Choi's theorem on completely positive maps

Remarks

Proof idea: $1 \Rightarrow 2$ is clear, and $2 \Rightarrow 3$ holds because $C_\Phi = \Phi_n(x)$ for the positive block matrix $x = [E_{ij}]_{i,j}$. For $3 \Rightarrow 4$, write $C_\Phi = \sum_k v_k v_k^*$ with at most $nm$ vectors $v_k \in \C^{nm}$, using the spectral decomposition, and reshape each $v_k$ into an $m \times n$ matrix $V_k$. For $4 \Rightarrow 1$, each map $a \mapsto V a V^*$ is completely positive, and sums of completely positive maps are completely positive.

Example: for the transpose $\Phi(a) = a^T$ on $M_2(\C)$, the Choi matrix is the swap matrix on $\C^2 \otimes \C^2$, which has eigenvalue $-1$. So the transpose is not completely positive, although it is positive. The matrices $V_k$ in clause 4 are called Kraus operators.

Going deeper

For a student

The theorem is the finite-dimensional form of the Stinespring dilation theorem. In quantum information, it gives the Kraus representation of quantum channels; the channel is trace-preserving exactly when $\sum_k V_k^* V_k = I$. The correspondence $\Phi \mapsto C_\Phi$ between maps and matrices is the Choi–Jamiołkowski isomorphism, and condition 2 shows that $n$-positivity already implies complete positivity for maps defined on $M_n(\C)$.