Theorem 4.23 — unital qubit channels are mixed unitary

For every function $\Phi:M_2(\mathbb C)\to M_2(\mathbb C)$, where matrices have rows and columns indexed by $\{0,1\}$, assume that there exist a natural number $m$ and matrices $E_j\in M_2(\mathbb C)$ indexed by $j\in\{0,\ldots,m-1\}$ such that $\sum_{j=0}^{m-1}E_j^*E_j=I_2$ and, for every $\rho\in M_2(\mathbb C)$, $\Phi(\rho)=\sum_{j=0}^{m-1}E_j\rho E_j^*$; assume also that $\Phi(I_2)=I_2$. Then there exist, in order, a natural number $k$, a function $p:\{0,\ldots,k-1\}\to\mathbb R$, and a function $V:\{0,\ldots,k-1\}\to U(2)$, where $U(2)$ consists of complex $2\times2$ matrices satisfying $V_j^*V_j=V_jV_j^*=I_2$, such that for every index $j$, $0\le p_j$ and $p_j\le1$, such that $\sum_{j=0}^{k-1}p_j=1$, and such that for every $X\in M_2(\mathbb C)$, $\Phi(X)=\sum_{j=0}^{k-1}(p_j:\mathbb C)\,V_jXV_j^*$. Here $*$ denotes conjugate transpose, each sum is over the indicated finite index set, and the formula holds for all complex matrices, without any restriction to states. Both $m=0$ and $k=0$ are included in their natural-number quantifiers, with empty sums equal to zero, but $m=0$ cannot satisfy the Kraus identity $0=I_2$ and $k=0$ cannot satisfy the probability normalization $0=1$.