Analyzing the composition of a channel with its adjoint in relation with an identical composition obtained for the channel's complement

Let us consider two quantum channels $$\Phi:M_d\rightarrow M_{d_1}$$ and $$\Phi_c:M_d\rightarrow M_{d_2}$$ that are complementary to each other, i.e., there exists an isometry $$V:\mathbb{C}^d\rightarrow \mathbb{C}^{d_1}\otimes \mathbb{C}^{d_2}$$ such that

$$\forall X\in M_d: \quad \Phi(X) = \text{Tr}_2 (VXV^\dagger) \quad\text{and}\quad \Phi_c(X) = \text{Tr}_1 (VXV^\dagger).$$

We define the adjoints $$\Phi^*: M_{d_1}\rightarrow M_{d}$$ and $$\Phi_c^*:M_{d_2}\rightarrow M_d$$ of these channels uniquely by the following relations: $$\forall X\in M_d,\, \forall Y\in M_{d_1},\, \forall Z\in M_{d_2}:$$

$$\text{Tr}[\Phi(X)Y]=\text{Tr}[X\Phi^*(Y)] \quad\text{and}\quad \text{Tr}[\Phi_c(X)Z]=\text{Tr}[X\Phi_c^*(Z)].$$

It is well-known that the adjoints of quantum channels are unital (i.e., they map the identity matrix to the identity matrix) and completely positive. I am interested in the following composed maps: $$\Phi^*\circ\Phi: M_d\rightarrow M_d \quad\text{and}\quad \Phi_c^*\circ\Phi_c: M_d\rightarrow M_d,$$ which are again guaranteed to be completely positive (notice that these maps are neither trace-preserving nor unital in general). If people have looked at these kinds of compositions before, I would love to get hold of a reference.

In particular, I want to study the supports (or ranges) of the images of quantum states under the above maps. If the channel $$\Phi$$ is also completely copositive, i.e. $$\Phi\circ T$$ is again a quantum channel and hence $$T\circ \Phi^*$$ is again unital and completely positive (where $$T:M_d\rightarrow M_d$$ is the transpose map), I claim that for all pure states $$\psi\psi^*\in M_d$$, the following inclusion holds $$\text{supp}\,\, [\Phi_c^* \circ \Phi_c](\psi\psi^*) \subseteq \text{supp}\,\, [\Phi^* \circ \Phi](\psi\psi^*).$$

Any help in proving/disproving the above claim would be greatly appreciated. Thanks!

Cross posted on math.SE

• I tried to have the title reflect what more specifically is being asked, though I'm not sure I succeeded, partly due to the question being not too focused. I think this could really be broken down into two questions that can be asked separately: (1) references about compositions of a channel with its adjoint, and (2) relation between such compositions taken for a channel and its complementary.
– glS
Mar 27 '21 at 11:34
• @glS Yup, that's why I kept the original title a bit more generic. Anyways, thanks for trying to make it more specific. I guess it kind of works. Mar 27 '21 at 16:49
• I have now changed it slightly to make it work even better. Mar 27 '21 at 16:57
• isn't the composition between channel and it adjoint? (i.e are "adjoint" and "complement" inverted in the current title?)
– glS
Mar 27 '21 at 21:32
• I can see why you are confused. Hopefully the new title is more clear. Mar 28 '21 at 3:37