Let $d$ be a natural number. Given $A_1,\dots,A_r\in M_d(\mathbb{C})$, define a linear operator $\Phi(A_1,\dots,A_r):M_d(\mathbb{C})\rightarrow M_d(\mathbb{C})$ by letting $\Phi(A_1,\dots,A_r)(X)=A_1XA_1^\ast+\dots+A_rXA_r^\ast$. We say that a linear operator $\mathcal{E}:M_d(\mathbb{C})\rightarrow M_d(\mathbb{C})$ is completely positive if $\mathcal{E}=\Phi(A_1,\dots,A_r)$ for some $A_1,\dots,A_r\in M_d(\mathbb{C})$.
Observation: $\text{Tr}(\Phi(A_1,\dots,A_r))=|\text{Tr}(A_1)|^2+\dots+|\text{Tr}(A_r)|^2$. Therefore, the trace of a completely positive superoperator is always a non-negative real number. Furthermore, $\text{Tr}(\Phi(A_1,\dots,A_r))=0$ precisely when $\text{Tr}(A_1)=\dots=\text{Tr}(A_r)=0$. More generally, $\text{Tr}(\Phi(A_1,\dots,A_r)^m)=0$ precisely when $\text{Tr}(A_{i_1}\dots A_{i_m})=0$ whenever $i_1,\dots,i_m\in\{1,\dots,r\}.$
Suppose that $n\geq 3$. Then is there any characterization of all completely positive operators $\mathcal{E}:M_d(\mathbb{C})\rightarrow M_d(\mathbb{C})$ where $\text{Tr}(\mathcal{E}^m)=0$ whenever $m\neq 0\mod n$? Equivalently, is there a characterization of all tuples $(A_1,\dots,A_r)\in M_d(\mathbb{C})^r$ where $\text{Tr}(A_{i_1}\dots A_{i_m})=0$ whenever $i_1,\dots,i_m\in\{1,\dots,r\}$ and $m\neq 0\mod n$? Does there exist a normal form for the collection of all tuples $(A_1,\dots,A_r)\in M_d(\mathbb{C})^r$ where $\text{Tr}(A_{i_1}\dots A_{i_m})=0$ whenever $i_1,\dots,i_m\in\{1,\dots,r\}$ and $m\neq 0\mod n$?
Motivation: Let $f$ be a homogeneous non-commutative polynomial of degree $n$ with complex coefficients (the coefficients can be random if you want), and let $A_1,\dots,A_r\in M_d(\mathbb{C})$ be matrices that maximize $$\frac{\rho(f(A_1,\dots,A_r))^{1/n}}{\rho(\Phi(A_1,\dots,A_r))^{1/2}}$$ where $\rho$ denotes the spectral radius. Or you may also maximize $$\frac{\rho(f(A_1,\dots,A_r))^{1/n}}{\|A_1A_1^\ast+\dots+A_rA_r^\ast\|_s^{1/2}}$$ where $1<s<\infty$ and $\|\cdot\|_s$ denotes the Schatten norm (if you would rather see less spectral radii and more norms). If everything works out right, we would get $\text{Tr}(\Phi(A_1,\dots,A_r)^m)=0$ whenever $m\neq 0\mod n$, so I would like to understand the tuples of matrices $(A_1,\dots,A_r)$ a bit more. The motivation for this optimization is that the matrices $(A_1,\dots,A_r)$ reveal structure behind the tensor $p(x_1,\dots,x_r)$.
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