Proposition: An injective channel without a CPTP left inverse

For $d\ge 2$ and $0<p<1$, the depolarizing channel

is injective and therefore mathematically determines its input from its exact output. It has no CPTP left inverse on the complete input space.

Proof. Its linear inverse is

Thus it is injective. Applied to a pure-state projector, this inverse has a strictly negative eigenvalue on every orthogonal direction. It is not positive.

Since ${\mathcal{N}}_p$ is an invertible linear map on the entire matrix space, any linear left inverse must be this inverse. A CPTP left inverse is therefore impossible.