Theorem 4.23 proof — compact convex geometry of unital channels

For every natural number $d$ with $0<d$, let $M_d(\mathbb C)$ be the space of complex $d\times d$ matrices indexed by $\{0,\ldots,d-1\}$, and let $U_d$ be the following subset of the space of all functions $M_d(\mathbb C)\to M_d(\mathbb C)$: a function $\Phi$ belongs to $U_d$ exactly when $\Phi(I_d)=I_d$ and there exist a natural number $m$ and matrices $E_0,\ldots,E_{m-1}\in M_d(\mathbb C)$ such that $\sum_{j=0}^{m-1}E_j^*E_j=I_d$ and, for every $\rho\in M_d(\mathbb C)$, $\Phi(\rho)=\sum_{j=0}^{m-1}E_j\rho E_j^*$, where $*$ denotes conjugate transpose and $I_d$ is the identity matrix. The witness $m$ is allowed to be zero, with sums then interpreted as zero, although the normalization equation excludes that case for the assumed positive $d$. Then $U_d$ is compact in the product topology on the space of all such functions, namely the topology of pointwise convergence, with each matrix space carrying the finite product of the usual topology on $\mathbb C$; $U_d$ is convex over $\mathbb R$ under pointwise addition and real scalar multiplication of functions (so for any $\Phi,\Psi\in U_d$ and any nonnegative real $a,b$ satisfying $a+b=1$, the function $a\Phi+b\Psi$ belongs to $U_d$); and the real linear span of $U_d$ inside this space of all functions is finite-dimensional as a real vector space.