Kraus representation of a channel

Remarks

The operators $K_k$ are called Kraus operators. In part 3, families of different lengths are compared after padding the shorter one with zero operators. Part 1 is part of Choi's theorem on completely positive maps. For the proof of 1, form the Choi matrix $C = \sum_{i,j} e_ie_j^* \otimes \Phi(e_ie_j^*)$, which is positive when $\Phi$ is completely positive. Write $C = \sum_k v_kv_k^*$ with at most $nm$ vectors $v_k$, from its spectral decomposition, and reshape each $v_k$ into a map $K_k$. The converse is direct, since each map $X \mapsto KXK^*$ is completely positive. Part 2 follows from $\tr \Phi(X) = \tr\big(\sum_k K_k^*K_k\, X\big)$ for all $X$. Example: the qubit dephasing channel $\Phi(X) = (1 - p)X + pZXZ$ has the Kraus operators $\sqrt{1 - p}\, I$ and $\sqrt{p}\, Z$, and indeed $(1 - p)I + pZ^2 = I$. The decomposition is a finite-dimensional form of the Stinespring dilation theorem.

Going deeper

For a student

The theorem turns the abstract condition of complete positivity into a concrete formula, through the Choi matrix. It is equivalent, in finite dimensions, to Stinespring's dilation: a channel is an isometry into a larger space followed by a partial trace. The Kraus operators are far from unique, as part 3 shows, and the minimal number of them is the rank of the Choi matrix. For quantum operations that may decrease the trace, the same holds with $\sum_k K_k^*K_k \le I$.