Theorem 4.23 proof — extreme unital qubit channels are unitary

For every function $\Phi:M_2(\mathbb C)\to M_2(\mathbb C)$, assume that $\Phi$ belongs to the set $S$ of unital channels and is an extreme point of $S$ over $\mathbb R$, with the following meanings: a function $\Psi$ belongs to $S$ precisely when $\Psi(I_2)=I_2$ and there exist a natural number $m$ and complex $2\times2$ matrices $E_0,\ldots,E_{m-1}$ such that $\sum_{j=0}^{m-1}E_j^*E_j=I_2$ and, for every complex $2\times2$ matrix $X$, $\Psi(X)=\sum_{j=0}^{m-1}E_jXE_j^*$; being an extreme point means that $\Phi\in S$ and, for every $\Psi,\Theta\in S$ and all real numbers $a,b>0$ with $a+b=1$, if $\Phi=a\Psi+b\Theta$ as functions (real scalar multiplication and addition being pointwise), then $\Psi=\Phi$ and $\Theta=\Phi$. Under this assumption there exists a single complex $2\times2$ unitary matrix $U$ (so $U^*U=UU^*=I_2$) such that, for every complex $2\times2$ matrix $X$, $\Phi(X)=UXU^*$. Here $M_2(\mathbb C)$ is the space of matrices whose rows and columns are indexed by $\{0,1\}$, $I_2$ is its identity matrix, and $A^*$ denotes conjugate transpose. The natural number $m$ is allowed to be zero in the defining existential quantifier, with empty sums interpreted as zero, although that choice cannot satisfy $\sum_jE_j^*E_j=I_2$; the universally quantified inputs $X$ are arbitrary matrices, without positivity, self-adjointness, or trace restrictions.