $S$ is finite-dimensional von Neumann entropy in nats (Finite-dimensional von Neumann entropy). The value does not depend on the chosen purification: after padding reference spaces, any two purifications differ by a unitary on $R$ (Purifications exist and are unitarily equivalent after padding); applying $\mathcal{N}\otimes {\mathrm{id}}_R$ preserves that unitary relation. A unitary maps an orthonormal eigenbasis to an orthonormal eigenbasis with the same eigenvalues, so entropy is unchanged (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely). Padding adds only zero eigenvalues, which do not contribute.

The chosen complement may vary with the dilation, but its output entropy on $\rho$ equals $S({\omega }_{BR})$: applying $V$ to the first tensor factor of the purifying vector gives a pure state on $B\otimes E\otimes R$. After reordering the factors, its complementary $BR$ and $E$ marginals have the same positive eigenvalues. The corresponding entropy identity is proved in Coherent information equals output entropy minus environment entropy.