Definition 4.2 — random-unitary channels preserve the identity

For every natural number $d$ with $0<d$, and every function $\Phi$ from the complex $d\times d$ matrices to themselves, suppose that there exist a natural number $k$, real numbers $p_j$ indexed by $j\in\{0,\ldots,k-1\}$, and complex unitary $d\times d$ matrices $V_j$ (so $V_j^*V_j=V_jV_j^*=I_d$), such that $0\le p_j\le1$ for every $j$, $\sum_{j=0}^{k-1}p_j=1$, and, for every complex $d\times d$ matrix $X$, $\Phi(X)=\sum_{j=0}^{k-1}p_jV_jXV_j^*$, where the real weights are embedded in the complex numbers and $V_j^*$ denotes conjugate transpose; then $\Phi(I_d)=I_d$. The dimension $d=0$ is excluded by the hypothesis; the existential quantifier permits $k=0$ syntactically, but the required weight sum would then be $0=1$, so no such witness satisfies the hypotheses; individual weights may be zero or one. No linearity or channel hypothesis on $\Phi$ is assumed separately from the displayed representation.